Primordial Cosmology Patrick Peter and Jean-Philippe Uzan Institut d' Astrophysique de Par-is
Translated by Jasna Bruj ic and Claudia de Rham
,
OXFORD UNIVERSITY PRESS
OXFORD UNIVERSITY PRESS
Great C la rendon Street, Oxford OX2 6DP Oxford Univers ity Press is a d epartment of t he University of Oxford. It furthers the University 's object ive of excellence in research , scholarship, and education by publishing worldwide in Oxford New York Auckla nd Cap e Town D a r es Sa laam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico C ity Nairobi New Delhi Shanghai Ta ipei Toronto With offices in A rgentina Austria Brazil Chile C zech Repu blic France Greece G u atem a la Hungary Italy Japan Pola nd Portuga l Singapore South K orea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered tra de mark of Oxford University Press in the UK a nd in certain other count ries Published in the United States by Oxford Universit y Press Inc. , New York Edition originale: "Cosmologie primordia le" © Editions Belin-Pa ris, 2005
English translation © Oxford U ni vers ity Press 2009 O uv rage publ ie avec Ie concours du Ministere fr an~ais ch a rge de la culture - Centre nationa l du livre The moral rights of the a uthors have b een assert ed D atabase right Ox ford University P ress (m aker) First Published in English 2009 All rights reserved. No p a rt of this publication may be reproduced , stored in a retrieval syst em, or tra nsmitted , in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law , or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside t he scope of the ab ove should be sent t o t he Rights Department , Oxford U niversity Press , at the address above You must not circulate this book in a ny other bind ing or cover a nd you must impose t he same condit ion on any acquirer B ritish Library Cataloguing in P ub lication D a t a Data available Library of Congress Cataloging in Publicat ion Data Data available Printed in G reat Britain on acid-free paper by C PI , Chippenham, Wilts ISB N 978- 0- 19-920991 - 0 1 3 5 7 9 10 8 6 4 2
Foreword to the English edition We live at a very special time. The universe was born about 14 billion years ago. We began to study it by very primitive tools only several t housand years ago. Less than a hundred years ago we learned about the existence of other galaxies. F inally, during the last two decades we entered the era of precision cosmology. We are able to study with incredible accuracy the structure of the observable part of the universe, finding imprints of what happened in the first milliseconds after the big bang. But this precision would not help much without the development of new tools which previously were unavailable to cosmologist s. For a long time we did not know how to study matter at densities much greater than nuclear density. This limited our ability to study physical processes in the very early universe. The st andard approach was to make various assumpt ions about superdense matter and then study consequences of these assumptions. The sit uation changed in the beginning of t he 1970s, when we learned that not only the temperature and density, but also t he properties of elementary particles in t he early universe were quite different from what we see now. In particular, according to the theory of t he cosmological phase transitions, during the first 10- 10 seconds after the big bang there was not much difference between weak and electromagnetic interactions. The next important step was t he discovery of the asymptotic freedom , which implied t hat the strength of interactions between elementary particles decreases at large density. This allowed us to investigate physical processes very close to the big bang, at densities almost eighty orders of magnitude higher than the nuclear density. These discoveries culminated in the invention of inflationary theory, which helped to explain why our universe is so large and flat , why it is homogeneous and isotropic, why its different parts started their expansion simultaneously. According to t his theory, the early universe experienced a period of exponentially r apid expansion (inflation) in a slowly changing vacuum-like state. All elementary particles surrounding us now were produced as a result of t he decay of this vacuum-like state at the end of inflation. Inflation stretched away all previously existing inhomogeneities, but simultaneously it produced new inhomogeneities from t iny quantum fluctuations amplified during the exponential growth of t he universe. These fluctuations served as seeds for the subsequent process of galaxy formation. In certain cases, these quantum fluctuations may become so large that they can be responsible not only for t he formation of galaxies, but also for the formation of new exponentially large parts of t he universe with different laws of low-energy physics operating in each of them . Instead of being a sphere, our universe became an eternally growing fractal, a multiverse consisting of different exponentially large parts. In the beginning, inflationary theory could seem too radical to be true , but during the last two decades many of its predictions were confirmed by cosmological observa-
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Foreword to the English edition
tions, and many cosmologists accepted it as a part of the new cosmological par Moreover, after the progress with vacuum st abilization in string theory achi few years ago, the picture of an eternally growing inflationary multiverse consis many exponentially large parts with different properties became a part of what called 'the string theory landscape'. It is clear that further development of cosmology will require knowledge o subjects which traditionally did not belong to the toolbox of a cosmologist. One know not only the general theory of relativity, but also the modern theory of elem particles, quantum field theory, and basic elements of string theory. An add challenge is to relate the new cosmological theory and the rapidly growing observational data. This book by Patrick P et er and J ean-Philippe Uzan successfully solves thi cult problem . It begins with an int roduction to general relativity and modern p physics, which received a powerful boost in the beginning of the 1970s. In Pa the book they describe cosmological theory of a homogeneous and of a sligh homogeneous universe. Then they describe inflationary cosmology. This includ classical theory of a homogeneous inflationary universe , t he theory of inflationa turbations, and t he theory of reheating, which is responsible for the formation elementary particles after inflation. The authors bring it all together in P art III, they discuss grand unification , baryogenesis , supersymmet ry and supergravity, logical phase transitions, various models of inflation, and elements of string This provides an excellent background for further independent studies of t heo and observational cosmology and of the recent work on inflation in string theo on the string t heory landscape. The book is well written , well illustrated and very user-friendly. I am sure will be appreciated not only by students who enter this exciting field of knowled also by experts in cosmology who will be happy to see a unique source of infor covering every important aspect of modern cosmology.
Andrei Linde Stanford University USA
Foreword to the French edition Scientific cosmology was one of the most fascinating intellectual adventures of the twentieth century, and it still is in the twenty-first. It began in the early part of the century when the theoretical work of Albert Einstein, Alexandre Friedmann, and Georges Lemaitre came together with the observational discoveries of Vesto Slipher, Henrietta Leavitt, and Edwin Hubble. But, for many years afterwards, the small quantity and low degree of precision of the observational data was insufficient grist to the mill for theorists, and cosmology made slow progress. This situation began to change during the latter part of the twentieth century. Some remarkable, and often unexpected, observational discoveries - like the detection of cosmic background radiation and the confirmation of the 'dark matter problem' - as well as the gradual increase in the precision of observational data, led to fruitful exchanges between theory and experiment. Interest in cosmology increased at the end of the century when new theoretical paradigms blossomed (e.g. cold dark matter and inflation) together with experimental discoveries that were b eginning to come thick and fast , accompanied by an incredible increase in precision (e.g. anisotropies of the microwave background, the discovery of the acceleration of the rate of expansion). At the present time , cosmology is one of the most active areas of science. In its rapid advance, it mobilizes all the resources of the latest theoretical physics (general relativity, the quantum theory of fields , superstring theory) as well as those of observation (large telescopes, satellites). The present work provides the reader with an invaluable means of access to state-of-the-art cosmology. It only requires very elementary knowledge of the basics (special relativity, non-relativistic quantum mechanics) , and explains in detail all the theoretical tools underpinning modern cosmological research: general relativity, the classic and quantum theories of fields , particle physics, the theories of grand unification. It even includes an introduction to the extensions of standard theories such as, for example, supersymmetric theories, the theory of strings , and brane cosmology. Furthermore, it teaches the reader - in great detail - about the main applications of the most recent theoretical frameworks for the description of fundamental objects of cosmology: the theory of cosmological perturbations , fluctuations in the background radiation , gravitational lenses , the generation of quantum fluctuation during inflation, and topological defects. I have no doubt that this remarkable book by Patrick Peter and Jean-Philippe Uzan will b e of enormous value to a wide readership: students who are beginning research into cosmology or into astroparticle physics; experienced researchers who wish to update their knowledge; and teachers who want a synthesis of the present state of our knowledge of cosmology. All of them will find a great amount of detailed information, presented in a way that is clear and easy to understand. Finally, this is a work written by two bright young researchers that not only transmits the knowledge
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that is necessary in order to understand modern theoretical cosmology and enable i to progress, but also communicates the enthusiasm generated by taking part in a exceptional intellectual adventure whose future promises to be as exciting as its past
Thibault Damour Institut des Hautes Etudes Scientifiques Academy of Sciences France
Acknowledgements We would like to thank our colleagues and collaborators, in particular t hose from the Astrophysics Instit ute in P aris (Institut d 'Astrophysique de Paris). The numerous discussions we shared with them have helped us in clarifying many important issues. We particularly want to thank K arim Benabed , Francis Bernardeau, Fran<;ois Bouchet , 1hstan Brunier , Christos Charmousis, Chris Clarkson, Alain Coc, Cedric Deffayet , Nathalie Deruelle, P eter Dunsby, Ruth Durrer , George Ellis, Gilles Esposito-Farese, Ted J acobson, Lev Kofman , David Langlois, J ean-Pierre Lasota, Roland Lehoucq, Martin Lemoine, Keith Olive, Thiago P ereira , Nelson Pinto-Neto, Alain Riazuelo , Christophe Ringeval, Carlo Schimd , Joe Silk, Ismael Tereno, Alberto Vallinotto, Ludovic van Waerbeke, Filippo Vernizzi and J effrey Weeks. This book would not have been of the same quality without the dedication of some of our colleagues in proofreading and edit ing of t he first manuscript. We thank Nabila Aghanim, Philippe Brax, Bernard Fort , Marc Lilley, J erome Martin, Yannick Mellier , Renaud Parentani , Cyril Pitrou, Simon Prunet, Alain Riazuelo, Jonathan Rocher , Carlo Schimd , Frederic Vincent, and Elisabeth Vangioni , Annick Lesne and Patrizia Castiglione for t heir comments and their help in putt ing together the final version, as well as Laurent Vigroux and t he Institut d 'Astrophysique de Paris for all the support in publishing this book. We would also like to thank Andrei Linde and Thibault Damour for giving us the honour of writing t he forewords t o the English and French editions respectively. We also largely benefited from t heir useful and improving comments.
Contents Foreword to the English edition
v
Foreword to the French edition
vii
Acknowledgements
ix
Introduction Cosmology: an ancient yet contemporary subject The specifics of cosmology Primordial cosmology Organization and objective of this b ook Warnings References PART I
1
1 1 2 2 3
5 6
OVERVIEW OF THE THEORETICAL BASIS
General relativity 1.1 Space-time and gravity 1.1.1 Absolute space and time of Newtonian physics 1.1.2 Space-time of special relativity 1.1.3 General relativity a nd curved space-time 1. 2 Elements of differential geometry 1.2.1 Manifolds and tensors 1.2.2 Geodesic equat ions and Christoffel symbols 1.2.3 Covariant derivative, parallel transport a nd Lie derivative 1.2.4 Curvature 1.2.5 Covariant approach 1.3 Equations of motion 1.3.1 E instein's equ ations 1.3.2 Conservation equations 1.3.3 ADM Hamiltonian formulation 1.4 Geodesics in a curved space-time 1.4.1 Conserved quantities along a geodesic 1.4.2 Geodesic deviation equation 1.5 Weak-field regime 1.5.1 Newtonian limit 1.5.2 Gravit ational waves in an empty space-time 1.6 Tests of general relativity 1.6.1 Test of t he equivalence principle 1.6.2 Tests in the Solar System
11 11 11 14 19 21 21 26 28 31 34 38 38 41 44 46 46 47 49 49 53 55 55 60
XII
2
Contents
1.6.3 Gravitational radiation 1.6.4 Need for t est s at astrophysical scales References
6 6 6
Overview of particle physics and the Standard Model 2.1 From classical to quantum 2.1.1 Analytical classical mechanics 2.1.2 Quantum physics 2.1.3 Quantum and relativistic mechanics 2.2 Canonical decompositions 2.2.1 Technical clarification 2.2.2 Real free field 2.2.3 The complex scalar field 2.3 Classification and properties of the elementary particles 2.4 Internal symmetries 2.4.1 Group theory 2.4.2 Generators 2.5 Symmetry breaking 2.5.1 Gauge group 2.5.2 Higgs mechanism 2.5.3 Non-Abelian case 2.6 The standard model SU(3)c X SU(2)L X U(l) y 2.6.1 The strong interaction (QCD) 2.6.2 E lectroweak interaction 2.6.3 A complete model 2.7 Discrete invariances 2.7. 1 Parity 2.7.2 Charge conjugation 2.7.3 Time reversal and CPT theorem References
6 6
PART II
3
6 7
8 8 8
9 9
9
10 10 10 10 10 10
11 11 11 11 11
12 12 12 12 12
THE MODERN STANDARD COSMOLOGICAL MODEL
The homogeneous Universe 3.1 The cosmological solution of Friedmann- Lemaitre 3.1.1 Constructing a Universe model 3.1. 2 Cosmologica l and Copernican principles 3.1.3 Cosmological principle and metric 3.1.4 Kinematics 3.1. 5 Space-time dynamics 3.2 Dynamics of Friedmann- Lemaitre space-times 3.2 .1 Some solutions 3.2.2 Dynamical evolut ion 3.2.3 Expansion and contraction 3.3 Time and distances 3.3.1 Age of the Universe and look-back time
12 12 12 13 13 13 13 14
14
14
14
15 15
Contents
4
xiii
3.3.2 Comoving r adial distance 3.3.3 Angular dist ances 3.3.4 Luminosity distance 3.3 .5 Volume and number counts 3.4 Behaviour at small redshifts 3.4.1 Deceleration paramet er 3.4.2 Expression of the time and distances 3.5 Horizons 3.5 .1 Event horizon 3.5.2 P article horizon 3.5.3 Global properties 3.6 Beyond the cosmological principle 3.6.1 Classifying space-times 3.6.2 Universe with non-homogeneous spatial sections 3.6.3 Universe with homogeneous spatial sections References
151 152 154 157 158 158 158 160 160 162 163 167 168 170 171 175
The standard Big-Bang model 4. 1 T he Hubble diagram and the age of t he Universe 4.1.1 The Hubble constant 4.1.2 The contribution of supernOV
177 177 178 182 186 188 188 197 201 204 205 206 208 211 215 216 217 221 226 226 226 228 234 235
xiv
5
Contents
The inhomogeneous Universe Newtonian perturbations 5.l.1 Case of a static space 5.l.2 Case of an expanding space 5.l.3 Predictions and observables 5.l.4 Towards the non-linear regime 5.2 Gauge invariant cosmological perturbation theory 5.2.1 Perturbed space-time 5.2.2 Description of matter 5.2.3 Choosing a gauge 5.2.4 Einstein equations: derivation 5.2.5 Einstein equations: SVT decomposition 5.2.6 Perturbed conservation equation for a fluid 5.2.7 Interpretation of the perturbation equations 5.3 Evolution 5.3.1 Vector and tensor modes 5.3.2 Evolution of the gravitational potential 5.3.3 Scalar modes in the adiabatic regime 5.3.4 Mixture of several fluids 5.4 Power spectrum of density fluctuations 5.4.1 Two equiva lent approaches 5.4.2 Different regimes 5.4.3 Some refinements 5.5 The large-scale structure of the Universe 5.5.1 Observing the large-scale structure 5.5.2 Structures in the non-linear regime References
23 23 23 24 24 24 25 25 25 25 26 26 26 26 26 26 26 27 27 28 28 28 29 29 29 29 30
The cosmic microwave background Origin of the cosmic microwave background anisotropies 6.l.1 Sachs- Wolfe formula 6.l.2 Angular power spectrum 6.2 Properties of the angular power spectrum 6.2.1 Large angular scales 6.2.2 Intermediate scales 6.2.3 Small scales 6.2.4 Other effects 6.3 Kinetic description 6.3.1 Perturbed Boltzmann equation 6.3 .2 Gauge invariant expressions 6.3.3 Thomson scattering and polarization 6.3.4 Numerical integration 6.4 Anisotropies of the cosmic microwave background 6.5 Effects of the parameters on the angular power spectrum 6.5.1 Cosmological parameters 6.5.2 Parameters describing the primordial physics
30 30 30 311 31 31 31 32 32 32 33 33 34 35 35 36 36 36
5.1
6
6.1
Contents
xv
References
371
7
Gravitational lensing and dark matter 7.1 Gravitational lensing and its applications 7.1.1 Gravitational lensing in the thin-lens regime 7.1.2 Lensing by galaxies and galaxy clusters 7.1.3 Gravitational distortion by the large-scale structure 7.1.4 Cosmic convergence 7.1.5 Cosmic shear 7.1.6 Measurement of the cosmic shear 7. 1. 7 Lensing on the cosmic microwave background 7.2 Evidence for the existence of dark matter 7.2.1 Dark matter in galaxies 7.2.2 Dark matter in clusters and groups of galaxies 7.2.3 Cosmological evidence 7.2.4 Summary 7.2.5 Candidates and constraints References
374 374 375 387 395 399 402 405 408 411 411 419 422 423 423 445
8
Inflation 8.1 Genesis of the paradigm 8. 1.1 Original motivations 8.1. 2 Resolution of the Big-Bang problems 8.1.3 F irst models of inflation 8.1.4 Inflation as a de Sitter phase 8.2 Dynamics of single-field inflation 8.2 .1 Equations of evolution 8.2.2 Slow-roll parameters 8.2.3 End of inflation 8.2 .4 Some examples 8.2 .5 Classification of inflationary models 8.3 Quantum fluctuations during inflation 8.3.1 Massless test scalar field in a de Sitter space-time 8.3 .2 Massive test field in de Sitter 8.3.3 Massive test field during slow-roll inflation 8.4 Quantum fluctuations of the inflaton 8.4 .1 Overview of t he questions to address and expected results 8.4.2 Perturbed quantities 8.4.3 Perturbation equations 8.4 .4 Evolution of t he long-wavelength modes 8.4.5 Junction conditions and their applications 8.4 .6 Quantization of the density perturbations 8.4.7 Gravitational waves 8.5 Perturbations in the slow-roll regime 8.5.1 An exact solution: power-law inflation 8.5.2 General case
450 451 45 1 452 454 455 45 7 457 459 463 465 469 470 470 476 477 478 479 479 481 483 485 488 492 494 495 496
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8.5.3 Relation to observations 8.5.4 Reconstruction of the potential 8.6 End of inflation and reheating 8.6.1 Perturbative reheating 8.6.2 Theory of preheating 8.7 Eternal inflation 8.7.1 Heuristic argument 8.7.2 Stochastic approach 8.8 Extensions 8.8 .1 Multifield inflation 8.8.2 Non-Gaussianity 8.8 .3 Trans-Planckian problem 8.9 Status of the paradigm References PART III 9
BEYO ND T HE STA NDA RD MO DELS
Grand unification and baryogenesis 9.1 Interactions 9.1.1 Superposition principle 9.1.2 N -particle states 9.2 Grand unification 9.2.1 Problems of the standard model 9.2.2 Unification of the coupling constants 9.2.3 Unification models 9.2.4 Consequences of grand unification 9.2 .5 Neutrino mass 9.3 Baryogenesis 9.3.1 Sakharov conditions for baryogenesis 9.3.2 Electroweak anomalies 9.3.3 Electroweak baryogenesis 9.3.4 Leptogenesis 9.3.5 Affleck- Dine mechanism References
10 Extensions of the theoretical framework 10.1 Scalar-tensor theory of gravity 10.1.1 Formulation 10.1.2 Local constraints 10.1.3 Cosmological aspects 10.1.4 Phenomenological aspects 10 .1.5 f(R) gravity and scalar-tensor theory 10.2 Quantum field theory in curved space-time 10.2.1 Quantum physics, classical gravity 10.2.2 P article creation 10.2.3 A complete example
Contents
10.3 Supersymmetry and supergravity 10.3.1 Technical generalities 10.3.2 Wess-Zumino model 10.3.3 Gauge field 10.3.4 Supersymmetry breaking 10.3.5 The minimal supersymmetric standard model (MSSM) 10.3.6 Supergravity References
xvii
604 604 610 614 615 618 621 625
11 Phase transitions and topological defects 11.1 Phase transitions 11 .1.1 Thermal field theory 11.1.2 Dynamics of the symmetry breaking 11.1.3 Formation of topological defects 11.2 Domain walls 11 .2.1 Correlation length 11.2.2 Static configurations 11.3 Vortices 11.3.1 Kibble mechanism for cosmic strings 11.3.2 Internal structure 11.4 Monopoles 11.5 Textures 11.6 Defects in general 11.6.1 Connectedness 11.6.2 Fundamental group 11.6.3 Homotopy group 11.6.4 Semi-topological defects 11.7 Walls in cosmology 11.7.1 Distribution and evolution 11.7.2 Observational constraints 11.8 Cosmological monopoles 11.8.1 GUT monopoles are unavoidable 11.8.2 The monopole problem 11.8.3 Possible solutions to the monopole problem 11.9 Cosmic strings 11.9.1 General properties 11.9.2 Gravitational effects of strings 11.9.3 Effect of a network on the microwave background 11.9.4 Other consequences of cosmic strings References
628 628 628 630 633 635 635 638 640 641 643 647 648 648 649 649 650 651 653 653 655 656 656 656 660 661 661 662 666 671 674
12 Cosmological extensions 12.1 Construction of inflationary models 12.1.1 A consistent model: supergravity 12 .1.2 F-term inflation 12.1.3 D-term inflation
678 678 678 681 683
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Contents
12.2 Cosmological constant and dark energy 12.2.1 The cosmological constant problem 12.2.2 The nature of dark energy 12.2.3 Quintessence 12.2.4 Other models 12.2.5 Other approaches 12.2.6 Parameterization of the equation of state 12.2.7 Implications for the formation of the large-scale structure 12.3 Varying constants 12.4 Topology of the Universe 12.4.1 Local and global structures 12.4.2 Mathematical introduction 12.4.3 Classification of the three-dimensional manifolds 12.4.4 Observational signatures References 13 Advanced topics 13.1 Extra dimensions: Kaluza- Klein theory 13.1. 1 General relativity in D dimensions 13.1.2 Projected equations in four dimensions 13.1.3 Dilaton and Einstein- Maxwell theory 13.1.4 Einstein frame 13.1.5 Compactification 13.2 A few words on string theory 13.2. 1 From particles to strings 13.2 .2 Superstrings 13.2.3 Open and closed strings 13.2.4 Dualities 13.2.5 Low-energy Lagrangians 13.2.6 Origin of three space-like dimensions 13.3 The Universe as a ' brane' 13.3. 1 Motivations 13. 3.2 Induced Einstein equations 13.3.3 The Randall- Sundrum model 13.3.4 Cosmological phenomenology 13.3.5 Possible extensions 13.3.6 The Universe as a defect 13.3.7 Models of induced gravity 13.4 Initial singularity and a bouncing Universe 13.4.1 Approaching the singularity 13.4. 2 The pre-Big-Bang scenario 13.4.3 The cyclic scenarios 13.4.4 Regular bounce and power spectrum References
684 685 687 690 697 701 705 70 711 716 716 71 718 726 733
73 73 73 739 741 742 743 744 74 746 749 752 753 755 758 758 761 76 76 76 769 771 772 772 775 781 785 793
Contents
XIX
Appendix A Numerical values A.1 Physical constants A.2 Astrophysical quantities A.3 Units A.3.1 Natural units A.3.2 Conversion factors A.4 Particle physics A.5 Cosmological quantities A.6 Electromagnetic spectrum References
796 796 796 797 797 798 798 799 799 803
Appendix B Special functions B.1 Euler functions B.2 Spherical harmonics B.2. 1 Definition B.2 .2 Expressions B.2.3 Properties B.2.4 Fourier transform B.2 .5 Useful integrals B.3 Bessel functions B.3 .1 Definition B.3.2 Asymptotic properties B.3.3 Special cases B.3.4 Spherical Bessel functions B.3.5 Some useful integrals B.4 Legendre polynomials B.4.1 Associated Legendre polynomials B.4 .2 Legendre polynomials B.5 Fourier transform and the eigenmodes of the Laplacian B .5.1 Fourier transform B.5.2 Power spectrum B.5.3 Cartesian coordinates B.5.4 Spherical coordinates References
804 804 805 805 805 805 806 807 808 808 808 809 809 810 811 811 811 812 812 812 813 813 815
Appendix C Useful cosmological quantities C.1 Background space C.l.1 Geometry C.l.2 Matter C.2 Perturbed quantities C.2.1 Geometry C.2.2 Matter
816 816 816 817 819 819 821
Index
823
Introduction Cosmology: an ancient yet contemporary subject Over the past few decades, the study of the Universe has undergone great transformations, both theoretical and observational. Cosmology, whose aim is to understand the global properties and large-scale structures of the Universe (their origin, evolution, characteristics ... ) has now entered an era of precision. For many years, cosmology had been a very speculative subject, relying as much on metaphysics as on physics. The development in 1915 of the theory of general relativity gave birth to a consistent theoretical framework, making it possible to mathematically formulate the notion of space and time. It was then possible, as early as in 1924, to formulate cosmological models grounded within this theory. Such models of the Universe, whose main characteristic is to be expanding, enabled many observational features such as the recession of the galaxies, as pointed out by Edwin Hubble, to b e understood. For about twenty years , cosmology confined itself to the description and the reconstruction of this expansion, as well as to debating whether this expansion was real. Within a second period, starting around 1948, studying physical processes in an expanding space led to the formulation of the hot Big Bang model. Taking this expansion seriously, one can deduce from the laws of nuclear physics that the Universe has a thermal history. In particular, the light nuclei must have been formed within the first minutes of the expansion, and there must exist an electromagnetic background radiation. Thanks to these developments, the model stands on three solid foundations, including the confirmation of the expansion. These conclusions only rely on the hypothesis that general relativity is a valid description of the dynamics of the Universe and on the validity of ordinary physics. No other alternative model has to date been able to reproduce these observations, especially not with so few ingredients. Since the 1980s, many developments have brought the Big Bang model to a new level. Not only does it describe the dynamics of the Universe on large scales, but it also attacks questions on the properties and the origin of the large-scale structures. Among the most recent progress, one can mention an explosion of observational data: - since the discovery of the cosmic microwave background in 1965, its observation has become more and more precise, from the initial discovery of the thermal anisotropies by the satellite COBE (COsmic Background Explorer) (1992) to the full-sky map by the satellite WMAP (Wilkinson Microwave Anisotropy Probe) in 2003; - the Hubble diagram, relating the recession speed of galaxies to their redshift, could be extended to large distances , mainly through the observation of distant supernovre;
2
Introduction
- the new galaxy cat alogues now have several thousands of obj ect s, which makes it possible t o map the t hree-dimensional distribut ion of matter ; - since 2000 , new observations have appeared , such as the cosmic shear. One can notice sever al major theoretical developments: - the formulation of the paradigm of inflation that links the origin of the large-scale structure of the Universe to high-energy physics; - the formulation of the cold dark-matter model that represents a framework for the study of the large-scale structure; - a consider able number of indications and observational evidence leading t o the postulat e of the existence of dark matter and dark energy. T he study of their nature will shap e research within t he coming decades.
The specifics of cosmology Despite these developments and the abundance of observations , cosmology nonetheless maintains a different status compared to the other sciences. Only one Universe is indeed observable, and , moreover, from a unique posit ion in space and time. The main part of these observations are therefore confined to our past light cone. Cosmology can t herefore not prevent the use of hypotheses t hat are impossible to check , such as the cosm ological crinciple, which has strong implications concerning cosmological space-time symmetries. The elaboration of a cosmological model relies mainly on t hree hypotheses: (1) the choice of a theory of gravity, (2) hypotheses concerning the na ture of the matt er present in t he Universe and (3) a symmetry hypothesis. One should stress that cosmological observat ions cannot be read independently of these theoretical hypotheses (Figs. 1 and 2) . Therefore the validity of a model cannot be proven , but one can look for a consist ency between observations and the theoretical fr amework wit hin which t hey are read. This does not prevent cosmology from excluding models that give predictions t hat are incompatible with observations, as in any other areas of physics.
Primordial cosmology Primordial cosmology, which will be discussed in t his book, relies on two pillars : numerous astronomical observations made by more and more capable instruments t hat reveal t he local structure of our Universe with an ever-increasing precision, and a set of high-energy physics t heories t hat are expect ed t o describe the dynamics of the primordial Universe. We t herefore have to deal wit h a two-way process (Fig. 2). On t he one hand (arrow labeled ' 1 '), using some high-energy physics hypotheses, one can build a phenomenological model for the primordial Universe. This model must provide the properties of the primordial fluctuations that give birth, through gravitational inst ability, t o the large-scale structure of the Universe. We should therefore be able to predict t he resulting propert ies of the Universe. Comparing a specific model wit h different sets of observations allows us to constrain it . On the other hand (arrow labeled '2'), t hese models bring solut ions to t he problems encountered while interpreting the observations: the nature of dark mat ter , of dark energy, etc.
Organization and objective of this book
3
Cosmological model • Gravity theory • Space- time • Matter content
I~'" "0'0, .
';>/"
Direct observations • Redshi ft • Count • Angles, magnitudes, ...
Indirect observations • Abundances • Ages • Density · CMS ,...
Fig. 1 The construction of a cosmological model and its link with observations. Most observations cannot be interpret ed outside this model. Thanks to these t heories, we hope to be able to reproduce the observations, and on the other hand , to use these observations to constrain the theories when they are extrapolated to experimentally inaccessible energies .
Organization and objective of this book The aim of this book is to provide t he tools for primordial cosmology, to describe its state and the current questions. It is organized into three parts. The first part reviews t he useful basis for the study of cosmology. The second part focuses on the standard cosmological model. It represents the heart of this book and we have tried to describe in detail all the technical aspects. The third part describes the extensions of this standard model and sheds light on the subjects currently under discussion. The first part reviews the required theoretical notions. Indeed , since standard cosmology is based on the theories of general relativity and on the standard model of particle physics, Chapters 1 and 2 will provide an overview of these two areas. We try to distinguish between what is well established and the murkier areas of the current theories. The different extensions of these standard frameworks play an important role for the elaboration of models of the primordial Universe. It is thus important to demarcate the limits of these areas . The second part presents the standard cosmological model. Chapters 3 and 4 draw up the most common cosmological solutions of general relativity and the hot Big Bang model. Chapter 3 is mainly mathematical while Chapter 4 focuses on the physics. These two parts consider only a homogeneous and isotropic Universe without taking account of t he structure present in t he Universe. This standard model will be our reference model. We will discuss its successes and its problems. The study of the large-scale structure of the Universe is developed in Chapter 5, which is the undeniable technical heart of primordial cosmology. Cosmological perturbation theory describes how initial perturbations evolve in an expanding Universe.
4
Introduction
High energies • Theoretica l arguments
• SUS¥, SUGRA
• Superstrings
Mode ls
1 2
Primordial cosmology
Solutions
1
Parameters
Observational cosmology • Large sca le stmctures
• Microwave background • Dark malter
Fig. 2 Primordial cosmology constructs phenomenological models t hat are inspired fro
high-energy physics in order to describe the Universe. It looks for mechanisms able t o describ the primordial phases of our Universe, t he origin of large-scale structures and of the matt e content of our Universe. Each model must produce observational predictions. In return, ob servational cosmology makes it possible to measure t he parameters of these models and t set, in the b est cases, some constraints on the initial t heories.
We will establish the equations of this theory and discuss its solutions within som regimes. Our book does not focus much on the observational aspects of cosmology, nev ertheless Chapters 6 and 7 are dedicated to the computation of two fundament observational predictions. Our aim is to make the reader able to compute these pre dictions within a given model. Chapter 6 describes the computation of the angula power spectrum of the cosmological microwave background temperature anisotropie and Chapter 7 focuses on gravitational lensing effects and on the proofs of the existenc of dark matter. Chapter 8 develops the inflation paradigm . This paradigm is fu ndamental to ex plain the origin of the large-scale structure and it appears to be not only consisten with all observations, but also to be the only mechanism so far capable of reproducin these observations. The hot Big Bang model associated with inflation provides ou reference model of the Universe. This chapter describes the general properties of infla tionary models and analyses in detail the computation of the primordial fluctuation spectrum , mainly within the slow-roll approximation. We will also address the rehea ing mechanisms that make t he transition between the inflationary Universe , the ho Big Bang, and the idea of eternal inflation.
Warnings
5
The third part addresses different extensions of these standard formalisms, for both the fundamental theories and the cosmological framework itself. Chapter 9 focuses on grand unified theories and on their consequences for the understanding of the origin of ordinary matter , the so-called baryogenesis . Chapter 10 describes different extensions of the theoretical framework that have consequences for cosmology. We first describe the scalar-tensor theories of gravity. These theories represent the simplest extension of general relativity and include, besides the graviton, a scalar interaction. These models are motivated by numerous high-energy theories, such as string theory, for which the low-energy action is usually a scalar-tensor theory. We then describe the basis of quantum field theory in curved space-time. This will place the computations made in Chapter 8 on more solid ground. Finally, we will describe supersymmetry and supergravity as extensions to the standard model of particle physics. These theories have many candidates for dark-matter particles and have many scalar fields , which makes them relevant for the elaboration of inflationary models. Chapter 11 addresses the issue of phase transitions during the primordial Universe and describes models of topological defects generated during phase transitions with symmetry breaking. These relics can play an important role during the early Universe even if we now know that they cannot be responsible for the origin of the large-scale structure as we observe it. The two last chapters describe the different implications of these extensions for cosmology. Chapter 12 draws the consequences from supersymmetry for the construction of inflationary models. We will then address the question of the origin of the acceleration of our Universe. Chapter 13 focuses on the phenomenology of string theory for the primordial Universe. These subjects are still currently debated by researchers and our aim is to sketch their motivations and characteristics.
Warnings In order to avoid an overflow of references in the text, we have made the choice of citing in this book mainly detailed reviews and texts that we judge to be clear and pedagogical or where a complete bibliography can be found . As for the numerical values of the cosmological parameters deduced from different analyses, these values should be taken with great care, as they depend: (1) on the set of chosen observations, (2) on the model we compare with these observations and for instance on the number of parameters allowed to vary, and (3) on which kind of statistical analysis has been used. The values we quote should be considered more as rough indications rather than the latest values to remember. We also point out that observations are developing rapidly such that some conclusions, for instance on dark matter , may change on the time scale of a year .
References For a discussion on the background of the formula tion of cosmological models:
[1] G.F .R. ELLIS, 'Cosmology and verifiability', Q. J1. A str. Soc. 16 , 245- 264 , 1975 [2] H . BONDI, Cosmology, Cambridge Monograph on Physics, 1961. [3] G .F.R. ELLIS, 'Issues in the philosophy of cosmology' , astro -ph/0602280. Introductory books for cosmology:
[4] E.R. HARRISON , Cosmology, the science of the Universe, Cambridge Universit Press, 2000. [5] J. BERNSTEIN , Introduction to cosmology , Englewood Cliffs, 1995. [6] A. LIDDLE, An introduction to modern cosmology, John Wiley and Sons, 1999. [7] B. RIDDEN , Introduction to cosmology, Addison Wesley, 2003. [8] S. WEINBERG, The first three minutes, Bantam Books, 1977. [9] E.R. HARRISON, Darkness at night: a riddle of the Universe, Harvard Universit Press, 1987. Books of the history of modern cosmology:
[10] H. KRAUGH, Cosmology and controversy, Princeton University Press, 1996 . [11] J .P. L UMINET, A lexandre Friedmann, Georges Lemaitre, Essais de cosmo1ogie Seuil, 1997. Reference books in cosmology:
[12] P.J. E. PEEBLES, Physical cosm ology, Princeton University Press, 1971. [13] P.J. E. PEEBLES, Principle of physical cosmology, Princeton University Press 1993. [14] S. WEINBERG, Gravitation and cosmology: principles and applications of the gen eral th eory of relativity, John Wiley and Sons, 1972. [15] E.W. KOLB and M.S. T URNER, The early Universe, Addison Wesley, 1993. [16] T. PADMANABHAN, Str uct ure form ation in the Universe, Cambridge Universit Press, 1993. [17] S. DODELSON , Mo dern cosmology, Academic Press, 2003. [18] J . RICH , Fundam en tals of cosmology, Springer, 200l. [19] J.A. PEACOK, Cosmological physics, Cambridge University Press, 1998. [20] A.R. LIDD LE and D.H. LYTH, Cosmological inflation and large-scale structure Cambridge University P ress, 2000. [2 1] V.F. MUKHANOV , Physical foundations of cosmology, Cambridge University Pres 2005.
References
7
For astrophysical complements: [22] T. PADMANABHAN , Theoretical astrophysics (3 volumes), Cambridge University Press, 2002. [23] J. BINNEY and S. TREMAINE, Galactic dynamics, Princeton University Press, 1987. [24] M.S. LONGAIR, Galaxy formation, Springer-Verlag, 1998.
Part I Overview of the theoretical basis
1 Genera I relativity Since gravity is the only long-range force t hat cannot be screened , the properties of the Universe on large scales will be essentially determined by this for ce. The goal of this chapter is not to present a complete and rigorous description of the theory of general relativity. Such detailed explanations exist in the literature and we refer the reader to them for more details; see Refs. [1- 5J. Here we want to recall the basic hypotheses and formulas of general relativity that are necessary for the rest of this book. We recall in Section 1.1 the evolution of the mathematical concept of space-time from Newtonian physics to general relativity. Section 1.2 summarizes the main definitions of the mathematical tools used in general relativity. We derive Einstein equations that determine the dynamics of space-time and the conservation equations that determine the evolut ion of matter in Section 1.3 and we show in Section 1. 5 how Newtonian gravity is recovered in a weak-field limit. Section 1.4 describes the kinetics in curved space-time and we summarize the various tests of general relativity in Section 1.6. The other fundamental interactions, such as electromagnetic, weak and strong interactions, are discussed in chapter 2.
1.1 1.1.1
Space-time and gravity Absolute space and time of Newtonian physics
In Newtonian physics, space is described by an absolute and immutable mathematical space. This space is E uclidean in 3 dimensions. We can therefore endow it with an origin and with 3 arbitrary reference axes, thus determining an absolute frame of reference. Time is also ideal a nd absolute. It is independent of the motion of any observer and plays the role of an external parameter. For any event P, there exists an int uitive notion of simultaneity (defined as a set of all events that occur at the same t ime). If we consider a second event Q there are three possible outcomes: (1) one can in principle go from P to Q, Q therefore belongs to the future of P , (2) one can in principle go from Q to P, Q therefore belongs to the past of P and (3) it is impossible to travel between P and Q, t he two events are t herefore simultaneous. T his causal structure of Newtonian space-time is represented in Fig. 1.1. Since space is assumed E uclidean, P ythagoras ' theorem would permit us t o calculate the distance between two neighbouring points (1.1 )
12
General relativity
Time
Fig. 1.1 The causal structure of Newtonian space-time. If Q can be related to P by trajectory Xi(t) with finite speed (dxi/dt < (0) , Q is in the past or the future of P, otherwis it is simultaneous. The set of all simultaneous events is a three-dimensional Euclidean space
in Cartesian coordinates. We can rewrite this distance in shorter form as 3
d£2 = L(dxif = L6ij dx i dx i= l
j
== 6ij dx i dx j ,
(1.2
ij
where 6ij is the Kronecker symbol equal to 1 if i = j and 0 otherwise, and th latin indices i , j ... = 1 .. . 3. This form introduces the Einstein summation conventio according to which we implicitly assume a sum over all repeated indices. For example if T and V are two vectors, T· V == Ti Vi = 6ij T i Vj = T1 V 1 + T2V2 + T 3 V 3 is th scalar product of t hose two vectors. The trajectory of any body is therefore given in the parametric form by xi (t). Th travel time from point A at tA to point B at t B is given by tA - t B and is independen of the trajectory between A and B. Laws of physics do not necessarily require Cartesian coordinates. One can , fo example, use spherical or cylindrical coordinates if they are better adapted to th problem at hand. Let us suppose that we have a system of coordinates (yi) relate to Cartesian coordinates (xj) by a relationship of the form x j (yi). The vector d of coordinates dx i in the Cartesian syst em has coordinat es dyi = (8yi/8xj)dxj; w therefore deduce that the distance between two neighbouring points, in any coordinat system , is
(1.3
gij being the metric of space in the new coordinates . For example, spherical coordinate (r , cp) are defined by
e,
x
= r sin ecos cp , y = r sin esin cp,
z
= r cos e.
Applying (1.3) , we deduce that the only non-zero components of the metric in spherica coordinates are
Space-time and gravity grr
= 1,
gee
= r2 ,
g
13
. 28 = r 2 SIn .
The line-element (1.2) takes the form
1.1.1.1
Galilean group
The choice of the reference frame is arbitrary. It is thus interesting to exhibit the frame transformations that preserve the form (1.1) of the line element dC 2 . It is equivalent to finding the coordinate transformations keeping the 3 axes of reference orthogonal. The set of these transformations is therefore given by the rigid rotations of these three axes and a translation of the origin (1.4) The rotation Aj can be time dependent, but it is rigid because it is independent of the position x j ; it is therefore the same for all points at a given time. Tj is a timedependent translation. The invariance under the action of these transformations (1.4) reflects the isotropy and the homogeneity of E uclidean space. Among all these rigid transformations , those of the Galilean group playa central role. The Galilean group is the subgroup of t ime-independent rotations and of translations linear in time, Ti = To-vit , where Td and Vi are constant vectors , hence preserving the equivalence between inertial frames. The laws of physics are assumed to be identical for all inertial reference frames , which means that they are invariant under the action of a transformation of the Galilean group (1.5) This group is composed of three rotations , three translations and three changes of the inertial reference frames. Taking also the possibility to change the origin of time, it is therefore ten parameters group (three angles , a vector with three translation coordinates, one velocity vector and a change in the origin of t ime). 1.1.1.2 Trajectory of a massive body The equations of motion of a body of mass m evolving in a gravitational potential ¢ can be obtained by determining the extremum of the Lagrangian constructed from the total energy of the particle (1.6) where we have considered that the inertial m ass, m[ , and the gravitational mass, m G , could a priori be different. The Euler- Lagrange equations [c5L / c5xi d(bL/bxi)/dt] are rewritten in any coordinate system as l ISee the following chapter for a brief reminder of analytical mechanics and Ref. [5] for a complete and detailed study.
14
General relativity
(1.7
In Cartesian coordinates, gij = Oij and ffj = O. We therefore recover the equations o motion in their classical form x = - V ¢;, when m[ = mG' Symbols ffj represent the fictitious forces (centrifugal, Coriolis ... ) that have to be taken into account when the reference frame is not inertial. Also , note that (1. 7) is independent of the mass of the body if m[ = m G ; all bodie fall identically, whatever their mass or chemical composition. This is what is called the weak equivalence principle or universality of free fall. Actually, in the Newtonian case it is a priori possible to introduce two different masses and the equality between the gravitational mass and the inertial mass seems fortuitous. Using pendulums, Newton (1686) showed that ImG - m[ I
m G+ m [
<
10-
3.
(1.8
The equality between the gravitational and the inertial mass was therefore only required up to one thousandth. However, Newton considered this equality to be the crux of his theory of gravitation and he dedicated the introduction of his Principia to it. As we will see, this property plays a central role in the construction of theories o gravitation. 1.1.2
Space-time of special relativity
The Maxwell laws of electromagnetism have the property of not being invariant unde the Galilean group (1.5). For example, the electric force generated by a charge at res is not invariant since a magnetic force appears in a reference frame where the source is in inertial motion (see, for example, Ref. [6]). On the other hand, Maxwell deduced from his theory that light is an electromagnetic wave whose velocity of propagation, c depends on the permittivity and the permeability of the medium. One had to introduce a fictitious medium in which these waves could propagate; the aether. The Maxwel equations would only be valid in this particular reference frame, identified with the absolute reference system of Newton. In a different inertial reference frame the speed o light should therefore have been measured as c ± v, which opened up the possibility o determining the absolute reference frame , that of the aether , at the price of abandoning the principle of Galilean relativity. 1.1 .2.1
Lorentz transformations
The experiments by Michelson and Morley (1887) refuted the above-mentioned hy pothesis; light was shown to have the same speed of propagation in any given inertia reference frame. In 1905, Einstein changed this perspective by reaffirming the principle of special relativity according to which all the laws of nature should have the same form , no matter what the inertial reference frame is (see for example Ref. [7] for the historical aspects). In particular, this should have been the case for the Maxwell equations, which implied that the speed of light should be the same in all inertial reference
Space-time and gravity
15
frames. Consequently, one deduced that the rules for changing the inertial frame can not be those of Galileo (1.5). The new group of transformations is the Lorentz group. An inertial observer can label an event by considering a rigid Euclidean reference frame, which allows him to determine the Cartesian coordinat es x, y , z . At every point in that grid, he can place a synchronized clock in such a way that all events are labelled by a quadruplet (t, x, y , z). A second observer in an inertial reference frame moving at a velocity v along the Ox axis with respect to the first observer can also construct such a coordinate system . It permits him to identify the same events by another quadruplet (tl ,xl , yl ,ZI). The two sets of coordinat es are relat ed by the Lorentz transformation , called a boost , ctl =
Xl
yl
=
=
ct - (v/c) x
J1 - v2 /c 2 ' X -
(v/c)ct
J1 - v2 /c 2 ' y,
Zl = Z.
(1.9)
These transformations have the property of reducing to Galilean transformations when v « c and keep the speed of light equal to c in all inertial reference frames .
1.1 .2.2 Minkowski space-time Special relativity postulates that all inertial observers are equivalent , the sets of coordinates determined by different observers have no intrinsic meaning. The invariant quantity upon changing the reference frame is obtained by considering a pseudo-Euclidean version of P ythagoras ' theorem. The line-element (1.1) is replaced by the line-element between two events, in Minkowski coordinates, of space-time (1.10) with x O == ct. The Greek indices vary between 0 and 3 and the Einst ein summation convention is extended to these values. We note that contrary to the Euclidean metric, ds2 can be negative or vanish for non-coincident events. In this fr amework, space and time are united into a space-time, the Minkowski space-time. This space-time is absolute, just as in the Newtonian framework. It is endowed with four orthonormal axes constituting the absolute Minkowski reference frame. The set of points such that ds 2 = 0 is known as the light cone and represents , for all points M in the space-time, the ensemble of points that can receive a light beam emitted in M (future light cone) or received in M (past light cone). This light cone determines the causal structure of Minkowski space-time illustrat ed in Fig. 1.2. Indeed, since the velocity of all bodies is smaller than c, only pairs of points (M, P ) such that ds 2 < 0, in other words such that M is at the interior of the light cone in P, can be joined by traj ectories ema nating from P. The dist ance is then time-like and the proper t ime T is generally introduced by the relation (1.11)
j
-----.. UNIVERSIDAD . ',
j
\
1
AU'TONOMA DE
MADPJO ru!3tlOr~ GEI'~a.~
16
General relativity
If ds 2 > 0, t he interval is said t o be space-like and the points of t he couple (M, P cannot be joined by any physical trajectory.
Fig. 1.2 T he causal structure of Minkowski space-t ime. The not ion of simultaneity is re placed by the notion of a light cone defined by d s 2 = O. T he worldline Xi>( A) emanating from P is found at t he interior of t he light cone (d s 2 < 0) . All points exterior to t he cone can neve be joined by a physical t rajectory leaving from P. They are separated from P by a sp ace-lik interval (d s 2 > 0) .
1.1 .2.3
Proper time
The trajectory of any massive part icles can be represented in the parametric form xl-' (A ) , where A is an arbit rary paramet er (increasing towards infinity) . We can repa rameterize t his worldline as a function of the proper time
where U I-' = dxl-' I dA is t he tangent vector to t he worldline. In t his new parameteriza tion , t he t angent vector is given by u l-' = dxl-' I dT and it satisfies
Let us now consider an observer 0 ' that moves wit h a velocity v with respect to a inertial reference frame, 5 , where an inertial observer 0 is locat ed . As 0 is motionless its t rajectory is given by Xi = const and its proper t ime is dT~= dt2.
The moving observer 0 ' has a tra jectory such that dx i I dt = reference frame 5. Its proper time is thus
Vi
with respect to th
Space-time and gravity
2 2 i ' 2 dTO' = dt - b;j dX dx J Ie =
(
17
V2 ) dt.
1 - e2
2
In conclusion, the proper times measured by two observers 0 and 0' , obtained by eliminating t he time coordinate, are related by
dTO' =
V~ 1 - ~ dTO·
(1.12)
Thus, contrary to the Newtonian case, t he duration of the journey measured by 0 is not the same as that measured by 0' , depending on their t raj ectories. If the clocks of o and 0' have been initially synchronized , they will not be synchronous at another meeting point. This is what is called the phenomenon of time dilation.
1.1 .2.4 Poincare group Just as in the Newtonian case, the metric (1.10) can be written in any coordinate system (ylt) that is related to Minkowski coordinates (XV) by a relation of the form XV (ylt) . A vector dx v in Minkowski space has the coordinates dylt = (aylt I ax 1.1 )dxl.l . We thus deduce that the line-element (1.10) has the form
(1.13) For example, we can consider the Rindler coordinat es, relat ed to Minkowski coordinates by
t = RsinhT,
x = R cosh T ,
y
=
y' ,
z = z' .
The line-element (1.13) then takes the form
The laws of physics should be, according to the principle of special relativity, independent of any preferred inertial reference frame. It is therefore important to exhibit transformations of the reference frame that preserve the form (1.10). These particular transformations constitute the Poincare group and take the form
(1.14) where the rotation matrices A ~ and the vector Tit do not depend on the coordinates. Note that this group of transformations is more constrained than the Galilean group in which the rotations could depend on time. This Poincare group consists of 3 rotations , three Lorentz boosts and four translations. It therefore has 10 degrees of freedom , which means that 10 values are needed to completely describe any transformation of the group.
18
General relativity
1.1.2.5
Poincare algebra
The Poincare group can be constructed using infinitesimal transformations. Let u first consider a simple transformation of vector TI-' = 5xl-' and write the transformatio (1.14) for a scalar function f(x) in the form f (yl-') = (1 - i5x cx Pcx ) f (xl-'). This relatio defining the generator PI-" we get PI-' = i0l-' by identification with a first-order Taylo expansion of the function f(y). Equivalently, a Lorentz transformation can be writte yl-' = xl-' + w~xv, with WVI-' = -wI-'V, and the generator JI-'V is obtained using th definition f (yl-') = (1 - iw cx {3 J cx {3 ) f (xl-') , which gives JI-'V = i (x l-'0V - xv0l-'), Other representations of the Poincare group are possible, but all of them respec the following commutation relations , which form the Poincare algebra:
(1.15 for the translations, and
(1.16
which is the Lorentz algebra, itself independently closed. Let us recall that the com mutator between two operators is defined by [A,B] = AB - BA.
1.1.2.6
Trajectory of a free body
In an analogous way to the Newtonian case, the trajectory of a massive free body ca be obtained by ext remising the action L
=
1
-g 2 I-'V (XA):i;I-':i;V ,
(1.17
where a dot above a variable represents a derivative with respect to A. Euler- Lagrang equations then take the form
(1.18
where ul-' = dxl-' Id A and ill-' = dul-' IdA. In Minkowski coordinates, we find that a inertial observer must follow a trajectory of constant velocity
(1.19
Note the analogy with the massive body case in (1.6) , but also the difference: in the cas of (1.17), it is not possible to simply introduce a t erm equivalent to the gravitationa potential.
1.1.3.1 Equivalence principle Einstein based his analysis on the fact that in Newtonian gravity all test objects fall in exactly the sarie way in an externai gravitational field, whatever their mass or chemical couipositiorr.
In Galilean and Newtonian physics, this universality of free fall conles from the eqrrality betwcen the inertial rriass arrd the gravitational mass, the equivalence principle [cf. (t.8)]. This equality may seeni accidental, but it is verified experimentally with high accuracy. In the Ner'vtonian casc. a smail deviation fiom this equality would not be catastrophic. Is this equivalence principle a good approximation, a simple coincidence or does it signal something more profound? Einsteiri corisiderecl the study of accelerated motion. At first glance, it could seem that objects are following clifferent laws of physics, since one rreeds to add fictitious forces (inertia, Coriollis...) to describe their dynarrics. For Einstein. these forces are just as real as the force of gravity since the iatter can also be eliminated in a free-fall reference fra,rne. Thus, an accelerated observer is subiect to the same laws of physics as those encor.rntered in the gravitational field, introduced in addition to the other forces. Everything happens as if one could free orreself from gravity by an astute choice of the reference frame. This is possible only if the equivalencc principle is strictly valid. Eirrsteirr thelefbre elevated it to the status ofa first principlc. This property teaches us that the 'confusion'
between gravity trnd acceleration is not accidental: it is a fundamental property of gravity. Note that tliis property only applies to gravity and to no other force of nature (in thc case of electrornagnetism, for instance. the acceleration in a given electric field rn'ould depend on thc ratio between charge anci rnass, which varies from particle to particle). The equivalence principle errsures that one can alu'ays find locally a reference frame in which the force of gravity is eliminated.
1.1.3.2 Tidal forces and space-time curvature
It
seerns as though gravit"l' has disappearecl. In fact, as we ernphasized, such an op* eration is possible only locally. As an example, let us consider a body A orbiting around the trarth at a distance rr. Its Newtonian acceleration is ao : -G*NI6rof r). A neiglibouring body B, orbitirrg according lo rt - rnl6r experiences a relative acceleration with respect to the body A
(d, _ 6ao, q41e r')\ri/ rt, _dtr,)
20
General relativity
at first order in Or . This relative acceleration cannot be cancelled out. If r A-or = body B approaches body A and if r A . Or = ±rAor, it moves away. Thus, a sm sphere in orbit would slowly deform into a type of ellipsoidal cigar , while conserv its volume. This distortion characterizes the tidal forces. We have seen that in the Newtonian and Minkowskian frameworks , the trajector a free particle corresponds to a path that minimizes t he length of a trajectory betw two points. Even though we can eliminate gravity locally, neighbouring trajector have a tendency to converge or diverge from each other. In a flat space-time, o one body can free itself from gravity and the fictitious forces. However, in cur space-time it is possible to consider that all the bodies are free and follow the line minimum length. Gravity and fictitious forces disappear for all bodies at the expe of postulating a new property of space. In this framework , all bodies follow the lines of shortest path and gravity has loc vanished for each one of them. However , since this is possible only locally, neighbour geodesics reveal the curvature of space-time. Gravity has therefore not disappeared is simply hidden in the curvature of the space-time, which explains the tidal forces a novel way. This heuristic example shows us that by introducing a curved space-time we eliminate all the fictitious forces related to the choice of a non-inertial reference fram at the expense of introducing a curvature for the space. The trajectory of all fr falling bodies, i. e. those subj ect to gravity alone, can be obtained as a geodesic, a of shortest path, in curved space-time. Accordingly, all reference frames, inertial not , are placed on the same footing, which allows us to reconcile gravity and spe relativity. This construction is however only possible because acceleration does depend on the mass of the particles in free fall, which provides a glimpse into relationship between t he universality of free fall and the geometrization of gravity.
1.1. 3.3
Gravity as a m anifestation of geom etry
Einstein's equivalence principle is at the heart of all metric theories of gravitati which includes amongst others the theory of general relativity. It is based on th conditions:
• the weak equivalence principle (or the universality of free fall) according to wh the trajectory of a neutral test body is independent of its internal structure a its composition. This body must have a negligible binding gravitational ene and be sufficiently small such that the inhomogeneities of the gravitational fi can be ignored; • the local position invariance according to which the result of all non-gravitatio experiments is independent of the point in space-time where the experiment to place; • the local Lorentz invariance according to which the results of non-gr avitatio experiments are independent of the motion of the laboratory as long as it is f falling.
One can argue (see, for example, Ref. [8]) that if the Einstein equivalence princi is valid then gravitation is the physical manifestation of a curved space-time, i.e
Elements of differential geometry
21
metric theory. Such a theory has t he following three properties: • the geometry of space-time is described by a metric, • free bodies follow the geodesics of that geometry, • in a local reference fr ame in free fall , the laws of physics t ake the same form as in special relativity. This implies that the contribution ofthe binding energies of the three non-gravitational interactions to the mass is the same for a gravitational mass and an inertial mass. It could be asked whether this property can be generalized to the gravitational binding energy itself. This leads t o the formulation of the strong equivalence principle. If the gravitational binding energy contribut es equally to the inertial and gravitational mass then it is possible t o eliminate t he external gravitational field. This can be achieved in a locally inert ial reference frame in which all the laws of Nature , gravitation included, have the same form as in the absence of an ext ernal field. This principle seems poorly defined and coarse: it implies a gravitational law that has as yet not been defined. A theory of gravitation that satisfies this principle is necessarily non-linear since the gravitational field 'weights' and generat es a secondary gravitational field , etc. General relativity and the Nordstr0m scalar theory are examples of theories that satisfy this principle (see Ref. [8]) . Since these principles are central t o the construction of general relativity, it is indeed importa nt to test them (see Section l.6).
1.2
Elements of differential geometry
The heuristic argument of t he preceding paragraph shows that we can describe the motion of a free-falling body (that is, a body on which no other force but gravity is acting) by a geodesic in a four-dimensional curved sp ace-time. Space-time is therefore described by a four-dimensional continuum so that one would need four values t o locate each event , just as in special relativity. In the Newtonian framework and in special relativity, this is assumed to be globally valid so that an ensemble of events can be mapped bijectively with JR.4. In general relativity we do not want t o make such a strong hypothesis about the structure of space-time before having determined it. This situation is analogous to the description of a sphere in two dimensions: we need two numbers to characterize the position of all points. Locally, the sphere looks like a plane JR.2 but globally it does not . The space-time of general relativity will t herefore be modelled as a four-dimensional manifold, that is, a space with four dimensions that locally looks like JR.4 but not necessarily globally (like in the case of a sphere in two dimensions). 1.2.1
Manifolds and tensors
Here, we define the notions of manifolds and tensors in a formal way. The two following sections can eventually be skipped and the reader can directly go t o the rules of tensorial calculus derived in the t hird section. 1.2.1.1 Manifolds To define a manifold , let us recall t hat an open set of JR.n is a set that can be defined as the finite union of open balls of JR. n . Such a ball of radius r centred at y = (Yl , . . . , Yn )
22
General relativity
is defined as the set of points x such that Ix - yl < T . A manifold is a space consisting of neighbourhoods that are locally like IR n an that can be continuously glued together. More precisely, a manifold M is a collectio {Od of sets that satisfy the following three properties:
1. for all points p of M, there exists at least one set Oi containing p , which mean
that {Oi} covers M, 2. for all i, there exists a homeomorphism 'lfJi from Oi into a subset Ui of IR n , tha is a continuous map that associates all points p of Oi to n-tuplet of real number that we shall call coordinates of p in the map 'lfJi, 3. if two sets Oi and OJ have a non-empty intersection, then the map 'lfJj 0 'lfJi that maps the points of 'lfJi( Oi n OJ) C Ui into 'lfJj (Oi n OJ) C Uj is an infinitel differentiable function on IRn.
Examples of manifolds are the Euclidean plane 1R2, which needs only one chart (iden tity), or a sphere that requires two charts.
M
Ij/j
Fig. 1.3 Illustration of the relationship between different charts of the same manifold. Lo cally, one can relate M and IR n but globally this is only possible if we use a collection o charts (an atlas) that cover all the manifold.
1.2.1.2
Vectors
The Euclidean space in Newtonian physics is a vector space. However, this structur is lost in curved spaces, such as that of a sphere, but can be recovered in the limit o
Elements of differential geometry
23
infinitesimal displacements. In lR. n there is a bijective map between vectors and directional derivatives and each vector y = (yl, ... , yn) defines a derivative yJ.1.(%xJ.1.), in any arbitrary coordinate system (xJ.1.). In the manifold M , consider the set F of Coo functions of M in R We can then define the tangent vector y at a point p in M as a map of F into lR., which is linear and satisfies the Leibniz rule 1. y (a ] + bg) = ay(J) + by(g ) for all ], 9 in F and all a, bin lR. , 2. y(Jg ) = ](p)y(g) + g(p)y(J).
The set of tangent vectors at point p forms a vector space Vp [since (Yl + Y2)(J) = Yl(J) + Y2(J) and (ay)(J) = ay(J)] of the same dimension as t he manifold. We can construct a basis {X J.1.} of Vp by considering the map from F to lR. defined by
We usually call XJ.1. simply %xJ.1. or 0J.1. ' such t hat all vectors y can be expressed, using the Einstein summation convention, in the form (l.20) where yJ.1. are the components of y. The tangent vector to a curve] with coordinates xJ.1. (T) is then given by the application
so that t = tIJ.0IJ. with tIJ. = dxIJ.(T)/dT . In a coordinate transformation of the form (l.21) we have, for all ] E F , 01/ ] = (OI/XIIJ.)(O~]). We thus deduce that the elements of the basis are transformed according to (l.22) so that the components of all vectors y transform according to (l.23) which is the standard formula for the coordinate transformation that we have used previously.
1.2.1.3 Tensors The notion of tensor is a generalization of that of a vector , when one considers quantities that have mult ilinear dependencies under an infinitesimal transformation.
24
General relativity
Fig. 1.4 The tangent sp ace of a two-dimensional sphere is a plane whose basis is given b ee = 8e and e'P = 8'P '
Given a vector space Vp, one can construct a dual space Vp* composed of the linea maps from Vp to R This new vector space has a basis ylJ.* that can be defined b specifying their action on the elements of a basis y IJ. of Vp
In particular, dx" represents the basis of Vp* associated with the basis OJ.' of Vp. dx should therefore satisfy dx" (01J.) = 6~
and it associates any vector y to its component y" defined by dx" (y) = y". In coordinate transformation , dxlJ. thus transforms as the components of a vector
(l.24
A tensor T of rank (r , q) is a multilinear map from (Vpy x (Vp*)q to lR such that
(l.25
where ® is the t ensor product. Such an object is said to be r times contravarian and q times covariant. In particular, a tensor of rank (0,0) is a scalar whose valu is independent of the system of coordinates, a tensor of rank (1 , 0) is a vector an a t ensor of rank (0,1) is a I-form. Using the transformation laws for vectors and 1 forms, we can deduce that in a coordinate transformation, the components of a tenso transform as
between two neighbouring points. The metric A should therefore be a symmetric and non-degenerate tensor of rank (0, 2). The line-element between two events is then given
by 1ds2
: grrd.rpdr"
(t.27)
,
where g* is the metric tensor; it is syrnmetric and has hence 10 components. From the previous results, we can deduce that it transforms as
0ro \tP , gp": Artt"
(1.28)
6*n9a0'
This expression is analogous to the one obtained in the Euclidean or Minkowski cases. In particular) we can deduce that the determinant of the metric g : detg1,,, transforms
o'-
[a"t
(#)]' '
(1.2e)
1.2.1.4 Tensor calculus Tensors are thus a generaiization of the notion of vectors. Tensors of the sarne rank can be added
nfl l; - sii i: + r1,"\,
,
and one can multiply tensors of different ranks to obtain a tensor of higher rank
Rll
y:
: sli !:r!:i,"1""
The indices can be contracted to form a tensor whose rank is lowered by
Ri:
i: sili] i{
2
s!,,,1,, ,1n,6i,,.
This operation reduces to sum over the repeated index. This allows us to define a scalar by contracting all the indices. In particular, TpV, - g*,VpT" represents the scalar product of two vectors. We conclude tha| Tlr'..,\' is a tensor if and only if its contraction with any tensor of the forrn RLtr...}, is an invariant. The metric can be r-rsed to raise or lower the indices Rltt...l-lrlt r,2...rc :
In Particular, Tt" - gtt'7,.
Rl,t
pr
rrrr...rngr,p
26
General relativity
A tensor is symmetric if it satisfies TI-'v = TVI-' and is antisymmetric if TI-'V = - TVI-' From any tensor, we can extract a symmetric and an antisymmetric tensor by
(1.30) This can be performed on one or several pairs of indices, or on all the indices. Fo instance , symmetrizing two indices of a vector of rank 4 gives 1
T(I-'laf3l v) =
"2 (Tw ,f3v + T vaf31-')
,
and a completely symmetric tensor is obtained by
In the particular case of a rank 2 tensor , it can always be decomposed as the sum o a traceless symmetric tensor , an antisymmetric t ensor and a trace
in four dimensions . We note that the completely antisymmetric tensor of rank n is given by
cI-'1l-'2 .. I-'n
(1.31 with Cl..n = A and c 1 following property caf3J-yc
aAVI-' _ -
.. n
,A ,v ,I-' -uf3 u J u-y -
= I j A. ,v ,I-' ,A uf3 uJu-y -
For n
,I-' ,A ,v uf3 uJu-y
=
4, the anti symmetric tensor has the
+ uf3 ,A ,I-' ,v + ,I-' ,v ,A + ,v ,A ,I-' uJu-y uf3 uJu-y uf3 uJu -y .
(1.32
Contracting this identity, we get
1.2.2
Geodesic equations and Christoffel symbols
Using the notions introduced in the previous section, we can go back to the trajectory of a free-falling body moving in a space- time with metric 91-'v· Such a traj ectory is a line of shortest length, such that its equation, Xl-'( A) , can be obtained by maximizing the action
s=
J
ds.
However, as already mentioned in the Minkowski case, ds can be negative or null for some traj ectories . It is therefore cleverer to maximize the action
s=
J
LdA ,
(1.34)
with xl-' = dxl-'jdA . This is equivalent to maximizing j'(ds 2j dA2)dA. We can show that extremizing these two actions comes to the same result if ds i- 0, but using the
Elements of differential geometry
second one allows us to get the correct result also for null geodesics 2 (ds Section 1.4.2). The Euler- Lagrange equations take the form
5L
d 5x'" - dA
27
= 0, see
(5L) 5i;'" .
Since
we have that
g/-,,,,x,,/-, -_ 2"1(o",g/-,vx' /-"V X - O{3g/-,,,,x.{3./-, X - o/-,g(3",X.(3./-,) x where we have symmetrized the term o{3 gw,i;{3i;/-'. Introducing the Christoffel symbols (1.35) and
'" r /-'v
= 9 A"'r A/-,V '
(1.36)
and contracting our equation with g"''', we get the geodesic equation
X",'
+ r/-'VAX.v X. A --
0
(1.37)
identical to (1.18) obtained in special relativity. As we have stressed, space-time curvature does not intervene in the equation of the trajectory of a single particle, but it does in the deviation of close trajectories. From the definition of the Christoffel symbols (1.35), we have, since the metric is symmetric, (1.38) as well as (1.39 ) Note that in four dimensions there are 4 x 10 = 40 Christoffel symbols, which is precisely the number of partial derivatives of the metric o",g/-,v , We can hence inverse the relation (1.35) to obtain, after some algebra, (1.40) Using the geodesic equation, we can convince ourselves that the Christoffel symbols are not t ensors since they transform as
o x '''' ox Cf oxP ox{3 ox'/-' ox,v "P
r'''' - - - - - - -r(3 /-' v -
02X'''' ox" ox P ox" ox P ox'll ox'v
(1.41)
under a coordinate transformation. 2Light-like geodesics are also called isotropic geodesics; we will avoid this terminology, which can be confusing . We will call them either null geodesics or light-like geodesics with no distinction.
28
General relativity
To finish , we consider the determinant of the metric
where 7r~e;:~ = + 1 if (af3ryb) is an even permutation of (p,l/PeJ) , -1 if it is an odd permutation and 0 otherwise. We thus obtain the derivative of the determinant
(1.42)
Another way to view this result is to write that dg = m/-'l.Idg/-,I.I' where m/-'I.I are the minors of the matrix of the g/-'I.I. The inverse of a matrix is g/-'I.I = (g /-,1.1) - 1 = m/-'I.I / g so we can deduce that oag = m/-'l.Ioag/-,1.I = gg/-'l.Ioag/-,I.I' We can then deduce the usefu property
(1.43) We can also point out another interesting property obtained using the fact tha = 4, and hence that oa (g/-'I.I g/-,I.I) = 0, that is
g/-'I.I g/-,I.I
(1.44) 1.2.3
Covariant derivative, parallel transport and Lie derivative
The laws of physics often take the form of partial differential equations. It is thus important to understand how a tensor field can be differentiated in a way that preserve its tensor properties.
1.2.3.1
Covariant derivative
The partial derivative of an arbitrary tensor field
is no longer a tensor, apart from the exceptional case of the derivative of a scalar tha is a I-form. To really understand this , let us consider the derivative of a vector T/and how it transforms under a coordinate transformation
(1.45)
The reason for this is that this operation makes the difference between the value o the t ensor at x a and x a + dx a and that at these points, the tensor has differen transformation laws. Consider a constant vector of the Euclidean plane, with coordinates TX = 1, TY = O. In polar coordinates, its coordinates depend on the position (TT = sin () , T e = cos ()) simply because the directions of the coordinate lines change from one point to the other. It follows that oeTT and oeTe do not vanish even if T is a constant. What we should define is a derivative such that the components of a constant vector do not change when it is transported from x a to x a + dxa.
1 Elements of differential geometry
29
The covariant derivative is the differentiation operator that parallel transports the tensor from x'" to x'" + dx'" b efore subtracting its value at x"'. Since this operation is linear, one can always define such a covariant derivative by
(l.46)
where the quantities C~'" should be determined. For that , we notice that (l.45) implies that these quantities should transform as (l.41) during a change of coordinate. Working locally in Minkowski space and noting that then \7 ",TJ.L = o",TIL, we conclude that
(l.47)
In the rest of this book, we will denote the covariant derivative either with the symbol 'V IL or by ; p, in subscript, with no distinction: \7 ILT" = T;~ . Similarly, we will use the notation 0ILT" = T,~ for the ordinary partial derivative. The relation (l.46) can be generalized to an arbitrary tensor
(l.48)
A consequence of this result is that \7 ", 9IL" = O",9IL" - r~IL90"" - r~,,90"/L" Using the expressions (l.36) for the Christoffel symbols and (l.40) for the derivative ofthe metric, we get \7 ",9IL"
= O.
(l.49)
To each metric, we can therefore associate a covariant derivative. Note that we can define various covariant derivatives but only one is compatible with the metric in the sense that it satisfies the property (l.49). 1.2.3.2 Parallel transport
The covariant derivative allows us to define the notion of parallel transport. A vector is parallel transported if DTIL == dx" \7 "TIL = O. We can use this condition to define the notion of parallel transport of a vector along a curve of equation x lL (T) by DTIL j DT = 0, i.e. by
(l.50)
where u lL = dxIL jdT. The example of the sphere presented in Fig. l.5 shows that the result of this operation depends on which path is followed . A special case of transport along a curve is the auto-parallel transport , i.e. a curve such that the tangent vector is transported parallel to itself; it should therefore satisfy
(l.51 )
This is simply the equation of the geodesic (l.37) so that we conclude that the geodesics (curves of shortest path) are therefore auto-parallel.
30
General relativity Path 1
Path 2
Fig. 1.5 (left) : On a 2-dimensional sphere, a vector can be transported parallel to itself a long a path 1 corresponding to a great circle until its antipodal point. We can reach the same point following the path 2 along the equator. The two vectors that we get do not coincide. (right): The Lie derivative evaluat es the change determined by an observer that goes from a point P to a point Q along the flow curves of the vector field ~!' transporting its local frame with it.
1.2.3.3
Differential operators
The covariant derivative allows us to define some mathematical operators such as the divergence and the curl. Using (l.35),we get that V f.LTf.L = of.LTf.L + r~pTP = of.LTf.L + TPo p In Fg, which can be used to express the divergence of any vector field by
V f.LTf.L
=
~0f.L (F9Tf.L)
y - g
.
(l.52)
Similarly, we can check that the divergence of any antisymmetric tensor is of the form
(l.53)
where we used the fact that the contraction of a symmetric tensor with an antisymmetric one vanishes. The d 'Alembertian operator is defined as 0 = V f.L Vf.L and can be applied to any kind of t ensor. In a Minkowski space-time, it reduces t o V f.L Vf.L = - 0; + ~ , where ~ = r5 ij OiOj is the usual three-dimensional Laplacian. Note that the relation (l.52) can be used to express the d ' Alembertian of any scalar by Vf.LVf.L¢
=
~ 0f.L (F9gf.L
y - g
V
ov ¢) ,
(l.54)
where we used the fact that V f.L¢ = 0f.L¢. Finally, t he analogue of the curl of a I-form Af.L is given by (l.55) Av;f.L - Af.L;v = A v,,, - Af.L ,v, since
r~v
is symmetric in /-LV.
Elements of differential geometry
31
1.2.3.4 Lie derivative Another way to construct a derivative is to consider the Lie derivative. For that we assume that there exists a vector field ~J.L and try to define how t he coordinates of a vector TJ.L will change for an observer that moves along the flow curves of the vector ~I' from a point P with coordinat es x'" to a point Q with coordinat es x'" + E~'" (where E is a small parameter) and that transports its system of coordinat es with itself (see Fig. l. 5). Using the coordinates defined in P at the point Q is equivalent t o performing a coordinate transformation in Q defined as x 'J.L = xl' ~ E~J.L , such that
and OX
V
rv
~ u x'J.L = u"~
to leading order in
E.
>:}V
+EU
~J.L '
The coordinat es of the vector TJ.L at Q will be given by
The Lie derivative is obtained by comparing this quantity with the vector TJ.L at P
£f, TJ.L
=
lim
~[T'J.L (Q) ~ TJ.L(P )],
€---> O E
so that (l.56) This relation can be generalized t o any t ensor by using the definition (l.56) and the relevant transformation laws p
£ €1',J.L1"· J.L P -_ <::,C"' o 0' 1',J.L1·"J.LP ""' 1',1'1 ""''' ·J.LpO0' <::'CJ.Li Vl"'V V l "'V ~ VI ' 'l/q q
q
i= l
q
+ ""'TJ.L1 "J.L p ~
VI ' .{3 "'V q
0V j <'"c(3 .
(l.57)
j= l
The Lie derivative plays an important role in the study of the symmetries of a Riemannian space. If £f,T,~ = 0 then T,~J.L ( X' ) = T;;(x') and the t ensor field is invariant under that change of coordinat es. In particular , for the metric t ensor , using (l.57) for a symmetric rank-2 tensor , the relations (1.40) and (l.38) , as well as the definition of the covariant derivative for a vector, we get (l. 58) 1.2.4
Curvature
1.2.4. 1 Curvat ure tensor As seen from F ig. l.5 , the parallel transport of a vector along a closed curve does not bring t he vector back to its initial stat e if the space is curved , contrary to what
32
Genera l relativity
dx~
Fig. 1.6 The vector T'" is parallel transported along the path PIP3 passing by either P2 or P4 to give , respectively, the vectors Tin and Ti43' These two vectors only coincide if the Riemann tensor vanishes.
happens in a Euclidean space. Just like for the geodesic deflection, this characterizes the curvature of space. Let us consider the parallel transport of a vector TI-" along the closed path shown in Fig. 1.6. From PI to P2 , the relation (1.50) implies that the variation of the vector TI-" is given by OI2TI-" = dx"' \7c;TI-" = dTI-" + r~aTadx"'. By iterating it, we get that the variation between PI and P 3 is given by 0123TI-" = dx"'dx,6 \7,6 \7 c; TI-" = d 2TI-" +2r~[/dx"'dT[/ +(8,6 r~a+ r~[/r~a)dx"'dx,6 Ta. The same variation passing through P 4 is given by OI43TI-" = dxc; dx,6 \7 '" \7 ,6TI-" . On comparing these two expressions, only the terms proportional to dx"'dx,6 are different and we get that (1.59) where the Riemann tensor is given by (1.60) The definition (1.60) allows one to conclude that the Riemann t ensor is antisymmetric in the permutation of the first or the second pair of indices (1.61) and that it is symmetric in the permutation of the two pairs (1.62) We can also show that it satisfies the identity (1.63) The two first symmetry properties imply that the Riemann tensor has as many components as a 6 x 6 symmetric matrix, ie. 21 components. The last relation (1.63) is
Elements of differential geometry
33
independent of the first ones and thus adds one extra constraint t o the number of components . The Riemann t ensor therefore has 20 independent components. Finally, the derivatives of the Riemann tensor enjoy a cyclic symmetry property that is (1.64) 3RaJ.L [va; J3l = R a J.Lva ;f3 + R a J.Laf3;v + RaJ.Lf3 v;a = 0, called the Bianchi identity. 1.2.4. 2 Ricci and Einstein tensors The symmetry properties of the Riemann t ensor imply that we can construct only one tensor by contraction of two indices. This is the Ricci tensor , which is symmetric and defined by (1.65) The Ricci t ensor has 10 independent components. Its trace defines a scalar , the scalar curvature, also known as the Ricci scalar (1.66) Contracting the Bianchi identities (1.64) , respectively, by g af3 and g af3 gJ.La implies that We can combine these two equations t o construct a conserved tensor for the covariant derivative GJ.LV = RJ.Lv -
1
2RgJ.Lv.
(1.67)
This tensor G J.LV is called the Einst ein t ensor. 1.2. 4.3
Killing vector fields
A vector that satisfies the Killing equation
is called a Killing vector; it keeps the metric invariant and therefore corresponds to a space-time symmetry. From the rela tion (1.59), we conclude that (1.58) implies that \1 a \1 f3~v = RJ.L af3 v~J.L" T his shows that from the value of ~v and \7 J.L~V at a given point one can determine the Killing vector field uniquely. One should then specify N values for ~v and (N - 1)/ 2 values for \7 J.L ~'/' keeping in mind it is ant isymmetric , so that there are at most N(N + 1)/2 linearly independent Killing vector fields. For N = 4, this is 10 Killing vectors, which is precisely the dimension of t he Poincare group of Minkowski space. Indeed , t his maximum number of Killing vector fields is not always reached in an arbitrary space-t ime since the Killing equations are not necessarily integrable for any choice of initial conditions; most space-times have no symmetries and no Killing vector fields!
34
General relativity
A space-time enjoying the maximum number of Killing vector fields is called a maximally symmetric space-time. It can be shown from the Killing equations and (1.59) that the Riemann tensor must then satisfy (see , e.g., Ref. [2] for a demonstration)
(N - l)RI-' ",j3v so that
N
R~ =
= R"'v6~
R N(N _ 1)
- R",j36~ ,
(R"'v6~ - R",j36~) ,
the Ricci scalar, R, being a constant. Maximally symmetric spaces are thus spaces of constant curvature. For N = 4, there are three maximally symmetric space-times: Minkowski , de Sitter and anti-de Sitter. See also the discussion in Section 3.1.3 o Chapter 3. 1.2.5
Covariant approach
We consider a family of time-like geodesics representing the worldlines of typical observers of the Universe (the fundamental observers). In the case of a fluid , we can think of this flow of geodesics as the set of worldline fluid elements. Each observer has a tangent vector to its worldline, ul-' = dxl-' /dT . This vector is time-like and satisfies 3
A more detailed description of this covariant approach can be found in Ref. [9].
1.2.5.1
3+ 1 decomposition
With the vector ul-' one can define two projection tensors
(1.68) along and perpendicular to the vector ul-', respectively. We have Uj)U~
= Uf;,
ut = 1,
Uj)u v
,~=
,!:u
and ,!:,~
=
,~,
3,
v
= ul-' , = o.
These two proj ectors allow one to define a natural notion of space and time for an observer 0 since the line-element takes the form
(1.69)
Any vector vI-' can be decomposed into 'time' and 'space' parts, relative to the vector ul-' as vI-' = vVUj) + vi, vi = vV,!: . In particular, a particle of mass m has momentum
3We now work in a unit system such that c = 1. This equation would otherwise take the form u!'u!' = _ c 2
Elements of differential geometry
35
The observer 0 will measure the energy of this particle to be (1. 70)
so that E2 - P.l!-,pi = m 2 . Note that if this particle does not follow a geodesic, then its equation of motion is (1.71)
where F!-' represents any non-gravitational force applied on the particle. Here , the last equality stems from the fact that v!-'v!-, = - 1.
1.2.5.2 Spatial and time derivatives We can now define a derivative along a worldline , tangent to T cx (3 ... !-,V...
u!-'
by
cx (3 .. . = uAV AT!-,v .. .
It corresponds to a derivative with respect to the proper time. We can also define a derivative in the hypersurface orthogonal to u!-' by
flAT::!!
=
1';' (1'~, I'g, ... ) b~' I'~' ...) V
A,
T::,'t,'.
This derivative is a three-dimensional covariant derivative associated to the spatial metric (we can indeed check that Vcxl'!-'v = 0). However, note that for an arbitrary function p , V[!-' Vv i P = U [!-,;vJ P (recall u!-' is geodesic). Thus, this term vanishes only if U[I';vl = O. The derivative V is well defined only under this condition. We may note that in particular VI'T!-' = V!-,TI' + uI'TI'.
1.2.5.3 Distortion of the geodesic flow We consider a vector flow u!-' describing, for instance, a flow of matter. The covariant derivative of the vector field u!-' can be decomposed as (1. 72)
The trace e = V!-'u!-' is called expansion and describes the isotropic distortion, a!-' v is a traceless symmetric tensor called shear (a!-,vu!-' = 0, at = 0) and describes the distortion of the flow of matter, w!-'v is an antisymmetric tensor called vorticity that describes the rotation of the matter flow (w!-,vu!-' = 0). The vorticity vector is defined by WU _ -
1
2u cxc
and satisfies by construction are, respectively, defined as
cxu!-'v
w!-'u!-,
_ (3 U w!-'v {:==} W!-'V - U c(3u!-'vw ,
=
Uv is the acceleration and vanishes if
w!-'vw!-'
u!-'
=
O. The shear and vorticity amplitude
follows a geodesic.
36
General relativity
It will be useful to introduce a characteristic scale S (T) defined by
~S = ~8 3 '
which determines S a long each worldline up to an overall factor. The volume of eac element of the fluid varies as S3. 1.2.5.4
Raychaudhuri equation
e
We now calculate = uV\lv(\l J.LuJ.L) using the decomposition (1.72) as well as (1.59 to commute the covariant derivatives. This leads to t he Raychaudhuri equation
e -- _~3 82 + il,!-Li.t J.L + 2(w 2 -
0"2) + VJ.L il.J.L - R J.LV uJ.LUV .
(1.73
This equation is at the heart of t he study of gravitational collapse. Two other equation describing the proj ection of the vorticity and of the shear can be obtained in a simila way (see Ref. [9]). This equation has an important consequence. Let us suppose that the vorticit vanishes (w = 0) and t hat uJ.L = 0 for all times. It then follows that if RJ.LvuJ.LuV > for all times, we have that + 8 2 /3 ::::; 0 so t hat
e
lIT
--> -
8(T) - 8 0
+- . 3
• If 8 0 < 0, i. e. if it is a converging flow , then at a given time T ::::; - 3/8 0, 1/ 8 must g through zero and 8 should therefore diverge. The geodesic flow is hence pathologica This signals the presence of a caustic or of a singularity. • If 8 0 > 0 at a given time, we can express the Raychaudhuri equation as a functio of S to obtain that S / S ::::; O. There must therefore exist a previous time TO < 3/8 such that S - t 0 when T - t TO. This translates into the exist ence of a singularity i the past. This is a sp ecial case of the theorems on space-time singularities that is discusse in detail in Ref. [10]. We give in the following table the definition of several energ conditions that will appear throughout this book.
Name null
Definition G J.LV kJ.LF > - 0 \fk J.L, kJ.LkJ.L = 0
Meaning Light is focused by matter
causal
G J.LV uVGJ.LPu P < - 0 \fuJ.L , uJ.LuJ.L = - 1
Energy does not flow faster than the speed of light
weak
GJ.LVuJ.LUV 2: 0 \fuJ.L, uJ.LuJ.L = - 1
The energy density is positive for any observer
dominant
causal and strong
strong
RJ.LvuJ.LuV 2: 0
S
Elements of differential geometry
1.2.5.5
37
Case of a vanishing vorticity
If w = 0 for all times , then the hypersurfaces that are locally perpendicular to the flow ulJ. can be globally extended. 'YIJ.V is then the spatial metric (i.e. three-dimensional Riemannian metric) induced by glJ.v on 2:; and 'V' is the covariant derivative associated with 'YIJ.V, 'V' a'YlJ.v = 0 (see Fig. 1.7). The 'bending' of 2:; in the four-dimensional spacetime M can be characterized by the change of direction of the normal ulJ. when moved on 2:;. This is associated to the extrinsic curvature tensor
(1.74) Using (1.72) , it reduces to KlJ.v
A Riemann 4 tensor, equality 'V' a 'V' {3Vv we get that
=
1 38'YlJ.v
+ (JIJ.V·
'V' from (1.59). Starting from the = 'Y~''Yg''Y~''Va'h~''Y~,'VpVa) for a I-form Va on 2:; (i.e. vaua = 0) , (3) R a {3lJ. v
can be associated to
where we have used the fact that 'Y~'Y~ 'V a'Yf = K avulJ. . When taking the antisymmetric part (over ex and (3), the second term vanishes so that we finally obtain the relation (1.75) where we have used 'Y~ua'V pVa = - K3va to express the two last terms. This is the Gauss relation. Contracted on the two indices {3 and v it gives
and taking the trace, we get the scalar Gauss relation (1. 76) It relates the intrinsic curvature of 2:; to curvature. To finish, the relation (1.59) for Codazzi relation 'YpIJ. n a 'Yaa' 'Y{3{3'RP aa' {3' -_
the Ricci scalar of M and the extrinsic the vector ulJ. projected on 2:; leads to the nv {3 KIJ. a - nv a KIJ. {3
,
which gives, after contraction, the contracted Codazzi relation (1. 77) 41n three dimensions , the Riemann tensor is fully determined from the Ricci tensor and the scalar curvature as
38
General relativity
Using t he expression of t he extrinsic curvature, the scalar Gauss relation (l.76 implies that the curvature of I; is given by
(l. 78
which is valid for all space-times such that w = O. Such a flow is called irrotational.
1.2.5.6 Decomposition of rank 2 tensors The existence of the vector field uIJ- allows one to decompose any symmetric of rank tensor in a unique way, in the form
(l. 79 with
In particular, we have
1.3 1.3.1
Equations of motion Einstein's equations
Similarly to electromagnetism, we would like to formulate second-order equations fo
9IJ- v ' For that, we need a Lagrangian that depends on the metric and its first derivative
only. The Lagrangian proposed by Einstein and Hilbert reduces to the scalar curvatur Other choices are possible, such as any function of the scalar curvature or term involving other invariants such as the Gauss- Bonnet t erm, or even more complicate theories of 'scalar-tensor' type; all these cases are discussed and studied in Chapter 10 The 'simplest' choice is in perfect agreement with all observations and defines th theory of general relativity. Any other choice will be considered as an extension of th standard case. Thus, we focus on the action
(l.80
where K, is a coefficient to be determined by requiring that the theory reduces t Newtonian gravity in the weak-field limit; this is the only free parameter of the theor .c mat is t he Lagrangian for the matter fields and A is a constant called the cosmologic constant. We will now present the variation of these two terms.
Equations of motion
39
Fig. 1.7 A vector flow up. defines an orthogonal surface at each point of the flow. We can then define a global notion of space and time for this flow if and only if its vorticity vanishes .
1.3.1.1
Einstein- Hilbert action
We start by varying the Einstein- Hilbert action
where we have just used the definition of the scalar curvature. Expanding this expression, we get
To evaluate the first term, we should vary the metric determinant. Using the result (l.42), we get (1.81)
which leads to the conclusion
The second term is more complicated to evaluate. For that , we work in a locally inertial coordinate system , i. e. where gp. v = fl /-'v and o,gp.v = 0 from which we deduce that
40
General relativity
r~V = 0 (but note that the derivatives of the Christoffel symbols do not vanish). W hence obtain
To go from the partial derivative to the covariant derivative, we used the fact tha t he result is a t ensor , and hence it should be the same in any frame. We can henc conclude t hat
1v9l-'VFgb RI-'vd4x =
Iv Fg [(9I-'Vbr~v);0" - (9I-'Vbr~v);I-'] d4x,
on any volume V. The integrand is therefore of the form Fg 1~ , which can be r expressed, using the property (l.52), as (Fg 11-') ,/-," This is therefore the integral of total derivative, which can be reduced to an integral on the boundary of the volum V so that it does not contribute to the equations of motion. We therefore obtain
b J RFgd 4x = J FgGl-'v bgl-'V d4x ,
(l.82
where the Einstein tensor GI-'V is defined in (l.67). The variation of the term in A trivial and gives
J 2AbFgd 4x = - J AFggl-'vbgl-'V d4x. 1.3.1.2 Stress-energy tensor The variation of the matter action with respect to the metric is given by 4 bJL mat yC::-d -g X
-
4 J b(LmatFg) Fgbgl-'V b9I-'V yC::-d -g x,
which we may write as
b J Lmat Fg d 4 x =
-~ J
FgTl-'vbgl-'V d4x,
(l. 83
where the energy-momentum tensor TI-'v, defined by
(l.84 is a rank 2 symmetric tensor.
1.3.1.3
Einstein 's eq uations
Using the results of the two previous sections , we obtain t he general form of E instein equations
(l.85
Note t hat there is another way to vary the Lagrangian , known as the Palatin method , in which the metric and the Christoffel symbols are considered as independen
Equations of motion
41
Extremizing the action with respect to the two fields then gives Einstein's equations and the condition that V f-Lg(){(3 = 0, which is equivalent to the definition (1.35). 1.3.2
Conservation equations
The Bianchi identities (1.64) impose and the fact that gf-Lv;a = 0, that
Cr: = 0, which implies, using Einstein 's equations (1.86)
This equation is satisfied by the total energy-momentum tensor (i.e. of all matter components). To go further , one should specify the form of the matter Lagrangian or of its energy-momentum tensor.
1.3.2.1
Covariant form
The most general form for the energy-momentum tensor is the one given in (1.79) valid for any symmetric energy-momentum rank 2 tensor ,
in which p = Tf-Lvuf-Lu V is t he energy density measured by an observer at rest with the fluid , P = Tf-LI/,f-L V/3 is the pressure, qf-L = - T(){(3u(){,(3f-L is t he energy flux relative to uf-L and 7r f-LV is the anisotropic pressure tensor (7rf-LV uf-L = 7rt = 0) . The conservation equation V f-LTf-LV = 0 can be projected along uf-L and ,t:, respectively, to obtain t he two conservation equations
Using t he decomposition (1.72) , we obtain - U v V f-LTf-LV = uf-L V f-LP + Vf-Lqf-L + (p + P)G +2qf-Luf-L + (Jt:7r~ and , ;Vf-LTf-L1/ = + V Ap +V(){7r aA +~GqA +((J~ +w~) qa +( p+ A P)u +7r(){AU(){. We get that
,;rjl/
(1.87)
These equations are valid for any energy-momentum t ensor and any space-time geometry.
1.3.2.2 Perfect fluid The perfect fluid is an especia lly interesting case for which 7rf-LV = 0 and qf-L = o. The two previous equations (1.87) and (1.88) reduce, respectively, to the matter continuity equation and to t he Euler eq uation
p + (p + P)G = 0,
(1.89)
42
General relativity
(p + P)UI-' + '\7 I-'P
=
O.
(1.90)
The term ul-' = u (X V (Xul-' represents an acceleration that vanishes for a pressureless fluid (P = 0). Particles of such a pressureless fluid will hence follow geodesics. For an irrotational flow , using (1. 85) to express the Einstein tensor, (1. 78) is of the form 2 (3) R = 2KP - 382 + 20"2 + 2A., (1.91) that we will call the generalized Friedmann equation. Some conditions are often imposed on the pressure and the energy. The energy conditions introduced previously can be rewritten as • • • • •
null: p ~ - P, causal: Ipi ~ IFI, null and causal: Ipi ~ IFI and p < 0 if and only if p = - P , weak: p + P ~ 0 and p ~ 0, strong: p + 3P ~ 0 and p + P ~ O.
1.3.2.3
Electromagnetism
Electromagnetism is described by the anti symmetric tensor FI-'II ' called the Faraday tensor and satisfying (1.92) FI-'I1 can be defined as the curl of the vector potentia l AI-' as
(1.93) The energy-momentum t ensor can be derived from the action (1.94) which gives (1.95) The conservation equations then reduce to Maxwell 's equations in the vacuum (1.96) This equation can be rewritten in t erms of t he potential vector AI-' as (1.97) where we chose the Lorentz gauge , defined by VI-'AI-' = 0 and where the Ricci t ensor arises from the commutation of the covariant derivatives. A source term may be added by introducing a coupling term AI-'jl-' in t he Lagrangian to obtain (1.98)
Equations of motion
43
An observer with tangent vector uJ-L can define an electric and a magnetic field, respectively, by (1.99) These vectors satisfy EJ-LuJ-L tensor as
= BJ-LuJ-L = 0 and we can express the electromagnetic field (1.100)
Using this expression, we can decompose the energy-momentum tensor as the one for a fluid TJ-LV
wit h E2
1
1
= 2(E 2 + B2)uJ-L u v + "6(E 2 + B2}-yJ-Lv + 2U(J-LEv)paAupEaBA + 7rJ-LV,
= EJ-L EJ-L' B2 = BJ-LBJ-L and (recall 7rJ-LV is traceless)
We can then deduce that the energy density measured by a co moving observer is Pern = ~(E2 + B2) and that the equation of state for radiation is Pern = 1Pem. The trajectory of any particle of charge e and mass m in an electromagnetic field is of the form (1.101 ) Finally, we can re-express Maxwell 's equations, V J-LFVJ-L Projecting along uJ-L and along ry~ we get, respectively,
VJ-LEJ-L .-yAj;;J-L _Ea.Apa u a. VpB a IJ-L
= j V, in terms of EJ-L and BJ-L.
- 2wJ-LBJ-L = c,
(1.102)
= _ .-yAJ.v aAEv+ Ea.APau a. (u p B a +p WE) Iv _ ~eEA+ 3 v a , (1.103)
with c = - jJ-Lu!-," The same procedure applied to the equation F [J-Lv;a.] respectively, VJ-LBJ-L + 2wJ-LBJ-L = 0, .-yABJ-L +Ea.Apa u a. V pE a IJ-L
= _~eBA+aABv _ Ea.APau a. (u p E a - p w B) 3 v a .
=
0 gives , (1. 104)
(1.105)
Written in this form , these four equations give a relativistic generalization of the four Maxwell's equations , valid for any space-time geometry.
1.3.2.4
Scalar field
The action of a scalar field ¢ evolving in a potential V (¢) is given by (1.106) Varying this action with respect to the metric , we get the energy-momentum tensor as defined in (1.84)
44
General relativity
(l.107 for which the conservation equation is simply the Klein- Gordon equation
(l.108
We notice that the energy-momentum tensor of a scalar field can be decompose similarly to that of a perfect fluid with velocity field
(and q!-'
= 0 , 7r!-'v = 0) and with energy density and pressure p¢
1
2
= 21/J + V(¢) ,
We can also note that 8
P¢
1 2 = -1/J 2
v.
= - (eJ; + V')N, w!-'V = 0, a!-, = - (o!-'1/J + eJ;u!-')N an
o"!-'v = -,;,~C'V(>, \l8)¢)N
+ l!-'v\l",[1/J-l\l"'¢]/3.
In particular, it follows that th
conservation equations reduce to
p+8(p+P) = 0
{==}
¢+8¢+V'=0,
for any Universe geometry.
1.3.3
ADM Hamiltonian formulation
To finish, we shall rewrite the gravitational action in the 1 + 3 form provided by th Hamiltonian analysis by Arnowitt, Deser, and Misner [4] . We consider space-times that can be foliated by a continuous family of three dimensional hypersurfaces , I: t . This means that there exists a smooth function £ o M whose gradient never vanishes so that each hypersurface is a surface of constant I: t
== {p
E
M,
£(p) = t},
\j t E
K Such space-times are called globally hyperbolic and they represent most space t imes of astrophysical and cosmological interest. We assume that each hypersurface o the foliation is space-like and that it covers M. If we split the line-element as
(l.109 then the normal vector to I: t has components n'" = (1, - Ni)/N , i.e. n", Calling t'" the vector defined by t"'\l ",t = 1, then
= N( - 1,0,0,0
and t'" = Nn'" + N"'. This gives a geometrical interpretation to Nand N i : N is th lapse function , N i the shift vector and lij the spatial metric. It is thus clear tha
Equations of motion
45
the components of the metric are gOO = - N 2 + 'Yij N i N j , gOi = N i and gij = 'Yij and the components of the inverse metric are goo = - N - 2 , gOi = N i / N 2 and gij = '"Y ij - NiNj /N 2. As t increases, we go from a hypersurface to another and we can view the four-dimensional space-time as the time evolution of a three-dimensional Riemannian space . One can then deduce a simple relation between the determinants g of gJ.!. v and 'Y of '"Yij. Using Cramer 's rule to compute the inverse metric, one gets that gOO = 'Y / g so that v=g = N v::Y. The Einst ein- Hilbert action (l.80) can be rewritten as
This action has to be considered as a functional of the variables q their time derivatives. The extrinsic curvature is K ij =
2~
('Yik D j N
k
+ 'Yjk D i Nk
=
( 'Yij,
N , N i ) and
- i'ij ) ,
so that the gravity sector derives from the Lagrangian density (l.110) Since this Lagrangia n does not depend on H and Hi, we conclude that t he lapse function and the shift vector are not dynamical variables and will be associat ed to two constraint equat ions. The only dynamical variable is therefore 'Yij and its conjugate moment um is (l.I11) It follows that the Hamiltonian density is given by 7i =
7rij i'ij
-£ and can be expressed
as 7i =
-v::Y [N
(3) R -
K ij K ij
+ K2) + 2N i (DiK
- DjKf)]
+ 2v::YD j (K Nj - KI N i ). The last t erm, being a divergence, does not contribute t o the integral, so t hat the Hamiltonian reduces to (l.112) with (l.113) ij i It is a functional of q = hij , N , N ) and p = 5£ / 5q = (7r , 0, 0). The scalar curvat ure (3) R is a function of 'Yij and its spatial derivatives and t he extrinsic curvature is a function of 'Yij and 7r ij obtained from the inversion of (l.111)
46
General relativity
1 (1
= Vi 2 'Ykl'Tr kl'Yij - 'Yik'Yjl1f kl) .
K ij
The Hamilton equations lead to two constraints , the 'energy constraint ' Co = 0 and the 'momentum constraint ' Ci = 0, and an evolution equation , which in the vacuum takes the form "'tij = 5H/ 51f ij = - 2NKij + D iN j + D j N i . It is b eyond the scope o this book to derive the Hamilton equation with matter. When matter is included, the constraint equations take the form (3) R
+K2-
D j K l - D iK
K ijKij = 161fGp,
= 81fG N Pi ,
(1.114 (1.115
with p == TJ.l. l/ n J.l. n l/ and P (3 = - (5~ + n (3 nJ.l.)TJ.l.C< n C< ' These relations can be obtained directly from the Einst ein equations and the Gauss relation (1.76) .
1.4 1.4.1
Geodesics in a curved space-time Conserved quantities along a geodesic
To each symmetry one can associate a Killing vector , ~J.l. , under which the space-time metric remains invariant . Such a vector satisfies the equation deduced from (1.58) \7 J.l.~1/ + \7 I/~J.l. = O. The exist ence of such vectors makes it possible to compute conserved quantities useful for the study of the solutions of Einst ein's equations. Consider a geodesic with tangent vector uJ.l.. One can check that the quantity uJ.l.~" is constant along the tra jectory. Indeed,
The first t erm vanishes from the geodesics equation and the second term is the con traction of a symmetric t ensor with an antisymmetric tensor. Similarly, if T J.l. 1/ is an energy-momentum t ensor then TJ.l. I/ ~1/ is a conserved vector
The first t erm vanishes from t he conservation equation and the second is the contraction of a symmetric t ensor with an antisymmetric tensor. This relation has an important implication. The vector j J.l. = TJ.l.I/~I/ A satisfies by construction
Integrating over a volume V , we can hence conclude that
The quantity
QO , defined by
Geodesics in a curved space-time
47
follows t he conservation equation
It follows that
1.4.2
aoQ o =
0 if the boundary term vanishes.
Geodesic deviation equation
We consider a family xI.L (A, s) of neighbouring geodesics, where A is the affine paramet er along the geodesics and s is a label. The vector axI.L nI.L = - -
as
measures the separation between two neighbouring geodesics. It can be chosen in t he plane orthogonal to the geodesic flow, i. e. such that uI.LnI.L = 0 (otherwise one can replace it by rtn V). One can check that
The relative acceleration between two neighbouring geodesics is defined by (1.116) Using t he previous property we obtain (1.117) To evaluate the first term, note t hat nI.L'V I.L (uv 'V vu"') = 0 since u I.L satisfies the geodesics equation. This means that (nI.L'V I.LU V) 'V vu'" + nI.Luv'V I.L 'V vuO: = 0 that can be rewritten , switching nI.L and uI.L in the first term and switching the dummy indices J.L and /.I in the second one, as (uI.L 'V I.LnV) 'V vu'" + nvuI.L'Vv'VI.Lu'" = O. Using t his result to evaluat e the second term of (1.117), we obtain, after commuting the two partial derivatives (1.118) This equation confirms the heuristic argument on deviation of nearby geodesics. Even if we work gravity locally vanishes) the Riemann tensor does of the Christoffel symbols a priori do not cancel) or defo cusing of neighbouring geodesics. 1.4. 2.1
the relation between curvature and in a free-falling referential (where not vanish (because t he derivatives and one cannot avoid the fo cusing
Covariant approach
We decompose the vector nI.L as nI.L = f.eI.L with eI.Le I.L = 1 (thus, ePeI.L = 0) and eI.LuI.L = O. We can t hen express ilP = uV'V vnI.L either as ilP = f.e. v + Ce v or as ilP = nV'V vuI.L. By comparing these two expressions and proj ecting on eI.L, we get
48
General relativity
(1.119 that we call the generalized Hubble law. As for the orthogonal projection, it gives
(1.120
which describes the change in the direction of observation of a distant galaxy, fo instance. 1.4.2.2
Optics and redshift
We now consider an electromagnetic wave with a potential vector of the form
= CJ.Lei'P.
AJ.L
The Lorentz gauge condition imposes CJ.L0J.L'P = O.
If we assume that the amplitude CJ.L varies slowly compared to the phase 'P and tha the wavelength is small compared to the space-time curvature length scale, we ca neglect the curvature term in (1.97) so that Maxwell's equations impose
\lwy J.L 'P = 0,
0J.L'PoJ.L'P = O.
For any function 'P, the vector kJ.L = oJ.L'P is orthogonal to the wave surface on whic 'P = const.; kJ.L is the wavevector. If we differentiate the second equation and use th fact that \l VOJ.L'P = \l J.LOV'P, we get that light is a transverse wave propagating alon null geodesics
= oJ.L'P, kJ.LkJ.L = 0, kJ.L\l J.Lk v = O. kJ.L
(1.121
In a Minkowski space-time, kJ.L = (- E, k) , where E is the photon energy and kit momentum. An observer that follows a trajectory with tangent vector uJ.L will measur an energy (or equivalently, a frequency) given by E
= l'iw = - kJ.LuJ.L"
A photon emitted with a frequency Wem will be received with a frequency Fig. 1.8). These two frequencies will be related by
W rec
(se
(1.122
This relation will be very important in cosmology in order to define the notion o redshift, z. The wavevector kJ.L can always be decomposed as kJ.L = E(uJ.L + eJ.L) dxJ.L /dp , with uJ.L eJ.L = 0 and eJ.LeJ.L = 1 and where p is the affine parameter along t h photon traj ectory. It follows that dE/dp = -kJ.L\l J.L(kvu V) and hence
. J.L J.L v) 1 d.\ _ 1 dE _ ( 1 e >':dp - -Edp- "3 +uJ.Le +CYJ.LV e e E.
(1.123
This expression points out the anisotropic contributions to the redshift coming from the shear. This expression is valid for any space-time.
Weak-field regime
Reception
Emission
Fig. 1.8 A photon emitted by a comoving observer with 4-velocity null geodesic and is received by a comoving observer with 4-velocity
1.5 1.5.1
49
U::
m ,
propagates along a
u~ec.
Weak-field regime Newtonian limit
An undetermined parameter appears in the Einstein equations, the constant K. To fix it, we consider the weak-field limit of the Einstein equations in order to underst and how Newtonian gravity is recovered. We expand t he space-t ime metric around a Minkowski space-time, in Minkowski coordinates, as
(1.124) where hJ.L1/ is a small perturbation. 1. 5. 1.1
Linearized Einstein equations
From t he previous decomposition, we can conclude that to first order in hJ.Ll/ ,
and that h J.L1/ = fJ o. J.LfJ(3J.L h o.(3 and we set h = h~. To linear order in hJ.L l/, the Christoffel symbols are given by 1 0.'\ [2 h '\(J.L,I/) r 0.J.LI/ = "ifJ
Accordingly, the Ricci tensor is of the form RJ.LI/ RJ.LI/
=
~
]
- h J.LI/,'\ .
=
80. r~1/
-
8J.Lr~I/'
[28,\8(J.Lhl/),\ - DhJ.L1/ - 8J.Ll/h] ,
so that
(1.125)
where 0 = fJ o.(380.8(3 is the d' Alembertian in Minkowski space. Contracting t he previous expression with fJJ.Ll/ , we get the scalar curvature
50
General relativity
R = h::" - Dh. The Einstein tensor is given by GJ.LV read
=
RJ.Lv - ~R7]J.Lv so that t he Einstein equation
(l.126 with
In particular , we can check that h
1.5.1.2
=
- h and that hJ.Lv
=
hJ.Lv - ~h7]J.Lv.
Gauge freedom
In general relativity, there exist a gauge freedom related to the arbitrary choice o coordinate systems: two different metrics can describe the same space-time. In the linearized limit, two perturbations can represent the same state if they are related by a change of coordinates. After a change of coordinates of t he form xJ.L --> xJ.L + ~J.L, the metric perturbation transforms as
where the Lie derivative is given by £f,7]J.LV = 2o(J.L~v). Using these transformations, the linearized Einstein equation (l.126), takes the remarkably simple form
if one chooses ~J.L such that D~J.L = ovhJ.Lv. This means that after the change of variable the metric perturbation satisfies the gauge condition
(l.128)
This gauge is known as the de Donder gauge or harmonic gauge. It is analogous to the Lorentz gauge in electromagnetism.
1.5.1.3
Solutions of Einstein's equations
In the harmonic gauge , the solution of Einstein's equations can be found using the
retarded potential method, just like in electromagnetism ,
,- ( )_ J ~
1J.Lv x , t - 271"
TJ.Lv(x ' , t -Ix - x'l/c)d3
IX
-
x, I
I
x .
The general solution is therefore of t he form h1 v ) = hJ.Lv + h1~), where h1~) is a solution of the homogeneous equations D h1~) = 0 with oVh1~) = O. g en
Weak-field regime
1.5.1.4
51
Geodesic equation
For a classical fluid, and to lowest order in the velocity, only the component Too gives a contribution (since P « pc 2 ). TOi and Tij should be taken into consideration only at the first post-Newtonian (lPN) order , i.e. at order v2 /c 2 . The form of the general solution of the linearized Einstein equations (l.127) implies that the only non-vanishing terms are hoo and hOi. They are given by 2 -
KC
hoo = 2;"
JIx _
p
3
I
xii d x ,
- Oih
KC -
27r
JIx -
PUi
xii
d 3 x.I
It follows that all diagonal components of h,.LV are equal to hoo / 2. We now consider the trajectory of a particle with velocity small compared to the speed of light (v / C « 1). As we have seen before, to lowest order in h the tangent vector to the worldline is given by
dx"
u'"
_Idx"
= dT ="
ill'
with Vi == dx i / dt and" = dT / dt = \/1 - v 2 / c2, so t hat we have u" = ,,- 1(C, vi). Using the relation d 2 x" / dt 2 = dbu")/dt =u"d" /dt +"du" / dt , the spatial component of the geodesic equation becomes d 2 x'
dt 2
dx'" dx,6
=
dx'" dx,6 dx i
r~i3 ill ill + r~,6 ill ill cit·
-
To linear order in h and v, this equation is of the form (l.129)
1.5.1.5
Determination of K
Let us work in a static gravitational field. The perturbation h"v only depends on the position Xi. Equation (l. 129) t hen simplifies to
The first term represents the coordinate acceleration. In the second term, we recognize the gradient of a scalar that we identify with the gravitational potential 2¢ = c2 h oo , so that we recover the Newtonian form 2
d X dt 2
i
= -oijo.'!' ,,+,,
We can therefore go back to the Einstein equations and compare them in this limit with t he Poisson equation that determines the gravitational potential in Newtonian mechanics. This allows one to identify the value of K such that the weak-field limit of relativity is consistent with Newtonian gravity. From the previous equation, we deduce that hoo = 4¢/c 2 . The energy-momentum tensor has for main component Too = pc2
52
General relativity
and the limit of the E instein equation is therefore b.¢ = Kpc 4 /2. Comparing thi equation wit h the Poisson equation b.¢ = 47rG N P, it follows that
(1.130
where G N is the Newton constant.
1.5. 1.6
Analogy with electromagnetism
In the previous expressions, the linearized equations of general relativity are ver similar to the equations of electromagnetism. One can take the analogy a step furthe by defining Ai = chOi . The geodesic equation can then be rewritten as
x= -
V¢-
A + v !\ (V !\ A) + 3¢v.
¢ = 0, the previous expression reduces to a form similar to t he one in electromagnetism. The Einstein equations are hence similar t the Maxwell equations if we introduce
In a static space-t ime, for which
1
9 = - V ¢ - -Ot A , c
B = V !\ A ,
V·B = O, V !\ 9
(1.131
1
= - - Ot B ,
c The following list summarizes some aspects of the analogy, which cannot unfortu nately be pushed much further.
Variables Gauge transfo rmations Gauge condition Field equations R estrictions on the other gauge transformations
electromagnetism
linearized relativity
All A ll ---+ All + 0IlX 0IlAIl = 0 D AIl = - j ll
h ll v h llv ---+ h llv - £ ~ T)llv + T)IlVOo<~O< , _ 0ll h llV = 0 D h llv = - 2KTllv D ~1l
DX = 0
=0
1.5. 1.7 SVT decomposition To go further in the study of perturbations, let us decompose h llv as
hoo = - 2A ,
hOi = Oi E + Hi,
hij = 2C6ij + 20 ij E + 2o(iEj ) + Eij
with
OiHi = 0,
OiEi = 0,
OiEij = 0,
EJ
= O.
The 10 components of the tensor h llv are decomposed into 4 scalar perturbation (A , E, C , E) , 2 vectors perturbations (E i , Hi) that have 6 - 2 = 4 independent com ponents and 1 tensor perturbation Eij that has 6 - 4 = 2 independent components
Weak- field regime
53
The important property of this decomposition, which will be used in the cosmological context in Chapter 5, is that these three types of perturbations decouple (see Ref. [11]) . As previously, we should take into account the effect of an arbitrary change of coordinates. In order to do so , we decompose the displacement vector field ~p. as ~o
~i
= - T,
= - ciL - V.
Under such coordinate transformation , the metric transforms as 9p.v [see (1.58)]. It follows that 5
A
-of
A
+ T,
B
-of
B
+t -
T,
C
-of
C,
E
-of
E
-of
9p.v - £E9p.v
+L
for the scalar modes, for the vector modes , and the tensor modes remain unchanged
The tensor modes are therefore gauge invariant since they do not depend on the choice of the coordinate syst em. This is not the case for the vector and scalar modes . However , we can define a combination of these modes that are gauge invariant. For the scalar modes, we define \Ii = - C,
and for the vector modes
i
=
Ei - Hi.
We therefore have defined 2 scalar quantities and 1 vector quantity (2 degrees of freedom) which are gauge invariant. All together , we therefore have 6 = 10 - 4 degrees of freedom, once the 4 arbitrary degrees of freedom related to the gauge choice have been absorbed. 1.5.2
Gravitational waves in an empty space- time
1.5.2.1 Propagation eq uations For the scalar modes, the Einstein equations G p.v = 0, imply that Ll\li = 0 (00 component ), Oi \[t = 0 (Oi component) and
=
\Ii
= 0,
which means that no scalar mode can propagate. For the vector modes, the component Oi implies that Ll~i = 0, the only regular solution of which is i = O. Just as for scalar modes, no vector modes can propagate. 5To first order in ~ , h l'v -. h l'v - £1;111'v and we just need to evaluate £<'100 = 2T , £(I)Oi Gi(T - L) -
£i a nd £~"I)ij =
- 2[;,LJij - 2G(i L j).
54
General relativity
The situation is different for tensor modes. The ZJ component of the Einstein equations implies that DEij = O. The only perturbations that can propagate in a Minkowski space-time are the gravitational waves and they satisfy (1.132)
The three conditions = W = i = 0 define a gauge equivalence class. We can choose a gauge in this family by imposing some conditions on the perturbations A , B, C , E and 13i , E i . For instance, setting E = B = 0 and Ei = 0, we define what is called a transverse and traceless (TT) gauge in which the metric is completely determined. In this case, the only non-vanishing component of hh~T) is Eij . 1.5.2.2
Plane waves
Let us consider a plane wave propagating along the axis Oz in the TT gauge. Since the wave is transverse and traceless, it satisfies hE'T) = h~~T) = h~~T) = O. Therefore, it has only two degrees of freedom (two polarizations) that should satisfy h~~T) = - h~~T) and h~~T) = h g T ). We hence can decompose it as follows
h~JT) = E +(et - z)ED
+ E xCet - Z)Elj,
with t he two polarization tensors being defined by
ED
=
(~000 ~1 ~) ,
Elj =
(~000 ~ ~)
The solution of the propagation equation is E+ /x = A+/x cos[k( z - ct) +cp], where cp is a phase. 1.5.2.3
Motion of a test particle
Consider the geodesic equation of a test particle in the gravitational field of a gravitational wave in the TT gauge. Since to leading order in perturbation we have roo = 0, a particle initially at rest will remain at rest . Of course, this does not mean that nothing happens , but rather that the frame of reference is co moving with the test particle. To see if anything happens , we should look at the relative motion of two neighbouring particles, which can be done using the geodesic deviation equation (1.ll8). The relative acceleration is given by al1- = -Rl1-vpauvnPua . To leading order in vic, we have ul1- = (1,0) and nl1- = (0, ni) , R iojo =
-a; h~JT) 12 so that
2
i
d n = ~02h(TT)inj. dt2 2 t J
Substituting the expression for h~JT) from the previous section, we can easily integrate this expression. Figure 1.9 illustrates the deformation of a ring of particles under
Tests of genera l relativity
55
the influence of a gravitational plane wave propagating along the axis z for each polarization case. t=T14
t =0
+
X
0 0
, /
,,
/
-
t=T12
,,
, \
I
I \
\
,,
I
,,
,
I /
,,
, \
I
0 0
t=JTI4
, /
,,
/
,
,,
,,
,,
I
I
Fig. 1.9 Effects of a gravitational plane wave propagating along the axis z on a ring of particles located in t he plane xy, depending on the wave polar ization . The four snapshots ar e parameterized by the period T corresponding to a gravitational wave with wavelength k - 1 .
1.6
Tests of general relativity
The cosmological models we will construct will strongly rely on the application of general relativity to the Universe as a whole. It is hence important to understand the limits of validity of our theory of gravitation. We summarize here the principles of the main tests of gravitation; all constraints are summarized in the t able at the end of this section. This question is presented in detail in Refs. [8, 12, 13]. 1.6.1 1.6.1.1
Test of the equivalence principle Uni versality of free fall
The mass of any body comes from different contributions (rest energy, electromagnetic or nuclear binding energy, etc.). If one of these energies contributed differently to the inertial mass than to the gravitational mass, we would observe a violation of the universality of free fall. To quantify this possible difference, we write (1.133) where for each interaction i, Ei is the associated binding energy, and ryi is a small dimensionless paramet er that quantifies the amplitude of the violation of the universality of free fall associated to this interaction. The acceleration of any test body in an ext ernal gravitational field is given by
56
General relativity
so that measuring the difference between the acceleration of two different bodies allow us to define the Eotvos parameter T) by
(l.134 to leading order in the parameter T):
T)i.
Two methods have mainly been used to constrain the value o
1. The first method uses the period T of a pendulum of length L ,
T = 2n
~I {f mG
-, 9
A
/
g
Fig. 1.10 Principle of Eotvos balance. 9 is the external gravitational field and w is th angular velocity of the measurement device. The unit vectors e J and eb are, respectively parallel to the torsion fibre and to AB.
Tests of general relativity
57
so that
IT1 - T21 . T1 +T2 2. The second method uses a torsion balance (see Fig. 1.10). Two objects of different compositions, A and B, are attached at the end of a horizontal beam of length 2L suspended on a torsion fibre. The balance has an angular velocity w, coming, for instance, from the Earth rotation; the force on each mass is therefore rJ=2
where c is the centrifugal acceleration. We can conclude from this expression that the wire tension is T = FA + F B and that the torque applied on the torsion fibre is M = Leb 1\ (F B - F A). There is therefore a torque M directed along the same direction as the tension. Performing the scalar product M . T , we get that L
M = rJ-1- -1(ebl\c).g. g+c This device was invented by the baron Roland von Eotvos (see Ref. [14]). In the original experiment , 9 corresponded to the Earth's gravity and the rotation was that of the Earth. The pendulum and torsion balance can test the weak equivalence principle because the gravitational binding energy of the test objects are negligible compared to the other energies. In order to test the strong equivalence principle, we should consider objects with an important gravitational binding energy such as celestial objects. This can be done by comparing the relative motion of the Earth and the Moon within the gravitational field of the Sun. There is therefore an extra acceleration with expression
oa -_
GN M 0 2 rJ R
( E (grav) 2
me
I
_ E(grav) I
Earth
me
2
)
eT
,
Moon
where e T is a unit vector directed from the Sun to the Earth and R the radius of t he Earth orbit. This implies that the orbit of the Moon around the Earth is polarized in the direction of the Sun, with maximal amplitude
Or m ax
30a
= - - - - -----,-
2 (WEarthWMoon)
Neglecting the gravitational energy of the Moon compared to that of the Earth, which is considered to be a homogeneous ball of radius R and mass m, it follows that E(grav) / me 2 = - 3Gm/ 5Re 2. A more precise computation gives E(grav ) / me 2 = - 4.6 x 10- 10 for the Earth and - 0.2 x 10- 10 for the Moon , from which we conclude that These measurements have been made possible, mainly owing to the Lunar Laser Ranging (LLR) experiment t hat measured the position of the Moon relative to the Earth with a precision of 1 cm over about thirty years.
58
General relativity
1.6.1.2 Local Lorentz in variance
There exist few clean tests of the local Lorentz invariance. In particular, many of the high-energy tests cannot usually distinguish the specific effects of Lorentz invariance breaking from the complications due to weak and strong interactions. The best constraint is the one obtained from the Hughes- Drever experiment [15]. The experiment observes the ground state of the lithium-7, with quantum number J = 3/2. In the presence of a magnetic field , this state splits into four equally spaced levels. Any perturbation associated with the existence of a preferred direction , would modify the spacing between the levels. This sets a limit on the potential anisotropy of the inertial mass om~j of the lithium nucleus that can be parameterized as
(1.135)
where OA is a dimensionless parameter that characterizes the amplitude of the anisotropy induced from the interaction A with binding energy EA. In the experiment with ij 2 t he lithium-7, we get c 1.7 x 10- 16 eV.
lom
1.6.1.3
1:s
Local position in variance
The first t est of local position invariance is related to the Einst ein effect, i. e. to the fact that clocks run more slowly in the presence of a gravitational field. Consider an atomic transition at point A emitting a photon towards an observer located in B. According to the previous sections , the photon is received with a redshift (l +z) = (U /J. k/J.) B / (U/J.k/J. )A, implying that /:).¢ z = ~,
where ¢ is the gravitational potential. If there is a space dependence, this relation should not be valid any longer. We would therefore expect /:).¢
z =(1 + a)-2 ' c
(1.136)
where a is a parameter that vanishes in general relativity. Several experiments have been performed to measure the redshift, either by sending atomic clocks in orbit or by observing solar spectra. Another method consists of testing the variation of the fundamental constants. Numerous tests have been performed mainly on the fine structure constant , on the electron to proton mass ratio, and on the gravitational constant. These experiments go from local scales to the early Universe. We will come back to these t ests in lat er chapters (see also the review Ref. [16]) .
1.6.1. 4
Fifth force
To finish , we point out tests on the potential presence of a fifth force. In these experiments , the gravita tional potential is supposed to be modified by a Yukawa potential of range .A
Tests of general relativity
59
(l.137) where (0'12,
0'12
characterizes the amplitude of the deviation. The constraints on the pair
A) are summarized in Fig. l.1l. In the laboratory, experiments using Cavendish
balances can test Newton's law up to a hundred micrometers [17]. The constraints go up to scales of the order of the Solar System size by studying the motion of celestial objects.
Lamoreaux
10'
Excluded region
10 4
10-4 10 -6
10 - 5
10-4
10-3
10-2
1-.(m)
10 - 2
10-4
10 -6
-a
Earth ± LAGEOS
10 - 8
10 - 2
LAGEOS ±
10 - 10 I
1-.(m)
Fig. 1.11 Constraints on the existence of a fifth forc e deriving from a Yukawa potential. The parameter 0: represents the parameter 0:12 of (1.137), which we assume to be the same for all bodies . The bottom graph represents a zoom on the smallest t est ed scales. The diagram presents the predicted deviations expected from some models, such as t he presence of extra dimensions, axions or dilaton, etc .. . These different models will be presented in later chapters of this book (quote from Ref. [18]).
60
General relativity
Tests in the Solar System
1.6.2
We will now concentrate on the t ests of the geometrical structure of space-time a of Einstein's equations. These tests rely mainly on the study of geodesics in the So System. We will focus on t he case of Schwarzschild space-time that describes the met generated by a massive body. We may, however, point out that many modern tes (such as LLR, LAGEOS or GPB experiments) , can only be described with aN-bo metric, i.e. beyond the Schwarzschild metric, which is beyond the scope of this bo (see Refs. [1- 4] for further details).
1.6.2.1
Schwarzschild space-time
The space-time in the Solar System can be described by a static solution with spheric symmetry of the form
(1.13
with d0 2 = de 2 + sin 2 edcp2 . The solution of the Einstein equations with a mass sour M localized at the origin is the Schwarzschild solution whose explicit form is 2 ( ds = -
2G M) 1- ~ N
2 dt +
2
(1 _2G M) dr
N
2 2 + r dO .
(1.13
c2 r Most of the tests in the Solar System rely on the study of geodesics of this metric.
1.6.2.2 Extension
To be more general and to be able to test both the geometrical structure of spa and the field equations, we will work in the weak-field approximation (G N M / c 2 r « which is justified since GNM0 /C2 = 1.4 X 10 3 m for the Sun). We assume that t metric is of the form ds 2 =
-
[
1-
G M 2_N_ c2r
+
2
(fJ PPN
2
-
G_ M2 I'PPN ) _N _ ] dt 2 c4 r2
+ (1 +
G M) 2 I'PPN_Nc2r
dr2 + r 2d0
(1.14 From the field equations of mot ion of general relativity and using (1.139) , we get th PPN I'PPN = fJ = 1, but there is a priori no reason for this to be the case within anoth metric theory of gravity. 1.6.2. 3
Light bending
The study of null geodesics can be performed directly in the general form (1.138) . B symmet ry, we can choose to concentrate on the solutions that lie in t he plane = 7r / The trajectory then reduces to {t(A) , r(A), cp( A)}. Since space-time is static and h a spherical symmetry, there must be two constants of motion. The one associat with time-translation invariance is identified with the energy E = Ut = gttdt/d
e
Tests of general relativity
61
while the second one, associated with the angular momentum rotation invariance, is L = u¢ = g
2d
For a null geodesic, we have also the extra condition
(1.141 )
.
kl-'kl-' =
0, which we can write as
Using the constants (1.141), we can deduce that the equation of the traj ectory is
= (E)2 e~l/(r)~I-'(r) ( ~)2 d
_
r2
e~l/(r) r2
(1.142)
The functions v(r) and J-i(r) can be identified with the corresponding terms in (1.140) . After introducing y = G N M/c 2 r and b = L /E, we therefore have
d2 y _ + y d
2
G M2 = 3,PPN y 2 + (l_,PPN)_N_ . b2 c4
To second order in M / b, the solution of this equation is given by the relation y = (G NM/bc) sin 00, we get
roc
which reduces to D.
~ iro
Actual source position
Observer
Fig. 1.12 Geometry of the trajectory of a photon d eflect ed by a massive body.
1.6.2.4
Shapiro Effect
The Shapiro effect , discovered in 1964, is an abnormal time delay when electromagnetic waves are emitted from Earth and reflected back by a satellite or a planet. Using the null geodesic equations derived in the previous section, we obtain
(r ~:
=
el-'(r) ~l/(r)
[1 -
el-'(r)
~~] ,
62
General relativity
the coordinate time a photon takes to go from r to ro is t herefore
t(r, ro) =
f
ro
r
Integrating this expression from rEarth to of change of the travel proper time dl:l.T
- - ::: -(1
dT
rplan et,
back and forth, we get that the r
+ ,PPN) IlS . h - 1
(1.1
assuming ro ::: 7 x 108 m. 1.6.2.5
Precession of the perihelion of Mercury
The planet trajectories are obtained using the time-like geodesics, such that
e-~(r) E2 = 1 + ev(r) (~ ) (:~ ) + (~ ) . 2
2
Introducing y = l / r and neglecting terms of order (G N M / rc 2)2 , we can integrate t equation to obtain the approximate solution
y(cp) =
G~M
(1
+ e cos {[ 1 _
(2
+ 2,PPN _
jJPPN)
G~~2] cp} ) ,
where e is the eccentricity. At each orbit, the perihelion advances by
l:I.cp =
27r(2
+ 2,
G 2 M2 = 27rG NM2 (2 £2 a(l - e )
PPN _ jJPPN)_N_
+ 2,PPN
_ jJPPN).
(1.1
For Mercury, we get 42.98" per century if , PPN = jJPPN = 1, which is consistent w observations. 1.6.2.6 Summary
The tests performed in the laboratory and in the Solar System give a bound to violation of the equivalence principle and to the deviations from the theory of gene relativity. We have only presented here some of these parameters, but the param terized post-Newtonian (PPN) formalism brings into play other parameters that beyond the purely Newtonian theory. This formalism has been developed in order take into account all kinds of possible deviations. The following table and Fig. 1 give a summary of the main constraints obtained in the Solar System (see Ref. [ for more details).
1.6.3
Gravitational radiation
Just as any charged object emits electromagnetic waves, a massive body can radi gravitational waves. In what follows, we review the properties of this radiation , with giving any proof. The existence and the properties of gravitational waves can be u to t est the theory of general relativity.
Tests of general relativity
63
Table 1.1 Solar-system constraints on the post-Newtonian parameters.
Parameters rJ (weak)
Constraints 10 - :3 5 x 10- 9 (-1.9 ±2 .5) x 10- 12 5.5 x 10(1 ± 15) x x l0 10
rJ (strong)
a
- ,j
- <:
2 x 10- 4 1.7 x 10 · 1(i eV
15m') . I
10- 23 10- 22 5 x 10- 18
6'
1.6.3.1
13
Details Pendulum Torsion balance Torsion balance (Be-Cu) Torsion balance (Earth-Moon) LLR Photon emitted (Moss bauer effect) H-Maser in orbit Lithium Strong interaction Electromagnetic interaction Weak interaction
Reference Newton (1686) [1 4] [19] [20] [21] [22] [23] [15]
Radiation of gravitational waves
The gravitational radiation emitted by an object is given by (TT)
hi)'
(x, t)
=
2
T) ,
2G N d Q kl ( -4-Pijkld 2 t - C T t c
where Pijk1(n) = (6 i k - n ink )(6jl - nknl) - ~(6ij - ninj)(6kl - nknt) and where Qkl is the quadrupole of the mass distribution Qkl(t)
=
J
p(x,t) ( XkXl -
~X26kl) d 3x .
The radiation is therefore sourced by the quadrupole of the matter distribution, and there is no monopole contribution (due to the mass conservation) or dipole (from the equivalence principle). The total power of the radiation is given by the Einstein quadrupole formula dE G N d 3Qkl d 3Q kl dt 5c 5 dt3 dt3 ' 1.6.3.2
Binary pulsar PSR 1913+16
Discovered in 1974 by Taylor and Hulse, this pulsar made it possible to test the formula for the gravitational radiation and prove the existence of gravitational radiation. Using Kepler' s law, it can be shown that if the energy of the binary system decreases , its period must t hen increase as
jdP) \dt
=
_1927r (27rG N 5c2 P
)5/3 m1m2
4 2 1+73e / 24 + 37e /96 m1+m2 (l _e 2 )7/ 2 '
where m1 and m2 are the two masses and e is the eccentricity. For the pulsar PSR 1913+16, m1 = 1.44M8 and m1 = 1.38M8 , e = 0.617 and P = 7 h 40 min so that
64
General relativity (( Lunar
\l Laser
Ranging
Y PPN
Precession of the perihelion of Mercury
1.004
Mars Radar Ranging
1.002 - - - - - - - - - - - - -
1
0.998
0.996
0.996
0.998
1
1.002
1.004
i3 PPN
Fig. 1.13 Summary of the constraints on the post-Newtonia n parameters in the Solar tem . The current constraints are 12,PPN - j3PPN - 11 < 3 x 10- 3 form the precession o perihelion of Mercury, 4j3PPN - , PPN - 3 = (-0.7 ± 1) x 10- 3 from the Lunar laser ran (LLR) experiment , b PPN - 11 < 4 x 10- 4 from the light bending (VLBI: Very Long Bas Interferometer) and b PPN - 11 = (2.1 ±2 .3) x 10- 5 from the measure of the time d elay o Cassini space probe signal.
(dP / dt ) = -2 .4 x 10- 12 . We may note that there is, however, a real discrepanc 14 (J" between the observed decrease and the theoretical prediction. This is because Doppler effect coming from the acceleration of the pulsar towards the centre o Galaxy should be taken into account . The theoretical works in Ref. [24] have ma possible to compute the equations of motion up to order v 5 / c5 and have led to idea how to t est general relativity in the strong-field limit [25] (see Ref. [26] for a revi They also prove that gravity propagates at the speed of light. 1.6.3.3
Direct detection
Many experiments are now in progress to detect gr avitational waves, either from i ferometry (LIGO, VIRGO, LISA, . . . ) or using Weber bars (see Ref. [27] for a revi In order to understand how difficult it is to det ect them, let us give an estima the amplitude of the gravitational waves. Consider for that a bar of length L and m M rotating with angular velocity w. The characteristic amplitude of the quadru is Q = lVI L2, so that dnQ/dt n rv lVI L 2w n We therefore have a typical amplit ude dE
dt
rv
G N M2L4 w 6 c5
Tests of general relativity
65
For a 500-ton mass and 20-m long bar rotating at 5 rad/s, we get h ~ 10- 38 at 50 m, and dE / dt ~ 10- 32 W! The radiation of ordinary objects is therefore undetectable. For a binary system of typical mass 1Mo, with an orbit of 10 6 km and an orbital period of order 10 h, we get dE/dt ~ 10 22 W. This power can even be increased and reach a typical value of dE/dt ~ c 5 /G N ~ 3.6 X 10 52 W for an ultrarelativistic source. 1.6.4
Need for tests at astrophysical scales
Most of the tests presented here have a characteristic scale of the order of the size of the Solar System or at most, of the size of our Galaxy for binary pulsars. In cosmology, we will extrapolate the law of gravitation to the Universe itself. This extrapolation requires some checks that we will present in detail in the rest of this book. Let us, nevertheless, remind ourselves of some facts worth keeping in mind (see Ref. [28] for more details): 1. At galaxy scales, the rotation curves indicate that the gravitational potential goes as ¢ ex In r on large scales. This behaviour is usually explained by the presence of dark matter, which has not been detected yet. Beyond a given scale, everything goes effectively as if gravity had a bidimensional behaviour. The study of the rotation curves, however, shows that if such a transition existed, this scale should depend on the galaxy mass (see Chapter 7 for more details on these relations). 2. For galaxy clusters of around 2 Mpc, the effects of gravitational lensing, which measures the gravitational potential from light bending, and the X-ray emission, which traces out the gas temperature, can be compared. This comparison has shown that the Poisson equation is valid up to a factor 2. Hence , it provides a test of general relativity (testing both the light bending and Einstein's equations in the weak-field limit). 3. At astrophysical and cosmological scales, there are no direct tests of the law of gravity. The growth of the large-scale structure of the Universe provides a test, but it mixes the properties of gravity with those of matter. For the theory to be consistent with the observations, dark matter should be present. 4. The observed recent accelerated expansion of the Universe (see Chapter 4) cannot be explained within the framework of general relativity with ordinary matter (among other conditions, matter with positive pressure; this follows from the Raychaudhuri equation). This observation is explained either through exotic matter (some candidates such as the cosmological constant, the quintessence , ... will be described in Chapter 12) or by modifying the laws of gravity at large scales. 5. Two tests are currently proposed. On the one hand , the Einstein equivalence principle can be tested by checking how constant the 'constants of Nature' are on large scales. On the other hand, the Poisson law can be tested up to scales of a hundred megaparsecs , by comparing the matter distribution in the galaxy catalogues with the weak lensing by large-scale structures (see Ref. [29]).
This shows the importance of testing gravity either in order to validate the standard approach that adds new exotic matter sectors (dark matter, dark energy), or in order to give new explanations for the Universe. We will come back to all these issues in Chapter 12.
References
[1] R HAKIM , An introduction to relativistic gravitation, Cambridge Univers Press, 1999. [2] H. STEPHANI, General relativity, Cambridge University Press, 1990. [3] S. WEINBERG, Gravitation and cosmology, John Willey and Sons, 1972. [4] R WALD , General relativity, Chicago University Press, 1984. [5] L. LANDAU and E. LIFSCHITZ, Course of theoretical physics, Volume 2, P ergam Press , 1976. [6] J.D . JACKSON , Classical electrody namics, Wiley, 1998. [7] J. EISENSTAEDT, Einstein et la relativite generale, CNRS Editions, 2003. [8] C .M. WILL , Theory and experiments in gravitational physics, Cambridge Univ sity Press, 1984. [9] G.F.R ELLIS, 'Relativistic cosmology' , in Proc. Inti. School of Physics 'Enri Fermi', Corso XLVII , R.K. Sachs (ed.), pp. 104- 179, Academic Press, 1971. [10] S.W. HAWKING and G .F.R ELLIS , The large scale struct ure of space-time, Ca bridge University Press, 1973. [11] J. BARDEEN, 'Gauge-invariant cosmologica l perturbations' , Phys. Rev. D 2 1882, 1981. [12] C.M. WILL , 'The confrontation between general relativity and experiment ', Livi Rev. Relativity 4, 4, 200l. [13] 1. CIUFOLINI and J.A. WHEELER, Gravitation and inertia, Princeton Univers Press, 1995. [14] RV. EOTvos, V. PEKAR and E. FEKETE, 'Beitrage zum Gesetze der Prop o tionalitat von Tragheit und Gravitat ', Ann. Physik 68 , 11, 1922. [15] V.W. HUGHES et al. , 'Upper limit for the anisotropy of inertial mass from nucle resonance experiments', Phys. R ev. Lett. 4, 342, 1960; RW. DREVER, 'A sear for the anisotropy of inertial mass using a free precession technique' , Phil. Ma 6, 683, 1961. [16] J.P. UZAN , 'The fundamental constants and their variation: observational a theoretical status', Rev. Mod . Phys. 75,403, 2003. [17] E. FISCHBACH and C . TALMADGE, 'Ten years of the fifth force' , in Dark matter cosmology, quantum measurem ents, experimental gravitation , Moriond procee ings, 443 (1996) , arXiv:hep - ph/9606249; E. ADELBERGER, B. HECKEL , and NELSON, 'Tests of the gravitational inverse square law', Ann. Rev. Nuc. Phy 53 , 77, 2003. [18J C.D. HOYLE et al. , 'Sub-millimeter tests of the gravitational inverse square law Phys. Rev. D 70 , 042004, 2004 . [19] Y. Su et al. , 'New tests of the universality of free fall ', Phys. Rev. D 50, 361 1994.
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[20] S. BAESSLER et al., 'Improved test of the equivalence principle for gravitational self-energy', Phys. Rev. Lett. 83, 3585, 1999. [2 1] J .O. DICKEY et al., 'Lunar laser ranging: a continuous legacy of the Apollo program', Science 265 , 482, 1994. [22] R.V. POUND and G.A. REBKA , 'Apparent weight of photons', Phys. Rev. Lett. 4 , 337, 1960. [23] R. VESSOT and M. LEVINE , 'A test of the equivalence principle using a spaceborne clock', J. Gen. Rel. Grav. 10, 181, 1979. [24] T. DAMOUR and N. D ERUELLE, 'General relativistic celestial mechanics of binary systems' , Ann. Inst. H. Poincare 43, 107, 1985; 44, 263 , 1986. [25] T. DAMOUR and J .H. TAYLOR, 'Strong field tests of relativistic gravity and binary pulsars ', Phys. R ev. D 45 , 1840, 1992. [26] I.H. STAIRS, 'Testing relativity with pulsar timing' , Living Rev. R elativity 5 , 1, 2003. [2 7] I. BLAIR, 'Detection of gravitational waves', Rep. Prog. Phys. 63, 1317, 2000. [28] J.-P. UZAN, 'Tests of gravity on large scales and variation of the constants', Int . J. Theor. Phys. 42, 1163, 2003. [29] J.-P. UZAN and F. BERNARDEAU, 'Lensing at cosmological scales: a test of higher dimensional gravity', Phys. Rev. D 64, 083004, 2000.
2
Overview of particle physics and the Standard Model
Although gravity structures space-time on large scales, numerous physical processes rely on the description of t he three other interactions, namely electromagnetism , the weak and strong nuclear forces. This chapter is dedicated to t he description of these interactions and of the particles on which they act . This standard model of particle physics describes ordinary matter , as observed in the laboratory. We shall see lat er that the cosmological observations concerning the existence of dark matter call for an extension of this framework. It is hence necessary to present it in detail. For this, we will work in this chapter in a flat space-time. Section 2.1 recalls the basis of analytical mechanics and the steps from classica mechanics to quantum mechanics. Section 2.2 then focuses on the special case of a scalar field. Being the simplest prototypal model, the free or self-interacting scala field illustrates in a complete way what can be learnt from quantum field theory. The following sections sket ch a general picture of the standard model of particle physics. Section 2.3 presents the spectrum of particles detected in accelerators, and Section 2.4 describes how , relying on the notion of symmetry, a classification can be sketched out. The Higgs mechanism will t hen be described in Section 2.5. Applying this mechanism to electroweak interactions will a llow us to obtain the standard mode of particle physics , presented in Section 2.6. F inally, Section 2.7 presents a study o discrete symmetries, followed by an overview on the current state of t his standard model. The ideas introduced in this chapter will be used repeatedly in the rest of the book. For instance, t he aspects of quantization and in particular t hose of scalar fields will be central to our discussion of inflation (Chapter 8) , and also when we touch on the cosmological constant problem (Chapter 12); the notion of symmetry breaking plays a crucial role in the formation of topological defects (Chapter 11 ) and in the grand unified theories. This chapter therefore lays the necessary basis for the study of cosmologically relevant topics such as inflation of course, but also of topologica defects, the origin of dark matter , baryogenesis, the cosmological constant problem etc.
2.1
From classical to quantum
It is now a little more t han a century since the principles of quantum mechanics started being used to describe the laws of Nature. This section summarizes t he steps from analytical classical mechanics to quantum mechanics.
From classical to quantum
2.1.1
69
Analytical classical mechanics
Formally, quantum mechanics can be seen as an extension of classical mechanics as formalized finally, e.g., by Lagrange in the second half of the eighteenth century and Hamilton at the beginning of the nineteenth century. The classical laws can be transformed into their quantum counterparts by applying the correspondence principle.
2.1.1.1
Analytical classical mechanics
The essence of classical mechanics (all the details of which can be found in numerous works , in Refs. [1,2] among others) can be summarized by the existence of a Lagrangian function L from which all the dynamics of a system can be derived. For a set of N interacting particles , this Lagrangian depends on the coordinates Xi (i = 1, ' " , N) of these particles, and on their velocities, Xi . We merge together the set of all these coordinates, which are not necessarily Cartesian, under the designation qi (generalized coordinates) , with the index i now varying from 1 to 3N, and the generalized velocities qi. The dynamics of this system can be obtained by minimizing the action
S=
ltfinal L [qi(t), qi(t), t] dt ,
tinitial , tfinal
= const. ,
(2.1)
tinitial
assuming that the value of the coordinates is fixed at the boundary of the integration domain [JS = 0 with Jqi(tinitial) = Jqi(tfinal) = 0]. The Euler- Lagrange equations are then obtained as (2.2) The first term of this equation brings into play the quantities pi == 8L / 8qi, which are the generalized momenta. It is possible to invert these relations to determine the generalized velocities in terms of the coordinates and momenta. Thanks to a Legendre transformation , all the dynamics can then be described by changing variables and using the generalized momenta instead of the velocities. For that, let us define the Hamilton function , or Hamiltonian H , by 3N
H [qi(t) ,pi(t) , t]
== LpiiJi - L {qi(t), qi [r (t)] ,t} ,
(2.3)
i=l
which allows us to write the second-order differential equations (2.2) as a first-order system, .i 8H . 8H
p = - 8qi '
qi = 8pi'
(2.4)
This system allows us to easily obtain the time derivative of any physical quantity by using the Poisson brackets defined by
(2.5)
70
Overview of particle physics and the Sta ndard Model
A and B being two quantities depending on qi and pi. Substituting for B the Hami nian H in the definition (2.5) , and using the equations of motion (2.4), the total t derivative of A can easily be seen to be dA
ill
=
8A
at + Li
(8A ' i fiiP
8A.)
+ ~qi q,
P
=
8A
at + {A , H}.
(
Finally, the Poisson brackets of the coordinates and momenta satisfy
(
This formalism introduces an antisymmetric operator between two quantities, such that {A,B} = - {B , A}. This property will allow us to make the transi formally but very easily to the quantum description. 2.1.1.2
Functional derivative
Classical field theory, which can be illustrat ed, for instance, by general relativity, be written using the same formalism as the analytical theory of point particles, i rely on field functionals rather than on functions. A functional Z [J], is defined as a map from a normalized linear space offunctio .c = {f (x); x E IR}, onto the set of complex (or real in some cases) numbers , Z : .c f-----+ C (or IR). The functional derivative is then defined in a similar way the partial derivative for a function of many variables. A functional can indeed understood as a function with an infinite number of variables. Its infinitesimal varia is
<5Z[J] =
<5Z[J] <5f(x) <5f(x)dx,
J
(
introducing the dummy variable x , this reduces to a definition similar to the on normal analysis , that is
<5Z[J(x)] 1 <5f() == lim - {Z[f(x) y
<--+ 0 E
+ E<5(X -
y) ]- Z[J(x)]},
(
where <5 is the Dirac distribution in one dimension. Note that for all computations, functions are supposed to be integrable and to vanish fast enough at large x in or for the surface terms to vanish in the integration by parts. The two definitions (2. 8) and (2.9) are equivalent and can both easily be shown satisfy the rules required for a derivative, i.e. the Leibniz rule
<5 <5f(x) (Z[J]W[J])
=
and the functional composition law ITechnically, this is called a Banach space.
<5Z[J] <5f(x) W[f]
<5W[J]
+ Z[J]<5f(x)
,
(2
From classical to quantum
5Z {W[j]} = 5f(x)
J
5Z[W] 5W[f] d 5W(y) 5f(x) y.
71
(2.11)
From the definition (2.9), it can easily be shown that
5f(x) 5f(y) = 5(x - y),
(2.12)
where the first function, f(x), is seen here as the trivial functional that maps any function into itself. We also have the useful property
_5 5f(y)
J
g
[df(X)] d = _~ I [df(y)] dx x dyg dy ,
where the prime indicates a simple derivative of the function with respect to its argument. Finally, it is interesting to show that , from the functional point of view, a function and its derivative are not independent. By choosing Z[j(x)] = f'(x), and using the definition (2 .8), we find that
5f I (y) =
J
5f'(y) 5f(x) 5f(x)dx =
J
f (y)] d [55f(x) dy 5f(x)dx
=
J
d [5(y - x)] 5f(x)dx, dy
and we can hence make the identification
5f'(y) = ~ [5(y - x)]. 5f(x) dy All of these relations can easily be generalized to an arbitrary number of dimensions. Most importantly, they allow us to write a classical field theory in a manner that bears close resemblance to the Hamiltonian formulation of a classical particle.
2.1.1.3
Lagrangian of a classical field
The equivalent of the action (2.1) for a scalar field is obtained from a Lagrangian density L[¢(x!")], as Sfield
=
J
L(t) dt =
JJ dt
d3 xL [¢(x, t), o!"¢(x , t) ].
(2.13)
Let us first consider L as a function of x and t. Using an ordinary derivative followed by an integration by parts in four dimensions gives rise to the Euler- Lagrange equations, equivalent to (2.2) for a scalar field ,
oL o¢(x a
)
o oL ox!" o[o!"¢(xa) ] '
(2.14)
This relation can also be obtained using a functional differentiation, i.e. viewing L as a functional of the field and its time derivative , in the following way. The time and spatial dep endencies of the Lagrangian can be separated explicit ly before the variation
72
Overview of particle physics and the Standard Model
oL =
J
3
d x
({
8L 8L } 8L . ) 8¢>(x, t) - V' 8 [V' ¢>(x, t)] o¢>(x , t) + 8¢(x, t) o¢>(x , t) ,
where an integration by parts was p erformed to get the second term. The definitio (2.8) for the functional derivative then implies t hat
oL =
J
d3 x
[
. t) ] , O¢>(o.c /¢>(X, t) + . OL O¢>(X, x,t 8¢>(x,t)
and t herefore, by identification, we obtain
o.c O¢>(X, t)
8L _ V' 8L 8¢>(x, t) 8 [V' ¢>(x, t)]
o.c
and
o¢(x, t)
8L 8¢(x, t)
Integrating these relations by parts, assuming that t he variation at the bounda vanishes, t his reduces to (2. 14).
2.1.1.4
Hamiltonian and neld Poisson brackets
Starting from the Lagrangian (2. 13), and using an approach completely similar t hat used in point-particle mechanics, we can first define the canonically conjuga field 7r(x, t) , which is equivalent to the generalized momentum [see (2.2)], by
_
oL(t) , o¢>(x, t)
7r(x , t) = .
(2. 1
so that the equation of motion is of t he same form as for point particles, name
ir(x , t) = oLjo¢>(x, t). In order to generalize (2 .3) to the scalar field case, t he Hamiltonian can be writte in the following form
H(t) =
J
7r(x, t)¢(x , t) d 3 x - L(t)
==
J
H (x , t) d 3 x,
(2.1
which hence defines t he Hamiltonian density, related to the Lagrangian density L , b the relation
H(x , t)
=
7r(x, t) ¢(x , t) - L(x , t).
(2.17
In t his for malism, the Euler- Lagrange equations (2. 14) get replaced by t he set Hamilton equations
(2 .1
This system of first-order equations is the equivalent of (2.4) for a scalar field.
From classical to quantum
73
The Poisson bracket of two functionals is constructed in the same way as for two classical quantities via the straightforward generalization of (2.5)
_J
{W[¢(x),7f(X)] ,Z[¢(x),7f(X)]} =
3
d x
[OW oZ oW OZ] O¢(x) 07f(X) - 07f(X) O¢(x) .
(2.19)
The Hamilton equations can hence provide the time evolution of any functional W by dW = oW dt
at
The conjugate momentum
7f
+
{WH}.
,
and the field ¢ obey the relations
{¢(X ,t) ,¢(X',t)} {¢(x , t) , 7f(X' , t)}
= =
{7f(x,t) ,7f(X',t)} 0(3) (x - x') ,
=
0,
(2 .20)
which generalizes (2.7). 2.1.1.5
Nrether Theorem and conservation laws
In 1918, E. Nrether enunciated the theorem according to which a conserved quantity can be associated to any transformation that leaves the action of a theory invariant. More precisely, this theorem is enunciated in the following way:
If the action of a given theory is invariant under the infinitesimal transformations of a symmetry group , then there exist as many conserved quantities (constant in time) as infinitesimal transformations associated to this symmetry group. Furthermore, these conserved quantities can be obtained from the Lagrangian density. As an example of this theorem , as well as for its own interest, let us mention the conservation of energy that results from the time translation invariance of the laws of physics (one invariance, one conserved quantity) , as well as the conservation of momentum and angular momentum, which arise, respectively, from the translation and the rotation invariances of space-time. In these two last cases, the invariance is vectorlike (three coordinates for the translation vector, or three angles for the rotation), and the conserved quantity remains of the same nature. All these invariances, which are usual in classical mechanics, follow from the conservation of a unique quantity, the energy-momentum tensor. This construction can be generalized to the case of general relativity (Chapter 1, Section 1.4). 2.1.1.6
Case of classical point particles
The classical mechanics of point particles is completely described by specifying a Lagrangian L(qi, (ii , t). If the physics that corresponds to this Lagrangian is timetranslation invariant, i. e. such that the result of any experiment involving this theory does not depend on the time at which it is made , it is then expected that
74
Overview of particle physics and the Standard Model
L(qi, qi, t - to) = L(qi, qi, t), for any arbitrary initial time to. This implies that t Lagrangian cannot explicitly depend on time , i. e. L(qi' qi, t) = L(qi, qi); it depen on time only through the variables qi. It follows that aLlot = 0, so that its to derivative with respect to time is
(2.2 The equat ions of motion (2.2) then imply that the function E, defined as
. oL
E= " q ·- - L , ~ to' i qi
(2.2
is conserved in time, i. e. dE I dt = O. One can easily be convinced that E is indeed energy if we take the example of a set of N point particles of mass m in a potential for which 3N
L ex =
1
2:= 2mqi2 -
(2.2
V (qj).
i= 1
The relation (2.22) then implies that
which corresponds to the classical expression for the total energy (kinetic plus p tential) of a system of point particles . Note that (2.22) is simply the Hamiltoni (2 .3). The inva riance under infinitesimal coordinates transformations and generalized locities, is obtained using the transformation qi f------+ Qi = qi + Eli, and qi f------+ Qi qi + Eji' where E is a small parameter (E « 1) and Ii is a vector that will be specifi later. The invariance under this transformation can be written as
Expanding the left-hand side of this relation to leading order in
E,
we get
Once again, the equations of motion (2.2) imply that
(2.2
and hence to the existence of a new conserved quantity. Equation (2.24) can be illustrated on the example (2.23) if one chooses translati in space as the coordinat e transformation (i.e. Ii = £i constants in Cartesian coor nates). For each of the three possible independent quantities £i (i.e. along the thr
From classical to quantum
75
possible axes), the quantity p = mq is conserved: this is the momentum. One can clearly see here that for a vector-like invariance (three independent quantities form a vector) corresponds to a conservation law that is itself vector-like. This is the meaning of Nocther's theorem. 2.1.1.7 Field theory case Let us now come back to the case of field theory, an example of which is the action (2. 13) , and let us consider a general coordinate transformation (2.25) Then the scalar field ¢ transforms as
which is a local variation. One can also define a global variation d through
which only takes into consideration the modification of the field itself at the point xc> , thus defining a total variation after the local variation is brought back to its original point. To first order in E, these two variations are related by (2.26 ) Note that the variation
d commutes with the derivative, unlike the variation 6. Indeed,
while
still to first order in E. The theory will be said to be invariant under the transformation x f--+ x' (to simplify, we write x == xC> when there is no ambiguity) if the action , i. e. the integration of t he Lagrangian density over a space-time volume V4 , remains unchanged during the operation. In other words , (2.27) where the integration is performed over the same volume of space-time, expressed in terms of the new coordinates (hence the notation VI in the second equality). The volume element is modified by the determinant of the J acobian matrix
76
Overview of particle physics and th e Standard Model
where the second equality comes from the expansion to first order in then b ecomes
E.
Equation (2.2
using the relation (2.26) between the operators 6 and d, this leads to
(2.2 Furthermore, we can expand d£ in this expression as
which is possible since d is an exact differential (d and expression and using the field equations (2.14), we get
1 V4
d 4 x d£ =
1 V4
d 4 x 0J.L
an commute). Integrating th
(0£J.L ¢> - ) ~ d ¢>
,
such that (2.28) b ecomes
Since V4 is arbitrary, this implies that the integrand vanishes, or in other word that the term in parenthesis is a conserved current JJ.L , i.e. that satisfies oJ.LJJ.L = T his current is explicitly given , using (2.26 ), by
(2 .2
for a transformation of the form (2.25). In this expression we have reintroduced th local variation 6¢>, which is often better controlled than the total variation d¢>. The previous discussion does not depend on the fact that ¢> is a scalar , i.e. on th way it transforms under a change of coordinates, nor does it depend on the fact th we have considered only one field. More generically, for a theory with N fields ¢ where a E [I , N ], the relation (2 .29) can be generalized to
(2.3
This relation is still valid if the indices of ¢>a form , for instance, a vector (¢>a or a tensor of higher rank.
->
AJ
From classical to quantum
77
The conservation of t he current J implies the existence of a quantity constant in t ime. To see t his , it is sufficient to evaluat e the integral of 0l-' JI-' on an arbitrary space volume V3 , namely
before using the theorem of Stokes- Ostrogradski to eliminate the second term
It t hus follows that
r
JV
oo J od 3 x
i.
=
dt
3
r
JV
JOd 3 x
= 0,
3
and t he integral of the time comp onent of the current is indeed conserved (see the generalization of Section 1. 4 from Chapter 1) . 2.1 .1.8 Field en ergy and momentum
Just as t ime and space t r anslation invariance imply the conservation of energy and moment um in point-particle mechanics , t he same invariance, expressed in t erms of the current (2.30) , generat es the conservation of the energy-momentum tensor. Under an infinitesimal t ransformat ion X'I-' = xl-' + dJ' in Minkowskian coordinat es, where the fl-' are fo ur constant quantit ies, we can see that the field remains unchanged , i.e. b¢ = O. Note that t his remains true for any vect or-like or tensor quantity of arbitrary rank , since aX' l-' / ox v = b~ for this specific case. Since the quantities fa are supposed to be constant, it is possible to read them off from the definition (2.30) . The conserved quantity is then a t ensor of r ank 2, since
which allows us to define a canonical energy-momentum t ensor
8j.J.a
(2 .31 ) This t ensor is conserved by construction , i.e. (2.32) For each value of a = 0, . . . , 3, corresponding to independent variations of fa, is associated a conserved quantity: they are the components of the energy-momentum vector p o. = (E / c, P) , given by
p o. =
~ c
which is hence conserved in time.
r d x8o<> (xf3) , 3
JV
3
(2 .33)
78
Overview of particle physics and the Standard Model
2. 1.1 .9 Symmetrization of the energy-momentum tensor
T he energy-momentum tensor (2.31 ) is not necessarily symmetric and thus does not necessarily correspond to t he t ensor (1.83) that appears in the Einst ein equations (1.85) However , it is always possible t o make it symmetric in an appropriate way, such that it remains conserved and such that the physical quantities we can derive from it are not affect ed. This allows us t o use its symmetric version as the source for the E inst ein equations. Let us define TP,cx == GP,cx + opGPP,cx .
This t ensor is conserved by construction if GPP,cx is antisymmetric in its first two indices, i.e. GP,pcx = _ GPp,cx . Furthermore, TP,cx leads to t he same conserved quantity as long as the fields decrease fast enough at infinity. We indeed have
(2.34) The second term of the last equality vanishes identically since GPP,cx is antisymmetric The last term can be expressed as an integral over the surface that limits V3 , i.e. oV3 which we send to infinity, where the fields are supposed to vanish . It follows that
(2 .35)
which we can identify with the momentum vector. This can b e illustrat ed by the example of free electromagnetism. In that case, the field ¢ get s replaced by the vect or potential AI-'" Its dynamics is determined by the Lagrangian density [ef. (1.94)]
which only contains a kinetic term , with Pcx(3 == ocx A (3 - o(3 A cx . The part of the energy-momentum tensor that is proportional to the metric is already symmetric. The second t erm of (2.31) is given by GP,cx - _ o£e. m. oC< A _ ~ p,cx F p cx(3 e .m . 00 A P 47) cx(3 p,
P
,
which can be expressed , after a simple rearrangement , as
Here again, only the last t erm prevents the tensor from being symmetric (note, furthermore, that it is not even gauge invariant).
From classical to quantum
79
Let us now look for a t ensor with t hree indices G PWY., ant isymmet ric in its first two indices, which will allow us t o remove the last t erm. In other words , it should satisfy opGPJ.LQ = FPJ.L o pAn Using Maxwell 's equations, t his can be easily solved to obtain GPJ.LQ = FPJ.L A The symmetrized energy-moment um t ensor is hence Q .
TJ.LV e .J11 .
=
FVP F J.L _ P
~",J.LV F FQ{3 4 '/ Q{3 ,
(2. 37)
which turns out to be the same as t he one obtained by variation wit h respect to t he metric (1.83) . 2.1.2
2.1 .2. 1
Quantum physics
Uncertain ty principle and Hilbert spaces
The great concept ual revolut ion of quantum mechanics [3J arose from its st atement that one cannot determine simultaneously with arbitrary precision the positions and momenta of any obj ects. T his means, for inst ance, t hat if one measures t he posit ion and moment um of a particle with precisions 6.q and 6.p , then , at best (in one dimension), 6.q. 6.p >
1 -n
- 2 '
(2. 38)
where n is Planck 's constant. T he existence of such an intrinsic limitation in t he possibility of knowing the stat e of a physical system obliges us to give up the not ion of a classical t rajectory. (See, however , Ref [4], which discusses t he Bohm- de Broglie ontological interpretation fo r which such notions are still meaningfuL ) T he quantum descript ion replaces the classical hypotheses, by assuming t hat any physical system is described by an element of a complex vector space, a space of states, i.e. configurations, the dimension of which depends on t he system. Let us imagine, for instance, that we would like to study a light beam polarized linearly, either vertically or horizont ally. If t his is t he only prop erty we describe on the syst em, we then have a system wit h two possible st ates, denoted as I f-T) and 11), in the 'kets' not ation introduced by Dirac; t he reverse notation allows us t o define a scalar product , fo r example (11 represents a ' bra', so that the set of t hem forms a 'bra-c-ket '. The vector space, in which t his system evolves, can have these two st ates as a basis, thus it is a two-dimensional space. Any element of this space can always be decomposed as 11lf ) = 01 f-T) + .81 1) with 0, f3 E
80
Overview of particle physics and the Standard Model
unaccountably infinite dimension. In this case (infinite dimension and endowed with a norm), it is called a Hilbert space.2 2.1.2.2
Operators and the superposition principle
In the quantum framework, the physical variables are replaced by operators that act on the vector space of states. For instance, for the stat e !q), which describes a particle at the position q, there exists a position operator, denoted as ij such that
(2.39)
In other words, the position state becomes an eigenstate of the posit ion operator. To avoid any risk of confusion and error in what follows, we will systematically denote a quantum quantity corresponding to a classical one by the same letter with the symbol , ' ': q f-+ ij, P f-+ p, etc. Any operator A should satisfy some conditions in order to correspond to a measurable quantity. In particular, since the space of states is complex, the eigenvalues o an observable, which can potentially be measured, should be required to be real. This implies that A is Hermitian , i. e. At :==TA* = A. Furthermore, since the eigenvalues Ok of this operator correspond to all possible values of the concerned property, the eigenstates !ak) of A should form an orthonormal basis of the space of states. In other words , any arbitrary physical state !\[! ) can b e decomposed as
(2.40)
where the discrete sum should be replaced by an integral in the case of a continuous infinity of degrees of freedom i.e. in the case of a Hilbert space of states. An operator A that satisfies these conditions is called an observable. The relation (2.40) directly leads to the central property of quant um theory, namely t he superposition principle, according to which several physical configurations can be superposed in a linear combination that is still a physical state; this is the case in (2.40). Moreover, a measurement of the observable A performed on the state !\[!) will necessarily give a unique eigenvalue Ok of this operator. Based on this principle, we can see that the dynamical equation satisfied by the physical state is necessarily linear. 2. 1.2.3 Non-commutativity
Any given classical physical quantity corresponds to an operator in the space of states, and the equations that control the dynamics must reproduce those of classical physics in the limit where we know they are valid. The easiest way to obtain the quantum laws is thus to derive them from the corresponding classical laws. Now, unlike the functions that define classical physics, operators act on a vector space and do not always commute . It is this property that is used to realize the correspondence.
We define the commutator between two operators A and B as [A, B] :== AB - BA. This obj ect has the same symmetry properties as the Poisson brackets introduced in
2The phrase ' Hilbert space' is a lso very often used for finite-dim ensiona l spaces, although this is mathemat ically incorrect.
From classical to quantum
81
(2.5). It turns out that we obtain an appropriate prescription if we make the substitution (2.41 ) {A , B} f-----+ - ~ [A , 13] .
In particular, t he position and momentum variables qi and pi are now variables that no longer commute and (2 .7) becomes
I [tii,qj]
= 0=
[pi,pJ]
and
(2.42)
This relation can be generalized to numerous cases (see for example the case of a scalar field later on). One can explicitly check here that in the representation where the position operator is expressed by (2.39), t he momentum operator then becomes
pi =
- infJ/fJqi .
2. 1.2.4
Predictability and statistics
The predictions of quantum mechanics are essentially probabilistic. Formally, t his translates into saying t hat if a physical system is in a state I"'), then t he probability Pw(<1» for it to be measured in a state 1<1» is proportional to the square of the scalar product (<1>1 "' ). Since only these probabilit ies are effectively predicted by the t heory, one can choose to normalize t hese physical states, which will always be assumed in what follows, i. e. ("'I"') = 1. With this convention, we have (2.43) Note that the normalization of a state is always possible, also because the scalar product of a state with itself is by definition positive in a vector space with positivedefinite norm (which is a pre-requisite here to define probabilities)
For an observable operator A, we have seen that the only values a measurement could give were those belonging to the spectrum (the set of possible eigenvalues) of t his operator. Suppose that we have a great number of particles prepared in the same physical state, for definiteness, we choose, for instance, the state I"') of the decomposition (2.40). The measurement of A on this set of particles will be given by the average of t he operator in t he state I"'), namely
with a variance around this average value given by
As we would expect, this variance vanishes when I"') is an eigenstate of A. The uncertainty relation for the observables A and 13 is obtained in a similar way [5]
by considering the operator
C = 613 (A - (A) ) + i6A (13 - (B) ) , where
from now
82
Overview of particle physics and the Standard Model
on (unless specified otherwise) the average is taken in an arbitrary state 11l1), an the redundant index has been suppressed. It is sufficient to notice that (ot 0) = II011l1)11 2 2: 0 and to expand this inequality to obtain
llAll13 2: ~ 1\ i [A, 13] ) I'
(2.44
which, once applied to the position and momentum operators, reduces to the Heisen berg inequality (2.38) , with an extra factor of, arising from t he fact that the coordi nates and momentum in orthogonal directions commute.
2.1 .2.5 Heisenberg and Schrodinger representations
Any physical state can always be normalized to unity. This allows for a probabilisti interpretation, which makes sense only if the dynamical evolution conserves the nor malization of the states. Denoting by (;(t) the t ime evolut ion operator, the fact tha the evolution is linear implies that the state 11l1(t)) evolves as
1111 (t ))
= (; (t
- to) 1111 (to) ) ,
(2.45
where (; satisfies
N (t) 1111 (t)) = N (to) 11l1(to) ) ¢=? (;t (; =
l.
(2.46
Thus the evolution operator (; is a unitary operator. In order to be physical, an quantum theory must satisfy this unitarity principle; this criterion allows us to rule ou some theories, usually very speculative, without even needing to resort to experiments Unitary transformations actually allow us to go further and to define some rep
d)
resentations i. e. some pairs ({An}, {11l1 ) t hat contain all possible operators and states of the theory. These representations are related to each other by
I ~h
{ An
-t
I~) k =, ~1 1l1 ) ~ ,
-t
An
= U An U
(2.4 7
.
These rela tions guarantee the conservation of the norm of t he states, and t hus of th probabilities: all predictions made in a given representation will be identical to thos obtained in another representation. There exist many useful representations, in particular, the Heisenberg picture i which the operators are time dependent but where the basis vectors, t he states, ar time independent. In this representation, thanks to (2.41), the classical equation i directly transposed to
dA
[ ,, ]
illcit = A , H ,
(2.48
where fl is the operator that corresponds to the Hamiltonian function of the classi cal system. To obtain (2.48) , we have assumed that the operator A did not depend explicitly on time. Such an explicit dependence can be taken into account by simpl adding the required term with a partial derivative. The Schrodinger picture corresponds to t he other extreme in the possibility o choices. In t his representation t he operators do not depend on time, and only t h
From classical to quantum
83
states are time dependent. The difference between these two representations is similar to the choice one can make when describing rotations. Either one considers an active rotation of the system , leaving the axis unchanged , or one can describe the same rotations in a passive way, considering that it is the axes that change (in the opposite direction) while the system itself does not move.
2.1.2.6
Schrodinger equation
In the Schrodinger representation, the quantum dynamics of the state, now depending on time, simply follows from the requirement that predictions, and thus probabilities, should be time-translation invariant. Indeed, (2.45) implies that the time propagator U(t - to) satisfies d d ' dt 1\IF(t)) = dt U(t - to) 1\IF(to )) . However, if the equation describing the dynamics of a state is linear and independent of the initial state and time to, then the right-hand side term in the previous relation should be proportional to the vector 1\IF(t) ). We can hence conclude that d ' dt U(t - to)
,
<X
U(t - to ).
This differential equation can be solved very easily to give
U(t - to) = exp
[- ~iI x (t -
to)] ,
(2.49)
where iI is a Hermitian operator. From t he time-translation invariance, this operator should be conserved , and should therefore be related to the energy. Actually it can only be the Hamiltonian. The form of (2.49) guarantees that the operator U is indeed unitary. Let us now consider again (2.45) with the solution (2.49), we obtain the Schrodinger equation
ifi~ I\IF (t)) dt
=
iI I\IF (t)) .
(2 .50)
Moreover, we can note t hat U is precisely the unitary operator one needs to shift from the Schrodinger representation to the Heisenberg one. The vector 1\IF(to)), which is effectively time independent, forms the basis of this representation. Equation (2.50) can also be understood as a formal correspondence rule between the Hamilton func tion , i.e. t he energy, and the operator ifidj dt.
2.1.2.7 Harmonic oscillator Before attacking the subj ects of relativistic quantum mechanics and field theory, let us first recall the results of a very simple system, the harmonic oscillator. This will allow us to put into practice the notions discussed previously. Furthermore, this syst em will also be useful in field theory since all t he fields , whose properties we shall use, can be expanded as sometimes infinite sets of such harmonic oscillators.
84
Overview of particle physics and the Standard Model
The Hamiltonian of a one-dimensional harmonic oscillator of mass m and angu frequency w is
(2.5
The Schrodinger equation (2 .50) for this Hamiltonian is obtained from the correspo dences discussed previously, namely H --+ ilid/ dt and p --+ - ilia / aq, . d'if; 2lidt
li2 a 2 'if; - 2m aq2
=--
+
1 2
2
- mw 'if;
(2.5
'
where we have defined the wavefunction 'if;(q , t) == (ql'if; (t)). This function is usua complex. Given the interpretation (2.43) of quantum mechanics, the square of modulus, 1'if;1 2 , represents the density of probability to find the particle at position q t ime t. The normalization of the state I'if;(t)) simply implies that the total probabili integrated over the whole space, is unity, so that 1'if;1 2 dq = l. The spectrum of the Hamiltonian, i.e. the set of the eigenvalues associated wi the operator il , is here obtained by solving the eigenvalue equation il'if; = E 'if; , equivalently by looking for the oscillation modes of the wavefunction of the fo 'if; = f(q) exp (-iEt/li). The equat ion we obtain is well known and its solutions a normalizable only if the energy is positive, which is physically acceptable. Furthermo t he derivative of t he solut ion at q = 0 is continuous only for a discrete set of values the energy, En. We define the operators a and at, called annihilation and creation operators, respe tively, (understood as being associated with each oscillation mode), by the relation
J
, Y2ii fmW(,q+2.ft) a== mw '
a, t -
f{#w ('q-2. ft ) . -
2li
mw'
(2.5
t hey are conjugate to each other but not Hermitian: they are not observables. Inverti these relations and using the commutation rules (2.42) , we find that [a , at ] = l. T Hamiltonian can then be expressed as
In this form, we can understand why the energies are positive-definite since
by the definition of the norm in a Hilbert space. Once we obtain a state In) of energy En, we can determine all t he others applying t he creation operator:
which shows that In + 1) = (n + 1) - 1/2a t ln ) is an eigenstate of the Hamiltonian wi the eigenvalue E n+ 1 = En + fiw. Similarly, In - 1) = n- 1 / 2 aln) is the eigenstate of
From classical to quantum
85
with the energy E n - 1 = En - nw. The state with lowest energy, the vacuum state 10), satisfies alO) = 0, and all the states can be obtained by
In) =
~ (att 10).
vn!
Thus, the vacuum state has a non-vanishing energy, Eo = ~nw , and the full spectrum is En = (n + ~) nw.
2.1.2.8
General potential
Let us now come back to the three-dimensional space. A particle evolving in a general potential V (x) can be expanded in a similar way in the basis of the eigenstates of the Hamiltonian, with eigenfunctions, '0E(XI-') satisfying (2.54) where the eigenvalue of the Hamiltonian E, is the energy of the state. For a harmonic oscillator , Vi1.o. ex x 2 , and (2.54) becomes invariant under the transformations x -> - x and p -> p. Therefore, the eigenstates then have vanishing average values of the position and momentum, i. e. (nlqln) = 0 = (nlpln). This reflects the fact that the potential has a minimum at the origin , increases with the distance (quantum analogue of a restoring force) and is symmetric. Thus, the wavefunctions are essentially localized at the origin and the requirement that they are normalizable demands that they vanish asymptotically. A general potential V(x) does not necessarily respect these conditions. In particular, it can be non-vanishing only in a finite region of space. This is the case, for instance, for the Coulomb potential of the electric interaction between an electron and a proton. This attractive potential is, in addition , negative. We therefore deduce that the eigenvalues are negative energies forming the different possible states of the hydrogen atom, spread on a discrete spectrum of normalizable wavefunctions. There is also a continuous component formed by all the positive energies, which corresponds to nonnormalizable wavefunctions. Actually, they are the wavefunctions of the momentum operator that can be expressed as
'0 p (xl-') ex
e i( p."' - wt ) ,
i.e. plane waves . We usually normalize them by artificially putting the particle in a box of finite volume V3 , and dividing the wavefunction by V3. Once all computations are performed nothing should depend any longer on V3 and one can take the limit V3 ---> 00. If this is not the case, using the wavefunctions of the Hamiltonian is delicate and one should resort to other methods and different representations. An important point, valid both in classical and in quantum mechanics, is that the potential V (x) must be determined by experiments, or should be imposed in one way or another. Its functional form is in the theoretical context completely arbitrary. The description of some interactions in t he framework of field theory will allow us to compute the form of this potential for many cases .
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Overview of particle physics and the Sta ndard Model
The eigenstates of the momentum can also be used as a basis for the representatio of free particles, i. e. without a potential. In this case, the creation operator at (p) act on the vacuum state to create a particle with moment um p , and the wavefunction i obtained using the Fourier transform of these states. It follows that
which allows us to define the field operator If, by
,'1-,( ) 'I'
X
=
J
3
d p i p·x (27f)3/ 2e ap , A
(2 .56
such that 1f;(x, t) = (O!If,(x)!1f; (t)). The operator If,(x) annihilates a particle at th point x, while its Hermitian conjugate If,t produces a particle at the same point. The generalized commutation relations for the creation and annihilation operator for momentum eigenstates, namely
(2.57
translate into
(2.58
Note that it is also possible to define a conjugate momentum IT( x) and a Hamiltonia formalism. We will come back to this in the context of the relativistic theory. 2.1.3
Quantum and relativistic mechanics
The application of the correspondence principle allows the immediate formulation o a relativistic quantum theory. Instead of using the classical expressions relating th total energy, the Hamiltonian , to the kinetic and potential energies, it is sufficient t consider the relativistic momentum , pi-' = mui-' , and the invariant Pi-'Pi-' = _ m 2 c2 which can be expressed as a function of the energy, E , and of the momentum , p , a Pi-'pi-' = _ E2/C 2 + p2 = - m 2 c 2 , to get E2
=
p
2 2 C
+ m 2 c4 .
The Klein- Gordon equation for the wave function ¢ can then be obtained using th correspondence E ---4 inOt and p ---4 - inV
(2.59
It will be shown in what follows how it is possible to take into account the propertie of special relativity to describe quantum field theory. An important part of thes developments can be generalized to curved space-times, in particular by replacing th partial derivatives oe. by covariant ones \7 e.' This is how we can consider the influenc
From classical to quantum
87
of gravity by means of general relativity to a large degree: the quantum field theory in curved space-time assumes that gravitation alters field theory only through the metric in which the (test) quantum fields evolve. From now on, natural units will be considered and we therefore fix 'It = c = l ' (see appendix A).
2.1.3. 1 Klein- Gordon and Dirac equations
In natural units , the Klein- Gordon equation , which describes the dynamics of a free scalar field , becomes (2.60) For a general self-interacting potential, the Lagrangian from which this equation is derived is given by (l.106). In the case of a massive free scalar field, we will take (2.61 ) i.e. a simple mass term. Its quantization will be discussed in detail in Section 2.2. A problem immediately appears in (2.60), that of the existence of negative-energy solut ions . Indeed, the classical relation E = p 2 + m 2 assumes that E > O. But the solutions of (2 .60) are simply the eigenstates of the momentum and the energy, which are ex: exp [i(p . x - Et)], but nothing indicates what the sign of the energy should be. In other words, both choices E = p 2 + m 2 are possible. But if the wavefunction rP corresponds to one of these states, then it contains less energy than the vacuum state itself. Thus, a theory containing such states is necessarily unstable. This is why we should go to the level of quantum field theory, for which the solution of (2.60) is not a wavefunction, as in the case of 'lj; for the Schrodinger equation , but an operator , similar to >it of (2.56). The difficulty we have just mentioned for the scalar field comes from the fact that its dynamical equation, unlike in the classical case, is second order in time, which is normal to respect the relativistic invariance. Another way to proceed is to postulate a first-order equation in time, and hence also in space. In order to respect the relativistic invariance, one should be able to write it as
J
±J
(2.62) which we can obtain by varying the Dirac Lagrangian (2.63) Since this equation , known as the Dirac equation, describes the dynamics of a relativistic particle its square should also reproduce the correspondence observed in the Klein- Gordon equation. We should therefore also have ('T)J.LI/ 8J.L81/ - m 2 )'lj; = 0, which implies that
88
Overview of particle physics and the Standard Model
This is only compatible with t he Klein- Gordon equation if the coefficients , " satisf
(2.64
T hus, the '" cannot be simple numbers. Actually, t hey should be matrices of rank a least equal to four . A useful representation for these ," is
,o=
(0 f) fO
a")
0 ( (j" 0
'
where each entry of these matrices is itself a 2 x 2 matrix, f = two-dimensional identity matrix, i. e. f = is 1
a =
(01) 10 '
a
2
=
(0 -i) 0 ' i
(~ ~).
aD
'
(2 .65
standing for th
T he a i are the Pauli matrices, tha
and
a
3
(1 0)
= 0-1 .
(2 .66
Finally, we have defined the vectors a" == (I , a i ) and (j" == (I , -ai ). For reasons t ha will become clear later , the matrices of (2.65) are called the chiral representation o the Clifford algebra, defined by (2.64). It is interesting to note that (2 .62) can also be obtained from the action
(2.67 Assuming that 'I/J , defined by
(2.68
and 'I/J are considered as two independent quantities , then t he variations must b performed simultaneously. T his action introduces the operator f--7
FoG == Fo(G) - o(F)G,
(2.69
for any two quantities F and G . In the case of (2.67) , F and G are spinors, but th definition applies also for any kind of field, for instance scalars.
2.1.3.2
Maxwell 's and Praca's theories
The scalar field ¢ is of spin 0 and the spinor field 'I/J of spin 1/2. T he fields know experimentally, among which we can find all the particles listed in Section 2.3, als include particles of spin 1 such as the photon, of zero mass, or the W ± and ZO vector of t he electroweak interaction. Such fields are described by a vector A" , the dynamic of which is governed by P roca's Lagrangian
(2.70
for a real field of mass M A . The Faraday tensor has the usual definition , F"v == 20I"Av
Canonical decompositions
89
Varying (2.70) using (2.14) for each component Ao< of the vector field, we find
and hence (2.71 ) This equation does not seem to be equivalent to the Klein- Gordon equation for each component of the field due to the right-hand side term. However, if we take the divergence 80< of this equation, the right-hand side term becomes a d'Alembertian , which simplifies with the one on the left, so that
Note that things are more complicated in the relativistic case, where the covariant derivatives do not commute [ef. (1.97)]. As can be seen here, if the vector field is massive, i.e. MA of. 0, then t he Lorentz gauge condition 8J.L AJ.L = 0 is automatically satisfied [the fact that Proca's theory (2.70) is not gauge invariant precisely because of this term is no surprise], and (2. 71) leads to a Klein- Gordon equation for each component, which is to be exp ected from the classical-quantum correspondence principle discussed around (2.59). The energy-momentum tensor fo r the massive vector field, once it has been suitably symmetrized, is identical to the one for the gauge field of electromagnetism (2.37) up to the mass term that has to be added. This leads to (2.72)
2.2
Canonical decompositions
The particular case of a scalar field, let it be real or complex, is discussed in detail in most textbooks on field theory; see, for instance, Refs. [6- 9]. We briefly review it in this section. For completeness, we also briefly describe the case of the vector and spinor fields. 2.2.1
Technical clarification
In what follows, we will systematically choose a symmetric definition of the Fourier transform and of its inverse that are defined by (B.58) and (B.59). 2.2.1.1
Lorentz invariant measure
One of the important properties of t he Dirac distribution is that
df
o [j(x) - a] = ~ ( dx a
where the
Xa
)-
1
o(x - xa),
(2.73)
X = Xa
represent the zeros of the argument, i.e. the solutions of f(xa) = a.
90
Overview of particle physics and the Standard Model
This property allows us to replace the Fourier integrals in 4 dimensions by three dimensional integrals when the fields over which we perform the integration are 'o shell ', i. e. their Fourier components satisfy the relation E2 == k;5 = k 2 - m 2 , the energ being posit ive-definite . In that case, it is possible to perform the integration over th time component ko with the previous relation by setting f(k o) = k;5 . This gives
with the definit ion
(2.75
and where e is the Heaviside step function. T he relation (2.74) defines a Lorent invariant measure despite being three-dimensional. Note that to obtain this measure it is necessary to impose the positive energy condition in order to keep only one term of (2 .73).
2.2.2
Real free fie ld
Equation (2.60) can be obtained by varying t he Lagrangian (l.106) in which we pos tulate an arbitrary potential V(¢) for t he scalar field ¢ . The conjugate field is then
7r(x) = so that the Hamiltonian density, H =
Of:
~ . -
J¢(x)
.
= ¢(x),
(2 .76
7r¢ - L , is given by
(2.77
General potentials V(¢) are useful in cosmology, for instance, when addressing t h inflation scenarios (Chapter 8).
2.2.2.1
Mode expansion
The eigenstates of the momentum operator are t he basis of every expansion in quantum field theory. To deduce that, we should first know the energy-momentum tensor , tha is 8 J.'
_ ~8
= 8J.'¢8
[~ (8¢)2 + V(¢) ] = TJ.'n
(2 .78
The last equality follows from the fact that the energy-momentum tensor of a scala field is already symmetric. Moreover, it coincides with t he tensor (l.1 07) obtained by varying (l.84) with respect to the metric.
Canonica l decompositions
91
The tensor (2.78) allows us to construct the four-momentum vector that is obtained by po. == TOO. , from which we indeed get, first the Hamiltonian density, which corresponds to the purely time-like component, i.e. pO == TOO = 1i, and then the associated momentum to the scalar field P = 7rV ¢. Because the presence of a general potential V (¢) can lead to some potential nonlinearities that can imply some unnecessary complications, here and in the following sections we restrict ourselves to the case of the free massive field for which V = ~m2¢2. Having found the momentum P, it is sufficient to perform the substitution of the correspondence principle and to replace P by P -+ -iV. The eigenmodes Uk(X) with eigenvalue k of the momentum operator thus satisfy -iVUk (x) = kUk (x). They are plane waves Uk(X) = Nkeik.x,
N k being a normalization factor to be determined later. It is now sufficient to expand the field on the basis of these eigenstates as
(2.79) Since ¢ satisfies the Klein- Gordon equation (2.60), we find that the coefficients ak(t) evolve as adt) + (k 2 + m 2 )ak(t) = 0, where we used the fact that V 2Uk (x) of this equation is simply
=
2 - k uk (x). Setting
wk = k 2 + m 2, the solution
with b;" = a - k since ¢ E lR. After some algebraic manipulations, and using the relations between the coefficients of the expansion, we find a more explicit four-dimensional expression, that is (2.80) where we recall that k . x == k . x - wkt. From (2.80) and from the invariant measure (2.74) we see that the field expansion, even though it was realized on the basis of three-dimensional states, is invariant, i.e. that ¢ is effectively a Lorentz scalar, as long as we choose the normalization (2.81) Moreover, this relation allows us to recover the usual commutation relations after quantization. Note that the conjugate momentum of the field is simply (2 .82) which we obtain in a similar manner as before. The factor ( - iWk) in the integrand of (2.82) implies that the latter is not explicitly a Lorentz scalar. This is not a problem since only the field should absolutely satisfy this property.
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Overview of particle physics and the Standard Model
2.2.2.2
Quantization
The field and its conjugate momentum can now be 'promoted' to the status of oper tors , which can be done by replacing the coefficients ak and a"k by the annihilation a creation operators, respectively, ak and The field Poisson brackets then becom the equal time commutations relations. If we impose
al .
(2.8 and all others to vanish, i. e. the usual commutation relations , we indeed obtain t canonical relations for the field
[¢(X,t),7T(X',t) ] =i5(3) (x-x') ,
(2 .8
a computation that is recommended to be performed explicitly. The relations (2.8 and (2.84) can only be respected if the normalization of the field modes is given (2.81) . The eigenmodes of the field operator are now defined by
u k (x , t)
=
N k ei( k .x- w /c t) ,
(2.8
so that the field operator is expressed as
(2.8
This relation can be inverted to express the creation and annihilation operators as function of the field , namely
(2.8 f--7
where the derivative 8 is defined by (2.69) . This analysis is then complet e as soon as we know the Hamiltonian, which is giv by
(2.8 We then get
(2.8
where we have used the commutation relation (2 .83) and defined Nk as the 'number particles in the state with momentum k ' operator, the significance of which will le us to the definition of the Fock space associated with this theory. A remark concerning (2.89) is necessary at this point. The second term, propo tional to 5(3) (0 ), is clearly divergent , even in the case where the momentum space finite, i. e. if there exists a cutoff k m ax above which one can no longer perform the com putation. In field theory, as discussed here, this infinite energy is not embarrassing
Canonical decompositions
93
it has no physical meaning since in practice only differences of energy are measurable. In practice, we define a normal ordering that automatically puts the creation operators on the left and the annihilation ones on the right, or by 'renormalizing ' the energy by simply suppressing the vacuum energy obtained this way. This possibility is unfortunately no longer an option as soon as the theory is coupled to gravity. The vacuum energy then assumes all its meaning and acts as a cosmological constant , which is in any case considerably more significant than the one we measure (Chapter 12). It is, in the current state of knowledge, the most serious problem that theoretical cosmology has to confront, and it comes from quantum fi eld theory!
2.2.2.3
Fock space
The complete quantum theory is obtained once we know the state of the particles. For that, we st art by defining the vacuum 10) as being the state annihilated by all the annihilation operators, i. e. aklO) = 0 'tIk E JR3 . We consider, furthermore, that this state is normalized, namely (010) = 1. A state with one particle is then constructed from this vacuum state by applying a creation operator to it. We thus have
(2.90)
Ik ) == 0,110) , normalized by
(klk' ) = (ola k al,10) =
5(3)
(k - k') ,
and we construct in this way a complete space of states, each state having n( k) particles in the state of momentum k . The definition (2.90) shows that a state with n(k) is an eigenstate of the operator Nk , since (2.91) It follows that any state describing a set of particles will be an element of
called a Fo ck space and having an unaccountable infinity of states. Finally, note that starting from the vacuum state 10) , it is possible to produce a particle at the point x simply by applying the field operator itself
lx,t ) =
q} (x , t) IO) =
J
d3k e- ik . x (271")3/2 J2Wk Ik),
(2.92)
which appears essentially as the Fourier transform of the one particle st ate with given momentum, hence its interpretation. Note that this equation uses the operator q) to generate a particle, which plays no important role here since the field is real and hence cfyt = cfy, but in the more general case of a complex field , the distinction is important.
2.2.2.4
Spherical waves exp ansion
The previous expansion is only a special case of the expansion in the basis of the momentum eigenstates expressed in Cartesian coordinates (x , y, z) . It is also possible
94
Overview of particle physics and the Standard Model
to perform this expansion in spherical coordinates (r, 0, rp) using spherical harmon In that case, we split the Laplacian into radial and angular parts t. = t.r + r - 2 t with t.r = 8; + (2 /r)8r and t.11 = 8~ + sin 2 08~. We express the eigenvalues
JP2
the Laplacian using the amplitude p == and two integer numbers £ E Na m E [-£, +£j . The scalar field ¢(x, t) can then be decomposed into the eigenmod ¢pem(x), of the Laplacian (2. which are given by
¢pem(x , t)
=
upe(r)Yem (0, rp).
Since the spherical harmonics are solutions of (B.8), (2.93) leads to an ordin differential equation for the radial part upe(r) , namely
2
d [ dr2
2d
+ ~ dr -
£(£+ 1)
r2
2]
(2.
+ P upe = 0,
the solution of which is a spherical Bessel function upe(r) ex: je(pr), (Appendix B). T normalization relations of the spherical harmonics (B.15) together with those of spherical Bessel functions (B .51) show that choosing
the modes of the scalar field are normalized , i.e.
(2 . Also note t hat with these convent ions, we have the completeness relation
roo
in
o
+e
L L 00
dp
Upe(r)Upe(r')y£~(O , rp) Yfm(e', rp')
= 5(3)
(x - x').
(2.
e=o m=-e
The scalar field operator is now expressed as
(2.
where the operators iipfm depend explicitly on time. Inserting the relation (2.97) the Klein- Gordon equation (2.60) , we get
22 (:t
w;
+ w~)
iipem(t)
=
0,
(2.
where = p2 + m 2 for each independent mode. The relation (2.98) shows that creation and annihilation operators, which could a priori depend on the three numb p , £ and m , have a time dependence that is only a function of the magnitude p of
Canonical decompositions
95
momentum. The solutions of equation (2.98) are complex exponentials, so that the total field takes the form
1 z= 00
¢(x, t) =
dp
o
Npupe(r) [Y£m(B,
+ Yi::n(B,
(2 .99)
e,m
and the momentum conjugate
(2.100) The remaining thing to do is to use the commutation relations (2.84) at equal time and to impose the same normalization as previously, namely Np = (2wp) ~ 1 / 2, to get the commutation relations for the creation and annihilation operators (2.101 )
The link between the Cartesian and spherical representations is obtained by expanding the plane wave in the basis of spherical waves (B.21). By identifying the coefficients of the term e ~ iwpt in both expansions (2.86) and (2.99), we obtain (2.102) where (Bp,
The complex scalar field
The generalization to the case of a complex scalar field, ¢ E
¢ = 3le(¢)
+ i~m(¢) ==
1
yf2(¢1
+ i¢2),
and we assume t hat the real and imaginary parts are independent. In a perfectly equivalent way, and this is the point of view we will adopt, we can also consider ¢ and ¢* as two independent fields. The Lagrangian obtained in both cases should be real, so that for a free massive field (2.103) Note the absence of the factor ~ in the mass term, compared with the Lagrangian (2 .61 ). This is due to the fact that since the two degrees of freedom of the field are
96
Overview of particle physics and the Standard Model
independent, the variation with respect to ¢ or ¢* does not give the factor 2 of real scalar field. The two conjugate momentums are
7T =
~~
= ¢* and
7T* = : ; = ¢,
which are indeed complex conjugates of one another and are hence independent. theory describing a complex scalar field is simply that of two real scalar fields of same mass, related to each other by the fact that they are the real and imagi parts of a complex number. The energy-momentum tensor is obtained by summing (2.31) over the indepen degrees of freedom
which gives
(2.
We find again the same result as the one obtained using the definition that comes f the variation with respect to the metric. The commutation relations at equal time between the field operators and t conjugate momentum are now
(2. The mode expansion is of t he form
(2. where we have introduced two different sets of operators, complex. Their commutation relations are now
ELk
and
bk , since the fie
(2.
all the other commutators vanish. With the normal ordering for the operators discussed in the case of a real sc field, the Hamiltonian takes a form equivalent to (2.89), that is
(2 .
which shows t hat a complex scalar field can be interpreted as a collection of kinds of different particles, but with the same mass, corresponding to the two of operators ELk and bk . The Fock space is then the tensor product of the two F spaces generated by these operators, with one mutual state, which is the vacuum annihilated by both ELk and bk .
Canonical decompositions
2.2.3.1
97
M axwell or Praca field
The Lagrangian of a massive complex vector field is obtained by generalizing Proca's Lagrangian (2. 71 ) as LP roca
=
-~FI.wpi.w + M~A:A" ,
(2.109)
where A" now has four complex components. As for the scalar field , the mass term no longer has the factor ~ characteristic of the real field. These types of fields can describe charged vector particles, such as t he W± bosons mediating the weak inter action. The expansion into creation and annihilation operators is analogous to (2.106), up to the fact that due to the gauge invariance for the Maxwell case with vanishing mass , or for the gauge constraint for the massive case of Proca [see the discussion below (2.71)], t he number of degrees of freedom is not equal to the number of field components (i.e. four ), but three for the massive field and two otherwise. This point leads to subtleties in the quantization that are discussed in Refs. [7- 9] to which we direct the reader. 2.2.3.2
Internal symmetries
The Lagrangian (2. 103) not only represents two independent scalar fields of t he same mass. Indeed, these two fields are the components of a complex number and the Lagrangian is thus invariant under the phase shift and
(2.110)
where ex E IR is an arbitrary constant. Expressed in terms of t he real and imaginary parts of ¢, the phase shift (2.110) corresponds to a global internal rotation. It is this invariance that imposes the stat es 'a' and ' b' to have the same mass. From Ncether's theorem , the existence of such an invariance should translate into the existence of a conserved quantity. To determine it let us work in the limit ex « 1, such that we can expand t he exponential in (2.110), leading to the infinitesimal transformation ¢' = (1 + iex)¢. This transformation being independent of the spatial coordinates, x, we have i" = 0 in (2 .29) , and the current
J
" _ . 8L . J. - 2 88,,¢ 'f'
-
. 8L ..J..* 88,,¢* 'f' ,
2
(2.111)
is conserved since in that case, we have o¢ = iex¢ and o¢* = -iex¢* (we have removed the irrelevant normalisation factor ex in this definition). The conserved 'charge ' is obtained by direct integration, i.e. (2. 112) After quantization , this charge leads to the operator (2. 11 3)
98
Overview of particle physics and the Standard Model
which is indeed time independent since its equation of motion , using the form (2.1 of the Hamiltonian , is
dQ dt
. [,Q, He , ]=
= -~
O.
Equation (2.113) can be easily interpreted by saying that particles associated w the operators a and b are particles of opposite charges; in reality they are antipartic of one another. The complex scalar field is only an example of internal symmetry, in this case of invariance under transformations of a U( l) group. The associated charge is, in gene called hypercharge, and corresponds to the electric charge only in the case where U(l) group is that of electromagnetism. The different existing particles of which will make an inventory in Section 2.3, indicate the existence of more symmetries, ba on larger symmetry groups.
2.2.3.3 Fermions and anticommutation relations
It is possible to expand the Dirac field (2.63) in a similar way as in the exp sion (2 .106) , with an important difference however: the field operators, and hen those associated to the annihilation and creation of particles, must respect antico mutation relations instead of commutation relations of the type (2.83) or (2.84). T consist ency problem related to this point is made explicit in Refs. [7,8], and is beyo the scope of this presentation. It will be sufficient to know that for fermions we ne to replace the Poisson brackets by anticommutation relations
and
(2. 1
where we used the fact that , from the Lagrangian (2.63), the conjugate momentum the spinor is given by
=
7r7j; -
O£Dirae _ .
o1/J
-
'"I,t
~'f/
.
(2.11
Note that these anticommutators, similarly to the commutators for bosonic fiel are taken at equal time. At this point, it is interesting to note that if we place oursel in t he context of field t heory at finite temperature (thermal field theory, see e Ref. [10]) , then these anticommutation relations are necessary so that the fermio satisfy the Pauli exclusion principle; moreover, they lead to the correct statisti distribution law, i. e. the Fermi- Dirac law (just as the commutation relations for bosons lead to the Bose- Einstein distribution) .
2.3
Classification and properties of the elementary particles
The general properties of elementary particles are gathered in Table 2.1 (see Ref. [11 When one looks at these properties, one finds structures, such as illustrated , for stance, in Fig. 2.1 showing the particular case of baryons of spin ~ and of posit parity (parity is defined later , here it is only an illustration).
Classification and properties of the elementary particles I
I
I
I
t,-
t, 0
t,+
t, ++
or-
0
0
0
o
2: -
2: 0
2:+
-1
I-
0
0
0
-2
r--
0
-3
-
~
99
-
1236 MeV
- 1385 MeV
so - 1532 MeV
0
Q-
0 I
-1
- 1672 MeV I
o
I
I
1
2
Fig. 2.1 An example of the structure observed between the elementary particles : the baryons of spin ~ and of positive parity. Table 2.1 Nomenclature of the elementary particles. The families based on quarks (q) have several hundreds of particles identified so far , whereas the leptons and the gauge fields are here presented in an exhaustive way in the context of the standard model. The gauge fields are written in the following way: I is the photon, the ZO and W ± bosons are the mediators of the weak interaction, and ga represent the 8 gluons of the strong interaction . Spin
Name
Examples
B aryons = qqq Ha drons
{
Mesons = qij
n
7r°'±, Ko'± ,J/'Ij;, Do , Bo , 1'/ , '
Leptons Gauge fields
This kind of diagram has made it possible to understand the existence of a very precise structure, the history of which is discussed in Ref. [12]. In what follows, we will only indicate the result of numerous years of experimental research that led to the so-called 'standard ' model of particle physics, its theory is the subject of Section 2.6.
2.3.0.4
Exchange bosons
The known particles come m three forms. First, we find the bosons of unit spin, mediating fundamental interactions: they are the gauge bosons, namely the photon " which mediates the electromagnetic interaction, the eight gluons ga, which mediate the
100
Overview of particle physics and the Standard Model
strong nuclear interaction , and the vectors ZO and W ± . The latter have the proper of being massive and are hence described by Proca's Lagrangian (2.109) , at least long as we ignore their interactions. The vectors ZO and W ± allow the weak nucle interaction to propagate, W ± are in addition electrically charged . According to the current theoretical ideas, to be complete, one should add to th table the graviton, hl-'v, massless 'particle' (in the framework of a quantum theo of gravity equivalent to that of the other interactions) of spin 2 responsible for t gravitational interaction (Chapter 1). In the absence of any satisfactory quantu theory of gravity, we do not consider at the moment the graviton as an elementa particle belonging to the standard model.
2.3. 0.5
Leptons
Of spin ~ , t he leptons, among which t he electron is the best known , are of negati electrical charge (e - , p,- and T- ) or vanishing (v e , vI-' and VT which are the associat neutrinos). The evolution of these particles is dict at ed by the Dirac equation, coupl or not to electromagnetism , depending on the case. They are also subject to the we interaction and can be grouped into three 'doublets' and
(::)
which have exactly the same properties with respect to this weak interaction. Actual the only characteristic that allows us to distinguish between these different double is the mass of the particles that constitute them. Moreover, among them, only t doublet of the electron is stable, the others, of higher mass, are subject to the weak a electromagnetic interactions, and can decay into, for instance, electrons and electro neutrinos. For a long time, the neutrinos have been considered to have a vanishing mass, b some recent experiments and measurements (see Ref. [2] of Chapter 9) have show that there must be a mixing between t he different species of neutrinos, which is on possible for massive particles.
2.3.0.6
Hadrons
Only one category of particles , albeit t he most numerous one, is subj ect to the stro interaction: it is the hadrons. They can be of half-integer spin, such as the proton a the neutron , and then belong to the category of the baryons. The mesons, of integ or vanishing spin, count among them , for instance, the 71"0 and its charged partne 71"±, which are described by the Klein- Gordon equation for the real scalar fields ( the 71"0) or the complex (for the 71"±). The surprising profusion of the hadrons, as well as the observed properties symmetry between the different elements of the families sharing some properties (sp and parity, for instance, as in the illustration of Fig. 2.1) have led to the understandi that these particles, unlike what we currently think of gauge bosons and lepton are not elementary particles, but are constit uted of 'quarks' a The latter are for t
3 According to the legend , this word was proposed by M urray C ell- Mann (Nobel Prize 1969) w would h ave read it in the book ' Finnegan's Wake ' by J ames J oyce.
Internal symmetries
101
moment seen as effectively elementary, of spin ~, and of electric charge + ~ for the quarks u, c and t, and for the quarks d, sand b. We can also group them into doublets for the weak interaction, with a similar structure to the one for the leptons , more precisely
-1
a nd This description, which is satisfying from the theoretical point of view, currently does not have any generally accepted explanation (even if many models exist) . The names of the quarks themselves come from this structure: at the time where only the first family was experimentally accessible, the chosen names were simply up (u) and down (d). Then , a third quark was needed to b e introduced, which was considered as strange (s). The second family was then completed by the charm (c). In the same philosophy, when the presence of the fifth quark started to appear, the name of beauty (b) was proposed, soon replaced by bottom, allowing the first family to be reproduced with the top (t). Just as for the leptons, the only difference between the quark families comes from their masses, as well as the fact that only the members of the first family are stable. The ones from the second and third families decay much faster than their leptonic counterparts since they do so through the strong interaction instead of the weak interaction, the latter having a much weaker amplitude , as its name suggests. All these particles have been produced in accelerators, but have never b een observed alone, which can be explained by the notion of confinement of strong interactions. This is how it is only possible to observe states containing three quarks , baryonic states of half-integer spin, and states composed of one quark and one antiquark, of integer spin: a meson. This point is not yet completely understood theoretically. Table 2.1 summarizes the nomenclature used to describe the diversity of elementary particles. The origin of the mass of all these particles, together with the important disparities observed between them , still pose many questions that are not all yet resolved. As we can see, the known particles reflect, or seem to reflect, some elements of symmetry. These are described by group theory, to which we now turn.
2.4
Internal symmetries
Contrary to leptons , which all appear in doublets , hadrons are found in numerous forms , suggesting that they are not elem entary. In reality, using a much smaller number of elementary particles, in this case quarks , and placing them in a relevant way in group representations, it is actually possible to combine them so that the other representations are perceived as new elementary particles. The role of group theory is precisely to lead us to this conclusion.
2.4.1
2.4.1.1
Group theory
Definitions
A group g is defined as a collection of objects {g;} , where the index i is not necessarily discrete , satisfying the following rules:
102
Overview of particle physics and the Standard Model
- There exists a composition law, denoted by 0, which can combine any two eleme of the group , the resulting combination being an element of the group: if gl E and g2 E g, then g3 = gl 0 g2 E g. Furthermore, this law is associative, gl 0 (g2 0 g3) = (gl 0 g2) 0 g3 = gl 0 g2 0 g3· - There exists a neutral element, denoted by 1, such that Vg E g, l og = go 1 = - Each element 9 Eg is associated with an inverse element , denoted by g - l, sat fying go g- l = g- l 0 9 = l.
Note that the law 0 in not necessarily commutative: when it is, i.e. if Vgl , g2 E gl 0 g2 = g2 0 gl, the group is then said to be Abelian. In the opposite case, the gro is usually said to be non-Abelian. The simplest group one can think of, besides the trivial group that only conta one element, is the discrete group Z2 that contains two elements, denoted by 1 (wh is the identity) and -1 with the ordinary multiplication law. This is an Abelian gro for which each element is its own inverse. For discrete groups, one can construct t complete table of all possible operations. 2.4.1.2
Lie Groups
Another well-known example of a group is that of translations that transforms a po in space x into x + a. In this case, any element of the group can be directly associat with a translation vector a: the composition law is vector addition, the inverse eleme of a is simply - a, and the identity is the translation by the null vector o. It is ag an Abelian group. Such a group contains an infinite number of elements, which can all b e identified the knowledge of three real numbers (the three coordinates of the translation vecto More generally, if it is possible to describe the elements of a group by a set of numb varying in a continuous interval (but not necessarily infinite), it is then a Lie grou A Lie group is also a manifold. The best-known non-Abelian group is probably that of rotations in three dime sions. Each rotation is determined by, for instance, the three Euler angles , or equiv lently, a unit vector (rotation axis) and the angle of rotation around this rotation ax The composition of two rotations is equivalent to a third rotation, which makes i group (since a rotation can always be inverted by performing the same rotation in t other way, and so a rotation of angle zero is the identity), but the order in which t rotations are performed is crucial and this group is not Abelian. This group, denot by SO(3) , is identical to the group formed by the orthogonal 3 x 3 matrices with u determinant. We can generalize it to the group SO(n) , which contains the orthogo n x n matrices with unit determinant. Each group is associated with an invariance. For instance , in the case of a rotatio the norm of a vector on which the rotation has been performed remains unchang during the transformation. A rotation in a generalized Minkowski space, i.e. havi p spatial dimensions and q time-like dimensions, must keep the form up.up. inva ant; it contains not only rotations in the p-dimensional space, but also pure Loren transformations (some ' boosts') . This group, generalizing SO(n) for arbitrary met signatures, describe p space-like coordinates and q time-like ones, and is denoted SO(p , q). The so-called Lorentz group, is then simply SO(3, 1). The translation gro
Internal symmetries
103
can be associated with the Lorentz group to form the Poincare group on which special relativity is based (see Chapter 1) . 2.4.2
2.4.2.1
Generators
Infinitesimal transformations
An infinitesimal transformation, T€, of a Lie group is a transformation close to the identity and hence can always be written as T€ '::' 1 + iEaAa. This defines the generators Aa , which are Hermitian matrices and depend on the nature of the performed transformation. We suppose here that the Lie group is described by p parameters, i. e. a = 1" " , po A finite transformation can be obtained from an infinitesimal transformation by exponent iation. Such a fini te transformation can always be written as the composition of a large number, tending to infinity, of infinitesimal transformations. If a represent the p necessary numbers for the finite transformation, then we can always write E = a lN, where N » 1, and performing N transformations will indeed lead to the finite transformation. In other words, we will have (2.117)
which is the general form of a finite transformation of the group. 2.4.2.2
Lie Algebra
In order for TOt the property
0
Tf3 to be a group transformation, T-y for instance, one should have
(2.118)
For infinitesimal transformations this gives to second order in the expansion of the exponentials eiOt Aeif3
A
= 1 + i (aa + f3a) Aa - ~ (aaab + f3af3b + 2a af3b) Aa Ab,
the logarithm of which, a lso expanded to second order [using the relation In(l E - ~ E2 + ... ], should be proportional to Aa . We find explicitly
a+
i (a a + f3a) A
~aaf3b
a [A , Ab]
=
+ E)
=
hc Ac .
This equality is in general only satisfied if the commutator of the>. satisfies (2.119)
This relation defines the algebr a associated with the Lie group , i.e. the set of operators of the form aa Aa. The numbers rabe are the structure constants of the group , and the Aa form its algebra. Note that since the generators are Hermitian operators, the structure constants are real numbers .
104
Overview of particle physics and the Standard Model
2.4.2.3
Representations
In order to be able to classify the particles in terms of symmetry groups , we need sort them depending on the representations of the group of the considered invarian An n-dimensional representation being a set {n a} of n X n matrices satisfying t commutation relations (2.119) of the algebra, the fields that describe the particles o given representation will be gathered in multiplets q. = {iP a}, a = 1,' . . ,n on whi the action of the group is
(2. 12
It is usual to say that iPa transforms under the representation n a , or more direct that iPa is 'in the representation n a'. If two representations are related to each oth by a similarity transformation , i. e.
na = unau t ,
where U is unitary (U t = U - 1 ) , then we say that they are equivalent. In other wor for all practical purposes, it is the same representation , such that the similarity tran formation does not provide any complementary information. If there is an invaria subspace in the representation , i. e. if the action of any element of the group on t vectors of this subspace leads to an element of the same subspace, then the represe tation is said to be reducible. In the opposite case, it is called irreducible. Only t irreducible representations are used to classify the particles in the representations the invariance groups. Moreover , if {na} forms a representation, then so does { _ (na)*}, which is cal the 'conjugate representation ' , [one can easily be convinced that these matrices a respect the algebra closure relation (2.119) since the structure constants of the gro are real] . If a representation and its conjugate are equivalent then we are deali with a so-called real representation. Otherwise, we say the representation is comple In order for a group to be able to potentially reproduce the internal symmetries particle physics, it is often required that it has some complex representations in whi fermions can be placed in a simple way. The representations that are the most relevant for quantum physics are those th preserve the norm defined by liP I2 == iP tiP,
of the multiplet iP . In order for the norm to be conserved during an arbitrary tran formation n a , we see that we need to have
(iP)tiP = iP'tiP' = (iP)t(TaYTa iP ,
for all transformations Ta. We should therefore require that (Ta)tTa = 1, i. e. th it is a unitary transformation. A representation satisfying this condition is a unita representation. Finally, the adjoint representation of a Lie group is formed using its structu constants. It is a representation of the same dimension as the order of the group , which the matrix elements of n a are
(2 .12
Internal symmetries
2.4.2.4
105
Caftan classification
Besides the space-time symmetries (Poincare group) , the invariance groups that are useful from the point of view of the classification of particles are the compact ones, i.e. for which the parameters vary on a finite interval, as if we had to deal with generalized rotations in an internal space (and in many cases, it is actually precisely the required invariance). Moreover, we wish to classify the particles along 'fundament al' invariances, i.e. with no substructure (we will see later that only one coupling constant comes into play in these groups, which makes them even more interesting). This is why we are interested in simple groups: if H is a subgroup of 9 and if Vg E 9 and Vh E H , the element go h 0 g - l is still in H, then H is an invariant subgroup of g. 9 is simple if there is no invariant subgroup in g, and semi-simple if it does not contain any Abelian invariant subgroup. One can show that a compact and semi-simple group is thus either forced to be simple or is the direct product of two or more simple groups. Using this philosophy, studying simple Lie groups is therefore sufficient to classify all semi-simple compact groups . Note also that the groups with structure S x [U(l)]n with S a semi-simple group and n an arbitrary integer, which are called reducible groups, have representations that can all be decomposed as a product of irreducible representations; t his property makes them especially interesting for the classification of elementary particles. 8ince the simple Lie groups are also subj ect to the constraint (2.119), they can be classified in terms of their structure constants . This classification was performed by E. Cartan at the beginning of the twentieth century. He proved that the simple Lie groups could be classified into 4 large categories, namely 80(2n) , 80(2n + 1), 8U(n + I) , and 8p(2n), to which one should add the so-called exceptional groups , which can not be classified into any of these categories, called G 2 , F 4, E 6 , E7 and Es· For the four infinite sequences, n is the rank of the group, i. e. the maximal number of generators that can be simultaneously diagonalised. For the exceptional groups, the rank is given by the index in the notation. For each group , the order, i.e. the number of generators, is indicated in Table 2.2.
Table 2.2 Order (number of generators) or dimension of the simple Lie groups of rank n according to the Cartan classification (the index of the exceptional groups is the rank). Group SU(n + 1) SO(2n + 1) Sp(2n) SO(2n)
G2
F4 E6 E7 Es
N generators
(n (n (n (n
~ ~
~ ~
1) 2) 3) 4)
n(n+2) n(2n + 1) n(2n + 1) n(2n -1) 14 52 78 133 248
106
Overview of particle physics and the Standard Model
The groups of the four infinite sequences can be understood using their fundamenta representation, i.e. the one of smallest dimension (from which, by direct product, a the other representations can be constructed) . The set of all complex unitary matrice of dimension n x n forms the group U(n) ('U' for 'unitary'), the subgroup restricted t the unit determinant matrices is 8U(n) ('8' for 'special'). It is the set of transformation that leave invariant the inner product of two vectors of a complex space of dimensio n. The transformations of this group leave invariant, among other things , the norm o the complex vectors and can be used in quantum mechanics to assure the conservatio of probabilities. The sequence 80(n) is constructed from the matrices that leave invariant the inne product of two real vectors. They are orthogonal matrices (hence the '0') with uni determinant (special) of dimension n x n. Finally, the group 8p(2n) regroups all th matrices M of dimension 2n x 2n that leave invariant the antisymmetric matrix
o
1
- 10
s=
o
1
-10
o
1
- 10
These matrices therefore satisfy MT S M = S, where MT is the transpose matrix of M Thus , the antisymmetric and quadratic form y T Sx for two real vectors of dimension 2 is conserved. The matrices M defined in this way are called the symplectic matrices thus the name of the group. The only transformations that allow us to conserve probabilities in quantum me chanics are those that are unitary. Can we hence deduce that only the groups of th type 8U (n) can playa role in physics, and that hence all theories should be based o such a group? Unfortunately, the answer to this question is negative: some representa tions of the groups classified by Cartan are unitary, as well as respecting the rules o their classification. We can therefore exclude a group as a possible internal invarianc only if we can show that it has no unitary representation. This is , for instance, th case of 80(11) , which is hence excluded as a grand unification group candidate. Thi is why it is indispensable to study in detail the representations of the possible Li groups.
2.5
Symmetry breaking
Not all the symmetries existing in Nature are manifest , and some underlying ones ar not immediately visible at low energy. This is , for instance, the case for the doublet of the type (e - , ve). If these particles really belong to the same representation of th gauge group of the weak interaction, why do they not have the same characteristic as well (mass , electric charge, for instance)? The answer to this question relies on th mechanism that enables us to break symmetries, and in particular the gauged ones.
Symmetry breaking
2.5.1
107
Gauge group
We are generally confronted with 'gauge' invariances, i.e. local invariances. This leads to the existence of a gauge boson, through which some interactions are performed: the presence of a gauged (local) symmetry in particle physics, as in gravity, implies the existence of an interaction.
2.5.1.1
Local symmetries
The simplest illustration of a local symmetry is given by electromagnetism. Its action is invariant under the transformation of the group U(l), which leads to the prediction of the existence of the photon, i. e. the term (1.94) of Chapter 1 in the total action. Consider a complex scalar field , cp(xl-') E C, invariant under the phase transformations cp - 7 cp' = e ic 4>C
(0l-'cp)*ol-'cp
-7
(ol-'cp')*ol-'cp'
= =
[(01-' - iC¢0l-'a) cp*] [(01-' + ic¢ol-'a) cp] (2.122) (0l-'cp)*ol-'cp + ic¢JI-'0l-'a + c~lcpI20I-'aol-'a ,
where the conserved current JI-' = -i(cp*ol-' cp - cpol-'cp*) is given by (2.111). We can note here that this term, proportional to the conserved current, leads, after integration by parts, to a surface term in the action and thus has no physical effects. Despite this fact, the action is still not invariant since the last term is remaining.
2.5.1.2
Gauge boson
The form (2 .122) suggests the replacement of the partial derivative by a gauge covariant derivative by introducing a new vector field GI-'. If we define (2.123) with 9 an arbitrary coupling constant, we find that this term transforms as
We can therefore simply impose GI-' to transform as
GI-'
-7
G'I-'
=
GI-' -
~ol-'a 9 ,
(2.124)
in order for the action to be again invariant , whatever the value of c¢. Since the action (l.94) with the Minkowski metric T) c
108
Overview of particle physics and the Standard Model
where GI-'V = ol-'CV - ovCI-" and we have generalized the potential of the scalar field for it to be an a priori arbitrary function of the field norm I¢I == j[¢f2. The action (2.125) is the basis of the scalar electrodynamics, an example of gauge theory. We should note that (2.125) does not have any mass term for the vector field CI-'. This comes from the fact that such a term, described by the Lagrangian of Proca (2.71) , is not invariant under the transformation (2.124). The action (2.125) may be generalized to the case where the gauge group is not Abelian, this is what is done in the context of the standard model of Section 2.6 Note already here that the gauge bosons introduced in the framework of non-Abelian theories can neither be massive, for the same reason that a mass term a la Proca is not gauge invariant.
2.5.1.3
Fermions
In a similar way, the action for the fermions (2.67) is made invariant under the local transformation 'Ij; ----> 'Ij;' = eicrcx'lj; by replacing the partial derivative by the gauge covariant derivative (2.126)
VIle can also see that an extra coupling term with the scalar field can be introduced. Finally, the action for the fermions, can be written in a general form as
(2.127)
with f an arbitrary constant, which, in addition of (2.125), allows us to describe a theory containing a fermion and a complex scalar field coupled via an intermediate vector. We can also see that the fermion can have a purely massive term in addition to its coupling with the scalar field , while respecting the invariance. 2.5.2
Higgs mechanism
The Higgs mechanism allows us to build a theory invariant under the transformations of a given group, while this invariance is not manifest. In other words, it allows us to impose an invariance with some properties to theories that do not seem to have it, as is, for instance, the case for theories having massive gauge bosons.
2.5.2.1
Higgs potential
We consider a complex scalar field ¢ and a gauge field AI-', the dynamics of which is described by the action (2. 125). For technical reasons that we will not discuss here, (of renormalization, see, for instance, Ref. [7]), the potential V(I ¢I ) cannot introduce powers of I¢I higher t han 4. We generally choose a quadratic term, i.e. we postulate a mass for the scalar field, and a quartic term, in AI¢1 4 . In the case of the Higgs mechanism, the chosen potential is given by
VHiggs (¢) = A ( I¢I 2 - Ti 2)2 ,
(2.128)
depicted in Fig. 2.2. The parameter Ti, having the dimension of a mass, is called a 'vacuum expectation value' (VEV).
Symmetry breaking
109
!Jle (1/1 / 17)
'i5'rn
(1/1 /17)
Fig. 2.2 The Higgs potential as a fun ction of the components (real and imaginary parts) of the Higgs field . The circle 11>12 = TJ2 indicates the set of possible minima.
For the theory to make sense, one should first define the vacuum. It can be obtained , at the classical level, by demanding the energy, i.e. the Hamiltonian, to be a minimum. A simple generalization of (2.88) gives in that case H(x , t)
= ~ [1>2 + (V¢) 2 + 2V(¢)]
,
(2.129)
which is the sum of two positive-definite terms and of the potential. As a consequence, the minimal energy is reached when the field is invariant with respect to the transformation of space and time, so that 1> = V ¢ = O. Furthermore, the potential should also be minimized , which leads to
(2.130) As illustrated in Fig. 2.2 , one of the solutions, here ¢ = 0, represents a local maximum of the potential, thus it is an unstable state of the theory. The second solution, i¢i = T), on the other hand, is the absolute minimum of the potential. This is the value the field will hence naturally take and that describes the vacuum st ate of the theory (thus the name of t he parameter T)) .
2.5.2.2 Goldstone boson The expansion of the field ¢ around its VEV is
¢ = [T) +
~h(X, t)]
ei8(X,t )/V2rJ,
(2.131)
110
Overview of particle physics and the Standard Model
e
where h and are two real fields. Noting be written as ,
mh
==
2,j),'rJ , the potential for h and
e
(2 .
and is independent of e. This can easily be understood by examining the poten the fluctuations h and e around the expectation value 'rJ of the field correspon fluctuations either along the radial direction (h) , thus changing the energy of configuration , or along the circle of the minima (e) , which by definition do not any energy. The field h is called the Higgs field , whereas e is the Goldstone boson
2.5.2. 3
Masses of the bosons and fermions
The degree of freedom corresponding to the Goldstone boson is not physical. Actu it is always possible to choose a gauge in which it is suppressed , since the only p where it comes in is in the kinematic term of the field cp, which can be expanded
(2.
Choosing the gauge in which CI-' ----> CI-' + (0l-' e) / (V2'rJ9 Cf ) , the field e comple disappears. After expanding the kinematic part (2.133) ofthe scalar field cp, one can see app ing a term similar t o Proca's one (2.70) for the vector field C w This implies that field has gained a mass during this process. After identification, we find Me = V2g The appear ance of a massive vector field can be explained by the loss of the G stone boson. Its corresp onding degree of freedom cannot have completely disappea and it has actually been completely absorbed ('eat en ', as it is sometimes writ by the gauge field. A massless vector field has indeed only two degrees of freed corresponding in t he classical theory to t he two t ransverse polarization modes ther circular or linear. On the other hand , a massive vector field evolves such its angular moment um has a vanishing projection , and therefore has three poss modes. The coupling of the scalar cp to the fermion in (2.127) leads to a modification of fermion 's mass, which is increased (or decreased , depending on the sign of the coup const ant f), by 6.mf = i'rJ. Thus, if the fermion was by symmetry initially mass as is the case in t he electroweak and strong st andard model, it will t hus acqui mass, in the same way as the gauge field. This is why fermions of extremely diffe masses can be placed in the same multiplet . We will see later , in the context of st andard model of the electroweak and strong interactions , how this process is app in a precise case, and is checked experimentally. 2.5.3
Non-Abelian case
T he Higgs mechanism can easily be generalized to an arbitrary group Q, which coul non-Abelian , but t hat we suppose is simple. We now consider a field in a representa
Symmetry breaking
111
of dimension n , i.e. a column vector ¢ composed of n real scalar fields ¢a, with a = 1, ... , n. The VEV is now a vector
~~
(Oi¢ iO)
~
C) ·
(2.134)
with 10) the vacuum st at e of the scalar fields. The individual VEVs , 7]a = (OI ¢a IO), satisfy L a7]; = 7]2 . The potential of ¢ is always of the form (2.128) , with I¢I = L a¢; , so that once again we have oV(¢) I =0
O¢a
> =TJ
'
and we define the quantum fields ¢a == ¢a - 7]a. The mass term then t akes the general form 1 02V ' , Lm ass
= - "2 O¢aO¢b¢a ¢b·
If we perform a global infinitesimal transformation with paramet er a c tential transforms as V(¢) --> V (¢) + SV(¢) , with
«
1, the po-
OV .oV C b SV (¢) = o¢a S¢a = Z o¢a (ac R) a ¢b, where we assume that ¢ transforms under the given representation with the matrices
RC For the pot ential to be invariant under the previous transformation, we need OV (¢ ) = 0, implying that
oV (Rc)b A. = 0 O¢a a 'f' b . Differentiating with respect to ¢d for ¢ = ry (for which t he first partial derivative of the potential cancels), leads to
I
0 2V o¢ o¢ a
d
> =TJ
(RCry)a = O.
There are now two possible ways to make these terms vanish , according to whether or not (R Cry) a vanishes. The set of the generators satisfying (RCry )a = 0 forms a subgroup 'H of 9 under which the vacuum is still invariant. Thus, the scheme of the symmetry breaking is 9 --> H. The expansion (2.131) around the VEV can be generalized t o 7]1
+ hI (x) (2.1 35)
o
112
Overview of particle physics and the Standard Model
assuming that only the p first values of
so that
(2 .
o
which defines the p physical Higgs fields of the theory. Under a general transformation with parameters aa, the gauge bosons Cal" tr form as the generalization of (2.124) to the non-Abelian case, more precisely
(2.
where 9 gives the coupling of the group and the r bca its structure constants. We sh stress t hat there is only one coupling constant , valid for all the fields submitted to required invariance. This property is only valid for a simple group or a simple g risen to any power with a transverse symmetry, which allows us to identify the diffe groups. In t he case of the Higgs mechanism, we use this relation for the transformat t hat only bring into play the n - (p + 1) Goldstone fields (h, i = p + 1, . . . ,n ins of the general paramet ers aa' To first order in Bi , the relation (2 .137) gives
Substituting t his expression, together with (2.136) in the kinetic term of
2.6
The standard model SU(3)c x SU(2) L x U(l)y
The standard model of non-gravitational interactions accounts for all the interact mentioned in Section 2.3 classified dep ending on the representations of the gr
The standard model SU(3)c X SU(2) L X U(1)y
113
SU(3)c for t he strong interactions, and SU(2)L x U(l)y for the electroweak ones. This last symmetry is broken by t he Higgs mechanism described in Section 2.5.2 along the following symmetry-breaking process SU(2)L x U(l)y
--+
U(l) elec,
leaving the photon m assless, as observed. 2.6.1
The strong interaction (QeD)
The description of the strong interaction is based on an invariance under the transformations of the group SU(3). Since t his group has 8 generators, its joint represent ation, in which t he gauge bosons G~ are placed , is of dimension 8: a = I , . . ·8. T he leptons are not subj ect to these interactions , and t hus appear as singlets of SU(3), which do not couple to the gauge boson. The quarks exist in three possible states, and are thus placed in a three-dimensional representation, denoted by 3. The quarks are therefore represented in column vectors with these three elements , Q ==
(ur) (dr) (sr) (Cr) (tr) (br) ug
'ub
,
dg
,
db
Sg,
cg
sb
cb
,
tg
,
tb
bg
.
(2.138)
bb
The index is called colour ,4 which has nothing to do with any actual coloured appearance. Inst ead , this terminology comes from an analogy: due to the fundamental properties of the strong interaction , a quark can never be observed alone but only in pairs of quark- antiquarks, or in sets of t hree quarks. In the latter combinations, the total colour cancels, in the same way as it is possible to combine the three fundamental colours to give white, which can be seen as t he absence of any colour. This is why t his SU(3)c group is called the colour group. The covariant derivative of each quark Q, with respect to the strong inter action is given by (2.139) where t he Aa are 8 m atrices forming the algebra of SU(3). We often take the Gell-Mann matrices (see, for instance, Refs. [7, 13] for more details) which satisfy the appropriate commutation relations of SU(3) , i.e. (2 .119) with the structure constants of SU(3) , denoted as r~. The coupling constant g3 of the group measures the intensity of the interaction. Experimentally, it has been measured that this coupling constant has a higher numerical value than those for the other interactions, t hus the name of strong interaction. By analogy with the colour states, this interaction is also called quantum chromo dynamics 'QCD'. The Lagrangian of QCD takes t he form .cQCD
=-
L
Q'"YJ1. D (3 )Q -
~GaJ1.vGaJ1.v,
(2.140)
quarks
4The index tha t t a kes the va lue r, g and b represents the ' fund amental ' colours red, green and blue.
114
Overview of particle physics and the Standard Model
2.6.2
Electroweak interaction
The electroweak interaction combines the properties of the weak interaction , com from the invariance under transformations of the group SU(2)L' and of electrom netism , based on U(l)elec, brought together as a unique interaction. All particles combined in singlets or doublets of SU(2)L' 2.6.2.1
Quarks
The quarks of each generation are supposed to be equivalent , so that the doublets
(2.1 where each fermion appears as an eigenstate of chirality with the definition 1
+
,5)'I/J,
(2.1
'l/JR== ( - 2where the matrix
,5is defined as - 10) ( 01'
One can explicitly check that (,5) 2 = 1, so that the operators P R,L == ~ (1 ± are proj ectors orthogonal to each other , i. e. they satisfy P ; ,L = PR,L and PRPL = Another useful property is to do with the anticommutation relations: anticommu with all , l-' , or (2.1 {,5"I-'} = 0,
,5
relation that will be very useful in the following sections. Finally, the quarks with right chirality are defined as singlets of SU(2)L:
2.6.2.2
Chirality
The proj ectors PR,L on the state of definite chirality 5 come into playas soon as we interested in the properties of the spinor with respect to the Lorentz transformatio In terms of the so-called chiral representation (2.65), t he projectors are found to t the following explicit form
1_,5 (10)
P L = -2- =
00
'
and
P = R
1+2,5 = (001' 0)
(2.1
In this form, the proj ection properties of these operators become transparent. N by the way that due to the fact that and anticommute (2.143), the joint spin 7jjL,R are defined using opposite proj ectors
,5
,0
5The op erator 'Y 5 is called chirality b ecause it indicates how its eigenstates transform unde parity transformation [6 , 9].
The standard model SU(3)c X SU(2)L X U(l)y
~L = ~ (I ~15 ) = ~PR
and
115
(2.145)
Thus, they satisfy (2.146) and and
(2 .147)
which confirms that a spinor existing only in one chirality state, cannot have any mass term in the Lagrangian. From the projectors (2.1 44) , one can deduce that the eigenstat es of the chirality are quadri-spinors and where ~ and X are two bi-spinors representing respectively the left and right components of the total spinor. The Dirac equation (2.62) can then be written as two coupled equations, the Weyl equations, for the bi-spinors
(at + ( j . V ) ~ = -imx, (at - ( j . V )X = -im~ ,
(2.148) (2.149)
where the mass m should now be understood as a coupling t erm between the two components. For a massless spinor , both equations decouple and each bi-spinor , ~ and X, called Weyl spinors, evolve independently. We will see lat er that these kinds of spinors have an especially important role to play in the formalism of supersymmetry.
2.6.2.3
Leptons
For a long time, neutrinos were thought to be represented by some Weyl tensors. Experimentally, t hey are only observed in the left state V L , leading to the conclusion that they should be massless. Since they only have two degrees of freedom , in order to put them in similar doublets as in (2.141), it is only possible to pair them with the left component of the corresponding lepton, leaving the right component in a singlet of SU(2)L' thus the index 'L' of the group . The structure is thus
(:eLL) , R, (~LL ) , e
~R '
(V;LL) ,
TR
,
(2.150)
which gathers in a similar scheme, the electron, the muon and the tau with their associated neutrinos. The covariant derivatives are defined as
D12) (:e L
)
== (01" -
ig2 B il"ai )
L
(:e L
)
•
(2. 151 )
L
(Ji (i = 1,2, 3) are the Pauli matrices (2.66), generators of the group SU(2) since the satisfy the algebra
[a i ,a j
]
= 2i Ei\ a k .
(2 .152 )
Similar to the strong interaction, the vector bosons B il" are in the adjoint representation of SU(2). The coupling constant of the group is here called g2 . The coefficients
116
Overview of particle physics and the Standard Model
ij E k
take numerical values +1 (resp. - 1) if (i,j,k) is an even (resp. odd) permutat of (1,2 , 3), and 0 otherwise (i .e. if two indices are the same) . The structure constant the group SU(2) are given by the rank 3 Levi-Civita tensor 6 , and this group is loc equivalent to that of rotations. Moreover, a charge with respect to the phase transformations is associated to e particle. This 'hypercharge' , to distinguish it from the electric charge Q, is deno by Y and we have YR = -2 for the right leptons, YL = -1 for the left lepto Y(qJ = ~ for the quark doublets (2.141) , of left chirality, and for the right chiral Y(UR , cR , t R) =! and Y(dR , SR , bR ) = -~. The doublet structure of (2.141) and (2.150) reminds us of that of spin ~ w its components 11) and 11)· This is why the doublet classification of SU(2)L is a called weak isospin, denoted by T to avoid any confusion with the spin S. The val T3 = ±~ are respectively given to the top and bottom component of the doub Using this analogy and bearing in mind that the electric charges of the quarks Qu ,c,t = + ~ and Qd,s ,b = -~, the ones for the massive leptons Qe,/-"T = - 1 and t neutrinos have no electric charge, we obtain the Gell-Mann- Nishijima relation
(2.1 Note that for this relation to hold for all particles, we need to set T = 0 for singlets. We can then add a covariant derivative t erm U(l)y for each field ,
where gl is the coupling constant associated with the group U(l)y. 2.6.3
A complete model
Putting together all the previous building blocks, a complete Lagrangian can be writ down to describe the three non-gravitational interactions and all their invariances
2.6.3.1
Kinetic terms
The kinetic terms of the standard model for the electroweak and strong interacti are thus given by
.c kin
=
- i-;!ry/-'
(a/-' -
- 4"1 (o/-,Gv -
- ~ (O/-,G av -
ig 3 G a /-,)...a
ov G/-,)
-
ig2Bi/-,ai - ig1YG/-,) 'ljJ
2
OvGa/-,
+ ifbcaGb/-,Gcv)
2
(2.154
6The Levi-Civita tensor components El , ... ,n = + 1
€
in n dimensions is the completely antisymmetric rank n tensor w
The standard model SU(3)c X SU(2)L X U(l)y
117
where t he action of the covariant derivative depends on which particle it acts. For inst ance, for the lepton doublet of the first family, using the hyper charges obtained earlier , t his gives,
since the leptons are not subj ect to the strong interaction. 2.6.3. 2 W± and ZO bosons and photon ; Weak angle Equation (2. 155) can explicit ly be expanded in left and right components as .c~ in
= - iv eL, I-'81-'ve L
-
ie,l-' (81-'
+ 2ig2 sin Bw A I-') e
- V2g2 (veL,I-'W:e R + eR,I-'W;veJ - L ' I-'Zl-' e L - - g2 -B - (-VeL' I-'ZI-' VeL - COS 2B WVe cos w
-
2 SIn . 2 BWV - e R' I-'Zl-'e ) , R
(2.156) where t he weak a ngle Bw is defined by t he relations
g1
-
g2
=
tanB w ·
(2. 157)
As for the vector fields, they are defined by
(2.158) and
Bw - sinBw) (B31-') ~ (B31-' ) == ( CO~Bw sinBw ) (AZ~,.. ) == (C~S sm Bw cos Bw CI-' CI-' - smBw cos Bw
(AZ~ )
,
,.. (2. 159)
AI-' being t he electromagnetic field. The Lagrangian (2.1 56) can be understood in the following way. First , we notice that the kinet ic term of the electron leads to a gauge coupling of the U( l ) type with the photon , since it can be written in the form
with
q = g2 sin Bw the electromagnetic coupling const ant and Q the electric charge operator , with value Qe = - 1 for the electron. This coupling is the same fo r the left and right degrees of
118
Overview of particle physics and the Standard Model
freedom of the electron. In addition to the coupling terms, an intermediate neut boson ZJ1. has appeared , coupling both right and left components of the electron a different way, and also adding a self-coupling term for the (left-handed) neutri Finally, charged intermediate bosons make interactions between neutrinos a electrons possible (the charges of t he W± come from the fact that each term of final Lagrangian should preserve the electric charge, which between a neutral neutr and an electron is only possible wit h the indicated signs for the charges). For a quark doublet , one can perform the same expansion, which gives us electric charges of the quarks, but leads to 36 different interaction terms for e generation of quarks once the triplet of colour is made explicit .
W:
2.6.3.3
Symmetry breaking
The Lagrangian (2.154) counts almost all possible terms that satisfy the invariance t he standard model and bringing into play t he particles known experimentally. D to its invariances, and in particular due to the weak isospin, it is not possible to wr mass terms for the leptons. As for the quarks, all elements of the same doublet SU(2)L should have the same mass, like for instance both quarks u and d, which is contradiction with the measurements. Thus t he model can only be made compati with experiments if the SU(2)L symmetry is broken. For this , we introduce a complex field doublet
the dynamics of which is governed by the potential (2.128)
The minimum of this potential is 14>112 + 14>2 12 = ry2. Assuming that the Higgs field not subject to the strong interaction, the kinetic term becomes
(2. 1 which can be expanded as
1&4>11 2 + 1&4>21 2 + v2g2 (.JJ1.- W+J1. + .JJ1.+ W - J1.) + (g2 B 3J1. + gl Yq, C J1.).Ji - (g2 B 3J1. - gl Yq, C J1.).Ji: + (g2B3J1. + glYq,CJ1.)214>112 + (g2B3J1. - gl Yq,CJ1.)2 14>21 2 + v2g2 (4)~4>2 + 4>14>;) [w-J1. (g2B3/l- + glYq,C/l-) - W+J1. (g2 B 3/l- + 2g2 (14)11 2 + 14>212) W:W-J1..
ID12 =
glYq, CJ1.)]
(2.16
In (2 .160) , we have introduced the Uy(l) hypercharge Yq, of the Higgs doublet; determine its value below.
The standard model SU(3)c X SU(2)L X U(1)y
119
The currents are defined by
and
JI-'- == i (cP~0l-'cP2 - cP20l-'cP~) , with J + = (J- )t = i (cPZ0l-'cPl - (h0l-'cPz) . We now work around the minimum of the potential, and perform a gauge transformation to suppress any of the three redundant degrees of freedom . In the unitary gauge, for which cPl = 0, and cP2 = 7) + hlJ2, i.e.
+ O
with
(~)
and
(2.162)
we are left with only the Higgs field, h, which is real. We then get
c :r =
- ~Ol-'hOl-'h - (7)2+ ~h2 + V27)h)
[(92B31-' - glY<jJ CI-')2
+ 29~W+I-'W;],
(2.163) where we can recognize the intermediate vector boson ZI-' provided that we fix the hypercharge of the Higgs scalar doublet to Yq, = 1. It is the only value that will preserve the electromagnetic invariance after the symmetry breaking, ensuring the photon defined by the relation (2.159) remains massless.
2.6.3.4
Masses of the intermedia te bosons
Equation (2.163) contains interaction terms between the Higgs boson, h, and the intermediate bosons, ZO and W±. The mass term of the latter can be extracted. Taking h = 0 in (2. 163) , we get
.c masses --
-
2g27) 2 2W+ W - 1-' I-'
-
7) 2 cosg22 Bw
Z I-' ZI-' =-
-
M2W W+W-I-' I-'
-
-21 M Z2ZI-' ZI-' ,
(2.164) so that we can make the identification
Mz =
and
..)2g27)
= Mw .
cosBw
cos Bw
(2.165)
This last relation allows us to derive the value of the weak angle Bw from the masses of the intermediate bosons. This has been done following their discovery at CERN in 1982 . Their actual masses are [11]
Mw = 80.419 ±0.056 GeV
and
Mz = 91.1882 ±0.0022 GeV,
which gives the weak mixing angle, namely sin 2 Bw ,...., 0.22 , that is approximatively ()w ~ 280. Note that no term of the type AI-'AI-' appears in the Lagrangian (2.164). The photon remains massless, in agreement with Maxwell 's electromagnetism.
120
Overview of particle physics an d the Sta ndard Model
2.6.3.5
Masses of the fermions
T he particles of right and left chirality belong to different representations of SUL (2 thus it is not possible to construct mass terms in a simple way just as for t he gau fields, one should impose that all the fermions are massless before the symmetry brea ing. Moreover , since the right neutrino should be absent from this theory, one shou postulate that the neutrino is massless, even after symmetry breaking. The most general coupling terms between the fermions and the Higgs field th satisfy the required invariances, are given by 7
(fleptonsfL . cpe R
+ f qT ii L
. ;PU R
+ fqlii L
.
cpd R
+ h. c. )
,
generations
(2.16 with ;;,. 'f'
. 2 A.*
=W
'f'
(
¢2 )
= -¢r '
where the matrix (j2 is defined by (2 .66), with hypercharge YJ) = -Y¢ = -1 , an where the sum is over the three different generations , so that the fields e R , u R an dR represent, respectively, (e, j.L , T)R ' (u , c, t) R and (d , s, b)R' In (2.166), the numbe iteptons, fqT and fq l, called the Yukawa coefficients, not only represent the coupling the quarks and leptons with the Higgs fields, but, once the symmetry is broken , als give the values of the m asses of the different fermions. One can see that the interactio of a given particle with the Higgs field is all the more important that it is massive. 2.6.3.6
Electromagnetic symmetry
Working in the unitary gauge defined by (2.162), we see that the vacuum configuratio CPo is identically annihilated by the electric charge operator Q given by the Gell-Mann Nishijima relation (2.153). Indeed, with
T3 =
~2(j 3 = ~2
(1 0) 0-1
'
and Y¢ = 1, we have
The annihilation of the vacuum state by the generator of electromagnetism impli that this state satisfies (2.167
and hence it is invariant under a transformation U(I)elec. Thus, the symmetry breakin is not total and the electromagnetic invariance is recovered, as required for the theor to be compatible with experiments .
70 ne can explicitly check that with the hypercharges given previously for all particles, these term are the only possible ones t h at are invariants under the tran sformation of U(1)y.
Discrete invariances
121
Note that even if here the relation (2.167) has been explicitly obtained in a unitary gauge, from the invariance of the initial gauge of the theory we can conclude , independently of the gauge , that this result will apply for a ny choice of gauge, i.e. that for any choice of the configuration of the Higgs doublet, as long as it is at the minimum of the potential.
2.7
Discrete invariances
In addition to the continuous symmetries, usually gauged ones, there exist discrete symmetries such as parity, charge conjugation and time-reversal invariance. The first one appears to be maximally violated by the weak inter actions, and the last one weakly, for instance , in the system J(o - Ko · In what follows, we will limit ourselves to the study of the scalar field, all results being generalizable to the fermionic and vector sectors [8]. For a classical field, a symmetry operation can be obtained in the following way. For a given transformation x f---7 f( x , a), where a represents a set of parameters defining the transformation , the field ¢( x) transforms as ¢'(X')
= A(a)¢(x) ,
(2. 168)
where A is a matrix if the scalar field ¢ has several components. A state I'lj;) in the Fock space, corresponding to the field operator ¢, transforms linearly i'lj;') = U(a)I'lj;), where ut = u - 1, enssuring that ('lj;I'lj;) = ('lj;' i'lj;'). We require that the matrix elements of the field operator satisfy the relation (2.168) ,
between two arbitrary states I'lj;l) and 1'lj;2) and the transformed states I'lj;D and which leads to ut(a)¢(x')U(a) = A(a)¢,
14)~) ,
i.e. to where the dummy variable x' has been replaced by x and expressed using the inverse transformation law. We can now see how these general transformations can be applied to different particular cases, useful in particle physics. 2.7.1
Parity
Let us assume that we perform a reflection transformation on the spatial coordinates
{
X
f---7
x' = -x,
t
f---7
t'
=
t.
(2.169)
If the physics remains unchanged under t his transformation (as is the case, for instance, for the Klein- Gordon equation t hat is of second order in the spatial derivatives) , t hen a classical field can only be modified by at most a phase factor
122
Overview of particle physics and the Standard Model
¢'(x') = sp¢(x), with Isp 12
=
1. For the field operator this translates into
p - l ¢(X , t)p =
sp¢( - x , t),
(2.1
where P is a unitary operator representing the parity. If ¢ E C, then the variation can be performed via a phase, i. e. sp = e iQ , a E However, for a real field (¢ E ~), the corresponding operator should be Hermitian, ¢t = ¢, and hence
pt ¢t (x , t)
(P-l) t =
p - l ¢ (X , t) p = s;¢( - x , t) ,
s;
from which we deduce that = sp. This, together with the relation Isp 12 = 1, imp t hat sp = ± 1. In this special case, the particle associated with the field will be s to be scalar if sp = + 1 and pseudo-scalar for Sp = -1 , i.e. when it has a negat intrinsic parity. The action of the parity operator on the Hilbert space can be obtained by expand the field in the basis (2 .106) (we consider from now on a complex field) and rewrit t he equation. We find
J
d3k [Pa pt ei(kx-wktl (21T)3 / 2V2wk k =
s p
J
3
d k
(21T)3 / 2V2Wk
+ Pb t pt e- i(k'X - wktl] k
[a e-i(k,x +Wktl k
+ bt ei(k'X -wktl] k
,
(2.1
where the sign of x in the plane-wave exponentials of the right-hand side have b inverted, in accordance with (2.170). Since t his last operation is equivalent, in exponentials , to the one that changes the sign of k , it gives the transformation la of the creation and annihilation operators
Pakpt = spa_ k and
Pblpt
= Spb~k'
(2.1
T hus, the parity operator reverses the sign of the momenta, which had to be expec since t hey are 3-vectors. For the case of a free complex scalar field, i.e. massive but with no interact with any other field , the Hamiltonian, given by the relation (2.108), transforms un parity as
(2. 1
since the integral is p erformed on all values of the vector k and Isp 12 = 1 whicheve the parity of the states we consider . Similarly, the field momentum, equal to
(2.1 transforms as
ppp-l =
_P,
which corresponds to the transformation law of a vector. At this point, it is interest to note from (2.173) that the parity operator commutes with the Hamiltonian, and th
Discrete invariances
123
the parity of a physical state is a constant of motion for the free complex scalar field. Some interactions can modify this conclusion: this is the case of the weak interactions. A parity can also be attributed to the vacuum, and we choose PIO) = 10) (the vacuum must remain identical to itself if the coordinates are inverted). Note that we can explicitly construct an operator P satisfying (2.172) for which this relation is verified, but it is also possible to give a negative intrinsic parity to the vacuum: this only complicates things a little and simply inverts all the fields parities. In other words , this is only a matter of convention. 2.7.2
Charge conjugation
Another discrete transformation as simple as parity can also be performed. This is charge conjugation C, which consists in replacing each particle in a given physical state by its antiparticle. In terms of operators, this amounts to transforming (p into (Pt and vice versa. We can therefore write (2 .175)
and
the second relation of (2. 175) , being simply the Hermitian conjugate of the first one. Just as with parity, in (2.175) , t he eigenvalue of Sc has a unit modulus since transforming twice in a row all the particles and their antiparticles is in the end equivalent to no transformation at all. This is equivalent to saying that C2 = 1. Furthermore, note that C should be unitary, so that finally, we have = C- 1 = C. In terms of the annihilation operators, we find
ct
(2.176) and the Hermitian conjugate relations for the creation operators . Assuming, as before, that the vacuum is an eigenstat e of the charge conjugation, with eigenvalue + 1, i.e. CIO) = 10), we can then determine the charge C of any physical state created with a determined number of particles. Here again, we find t hat for a free scalar field , the charge conjugation is a constant of motion, leaving the Hamiltonian (as well as the momentum) unchanged. However, the conserved charge Q of (2.11 3) changes sign, as one could expect,
CQC- 1 = - Q, so that an eigenstate of the conserved charge Qcannot simultaneously be an eigenstate of the charge conjugation. The time conservation of the latter is thus only useful for physical states with a global vanishing charge. = 0, and the ones that violate parity conserMost of the interactions satisfy
[iI,C]
vation, such as t he weak interactions, more often than not, leave the combination CP invariant . In the section dedicated to baryogenesis in Chapter 9, we will discuss the cases where even this constraint is not valid. However, in all field theories analogous to the one presented here, a discrete symmetry is satisfied, it is the combination CPT, where the last symmetry, the t ime-reversal invariance, requires some technical details that are given in what follows .
124
2.7.3
Overview of particle physics and the Standard Model
Time reversal and CPT theorem
The last discrete invariance, valid for instance for a free scalar field, is that related t time reversal T: {
X
f---+
x' = x ,
t
f---+
t' = - to
(2 .177
Unlike the two previous symmetries, which are unitary operators, this symmetry ca be implemented in the Fock sp~ce of states only in the form of an antiunitary operator An operator T is antiunitary if it is antilinear, i.e. such that
(2.178
and if it satisfies TTt = TtT = l. Thanks to this last operator , one can prove [14] the following very general theorem If a local quantum field theory can be represented by a Hermitian Lagrangian .c(XD that is invariant under proper Lorentz transformations (i.e. without the space an time inversions) and if its field operators satisfy the spin-statistics theorem,9 then w have
(2.179
Due to this relation (2.179), the action integral S = J d 4 x..c is invariant under th transformation CPT. Thus, the equations of motion, which are obtained by varyin the action, and the canonical commutation relations, are also unchanged under th transformation. We can also deduce that the Hamiltonian of the given theory mu commute with CPT,
[CPT, iT] = 0,
and hence any realistic theory, i.e. satisfying the hypothesis of this theorem, has th invariance. This result is often written in the symbolic form 'CPT = 1' . A non-trivial consequence of this theorem is the following: for any theory satisfyin it , i.e. for any realistic theory, the particles and antiparticles should have exactly th same masses and the same lifetimes. This consequence is verified with a very hig precision since, for example, for the Ko/ Ko system, we find [ll]iMKo - MRoi/MKo 10- 18 . For most of the other particles, we find typical values of order of ib.M i/M 10- 5 , and constraints on the differences of lifetimes of order of ib.Ti/T < 10- 3 .
8This happens b ecause of technical reasons related to the fact that the time-reversal operator mu also reverse the roles of the initial and final configurations [8] . 9The spin-statistics theorem , itself also very general in field theory, specifies that the particl associated to integer-spin fields (for instance bosons, scalars and vectors) have a statistics describe by the Bose- Einstein distribution function , whereas the ones corresponding to half-integer fiel (fermions, Dirac spinor, for instance) , are d escr ibed by Fermi- Dirac statistics. This theorem on relies on microcausality. It is discussed in detail in Ref. [15] .
Discrete invariances
125
The standard model of particle physics presented in this chapter has so far never displayed any flaws from the point of view of experimental physics. However no one sincerely thinks that it describes all interactions between the elementary particles in a final way. There are many reasons for this, which are as much experimental as theoretical. From an experimenta l point of view, the recent discovery of the existence of a mass for the neutrinos and the discussion concerning the chirality (Section 2.6.2) indicate that they cannot uniquely exist in the left chirality as attributed by the standard model. There must therefore exist a right neutrino , which has different properties from the left one since it would have otherwise already been detected experimentally. They therefore need to have a very large mass, in order to explain why no accelerator has already produced it. At the experimental level, there is still an unknown . Since the discovery of t he 'top' quark , all particles of the standard model are identified except one , which is maybe the most important since the principle of symmetry breaking relies on it, namely the Higgs boson . As long as it has not been produced directly in an accelerator , it will remain hypothetical, and the model will not have been fully verified. Now from a theoretical point of view, t he situation is even worse. First, this model, although it unifies in a quantum framework the three non-gravitational interactions (electromagnetism , weak and strong) , makes no mention of the last interaction, gravity, despite t he fact t hat one of its pillars, the Higgs mechanism, is supposed to be at the origin of the mass of elementary particles. Interestingly, these masses are also the gravitational charges of these particles . Moreover, t he standard model relies on three different symmetry groups, wit h t hree coupling constants having no links between them, which is not justified by any fundamental principle. Finally, and this may be the most serious, there are 19 free parameters in the theory, which can only be measured. T his proliferation of parameters is often considered as t he sign t hat it is not a fundamental theory but actually a low-energy effective theory. We will see later that in many extensions, the number of free parameters is seriously reduced, and that it is in principle possible to calculate t he 19 arbitrary coefficients that appear in t he general Lagrangian in terms of a much sm aller set of parameters.
References
[1] L. D. LANDAU and E. LIFSCHITZ, Mechanics, Pergamon Press, 1976. [2] N. D ERUELLE and J. P. UZAN , Mecanique et gravitation newtoniennes, Vuibe 2006 . [3] C. COHEN-TANNOUDJI , B. DIU and F. LALOE, Quantum mechanics, John Wil 1992. [4] P. R. HOLLAND, The quantum theory of motion, Cambridge University Pre 1993. [5] J. SCHWINGER, Quantum mechanics, Springer, 200l. [6] F. HALZEN and A. D. MARTIN, Quarks & leptons, John Wiley and Sons, 198 [7] M. A . PESKIN and D . V. SCHROEDER, An introduction to quantum field theo Addison-Wesley, 1995. [8] W . GREINER and J. REINHARDT , Field quantization, Springer-Verlag, 1996. [9] L. H. RYDER, Quantum field theory, Cambridge University Press, 1987. [10] J. 1. Kapusta, Finite-temperature field theory, Cambridge Monographs on Ma ematical Physics, 1994 [11] W. -M. YAO et al., 'The review of particle physics', Journal of Physics, G 33 2006; see also the website http ://pdg .lbl. gov / that gives the same informati Updated almost every two years to take into account new experiments and th results. [12] E. SEGRE, From X-Rays to quarks: modern physicists and their discoveries, Fr man , 1980. [13] H. GEORGI, Lie Algebras in particle physics, Westview Press, 1999. [14] W. PAULI, in Niels Bohr and the development of physics, Pergamon Press, 19 G. LUDERS, Ann . Phys. (NY) 2 , 1, 1957. [15] B. DIU et ai. , Physique statistique, Hermann, 1996.
Part II The modern standard cosmological model
3 The homogeneous Universe The first cosmological solution of Einstein's equation was given by Einstein himself in 1917. At the expense of introducing a cosmological constant, he was able to construct a closed and static space. The general cosmological solutions were discovered independently by Alexandre Friedmann and Georges Lemaitre in 1922 and 1927. This chapter details t he derivation of these solutions from Einstein's equations and studies their kinematics and dynamics before defining the various observational quantities that will be used in this book. It ends on some considerations on lesssymmetric space-times. For complementary developments , see Refs . [1-7].
3.1 3.1.1
The cosmological solution of Friedmann-Lemaitre Constructing a Universe model
Within the framework of general relativity, one can try to answer the question of which solution of Einstein's equation describes the Universe we observe, or more modestly, which solution is an idealized (but good) model of our Universe. To carry out this program, we need enough input from astrophysical and cosmological observations. This program is indeed difficult to tackle and its solution will, in particular, depend on the amount of available data. But , as we noted in the introduction, even with ideal data, there are limitations to the answers one can give to this problem. These limitations arise mainly because we observe only a finite part of one Universe (it is a unique object and we cannot discuss its probable nature by comparing it to similar objects) from a given space-time position (and we are unable to choose this point) . Astrophysical data are mainly localized on our past light cone and around our past worldline for geophysical data. They are thus localized on (just a portion of) a 3-dimensional hyp ersurface. First, it may be that various space-times are compatible with the same data and second, the interpretation of these data is not independent of the space-time structure. Thus , we must distinguish between the observable Universe (for which , by definition , we have data) and the Universe, which includes regions we cannot directly influence or experiment. The inference of the geometrical properties of the Universe from the observable Universe cannot b e achieved without hypothesis and philosophical prejudices. The standard cosmological model, in its simplest form, relies on four central hypotheses. HI Gravitation is well described by general relativity. Indeed , as discussed in Chapter 1, general relativity is well t ested on small scales (e.g. , Solar System scales),
130
The homogeneous Universe
but we cannot exclude that it is not t he case on large scales or at early times (se e.g., Chapter 12 for examples of theories that dynamically become more and mo similar to general relat ivity during t he evolut ion of the Universe). In particula t he E instein equivalence principle implies that the laws of physics can be extrap lated at all times (see Chapter 1). We will thus keep applying, as long as possibl the standard laws of physics, in particular for non-gravitational interactions. H2 Hypothesis on the nature of matter. We do not examine the space-time itself or i matter distribution. Rather , we observe particular objects (e.g. , galaxies, ... ) t h are supposed to be representative of t he total matter content . The influence of th variations of the properties of individual objects on the inference of the matte content of the Universe has to be quantified. On large scales, we will assume t h matter is described by a mixture of a pressureless fluid and radiation, allowin also for a cosmological constant. In particular , we assume that there is no matt outside the standard model of particle physics described in Chapter 2. The perfe fluid hypothesis will be shown to derive from the hypothesis H3. H3 Uniformity principles. The previous hypotheses are not sufficient to solve th E instein equations. We must assume some symmetries for t he space-time. Th hypothesis and their consequences are discussed in Section 3. 1.2 H4 Global struct ure. Hypotheses H1- H3 enable us, as we shall see, to solve Einstein equation and to determine t he lo cal structure of the Universe model. Howeve there may exist various manifolds with identical local geometry and differe global topology, which correspond to the choice of various boundary condition For a globally hyperbolic space-time, the choice of the topology reduces to th choice of the topology of the spatial sections.
As we shall see, t hese hypotheses allow us to construct very successful Univer models but there are reasons to challenge everyone of t hese four assumptions. Th will be discussed mainly in Chapter 12. 3.1.2
Cosmological and Copernican principles
Galaxies seem to be spread isotropically around us, just as the cosmic microwav background radiation. This indicates that the space-time describing the observab Universe should have a spherical symmetry around us. T here are two possible expl nations: either our Galaxy lies in a special place in the Universe so t hat it is no homogeneous and must have a centre. Or the space-time is homogeneous; every poin is then similar and the Universe is isotropic around each point (see F ig. 3.1 ). To construct cosmological models, we can distinguish two uniformity principle T he cosmological principle supposes t hat t he Universe is spatially isotropic and homo geneous. In particular, this implies that any observer sees an isotropic Universe aroun him. We can distinguish it from t he Copernican principle that merely states t hat w do not lie in a special place (the centre) of the Universe. This principle, together wit the isotropic hypothesis, actually implies t he cosmological principle. The cosmological principle makes definite predictions about all unobservable r gions beyond the observable Universe. It completely determines t he entire structu of the Universe, even for regions that cannot be observed. From this point of view this hypothesis, which cannot be tested, is very strong. T he Copernican principle ha
The cosmological solution of Friedmann-Lemaltre
{:] p
{:] Q
131
.......... .. .... ..-. ... .. .. .. ...... ........ .... ··· .. . .. •.. . .... ··.. . -.. U .- ....... . . . . . . .. -. -. .. .... ....... .. ........ {:] p
• • Q.
•
Fig. 3.1 A point distribution , statistically isotropic around every point (left) and around a unique point (P) (right). In the second version, P and Q are not equivalent. The cosmological principle excludes such kinds of solutions, which would assume that we lie in a sp ecial place in the Universe . From Ref. [l J of the introduction.
more modest consequences and leads to the same conclusions but only for the observable Universe where isotropy has been verified. It does not make any prediction on the structure of the Universe for unobserved regions (in particular, space could be homogeneous and nOn isotropic On scales larger than the observable Universe). Some cosmological spaces that do not satisfy this cosmological principle are described in Section 3.6. From a pragmatic point of view, these uniformity principles should be considered in a statistical sense, i. e. for a Universe smoothed On dist ances greater than its larger structures. Thus, they implicitly involve a smoothing scale and a smoothing procedure to which we shall return. They lead to the Friedmann- Lemaitre solut ions that are at the basis of modern cosmology. Recently, a test of the Copernican principle was proposed [8J. 3.1.3
Cosmological principle and metric
Homogeneity and isotropy are two distinct notions that we should define more precisely. The homogeneity of space means t hat at every moment, each point of space is similar to any other One. More precisely, a space-time is (spatially) homogeneous if there exists a one-parameter family of space-like hypersurfaces, I: t , foliating the spacetime such that for any t and for any pair of points (P, Q) of I: t , t here exists an isometry of the space-t ime metric taking Pinto Q (see Fig. 3.2). Isotropy means that at every point the Universe can be seen as isotropic. More precisely, a space-time is (spatially) isotropic at each point if there exists a congruence of time-like worldlines with tangent vector uJ.L (called isotropic observers) such that at any point P and for any pairs of unit spatial tangent vectors ei, e~ E Vp (see Chapter 1), there is an isometry of the metric that leaves P and uJ.L at P fixed and that rotates the vectors ei into e~. Thus, it is impossible to construct a preferred tangent vector perpendicular to uJ.L.
132
The homogeneous Universe
/ /
~
... Q
p. III
..
..
·x7 ./
p
Fig. 3.2 (left): A homogeneous space-time can be foliated by a family of homogeneous spaces (right) : for an isotropic space, there is an isometry that rotates any unit spatial tangent vecto ei to any other e~, while the vector u lL remains invariant . From Ref. [9J.
We recagnize that for a hamageneaus and isatropic space-time, the surfaces of ho mogeneity, ~t, must be orthogonal to the tangent vector uj.t of the worldline of the isotropic observers , that are thus called fundamental observers. (Proof If it were no the case; then, assuming that the homogeneous space and the cangruence of isotropi observers are unique, the failure of the tangent subspace perpendicular to uj.t to coin cide with ~t wauld enable us ta construct a geometrically privileged spatial direction This is in cantradiction with isotropy.) The metric gj.tv induces a spatial metric :rj.tv(t) on each hypersurface ~t · As w have just seen, there must exist an isametry of :rj.tv bringing any point of ~t into another point of ~t and it must be impossible to construct a preferred vector from :r,w. As one can show, this implies that the Riemann tensor canstructed from :rj.tv fo the three-dimensianal spaces ~t must be ..of the farm I [2, 9, 10]
where K has , because ..of homogeneity, the same value at any paint ..of ~t, and can ..onl be a functian ..of t, K(t) . A space satisfying this property is called a maximally sym metric space (as we shall see it has the maximum number ..of Killing vectars allawed by its dimensian) and is a space of constant curvature. Two spaces ..of canst ant curvatur with the same dimensian, signature and the same value ..of K are isametric. Thus, it) sufficient ta enumerate the three-dimensianal spaces ~orresJ::>0nding ta all values ..of K We classify these spaces according ta the sign of K . If K is pasitive, ~t is a three dimensional sphere ..of radius R(t) , with embedding equatian x 2 + y2 + z2 + w 2 = R2(t in a faur-dimensianal Euclidean space. In spherical caordinates,
1 As shown in Ref. [9], this conclusion can be drawn from the isotropy a rgument. Consider (3) R ILv cx { as the components of a linear map, L, of the vector space of 2-forms into itself. Since L is symmetric it can b e diagonalized. If its eigenvalues were not equal, then one could construct a preferred 2-form and then a preferred direction. Thus , isotropy implies that L is proportional to the identity and thu (3) R cx{3 = 2K b cx b{3 by symmetry. ILl.'
[IL
vi
The cosmological solution of Friedmann-Lemai'tre
x
=
RcosX,
y
=
Rsin xcos B,
z
=
R sin x sinB cos
w
133
Rsin xsinBsin
=
t he E uclidean metric ds 2 = dx 2 + dy2 + dz 2 + dw 2 induces the three-dimensional metric on I;t If K
= 0, I;t is a three-dimensional Euclidean space with metric
If K is negative, I;t is a t hree-dimensional hyperboloid , with embedding equation _w 2 + x 2 + y2 + Z2 = _ R 2(t). In spherical coordinates ,
x
=
R cosh X,
y
=
R sinh xcos B,
z
=
R sinh xsinB cos
w
=
R sinh xsinB sin
the Minkowski metric ds 2 = - dw 2 + dx 2 + dy2 + d z 2 induces the three-dimensional metric on I; t dS(3) = R 2(t ) [dX2 + sinh2 X (dB 2 + sin 2 Bd
3.1. 3.1
Four-dimensional m etric
The worldlines of the fundamental observers are orthogonal to the hypersurfaces I;t. The coordinates syst em can be transported from one hypersurface to another by means of this congruence of worldlines. The general form of the space-time metric is imposed by these symmetry considerations 91.1.1/ = - uJ.L UV + ~J.Lv(t), where, for each value of t , ~J.LV is the metric of a three-dimensional sphere, Euclidean space, or hyperboloid. It follows that the line-element of the cosmological space t akes the form ds 2 = - (U J.L dxJ.L)2 +~J.L v( t)dxJ.Ldxv . Setting uJ.LdxJ.L == dt , t he coordinat e t is t he proper time measured by the fundamental observers (i. e. co moving wit h uJ.L) and we get the general form of the space-t ime metric as (3.1 )
t being the cosmic time, a(t ) t he scale factor and
lij the metric of constant t ime hypersurfaces, I;t , in comoving coordinat es. 2 Latin indices i, j , . .. run from 1 t o 3. The conformal t ime 7) , defined by
2This implies t hat ;Yij (xk, t ) = a 2 'Yij (x k ), so that the sp atia l met r ic satisfies (3) R ijkl
and K(t )
= K/a 2 (t).
= 2K 'Yk(i'Yj]I ,
134
The homogeneous Universe
(3.
dt = a(t)d7] , can also be introduced to recast the metric (3.1) in the form
(3. T he spatial metric, in comoving spherical coordinates, takes the general for m
(3.
where X is a radial coordinate and dn 2 = dB 2 + sin 2 Bd'P2 is the infinitesimal so angle, 'P runs from 0 to 27r and B from - 7r to 7r. T he quantity runs from 0 to when K > 0, and from 0 to +00 otherwise. The function f K depends on the curvatu K (which now is a pure number) as
vTKTx
K - l / 2sin
fK( X) =
(VRx)
K
>
0
X
K =O { (- K) -1/ 2 sinh(v'- Kx) K < O.
(3.
It relates the surface S(X) of a comoving sphere to its radius X by S(X) = 47rf'k( that reduces to t he E uclidean expression for distances small compared to t he curvatu (i.e. for X « II Using the radial coordinate r = fK( X), (3 .1) can also take t for m
vTKT)·
(3.
This metric, or the equivalent forms (3 .1) or (3.3) , is called the Friedmann- Lemait metric. T his construction shows how the cosmological principle, and thus the symmet assumptions that follow, has allowed us to reduce the ten arbitrary functions of t space-time metric to a single function of one variable, a(t ), and a pure number , K.
3.1 .3.2
Geom etrical quantities
The expression for the Christoffel symbols can be obtained , using the definitions Chapter 1 and the metric (3.1). Their only non-vanishing components are f
o ij
= H a2'Yij,
fjo. = HOj,.
i
_
f jk -
(3)
rijk ,
(3 .
where the (3)fjk are the Christoffel symbols of the spatial metric 'Yij . The quanti H = ala is t he Hubble parameter , and a dot represents the derivative wit h respect the cosmic time t. The explicit form of (3)fjk is not needed to compute the component of the Ric t ensor because the spatial metric 'Yij satisfies (3) R ij kl = K hik'Yjl - 'Yinjk) , so th
The cosmological solution of Friedmann-Lemai'tre
135
(3) Rij = 2K lij and (3) R = 6K. It follows that the only non-vanishing components of the Ricci tensor are 3
Roo =
a a
-3~,
R ij
=
(
2H
2
K) a
+ ~ii + 2 a 2
2
(3.8)
lij,
from which we deduce the expression for the scalar curvature
R=6 ( H 2
K) ,
ii a
(3.9)
+~+2
a
and for the non-vanishing components of the Einstein tensor
Goo 3.1.3.3
=
3 ( H2
+
~) ,
G ij
=-
(
H
2
+ 2~ii +
K) a
a2
2
lij.
(3.10)
Killing vectors
As explained in Chapter I, to each symmetry of a space-time is associated a Killing vector field,~ , satisfying (1.58). We have seen that for a space-time of dimension n , the maximal number of independent Killing vectors, and thus of symmetries, is n(n + 1)/2. For the Friedmann- Lemaitre space-times, the spatial sections are maximally symmetric so that we expect to obtain 6 Killing vectors. They can be explicitly determined by considering the Weinberg coordinates
in terms of which the spatial metric, and its inverse , take the form
(3.11) with r2 == 6mn xmxn. The Killing equation, \7 I-'~V + \7 V~I-' = 0, gives for the OO-component, ~o = 0, so that ~o is a function of the spatial coordinates only, ~o = F(xk). Then, the Oi and ij components take the form ~i = gki Dk~O and likDj~k + IjkDie + 2H lij~O = 0, where Di is the covariant derivative associated to lij. They can be combined to get
Only iI a 2 may possibly depend on time in this equation. If this is the case, iI a 2 is not a pure number , then this implies that F = O. We thus have six solutions to the Killing equations: 3 As
fbpf?j
an example, consider the computation of R ij . It can b e decomposed as Rij = oor?j - OJ
~
r3jrf", + (3) R ij , where
r?o +
all terms involving no p, = 0 component have been gathered to
identify the Ricci tensor of the spatial metric. We then use that
(3) Ri j =
2K "Iij = 2(K / a 2 )gij.
136
The homogeneous Universe
• three Killing vectors associat ed to spatial translations
r = 1, . . . , 3,
(3.12)
• three Killing vectors associated to spatial rotations
r , S = 1 ... , 3.
(3.13)
When Ha 2 does not depend on time, Ha 2 = K, as we shall see from the Friedman equations. This occurs for the Minkowski, de Sitter and anti-de Sitter spaces. It follow that there are four additional solutions associat ed to time translation and Loren boost s. They are , respectively, given by
(3 .14 and
L /-L[rJ ..
L o[rJ -_
for K = 0, r . X ,
(3.15 for K = ±1 ,
in t erms of the conformal time TJ. Because of the expansion of the Universe , these four vectors are no longer Killin vectors of the Friedmann- Lemaitre space-time. They now satisfy the equation V /-L ~lI V lI~/-L = (V (X~(X)g/-L lI /2 and are called conformal Killing vectors. The symmetries in volving t ime (i.e. time translation and Lorentz boosts) are lost due to the fact tha the cosmological space-times are not static but they are still conformal to a stat space-time, a property related to the existence of these four conformal Killing vector 3.1.4
Kinematics
Without knowing the evolution law for the scale factor, a(t) , a few general consequence can be obtained simply from the form of the Friedmann-Lemait re metric. 3.1.4.1
Fundam en tal observers
Notice first of all that in the metric (3.1), the coordinates xi are comoving. Th worldline of a fund amental observer with traj ect ory orthogonal to the hypersurface I;t is given by x O = t and X i =constant (one can check that this is indeed a time-lik geodesic). As a consequence , this observer has a four-velocity
(3 .16 As seen earlier , this metric can be re-expressed in the form
(3 .17
The cosmological solution of Friedmann- Lemaitre
137
where 9 /-Lv = g/-LV + u/-LU V is the projector t ensor introduced in Section l.2.5. The expansion of the Universe thus allows a comoving observer to naturally decompose any vector p/-L into a time-like and space-like component as (3 .18) In particular, all tools introduced in sections l.2.5 and l. 3.2 of the first chapter can be used. 3.1 .4.2 Hubble law Let us now consider two comoving observers with traj ectories given by x = x = X2. In the physical space, their separation is given by r12
Xl
and
= a(t)(xl - X2)'
It follows that
(3 .19) The farther the observers are from each other , the greater is their relative velocity. T his is what is usually called the Hubble law. Notice t hat H depends a priori on time and can only be approximat ed as a constant for obj ect s whose distance is small in comparison to the Hubble radius. Note that this law is a special case of the generalized Hubble law (l. 119) with e = 3H and (J/-LV = 0, as imposed by homogeneity and isotropy. 3. 1.4.3
Recession velocity and proper velocity
Since the physical distance r is related to t he co moving distance x wit h the relation r = a(t)x, we deduce that
r = Hr + a(t)x ==
V rec
+ vpro p er '
(3.20)
The recession velocity, V rec , is a t ime-dependent quantity and represents t he component of the velocity relat ed to the Hubble flow. As for v proper, it is the proper velocity of the object with respect to the cosmological reference frame privileged by the expansion. 3.1.4.4
Properties of ligh t
In the geometrical optics limit , it was shown that the traj ectory of a light ray, X"' (A) , is given by a null geodesic with wavevector k'" = dx'" / dA satisfying (3 .21 ) Using the general decomposition (3. 18), we obtain
where E = - k/-Lu/-L represents the energy of a photon measured by the comoving observer and p/-L its momentum (p/-Lu/-L = 0). It follows that
138
Th e homogeneous Universe _E2
+ a2iij p i p j = 0,
which is simply the generalization of the well-known relation between the energy of obj ect , its momentum and its mass. Moreover, the projection along ul-' of the geode equation gives . Oij . 2 i' EE + f ijP P = EE + H a iij P rY = 0,
so that t/E = - H and hence E = ko la , ko being a constant. The energy, a hence t he frequency 1/ oc E , of any photon varies as the inverse of t he scale fact Its wavelength varies as the scale factor. In an expanding space, wavelengths are th redshifted achromatically. T he redshift defined by the general equation (l.122) is t herefore given by 1+ z
=
(ul-'kl-')emission
(ul-' kl-' )observation
a( t o b servation) a( temission) .
(3. 22
The redshift can be measured from the comparison of an observed spectrum to laboratory spectrum , from which one can deduce how much the Universe has expand since it was emitted. We stress at this point that most observations give an angu position on the celestial sphere and a redshift. The radial distance cannot be measur directly but is obtained from the redshift and requires the knowledge of the expansi law of the Universe, that is a(t). 3.1.4.5
Beh aviour of a m assive test particle
Consider now the trajectory of a massive t est particle with 4-velocity vI-' . It satisfie
(3.2
can be split as vI-' = aul-' + pI-', where ul-' is t he 4-velocity of t he fundamen observers and pI-' satisfies ul-'pl-' = O. It follows from ul-'ul-' = vl-'vl-' = - 1 t hat pl-'pl-' p2 = - 1 +a 2 . The geodesic equation fo r vI-' then leads to -aa = (8/3)11-'''PI-'P'' = H so that aa + H( - 1 + ( 2 ) = 0 from which we obtain that - 1 + a 2 oc a- 2 . We conclude that p2 oc a - 2 so that a --+ 1 and vI-' --+ ul-'. The proper velocity any massive t est part icle is damped as a -I and its worldline tends to adjust to t Hubble flow , that is to the worldlines of the fundamental observers. vI-'
3.1.5
Space-time dynamics
Up to now we have described the implications of the Robertson- Walker geometry. investigate the dyna mics of these space-times, we must obtain the equations of moti deriving from the E instein equation. When these equations are fulfilled , we refer these solutions as Friedmann- Lemaitre space-t imes. 3.1. 5.1
Energy-moment um tensor
The energy-momentum tensor, TI-'''' is a symmetric tensor of rank 2. The possi forms this tensor can take are reduced by t he space-time symmetries. Indeed , t most general form compatible with symmetries is
The cosmological solution of Friedmann-Lemaitre
139
B represents the only spatial component (for the fundamental observers); it is therefore identified with the pressure that is thus seen to reduce to a unique number for a homogeneous and isotropic space. A = TjJ,vu jJ,u v == p is the energy density measured by the fundamental observers. In conclusion, the most general form of the energymomentum tensor is that of a perfect fluid (3.24) The energy density, p, and the pressure, P , only depend on time. This form is completely fixed by the cosmological principle and is not an independent choice. Thus, the only remaining freedom is the choice of the equation of state, that is the relation between the pressure and the energy density.
3.1.5.2 Friedm ann and conservation equations Using the expressions for the E instein t ensor and the form (3.24) of the energymomentum tensor, we can easily deduce that the two Einstein equations reduce to two independent equations
(3.25)
(3 .26 ) recalling [see (1.130)] that r;, == 87rG N • As for the matter conservation equation (\7 jJ,TjJ,v = 0) , it reduces to a single equation
I p + 3H (p + P) = o. I
(3.27)
We can convince ourselves that t he three equations (3.25) , (3.26) and (3 .27) are not independent [indeed, by differentiating (3.25) with respect to time and expressing (i / a using (3 .26) and (3 .25) to eliminate H2 , we find (3.27)]. This is a consequence of the Bianchi identities.
3.1.5.3
Covariant approach
In the covariant approach, a Robertson- Walker space is defined by the conditions V/-Lf = 0 for any scalar function f , which implies that ilP = 0, and CJ jJ,V = wjJ, = 0, so that \7 jJ,u v = 8 9jJ,v/3 . Space being isotropic , we can set S = a, and thus H = 8/3 and the Raychaudhuri equation (1.73) gives (i
1 r;, ujJ,u v = - -(p 2 jJ,V 2
3- = -- R a
+ 3P ) + A'
140
The homogeneous Universe
and the generalized Friedmann equation (1. 78) gives 4 ~ 6K 2 (3) R = - 2 = 2KP + 2A - _ 8 2 = 2KP + 2A - 6H2. a 3
We recover the Friedmann equations, but the regime of validity of the Raychaudh and generalized Friedmann equations is larger since they also apply to non-isotro Universes , such as Bianchi Universes (see below).
3.1 .5.4
Equations in con form al time
It will sometimes be convenient to work in t erms of the conformal time. Rememberi that dt = ad7] , we find that the derivative of any quantity X with respect to 7], X', related to that with respect to t by XI = aX. It will be convenient to remember th
X" - HX I = a 2 X.
XI=aX ,
al / a has b een introduced, which satisfies
The comoving Hubble constant H particular
a" aii = - - H 2
H = aH,
a
'
a2if = HI - H2.
We deduce that the Friedmann and conservation equations [(3.25) to (3.27)] take t form
(3.2
I
H =
K
2
-"6 a
(p
A
2
+ 3P) +"3a ,
I pI + 3H (p + P) = o. I
(3.2
(3.3
Notice also that if the scale factor is a power law in cosmic time , then
a(t) ex t n as long as n
3.1.5.5
i=
{==}
a(7]) ex 7] ''.'.n,
(3.3
1.
Eq uation of state
We hence have two independent equations for three unknowns (the scale factor , t energy density and the pressure). We therefore need some additional information solve this system. For that, the most convenient way is to give an equation of sta for the matter in the form P=wp. (3.3
For instance, pressureless matter will be described by w = 0 whereas for radiation , will have w = 1/ 3. For the cosmological constant, A, which corresponds to a consta
4Note t h at here (3) R refers to the Gaussian curvature of the hypersurfaces L;t with metric ~I This explains why (3) R = 6K Ia 2 .
The cosmological solution of Friedmann-Lemai'tre
141
energy density, (3 .27) implies P = - p and thus w = -l. The resolution of (3.27) implies that for any constant w, p ex
a- 3 (l+w).
(3.33)
Comparing this general result with equations (3.25) and (3.26), we notice that the curvature term can be assimilated with a fluid with equation of state w = -1/3. By differentiating w with respect to time and expressing pi with the conservation equation (3.30) we get Wi = -3H(1 + w)(c; - w), (3.34) where
Cs
is the speed of sound, defined by
pi
cS2 =pi For a barotropic fluid w
=
(3.35)
c;, so that w is a constant.
3.1.5.6 Reduced form It is useful to rewrite the Friedmann equations in a dimensionless form in terms of reduced quantities. For that, we introduce the energy density parameters
(3.36) respectively, for the matter , the cosmological constant and the curvature. The matter term can be decomposed as a sum of components with different equations of state as o = I:x Dx with
(3.37) and we will denote by D xo its value today. The first Friedmann equation (3.25) then takes the form of a constraint
(3.38)
Using the solutions of the conservation equation, it follows that
_
Dx - D xo for a constant equation of state 5 as 5 For
a
general
Oxo (Hal H)2 exp [- 3
time-dependent
(~) -3(1+WXO) ao
Wx
= Wxo, so that
equ ation
J:o (1 + wx)d In a].
(Ho)2 H '
of state,
E(a) == it
can
H/H
o
be
can be expressed
checked
that
Ox
142
The homogeneous Universe
(3.3
°
Quantities with an index are evaluat ed today. Thus, to will be the dynamical age the Universe and ao = aCto) . Notice t hat E only depends on x = a/ao = 1/( 1 + Depending on the case, it will be denoted by E(a), E(x) or E( z). For the matter , t he following different components can be distinguished Tot al matter
0
Different equations of state
Om ~
Db
Different species
Dc
where the total matter component is decomposed into matter (w = 0) and radiati (w = 1/3) and then, respectively, into baryons (b), cold dark matter (c) , photons h and neutrinos (v). Of course , other species can be added, whether as a species with new equation of state or as a new component.
3.1 .5.7
Units and conventions
Two conventions are often used for the choice of dimensions. The decomposition any physical length as the product of t he scale factor and a co moving length , C(phys) a(t) C(com) can indeed lead to an ambiguity since we can arbitrarily choose either t scale factor or t he co moving length to have dimension of length. We will use bo conventions according to the situation at hand.
- The scale factor can be chosen to have dimension of length , and X to be dime sionless. This implies that K can be normalized to K = 0, ± l. The value of can t hen be determined by evaluating the Friedmann equations today.
I [aJ ~ [, and [xl
LX
=x,
K = 0, ± 1,
1
ao = Ho
1
J 11- Dol
.
For a Universe such t hat Do = 1 (i. e. K = 0) , ao is arbitrary, which reflects t invariance of the Euclidean space under dilatation. - The scale factor can be chosen to be dimensionless. X will then have dimensi of length , and thus K the dimension of inverse length squared. It is t hen oft convenient to set ao = 1.
[aJ = 1 and [xJ
=
L: K = (0, ±1)/ R~,
We recover that for Do = 1 (i.e. K charact eristic curvature scale.
1
Rc = - aoHo 0) , Rc
= 00 .
1
Jl1 - Do l
.
A Euclidean space has
I 1, the conservation equation implies that p o( a the Friedmann equation (3.25) implies H2 x o-3(1+ru). It follows that a By integrating this relation, we obtain, As seen earlier, if uL
so o<
2
(3.40)
a(l) x ,3{1+u) From this expression, the conformal time can be computed by integrating
giving 4
o li#b if w I -1/3. o(r1)
that
a-Q+3u)/2.
dl :
dt I
a(t)
,
We cleduce that
21 x q' ts;
if
al
(3 41)
;.
and a(4) o( exp ? otherwise.
3.2.1.2
K:0andu:-\
If the matter density is dominated by a cosmological constant, we have H2 x p x so that a x a. By integrating this relation, we get a(t) x eHt
a0,
(3.42)
This space has an accelerated expansion and is called de Sitter space. In terms of the conformal time, the scale factor is of the form
aQi)x-h,
4<0.
This space can also be written in such a way that Chapter 8). ds2
3.2.1.3
:
-dt2
. *+f!
it
(3.43)
has spherical spatiai sections (see
(dx, +sin21d02)
.
(3 44)
K:IIandu+-Il3
It is more convenient to work in conformal time to obtain
{"0i,t(:d}.
Using p
:
Qr_ Jlr: , {)sl, (g) ll \osl
in units such that l1{l
n,(t) in the parametric form
po(alas)-3(1+u) and (3.25), we get
: 1 This equation
t'-t''
_*.
can be integrated to give
144
The homogeneous Universe
(aoa) = (11 -0 00
)1 / 20: {(Sinh o.'TJ)1 / 0: if 1 (sin o.'TJ) 1/ 0: if 0
K = - 1, K = + 1,
(3.4
with 20. = 1 +3w. This solution is valid for all constant w of. - 1/3. The case w = - 1 corresponds to a Universe with no matter since matter can then be absorbed into curvature term. 3.2.1.4
K = 0, w = 0 and A of. 0
Let us mention the solution for a spatially Euclidean model dominated by a pressurel fluid and a cosmological constant [11]
a(t)
=
(1 ) OAO -
1
1/ 3
sinh
2 3 /
(3Tt) '
(3.4
where we a = HOVOAO. This solution is relevant for describing the late time evolut of a flat ACDM model. 3.2.1.5
Other solutions
We present in Fig. 3.3 the profile of the scale-factor evolution in a large number of si ations, including for different kinds of matter, curvature and value of the cosmologi constant. We may notice the presence of periods of slower evolution (a 'hesitating' phase) various lengths depending on the values of the cosmological constant (see, e.g., ca F, L or K). During these phases, the Universe gets closer to an Einstein static Unive (i.e. a = 0) for which
Thus, this is a spherical Universe for which the gravitational attraction is just co pensated by the repulsive effect of the cosmological constant. The behaviour of th solutions depends mainly on the relative values of the cosmological constant and A Another interesting solution is the one obtained for an empty Universe with hyp bolic spatial sections (K = - 1) . This space, called the Milne space, has t he follow metric (3.4
It can be shown that this metric corresponds to a Minkowski metric by an appropri change of coordinates (T = t cosh X, R = t sinh X leads to the Minkowski metric spherical coordinates) , but this space only covers a quarter of the Minkowski spa time.
3.2.2
Dynamical evolution
In order to study the general dynamics of the solutions of t he Friedmann equatio it is interesting to rewrite the system (3.25) and (3.26) in the form of a dynami system for t he quantities 0 , OA and OK (see Refs . [12- 14]).
Dynamics of Friedmann-Lemaitre space-times 3.0
0
<
*·····
c\\\~..
\,>o\}~.. ····
2.0
145
LK
~............
H
1.0
OJ
..................... ...G
0.0
B
A
D
oj
- 1.0 0.0
1.0
2.0
3.0
QmO
3.0
3.0
a
a ao
3.0
2.0 a
ao 1.0
0.0 -3 -2 -1
0
1
2
1.0
0.0 - 3 -2 -1
3
Hot
2.0
0
1
2
0.0 -3 -2 -1
3
Hot
3.0
a
a
ao
A =0
2.0
2
3
a
\
......
1.0
1
1
2
3
1
2
3
3.0
ao 1.0
0
Hot
3.0 Hesi tating ~ (Spherical:
0
2.0
0.0 -3 -2 -1
Hot
0
1
Hot
2
2.0
Hyperbolic
ao 1.0
...:
3
0.0 - 3 -2 -1
0
Hot
Fig. 3.3 Depending on the values of the cosmological parameters (upper panel) , the scale factor can have very different evolutions.
3.2.2.1 Using
Dynamical system
if =
ii/a - H2 and expressing ii/a with (3.26) and H2 with (3.25) leads to H
H2 = -(1 + q) .
(3.48)
q is the deceleration parameter defined by
(3.49) with 'Y == w + 1. The system of Friedmann equations can thus be rewritten using the new time variable p == In( a/ ao). The derivative, XI, with respect to p of any quantity X is then given by XI = X/H.
146
The homogeneous Universe
The evolution equation for the Hubble parameter (3.48) then becomes
+ q)H.
H' = -(1
(3.5
n
If we differentiate n, n A and K with respect to p and use (3.50) to express H noticing that a' = a and that (3.30) can be used to obtain pi = - 3,p, we then obta the system
n ' = (2q + 2 - 3, )n , n~
(3.5
= 2(1 + q)nA '
n~ = 2qn K
(3.5
(3.5
.
It is useless to keep all of these equations since (i) H does not enter in (3.51)- (3.5 can be deduced algebraically using (3.38). and (ii)
n
The dynamics of Friedmann- Lemaitre space-times can thus be studied by considering the dynamical system
(3.54) where q is a function of
nA
and
nK
given by (3.49).
This system associates a unique tangent vector to each point so that only o integral curve passes through each point of the phase space where the tangent vect is defined. If two trajectories crossed each other, the tangent vector at this point wou not be unique. Points where the tangent vector is not defined are called fixed point 3.2.2.2 Determination of the fixed points The fixed points are defined by the conditions solutions of (1 + q)nA = 0,
n~
= 0 and
n~
o and
are thu
(3.5
They represent the equilibrium positions (stable or not). This equation has thr solutions (nK,n A) E {(O,O),(O, l) ,(l, O)} , (3.5 and each of these solutions represent a Universe with different characteristics.
- (n K , n A)
= (0, 0): the Einstein- de Si tter space (EdS). This is a space-time wi no curvature (Euclidean spatial sections) and no cosmological constant. - (nK' n A ) = (0, 1): the de Sitter space (dS). This is a space with no matter b with a cosmological constant and the spatial sections have a positive curvatur This Universe is in eternal exponential expansion. - (nK , n A) = (1 , 0): the Milne space (M). This is an empty space with no cosm logical constant but with hyperbolic spatial sections (K < 0).
Dynamics of Friedmann-Lemaitre space-times
147
3.2.2.3 Stability of the fixed points To determine if these fixed points are attractors (A), saddle points (S), or repellers (R), we should study the evolution of a small perturbation around each equilibrium position. For this we set
== 0,K +WK, == 0,1\ + W1\,
OK 01\
(3.57) (3.58)
with (0,1\, 0,K) the coordinates of a fixed point and (W1\' WK) a small deviation. Rewriting the system (3.54) in the form
(3.59) where FK and F1\ are two functions determined by the system (3 .54) that vanish at the fixed point , it can be expanded to linear order as
~j ' WK
_ p_ _
W
-
(!tA,!tK)
~WKj W
with
p(IT IT ) A,
K
==
(;~: ~~:) aF1\
(3.60)
aF1\
aO K a01\
(ITA ,IT K
)
The stability of a fixed point depends on the sign of the two eigenvalues (A1,2) of the matrix P(ITA,IT K ) ' If both eigenvalues are positive (resp., negative), the fixed point is unstable (resp ., stable). If the two eigenvalues have different sign, it is a saddle point. The eigenvector U.\ 1,2 then gives the stable and unstable directions . - EdS: we have PEdS
= (
3"1 - 2 0 ) 0 3"1 '
with eigenvalues 3"1 - 2 and 3"1- EdS is thus an attractor for "1 Ej- 00, 0[, a saddle point for "1 Ej O, 2/3[ and a repeller for "1 Ej2/3, +00[. The associated eigendirections are U(3,. - 2) = (1,0). - dS: we have PdS
=
(2 =~'Y -~'Y
) ,
with eigenvalues - 2 and - 3"1- If "1 Ej - 00, O[ then dS is a saddle point and for "1 EjO, +00[, it is an attractor. The two eigendirections are given by U(-2)
= (1 , -1).
- M: we have PM
_ (2 - 3"1 - 3"1 ) -
0
2
'
with eigenvalues 2 and 2 - 3"1- If "1 Ej - 00,2/3[ then M is a repeller and if "1 Ej2/3, +00[, it is a saddle point. The two associated eigenvectors are U (2)
= (1 , - 1).
148
The homogeneous Universe
These results are summarised in Table 3.1 and in Figs. 3.4 to 3.6. Table 3.1 Stability of the fixed points as a function of the value of the equation of state, , = w+ 1, (A: attractor, R: repeller, and S: saddle point). ] -
EdS dS M
00,
o[
°
A
N.A.
S S
A R
]0,2/3[
2/3
]2/3, +oo[
S A R
NA A R
R
A S
0.5
E
C
0.5
- 0.5
- 1~
- 0.5
__~~~__~__~__~~__- J 1.5 o
0 -0.5
0
0.5
1.5
QA
Fig. 3.4 Dynamical evolution in the plane (r:lK , r:l A) (left) and (r:lm , r:lA) (right) for, = (w = 0). From Ref. [14].
3.2.3
Expansion and contraction
When the total energy density vanishes, the expansion rate of the Universe can chan sign. For a Universe with non-flat spatial sections and with matter, radiation and cosmological constant , there are four possibilities (see Fig. 3.3) . The expansion lasts eternally starting from the Big Bang. This is the case if K or - 1 for a Universe without cosmological constant.
=
The expansion is followed by a contracting phase leading to a big crunch. This the case for a Universe with K = + 1 with no cosmological constant.
The expansion lasts eternally but was preceded by a contracting phase; the must therefore have been a bounce. This situation can occur when the Univer has spherical spatial sections (K = + 1). - The Universe undergoes a series of contraction and expansion phases; this is oscillating Universe .
149
Dynamics of Friedmann-Lemaitre space-times 1
1.4 Qm
0.8
1.2
1
0.6
" 0.8
cf°.4
cf
0.6 0.2 0.4 0 - 0. 2 - 0. 2
0.2 O ~
0
0.2
0 .4
0 .6
0.8
1
__
~~~~~-L_ _~-w~~
-0.4 -0.2
0
QA
0.2 0.4 QA
0. 6
0.8
Fig. 3.5 Dynamical evolution in the plane (O K , Ot>.) (left) and (Om , Ot>.) (right) for 'Y From Ref. [14].
1
=
1/3 .
E
c:
- 0.2
0
0.2 0.4 QA
0.6
0.8
Fig. 3.6 Dynamical evolution in the plane (OK , Ot>. ) (left) and (Om , Ot>. ) (right) for 'Y = - l. From Ref. [14].
The separation between a Universe with infinite expansion and a Universe with a big crunch is obtained when
2
nAO = 1 - nmo + :3 cos [arct an (S;;;~) - 7r] )
(3.61 )
with SmO = JI2n mo - 11. The separation between a Universe with infinite expansion and a bouncing Universe is obtained when
150
The homogeneous Universe
nmO = 0, or 1
_
o ::; nmo ::; 2'
nAO - 1 - nmO
3 [ + 2(n 1110 )1 /3 (1
- n mo
+ '::'~mO )1 /3
+ (1 - nmo - :=:mo) 1/3] , nAO = 1 - n mo
3.3
2
+ 3" cos [arctan (:=:;;;6) + 7r].
(3.6
Time and distances
The characteristic distance and time scales of any Friedmann- Lemaitre Universe a fixed by the value of the Hubble constant. We define t he Hubble time and distance 1
c
tH = H'
DH = H'
(3.6
The sphere with radius equal to the Hubble radius is called the Hubble sphere. T order of magnit ude of these quantities is obtained by expressing the current value t he Hubble parameter in the following way
(3.6 with h typically of the order of 0.7. We will come back later to the measurement of t he parameters introduced in t his chapter. We obtain t hat
DHo = 9.26 h- 1 x 10 25 m ~ 3000 h - 1 Mpc ,
tHo = 9.78 h - 1 x 10 9 years.
(3.6
All the quantities that will be introduced in this section (distance and time) will expressed in terms of E( z ) defined in (3.39). As can be seen in Fig. 3.7, this functi has some degeneracies that change with t he redshift. By combining observations different redshifts, one can lift these degeneracies and thus use them jointly to bett constrain the cosmological parameters.
3.3.1
Age of the Universe and look-back time
From the definition of the Hubble parameter, it follows that dt = da /a H . Thus, t expression (3 .39) for the Hubble constant gives
da dt = tHo aE(a)'
The age of the Universe is thus obtained by integrating this equality between a = and a = ao, or equivalently between z = 0 and z = 00,
r dx tHo J xE(x)
roo
1
to
=
o
=
tHo Jo
dz
(1
+ z)E(z)'
(3.6
This function is represented for various sets of cosmological parameters in Fig. 3.8.
Time and distances 0.9
0.')<=== == =-=---:--.
0.8
o.
0.7
151
o
0
a<
c:< 0.6 0.5 0.4 0.3 0.1
0.2
0.3
0.4
0.5
QmO
Fig. 3.7 Isocont ours of the funct ion E( z ) in terms of the cosmological parameter (Dmo , DAO) for a J\CDM modeL T he degeneracies at z = 1 (left), z = 3 (middle) and z = 10 (right) are not the same. The code of grey shade is arbitrary.
The analogous expression in conformal time is 1
'T}o
=
tHo
10 x2~x) ·
(3.67)
In a similar way, the age of the Universe at t he t ime when a photon with redshift
z. was emitted is defined by
roo
t( z.) = tHo Jz. (1
dz
+ z)E(z) ·
(3.68)
The look-back time is the difference between the age of the Universe and its age when the photon was emitted by the source. The previous relations lead to
llt( z*) = to - t( z*) = tHo
3.3.2
Jro ' (1 + dz z)E(z )·
(3 .69)
Comoving radial distance
The co moving radial distance X of an object of redshift z t hat is observed by an observer located at X = 0, is obtained by integrating along a radial null geodesic. The form of the metric (3.4) gives dX = dt /a, so that dz
H( z) . It follows that
aoX( z*)
=
DHo
Jr' o
dz
E( z )·
(3.70)
The dependance on ao simply reflects the arbitrary choice of the system of units. This function is represented for various sets of cosmological parameters in Fig. 3.9.
152
The homogeneous Universe
9
11
10
8 9
1.0
1.5
2.0
2.5
QmO
Fig. 3.8 Isocont ours of the functi on to as a fun ction of the cosmological parameters. curve indicat es t he value of to in units of 10 9 years. The grey zone represents the zo paramet ers for which there is no Big Bang.
For a fiat Universe with no cosmological const ant, we find, for instance,
(
For a Universe with a non-vanishing curvature, we can give the expression for radial dist ance in units of the curvature radius
( 3.3.3
3. 3.3.1
Angular distances
Comoving angular diam eter
The co moving angular diameter relates the comoving transverse size of an obj ect the solid angle under which it is observed . It is defined by the relation
between the solid angle and the apparent surface of an obj ect. Remembering t h comoving sphere centred a t X = 0 and with comoving radius X has a comoving su s (co m ) = 47f jk (X ), we deduce that
Rang (z) = jf( [X( z)] .
(
Time and distances
153
................__... -_.. ... .... .
..:.: :.: ..
-:":;'
3
5
4
Redshift z
Redshift z
Fig. 3.9 (left): Age and look-back time in units of the Hubble time, t( z )j tHo and 6t( z )j tHo as a function of z for three cosmological models defined by (Dmo, DAO ) = (1 , 0) , (0.05 , 0) and (0.2 , O.S), respectively, as plain , d otted and d ashed lines. (right) : Comoving radial distance in uni ts of t he Hubble length , aox(z ) j DHo, as a fun ction of z for the same cosmological models.
Notice that t his function is not necessarily increasing wit h z . Indeed , if K = +1 , then two objects with the same comoving size, located at X and 7r - X will have the same comoving angular diameter.
3.3.3.2 Angular dis tance The angular dist ance is the not ion that generalizes the concept of parallax. It is defined , for a geodesic bundle converging at t he point where the observer is, as the ratio between the transverse physical size of an object and the solid angle under which it is observed . The physical area of the source is related to the co moving one by dS p hys = a 2 dS com . With this defini t ion phy s com dSso 2 urce 2 dssource D A= 2 = a 2 dD o b s dD obs This quantity will also be denoted by T o . Using the definition of the co moving angular diameter , int roduced in the previous section , we obtain
aoRang(z ) l+z
(3.74)
3.3.3.3 R eciprocity theorem In the same way, we can define the angular distance from the source. It is defined using a geodesic bundle diverging from the source as the ratio between the physical transverse size of t he observer and the solid angle under which it would be observed from the source, phys _ .2 d n 2 dS o bs
-
rs
~ [,s o urc e'
154
The homogeneous Universe
It can be shown that if the tra jectories of the photons are null geodesics and if t equation for the geodesic deviation is valid, then there exists a relation between T s a To
(3.7
called the r eciprocity theorem (see Ref. [15] for a demonstration) . This relation cann be directly checked since Ts cannot be observed directly. 3 .3 .4
Luminosity distance
3.3.4.1
Defini tion
The luminosity distance relates the luminosity of a source, locat ed at a comoving rad dist ance X from the observer , to the observed flux as A.
_
'Po bs -
Lsource (X) 47rD[
(3.7
To relat e the luminosity of the source and the observed luminosity, notice that t luminosity of t he source is given by the emitted energy per unit of emission tim L source = !:::"Eem/ !:::"t em . Due to the expansion of the Universe, !:::" Eobs = !:::" Eem a/ and !:::"to b s = !:::"temaO /a (because !:::,.'rJobs = !:::"'rJem for two null geodesics) so that L obs L so u rce (l + z )-2 The flux received by the observer at X = 0 is given by
using
S (phys) = a6 s (com).
Identifying this with the definition (3.76), it follows t hat
I DL( z ) =
ao(1 + z )fJ( [X (z)].
I
(3.7
The luminosity of the source is indeed unknown. However , by comparing some sour to a reference source for which the redshift z. and the distance are known, we obta tha t
By extracting a family of astrophysical obj ects, called s tandard candles, wit h t he sa luminosity, the ratio of their fluxes can be used to measure the luminosity dist ance To finish , let us not e that deriving (3.77) allows us to extract the Hubble rat e a function of redshift as 1
H( z )
[
1 + DJ(oHg
(~) 2]-1/2~ (~) . 1+ z
dz
1+ z
Time and distances
,
" ... - - -
~ ~
155
15
... :..- . .......... .. ......... .
" ..... .
,''' " f
...
I,:
I :'
"
I:' I:'
2 3 Redshift z
4
5 Redshiftz
Fig. 3.10 (left): The angular distance D A (z) / DHo as a function of the redshift z for three cosmological models, defined by (n mo , n AO ) = (1,0), (0.05, 0) and (0.2,0.8), respectively, as plain, dotted and dashed lines. (right): The luminosity distance Ddz) / DHo as a function of the redshift z for the three same cosmological models.
3.3.4.2 Distances duality relation If the number of photons is conserved, which is usually the case if the Maxwell equations are valid, and if the absorption from the medium in which they propagate is small, then (for a proof, see Ref. [15]) DL
= rs(1 + z),
so that (3 .75) implies a duality relation between the angular and luminosity distances (3.78) Unlike (3 .75) , this relation can be tested experimentally [16]. 3.3.4.3
Distance modulus
The luminosity distance is usually expressed in terms of the distance modulus, m - M, defined such that it vanishes if the source is located conventionally 6 at 10 pc, m - M
= - 2.51og [¢(z)/¢(10pc)] .
(3.79)
Using (3.76) and noticing that at 10 pc, the luminosity distance is also 10 pc, we obtain m - M = 5 1og[DL(z)/1O pc]. Using the expression 10 pc = DHo/3 X 10 8 h , it can be deduced that m - M
= 25 + 5 log [3000 DL ] - 5 log h. DHo
(3.80)
6Note that the -2.5 factor is a rbitrary a nd was chosen to match the magnitudes of star s initia lly defined by Hipparcos .
156
The homogeneous Universe
m is the apparent magnitude and the absolute magnitude M is defined by comparis to the Solar luminosity M = - 2.5 log ( : 8 )
(3.8
+ 4.76
where L 8 ~ 3.8 x 10 26 W is the Sun luminosity and Mo = 4.76 its magnitude. Wi such a definition , the brighter an object is, the weaker is its magnitude.
3.3.4.4
Magnitudes
The luminosities considered previously were the luminosities integrated over the enti electromagnetic spectrum, defining the bolometric magnitude. It is difficult to measu these bolometric magnitudes and, in general, some filters are used, selecting a speci frequency band. The magnitude Mx in the band X is then defined by
Mx = -2.5 10g ( Lx ) L8X
+ Mo x.
(3 .8
An example of different frequency bands used in astronomy is given in the table belo Band
Central
Width (run)
MO
66
5.61
wavelength (nrn)
365
U (u ltraviolet)
B (blue)
445
94
5.48
V (visible)
551
88
4.64
R (red)
658
138
4.42
I (infra-red)
806
149
4.08
J
1200
213
3.64
K
2190
390
3.28
00
4.76
Bolometric
The flux detected in the frequency band 6.vx was emitted in the frequency ban (1 + z )6.vx . The flux received per frequency unit is thus A.
() - (1+ )L v [v(l+z)] z 47rDl '
'l'X v
(3.8
where Lv is the source luminosity per frequency band. It follows that
¢x (v)
=
(1 +D~) 47r L
r
Lv[v(l
+ z)]dv.
(3.8
} tlvx
The correction induced by the deformation of the spectrum does not appear when t source is at 10 pc so that the distance modulus in the band X defined by mx - Mx - 2.5 log[¢x/¢x(10 pc)], is given by
mx - Mx = 510g ( DL ) 10pc
+ K.
(3.8
Time and distances
157
The K-correction represents the distortion of the spectrum due to the cosmological expansion K
= -2.51og(1 + z) - 2.51og {
Lvx Lv[v(l
J
+ Z)]dV}
[]
D.vx Lv v dv
.
(3.86)
Finally, note that other effects of astrophysical origin, such as the absorption by the interstellar or intergalactic medium, can affect the apparent magnitude. We shall discuss them in the next chapter. 3.3.5
Volume and number counts
The geometric properties of space-time also playa role in the determination of the distribution of a family of objects (for instance, galaxies) in a given redshift band. Let us consider such a family with proper number density n(z) and proper radius r*. First, we can determine the probability for a line of sight to intersect such a galaxy with redshift lying between z and z+dz. This probability is proportional to the surface area of the galaxy, to the density, and to the distance light propagated in this redshift interval,
(3.87) The probability to intersect an object with redshift smaller than z , called the optical depth, is then given by
1 Z
T(Z)
=
2 7rr*DHo
o
n(z) (
dz) (). l+zEz
In particular, for galaxies diluted by the cosmic expansion, n(z) Universe with 0 = 1 and OA = 0 we find
(3.88)
= no(1 + z)3, in a
(3.89) If no rv 0.02h 3 Mpc~3 and r* rv 10h ~ 1 kpc, T rv 4% at z = 1 and 100% at z rv 20. The proper volume contained in a solid angle d0 2 and between the redshifts z and z + dz is the product of the area D~.d02 and the length D Ho dzj(l + z)E(z), so that
~ - D d0 2 dz -
Ho
(1
iI[x(z)] + z)3 E(z)'
(3.90)
The number of galaxies in a solid angle d0 2 and in a redshift band dz is then dN d02dz
iI[x(z)]
= DHon(z) (1 + z)3 E(z)'
Galaxies number counts are among the oldest cosmological tests (see Ref. [1]).
(3.91 )
158
The homogeneous Un iverse
3.4 3.4.1
Behaviour at small redshifts D ecelerat ion parameter
We can expand the scale factor around its value today as
a(t) with 0 < to - t
«
=
1 2 ao [ 1 + Ho(t - to) - 2qoHo(t - to) 2
+ ...] ,
(3.9
to. The factor qo is the deceleration parameter7 and is defined by qo
== -
a!2It=to '
(3 .9
so that the Universe accelerates if qo < 0 and decelerates otherwise. Using the Frie mann equations to express 0,0 / ao , we easily find t hat
for a ACDM model, since the contribution from radiation is negligible today. For a arbitrary matter composition, the deceleration parameter is given by
(3.9
Thus , in order to have qo < 0, there should be at least one cosmic fluid (dominatin today) with equation of state Wx < - 1/3 . We stress that t he expansion (3 .92) on relies on the hypothesis that the Universe is described by a Friedmann- Lemaitre spac time. In particular , it makes no assumptions on t he matter content of the Univers contrary to (3 .94). 3 .4.2
3.4.2.1
Expression of the time and distances
Small redshift expansions
At small redshifts, we can expand the function E(z) as
neglecting t he contribution of radiation. Thus, we conclude t hat the look-back t ime
6.t( z ) =
tHo
[1 - ~(qO + 2)Z] z + 0( z3 ).
(3.9
The radial distance takes the form
7The choice of sign is historical. At the time qO was introduced it was thought that the expansi of the Universe was d ecelerat ing so t hat qO > 0 was exp ected. C hapter 4 details why we now thi t h at qO < O.
Behaviour at small redshifts
159
the comoving angular distance,
Rang(Z)
=
DHo ao
[1 - ~(qO + I)Z] Z + O(z3), 2
(3.97)
the angular distance, (3.98) and the luminosity distance, (3.99) Locally (z « 1) all the distances reduce to DHoz and the deviations only depend on the value of the deceleration parameter. A distance-redshift diagram will thus be sensitive to Ho in its region z « 1 (slope of the diagram), to qo mainly in its intermediate region (z ~ 1) and to the full set of cosmological parameters for larger z. This illustrates the evolution of the degeneracy of the function E(z) represented in Fig. 3.7. 3.4.2.2
Time drift of the redshift
Suppose we observe the same galaxy at to and to redshift given by
+ Mo.
There will be a change in its
Now, null geodesics are straight lines in conformal coordinates so that br] = br]o Malao = btla from which we deduce that bzl(1 + z) = [Ho - H]br]o to first order in 61)0 and thus
bz = Ho [(1 uto
£
+ z)
- E(z)] ,
(3.100)
where E is defined by (3.39). This gives the time evolution of the redshift of any object at redshift z as a function of the cosmological parameters. At low redshift, it reduces to bzlbt = -qoHoz + O(z2). Indeed, this drift is extremely small and of the order of az I at ~ 10- 8 I century. However, it gives a direct information on E(z) and thus on the cosmological parameters (see Refs . [17,18]). Note that the redshifts are free of the evolutionary properties of the sources, and that a direct detection of their drift would represent a new (and yet never exploited) and direct approach in observational cosmology to constrain the evolution history of the Universe, alternative to those based on luminosity or diameter distances. It can be of importance in the study of the dark-energy problem (see Chapter 12).
160
3.5
The homogeneous Universe
Horizons
An horizon is a frontier t hat separates observable events from non-observable on T wo very different concepts of horizon have t o be distinguished in cosmology ( Refs. [10, 19]) . The different quantities defined in t his section are represented in Figs. and 3.12. 15
~
Geodesic of
>.
'"0
:::-
a particle _ entering into our particle horizon today _
10
~
0)
B
0. 8 0.6
(f'
:3
t)
"
. _ 0.4
.~
0)
0;
"
0
CIJ
U
- 0.2 0 0
10
20
40
30
50
Comovin g di stance, aoX, ( 109 light-years) 15
LO
~
>. 0
~
:::-
0. 8
IO
~
" E
~
"
.~
0.6
'" ..'": .9
~
"
5
0)
0;
"
CIJ
0
U
20
30
40
50
Physica l di stance, ax, (10 9 light-years)
Fig. 3.11 The particle horizon at a given time (3. 102) is defined at different moments the geod esic of the most distant observable comoving particles at this moment (plain verti lines). T his surface can be visualized as the intersection of the particle horizon (3.103) w a constant-time hypersurface . The equation of this surface is given by X = ¢(t) in comov coordinates (top) and by X = u(t) in physical coordinates . This diagram represents a U verse with t he following cosmological parameters (!lmo , !lAO) = (0. 3, 0.7) and h = 0.7. Fr Ref. [20].
3.5.1
Event horizon
For an observer 0 , the event horizon is the hypersurface that divides all events in two families: the ones that have been , are, or will be observable by and the on that are for ever outside the observational perimeter of 0. A necessary and sufficient condition fo r the existence of an event horizon is t h the integral OO dt
°
J
a(t)
Horizons
161
Physical distance (10 9 light-years)
Comoving distance, GoX (10 9 light-years) 60
~
50
;>,
,
lOpO
,,'
3.0
o~/
2.0
~~,
b-~~--~----~----:~r7~~-*~--~~____~__~~OT'~'______~1.0
40
0. 8
30
04
~
0
¥
'-'
oJ
.~
OJ
E = 0
<2
€
20
0.1
U
"5"
'"
10 LLL-_'------'L--'---_"----_~_-'.L-_
- 40
_"'____
__"__
__"__
_"__
- 20 20 o Como ving di stance, GoX ([0 9 light-years)
___L_"____L_
40
_i:>....:J
0.01 0.00 I
60
Fig. 3.12 Universe diagrams for a Friedmann- Lemaitre space with paramet ers OmO = 0.3, [lAO = 0.7 and h = 0.7 . The first two diagrams represent , respectively, the physical distance, a(t)x, and the comoving distance , X, in terms of the cosmic time, t (left scale) or in terms of the scale factor, normalized to 1 today (right scale). The last diagram represents the comoving distance in terms of the conformal time, 7). The dotted lines represent the worldlines of comoving observers and our world line is the central vertical line. Above each of these lines is indicated the redshift at which a galaxy on this world line becomes visible for the central observer. We represent the past light cone for the present central observer and the event horizon corresponding to a similar light cone originated from time-like infinity (plain bold line) . We also show t he Hubble sphere, defined by (3.63) (plain light line) and the particle horizon , defined by (3 .103) (clashed line) . From Ref. [20].
is convergent . If this integr a l converges , then at a ny time to , there exists a worldline
1
00
X = Xo =
to
dt
a(t) ,
such that the photon emitted at to in Xo towa rds the origin , reach es X
=
0 at t
= +00.
162
The homogeneous Universe
Any photon emitted at to in X > Xo never reaches the origin and any photon emitt at to in X < XO reaches the origin in a finite time. The de Sitter space, for which a = exp(Ht ), has such an event horizon, since
=
Xo
=
dt e- Hto exp H t = ~ <
1 to
00.
As a second example, let us consider Universes wit h scale-factor evolution of the for a(t ) = tn. The integral
J=
c ndt
converges if and only if n > l. Recalling that for a flat Universe (K = 0) n is relat to the parameter w of t he equation of st at e by 3n = 2/ (1 + w), we find that an eve horizon exist s if 1 (3 .10 w < - '3 {==} p + 3P < 0, i.e. if the strong energy condition is violated. 3.5.2
Particle horizon
For an observer 0 at to, t he particle horizon is t he surface of the hypersurface t = dividing all particles of the Universe into two non-empty families : the ones that ha already been observed by 0 at the time to and the ones that have not. For ea time to, it is thus t he intersection between the geodesics of the most dist ant comovi par ticles that can be observed, with the hypersurface t = to. Hence, it is a tw dimensional space-like surface, which reduces to a sphere of centre 0 for an isotrop and homogeneous Universe. A necessary and sufficient condition for the exist ence of a particle horizon is th the following integral be convergent:
· 1
dt
dt a(t) '
J_=
or
a a(t )
depending on whether or not a(t) continues for negative values of t . Indeed , at any time t o, any particle such that X > a - 1dt has not yet be observed by an observer 0 at the origin. The two-dimensional surface
J;o
X = ¢(to ) =
i
dt
to
a
-
a(t)
(3.10
defines t he particle horizon at a given time to and thus divides all particles into tw sub-families: the ones t hat have been observed at to or before to [X :s: ¢(to )] and th ones t hat have not yet been observed [X > ¢(to) ]. This surface is called the partic horizon of 0 a t to. It can be seen as the section at t = to of t he space-time surfa (J = ¢ (t ). We thus define the particle horizon as the hypersurface X
=
(J
= ¢(t ).
(3 .10
The hypersurface (3 .103) is the fu t ure light cone emitted from the position of t observer at t = O. Since a(t ) is a positive function , when ¢(t ) exist s, it is an increasi
Horizons
163
function of t so that as time elapses, more and more particles are visible from O. In conformal time, t he particle horizon is a cone and the part icle horizon at to is a sphere. If ¢(t ) is finite when t t ends t o infinity, then the Universe has an event horizon since only the particles with X ::; ¢ (oo) will be accessible to the observer O. This is represented in Fig. 3.11 where the vertical lines represent the particle horizons at various times, whereas the d ashed line represents the surface defined by X = ¢(t) . As an example, let us consider Universes wit h scale- factor evolution of the form a( t ) = tn. The integral
lC
n
dt
converges if and only if n < 1. Since for a flat Universe (K = 0) , 3n = 2/ (1 + w) , it follows that there exists a particle horizon if 1
w > -3
~ p
+ 3P > O.
(3.104)
From the conditions (3.101) and (3 .104), we thus conclude that a Universe containing a single fluid with constant equation of st ate can have either an event horizon or particle horizon, but not both at the same time. The two types of horizons are thus mutually exclusive in this particular case. The equation (3.102) gives the co moving radius of the particle horizon. Its physical diamet er at a time t 2 for a n event that occurred at tl < t 2 is thus (3.105) It can be checked that if t l and t 2 are two events from an era dominated by a fluid with const ant equation of st at e w , then _ 6(1 + w)
D p h (t 1 , t 2) -
[ 1 +3w t2 1 -
(h) t2
l
(1+3W / (3+3W l ]
.
(3.106)
If tl « t2 t hen t he horizon diameter is proportional to t he Hubble radius (3.63) at the time t 2
(3.107) The particle horizon is at t he origin of the 'horizon problem ' of the st andard Big-Bang model (see Section 4.5.2 of Chapter 4). 3.5.3
Global properties
To study the global structures, and in particular at infinity, of the cosmological spacetimes , we shall use the representation introduced by P enrose (see R ef. [10] for a general study). It is based on the idea of constructing, for any manifold M with metric g/J>v , another manifold ;\It with a boundary 3 and metric 9/J>v = W g/J>v such that M is conformal to the interior of ;\It , and so that the 'infinity' of M is represented by the 'finite' hypersurface 3. The last property implies that W vanishes on 3. All asymptotic properties of M can be investigated by studying 3 (see Ref. [21 ]) .
164
The homogeneous Universe
3.5.3.1
The Minkowski space-time as an example
The Minkowski metric (1.10) in spherical coordinates can be written in terms of t advanced and retarded null coordinates, respectively, defined by v = t + r and u = t as 1 ds 2 = - dvdu + 4(v - u) 2d0 2,
where v ?: u and v and u range from -00 to +00 . No terms of t he form dv 2 or d appear because the surfaces of constant v (or constant u) are null surfaces. We can make these coordinates vary in a finite interval by introducing tan V = v ,
tanU = u ,
so that - 71"/ 2 < U :S V < 71"/2. Then , introducing the coordinates T and R by
T = U + V,
R = V -U,
with - 71" < T + R < 71" ,
- 71" < T - R < 71" ,
(3. 10
the Minkowski metric turns out to be conformal to the metric 9 given by ds 2 = - dT 2 + dR 2 + sin 2 R d0 2,
with conformal factor W = {2 sin[(R + T) / 2] sin[(T - R) / 2]} - 2. The Minkowski spa time is thus conformal to the region (3.1 08) of the Einstein static space-time (a cyl der S3 x IR. that represents a st atic space-t ime with spherical spatial sect ions). T boundary of this region therefore represents the conformal structure of infinity of t Minkowski space-time. This boundary can be decomposed into - two 3-dimensional null hypersurfaces , .:7+ and .:7- defined by
(3 .10
or equivalently by T = ±( 71" - R) with R E [0,71"]. T he image of a null geode originates on .:7- and t erminates at .:7+, which represent future and past nu infinity. - two points i+ and i - defined by U = V =
±-=2'
(3 .11
or equivalently by R = 0 and T = ± 71". The image of a t ime-like geodesic origina at i - and terminates at i + . They represent future and past time-like infinity, th is, respectively, t he start- and end-point of all time-like geodesics. - one point iO defined by U = - V= --= (3.11 2' or equivalently by R = 71" and T = o. It is t he start- and end-point of all space-l geodesics so that it represents spatial infinity.
Horizons
165
The fact that i± and iO are single points follows from the fact that sin R = O. These are coordinate singularities of the same type as the one encountered at the origin of polar coordinates. The manifold Nt is regular at these points. J - is a future null cone with vertex i- and it refocuses to a point iO that is spatially diametrically opposite to i - . The future null cone of iO is J +, which refocuses at i+. Note that the boundary is determined by the space-time and is unique but that the conformal ext ension space-time (here the Einstein static space-time) is not fixed by the original metric and is not unique since another conformal transformation could have been chosen. For any spherically symmetric space-time, the region (3.108) can be represented in the (T, R) plane by ignoring the angular coordinates so that each point represents a sphere S2 In this representation the Minkowski space-time is a square lozenge (see Fig. 3.13) . Null geodesics are represented by straight lines at ±45 deg running from J - to J + .
i+
i+
J+
(1) =
+ on )
:
I
I I
I I I
I
I I
I
J -
(I) = 0)
J -
(I) = -
00.')
I
Fig. 3.13 Conformal diagrams of the Minkowski space-time (left) and Friedmann-Lemaitre space-times with Euclidean spatial sections with A = 0 and P > 0 (middle) and de Sitter space (right) .
3.5.3.2
Friedmann-Lemaltre space-times with K = 0
When written in terms of the conformal time, Friedmann- Lemaitre space-times with Euclidean spatial sections are conformal to Minkowski. It follows that they map on a part of region (3 .108) representing Minkowski space-time in the Einstein static Universe. The actual region is determined by the range of variation of 7]. For A = 0 and P > 0, 0 < 7] < 00 so that it is conformal to the upper half of the Minkowski diamond defined by T > 0 (see Fig. 3.13) with a singularity boundary, T = O.
166
The homogeneous Universe
3.5.3.3 de Sit ter space-time
The metric of t he de Sitter space-time is given by (3 .44) with t varying in R It spherical spatial sections and is conformal to the whole Einstein static space-time. study its infinity we define the coordinates
R =X,
T = 2 arctan
(eHt )
,
(3.1
with 0 < R < 7r and 0 < T < 7r. It is then conformal to a square region of the Eins static space with conformal factor W = H - 2 cosh2(Ht) (see Fig. 3.13). We see that this conformal diagram has a space-like infinity both for time-like null geodesics, in the future and in the past.
3. 5.3.4
Friedmann- Lem aitre space-times with K = ± 1
Friedmann- Lemaitre space-times wit h spherical spatial sections (K = + 1) are con mal to the Einstein static space-time (with 17 --7 T and X --7 R). They are thus map into the part of this space-time determined by the allowed values for 17· There are three general possibilities. When A = 0, 17 varies from 0 to 7r if P = 0 from 0 to a < 7r when P > O. The Friedmann-Lemaitre space-time is thus confor to a square region of the Einst ein st atic space-time so that both J + and J space-like and represent two singularities. When A i- 0, 17 can vary from 0 t o 00 the hesitating Universes (such as examples Nand M in Fig. 3.3) or from - 00 to for t he b ouncing Universes (such as examples G and H in Fig. 3.3). The Friedma Lem aitre space- time is thus conformal to either half of or the entire Einstein st space-time. As in the case of the de Sitter space, the conformal region will be a squ of the Einstein static space and J + and J - are also space-like but do not necess represent a singularity. In the case of Fr iedmann- Lemaitre space-times with hyperbolic spatial secti (K = -1 ), the metric can be shown to be conformal to part of t he region (3.108) means of the coordinat e transformation
T= R =
arctan [tanh ( 17 : arctan [tanh ( 17 :
X)] + arct an [t anh (17 ; X)] , X)] _arct an [t anh ( 17 ; X)] .
Again, the exact shape of this region depends on the matter content (equation of s and A). We see in these examples that some parts of the boundary correspond to the B Bang singularity a = O. When P > 0 and A = 0, the initial singularity is space-l which corresponds t o the existence of a particle horizon. When K = + 1 and A = the future boundary is space-like, which signals the exist ence of event horizons for fundamental observers. For most physically interesting space-times, J + and J - are either space-like null , which is closely related to the existence of one of the two types of horizon descri above.
Beyond the cosmological principle
167
If .]- is space-like, the worldlines of the fundamental observers do not all meet on .]- at the same point . For any particular observer and event on its worldline close to J -, the past light cone of this event will not intercept all the particles in the Universe before it reaches .]- and there will be a particle horizon. If .]- is null , it is expected that all the worldlines of the fundamental observers pass through the vertex i- so that the past light cone of a point P will intercept all the worldlines and there is no particle horizon. If .]+ is space-like, the worldline of any fundamental terminates on a point 0 of J+ and the past light cone of 0 divides the Universe into events that can be seen by the observer and events that he can never see; there is en event horizon. If .]+ is null , all the worldlines will pass through the vertex i+ , so that 0 = i+ and its past light cone is .]+ and there is no event horizon. In conclusion, .]- space-like <===>- existence of a particle horizon, .]+ space-like <===>- existence of an event horizon.
,....
J + {}I
0
=;: I
k
1
II
II I ;><
J+
...
( 1/ = + ,~ ,c- )
a
II ;><
1 1
1 ;>< 1 1 II
I
"1 1
..J.... I
J -
(II= 0 )
J+{ I/= +'X ·I I
I
....
1 1
..
(I I = 0 )
a
II
,,
, I
;><
1 1
;><
II "1
1 1
I I
J - (17 =
- cc· )
Fig. 3.14 Conformal diagrams of the Friedmann- Lemaitre space-times with spherical spatial sections with P > 0 for A = 0 (left), and A =F 0: hesitating case (middle) and bouncing case (right). T he dotted line represents a singularity. The three diagrams differ only by the nature of J±, even though they are all space-like surfaces .
3.6
Beyond the cosmological principle
The Friedmann- Lemaitre space-times have spatial sections with constant curvature, i.e. three-dimensional homogeneous and isotropic surfaces. These symmetries are characterised by six Killing vectors (Chapter 1) that completely fix the geometry of the spatial metric , up to a number, K. Reciprocally, we can construct some solutions by restraining the number of symmetries and by imposing the structure of their Lie algebra. Numerous solutions of Einst ein's equations are known in this case and have been classified [22- 24].
168
The homogeneous Universe
We give a general description of the classification of cosmological space-times cording to t heir symmetries and t hen we present two cosmological solutions in or to illustrate the richness that this provides. 3.6.1
Classifying space-times
Even t hough the previous discussion focused on a particular class of cosmological so tions with maximally symmetric spatial sections , most solutions of E instein's equati have no symmetry at all. This simplification was motivated by the cosmological pr ciple but it is important to have an idea of exact solutions with less symmetry particular to construct counterexamples to some conclusions or t est their robustne
3.6.1.1
Symmetries and Killing vectors
Symmetries of a given space are t ransformations of this space into itself leaving metric unchanged as well as all its geometrical properties . As explained in Chapte a continuous symmetry can be characterized by a Killing vector field , ~ , satisfying
The group of isometries of a Riemannian manifold forms a Lie group (that is a gro that is also a smooth m anifold with a group operation xy - l that is a smooth m into t he manifold itself; see Chapter 2). The set of Killing vector fields generators t hese isometries forms the associated Lie alge bra: it is a vector space, since t he lin combination of any two Killing vectors is also a Killing vector, of finite dimensio since the maximum number of Killing vectors for a space of dimension n is n(n + 1) reached for maximally symmetric spaces. The product operation is defined by commutator of two Killing vectors
(3 .1 of coordinates [~a, ~b]'" = (8f3~b) ~g [~a, [~b , ~cll
+
[~b, [~c,~all
+
[~c , [~a , ~b ll
-
(8f3~~n~f and it satisfies the J acobi ident
= O.
It is easily checked that if ~a and ~b are two Killing vectors then [~a, ~bl is als Killing vector (since £[l;a, l;blgj.LV = 8!L[~a , ~blv + 8v[~b,~al!L = 0). Considering a ba {~a}a = l..r of this algebra , any Killing vector can be decomposed on this basis a linear combination with constant coefficients. It follows t hat
(3 .1
C~b are t he structure constan ts of the Lie algebra. They are antisymmetr -Cba ' and must satisfy, from the J acobi ident ity C~[bC~d) = O. These relati are integrability conditions t hat ensure that t he Lie algebra is defined in a consist way.
where C~b =
3.6.1.2
Defini tions
T he action of a group of isometries defines orbits in the space where it acts and dimensionality of these orbits characterizes t he symmetries.
Beyond the cosmological principle
169
T he orbit of a point P is the set of all points into which P is transformed by the action of the isometries of the space. The orbits are thus invariant sets. An invariant variety is a set that is globally left invariant by the action of the group of isometries. It will be larger than all t he orbits it contains. T he orbits are t he smallest invariant subspaces (for all the isometries in the group) . An orbit that reduces to a single point is a fixed point. It is a point where all Killing vectors vanish. We distinguish general points where the dimension of Killing Lie algebra takes the value it has almost everywhere from special p oints, such as t he fixed points, where the dimension is lower. The group of isometries is transitive on a space S if it can move any point of S into any other point of S. Orbits are the largest spaces through each point on which the group is transitive and can be called surfaces of transitivity. Their dimension, s, has to be smaller than or equal to t he dimension of t he group of translation, s = dim(surfaceoftransitivity)
~
n.
The isotropy group, at a point P, is the subgroup of isometries letting that particular point be fixed. It is generat ed by the Killing vectors that vanish at P. Its dimension q is smaller t han the dimension of the group of rotations, n(n - 1) / 2, q
= dim(isotropygroup)
~
1 "2n(n -1).
The dimension of the group of isometries , r , is the sum of the dimensions of the isotropy group and of t he surface of transitivity. It follows t hat r has to be smaller t han the maximum dimension of the Killing Lie algebra, that is n( n + 1) / 2, that is for a space of constant curvature. r = dim(groupofisometries) = s
+ q ~ ~n(n + 1).
The group of isometries of a space is thus characterized by two of the t hree numbers (S, q,T) and by the structure constants of t he group (C~b)' 3.6.1.3
Classification of cosmological models
For an = 4 dimensiona l space-t imes, r ~ 10 and s can run from 0 to 4. The dimension of the group of isometries being a global property of t he space, r must not change over space. Indeed, the real Universe has no symmetry (r = 0). For cosmological spaces, we assume that p + P > 0, which implies that q E {3, I , O} because the congruence of t he worldline of the fundamental observers is invariant, so that the isotropy group at each point must be a subgroup acting orthogonally to uf.' and it can be shown t hat there is no subgroup of 0(3) of dimension 2. Furthermore, q can vary over space but not over an orbit and it can be greater at special points where s is smaller. The space will be said to be isotropic for q = 3 (this is the case of the Friedmann- Lemaitre spaces), locally invariant under rotation for q = 1 (there is a preferred spatial direction) and anisotropic for q = O. Table 3.2 summarizes all t he possibilities according to q and s. For more details, see Refs. [7,22- 24].
The homogeneous Universe
170
Table 3.2 Classification of cosmological models according to their isotropy and homogeneity properties . We recall that r = q + s :::; 10.
q= 6
8= 3
8= 2
spatially homogeneous
inhomogeneous
Minkowski , de Sitter anti-de Sitter
none
none
Einstein static
Friedma nn- Lemaitre
none
q= 1
Bianchi Kantowski- Sachs
Lemaitre- Tolman- Bondi
q= O
Bianchi
q= 3
3.6.2
8= 4 space-time homogeneous
Universe with non-homogeneous spatial sections
3.6.2.1 Lemaitre- Tolman-Bondi space-time
Among all the non-homogeneous spaces, the Lemaitre-Tolman- Bondi space-tim the simplest. It enjoys a spherical symmetry, so that s = 2, q = 1 and r = 3 (but n that the origin is a special point with q = 3 and s = 0). Its metric takes the gen form ds 2 = -dt 2 + S2(t , r)dr2 + R2(t , r)dS1 2. (3. 1 The Einstein equations for a fluid of dust (P = 0) reduce to 2
S (r, t)
= 1_
(R')2 K(r)r2
R? = 2M(r) _ K(r)r2
R M'(r) 47rG N P(r, t) = R2 R' '
(3.1
(3 .1
(3.1
where a prime and a dot represent the derivatives with respect to rand t , respectiv M(r) and K(r) are two integration functions. The Friedmann- Lemaitre space-ti correspond to the special case R = a(t)r and K(r) = K. These equations can integrated to give the solution
R(rJ , r)
=
M(r) d
M(r) t(rJ , r) - to(r) = (2E) 3/2
(3. 1
where 2E = (2E , 1, - 2E) ,
Beyond the cosmological principle
3.6.2.2
171
The Swiss-cheese model
This family of inhomogeneous space-times is constructed by cutting out spherical regions from a Friedmann-Lemaitre Universe and filling the voids by a spherically symmetric solution of the Einstein equation, for example, a Schwarzschild solution. The two space-times are then glued along a 3-dimensional time-like hypersurface. This implies constraints on the two space-times since they have to obey junction conditions for the total space-time to be well behaved (see Section 8.4.5 of Chapter 8 for a description of these conditions). For instance, it can be shown that it implies that the central mass of the Schwarzschild space-time, the density and scale factor of the Friedmann- Lemaitre space-time and the radius of t he spherical void must be related by MSchw = (471" /3)[ pa 3 R 3 ]FL . Thus, we can construct a spherical void with an overdensity at the centre. The junction conditions impose that the over- and underdensities average to the mean density of the Friedmann- Lemaitre space-time. It follows that the mass in the central void cannot have any effect on matter outside the void, and in particular on the expansion rate: its gravitational effect is screened. 3.6.3
Universe with homogeneous spatial sections
The Universe models with s = 3 are central in theoretical cosmology because they mathematically realize the cosmological principle (they have homogeneous spatial sections). We have described at length t he case of isotropic space-times (q = 3; r = 6) in this chapter . Two classes of non-isotropic space-times can be constructed: (i) the family of solutions with a local rotation symmetry (q = 1; r = 4), which contains the family of Kantowski- Sachs Universes and some Bianchi Universes, and (ii) the family of anisotropic solutions (q = 0; r = 3) , which contains the other Bianchi Universes. 3.6.3.1 Generalities on Bianchi Universes To classify all spaces with s = 3, we need to determine the structure constants, C~b' of the group t hat is simply transitive on space-like sections. The C~b have 9 independent components and can thus be mapp ed onto t he 3 x 3 matrix Nab defined by N cd =
~2 Cab Eabd , C
where cabd is the completely antisymmetric tensor. Splitting Ncd into its symmetric and antisymmetric parts as Ncd = ncd +E cdb A b , we obtain that the structure constants are C~b = Edab ncd + 6gAa - 6g A b· One can always choose the basis of the Lie algebra such that nab is diagonal [nab = diag(nl,n2 , n3)]. A rotation allows us to set Ab = (a,O,O) while keeping nab diagonal. The Jacobi identity implies that nab Ab = 0, so that we have one of the three possibilities: (i) a = and nl y'c 0, (ii) a y'c and nl = 0, or (iii) a = and nl = 0. By rescaling the Killing vectors, we can finally set the nonvanishing n a to ± 1 but in general it is impossible to rescale both a and na. In conclusion
°
°
°
(3.120)
172
The homogeneous Universe
Table 3.3 gives the eleven groups of isometries that are distributed among the ni Bianchi types I to I X. They are divided into two classes: class A if a = 0 and class otherwise. In some cases, these groups allow higher symmetry. For instance, Bianch and V I 10 can be isotropic (leading to K = 0 Friedmann- Lemaitre space-times) as w as Bianchi IX (leading to K = +1 Friedmann- Lemaitre space-times) and Bianchi and Bianchi VIla (leading to K = - 1 Friedmann-Lemaitre space-times).
Table 3.3 The structure constants of the 11 groups of isometries and the definition of the 9 associated Bianchi types . Class
A
B
Bianchi type
I
II
VIla
VIa
IX
VIII
V
III
IV
VIla
VIa
a
0
0
0
0
0
0
1
1
1
a
a
nl
0
0
0
0
1
1
0
0
0
0
0
n2
0
0
1
1
1
1
0
0
1
1
1
n3
0
1
1
-1
1
- 1
0
1
- 1
1
-1
3.6.3.2 Example: Bianchi I Universe
The simplest Universe with anisotropic expansion is the Bianchi I Universe. It is homogeneous space-time so that the projected gradient of any scalar function vanish 'V'vi = O. This implies, in the notations of Chapter 1, that aIL = 0 (because 'V' ILP = and the vorticity is also vanishing, w ILV = 0). It follows that it is completely describ by 8(t) , a(t) , p(t), and P(t). The Raychaudhuri equation (1. 73) gives
The spatial sections are assumed to be Euclidean so that t he generalized Friedmann equation (1.78) reduces to
(3) R =
0, which implies th
The shear evolution equation is derived from the Gauss-Codazzi relation, using t fact that (3) Rab = 0, and is given by
In terms of the cosmic time derivatives , this rewrites as
In co moving coordinates , the metric of this space-time takes the form
(3. 12
Beyond the cosmological principle
173
The average of the directional expansion rates is S = (XY Z)1/3 and is related to by 30 = S/ S. Factorizing S, t he metric can be recast as
e
(3 .122) and the metric on constant-time hypersurfaces can be decomposed as 'Y ..
-
i2J -
where the
f3i
have to satisfy
L
f3i =
O.
e2 (3i (t) u>tJ. . '
"tij
(3.123)
is traceless and related to the shear by
The equation of evolution of the shear implies that (3.124) where ~j a constant tensor . This implies that 0- 2 == (Jij (J i j = ~2 / S6, where ~2 = ~j~L and thus that the generalized Friedmann equation (l.78) takes t he form (3.125) For a fluid with equation of state P = wp , P IX S-3(w+l) . T his implies that for any value of ~, the shear always dominates the primordial evolut ion of the Universe. It can also be checked t hat the Raychaudhuri equation takes the form (3.126) The evolution of the three scale factors can be shown to be obtained by setting (3.127) with W(t) =
dt
J
(3.128)
S3 .
The coefficients Bi must satisfy the constraint (3. 129) This relation is trivially verified with 27r .
a·=a+ -3 2' t where a is a constant. For instance, if w = 0, we obtain the solution
(3.130)
174
The homogeneous Universe
5 = (
~Mt2 + /f2:.t )
1/ 3
w = - /[~ In (3M + 2V62:./ t) ,
,
(3
once setting Kp = M / 5 3. At lat e times, the Universe becomes more and more isot and t ends towards an E inst ein-de Sitter space-time, but close to the Big Bang shear dominat es, 5 ----> (V32:.t)1 /3 and
X(t)
---->
X ot (1+2s in O:J)/3,
Y(t)
---->
Yo t (1+ 2 sin O: 2 )/3 ,
Z(t)
----> Zot(1 + 2 s in O: 3 )/ 3
(3 If 0: i- 7r / 2, two of the powers are positive and the third one is negative. Si behaviours close to the singularity will be discussed in Section 13.4.1 of Chapte As another example, consider the case of a pure cosmological constant. Then Friedmann equation can take the form
It follows that
5(t)
=
5. [sinh(3H. t)]1/3 .
Again, at late time, it converges toward the de Sitter scale factor 5 at early time it behaves as t 1 / 3 .
rv
exp(H. t)
References [1] P.J.E. PEEBLES , Principle of physical cosmology, Princeton University Press, 1993. [2] S. WEINBERG , Gravitation and cosmology: principles and applications of the general theory of relativity, John Wiley and Sons, 1972. [3] J. RICH, Fundam entals of cosmology, Springer-Verlag , 200l. [4] E. HARISSON, Cosmology, Cambridge University Press, 1981. [5] B. RIDDEN, Introduction to cosmology, Addison Wesley, 2003. [6] R.c. TOLMAN, R elativity, thermodynamics and cosmology, Oxford University Press, 1934. [7] G.F.R ELLIS and H. VAN ELST, 'Cosmological models ', in Theoretical and observational cosmology, Kl uwer , Dordrecht, 1999. [8] J .P. UZAN , C. CLARKSON and G.F.R ELLIS, 'Time drift of the cosmological redshift as a test of the Copernican principle' , Phys. Rev. Lett. 100, 191303, 2008. [9] R. WALD, Gravitation, Chicago University Press, 1984. [10] S. HAWKING and G.F.R ELLIS , The large scale structure of space-time, Cambridge University Press, 1973. [11] A.D. CHERNIN, D.l. NAGIRNER and S.V. STARIKOVA , 'Growth rate of cosmological perturbations in standard model: explicit analytical solution ' , Astron. Astrophys. 399 , 19, 2003. [12] J. EHLERS and W. RI NDLER, 'A phase space representation of Friedmann- Lemaitre Universes containing both dust and radiation and the inevitability of a Big Bang', Month. Not. R. Astron. Soc. 238 , 503 , 1989. [13] G.F.R ELLIS and J . WAINWRIGHT (eds.) , The dynamical system approach to cosmology, Cambridge University Press, 1996. [14] J.-P. UZAN and R LEHOUCQ, 'A dynamical study of the Friedmann equations', Eur. J. Phys. 22 , 371 , 200l. [15] G.F.R ELLIS , 'Relativistic cosmology' , in R elativity and cosmology, RK SACHS (ed.) , Academic Press, 1971. [16] J.-P. UZAN, N. AGHANIM and Y. MELLIER, 'The distance duality relation from X-ray and SZ observartions of clusters', Phys. Rev. D 70 , 083533, 2004. [17] M. DAVIES and L. MAY, 'New observations of the radio absorption line in 3C 286, with potential application to direct measurement of cosmological deceleration', Astrophys. J. 219 , 1, 1978. [18] A. LO EB, 'Direct measurement of cosmological parameters from the cosmic deceleration of extragalactic objects ', Astrophys. J. L ett. 499 , Ll11 , 1998. [19] W. RINDLER, 'Visual horizons in world-models', Month. Not. R. Astron. Soc. 116, 622, 1953.
176
References
[20] T.M. DAVIS and C.H. LINEWEAVER, Expanding confusion: common miscon tions on cosmological horizons and the superluminal expansion of th e Univ [astro-ph/0310S0S] .
[21] R. PENROSE, 'Conformal treatment of infinity' , in Relativity, groups and topo Les Houches 1963, pp. 561 C. DE WITT, and B. DE WITT (eds.) , Gordon Breach, 1964. [22] A. KRASINSKY , Physics in an inhomogeneo us Universe , Cambridge Unive Press, 1996. [23] D. KRAMER, H. STEPHANI, M.A.H. MCCALLUM and E. HERTL, Exact solu of Einstein's field equations, Cambridge University Press, 1980. [24] M.P. RYAN, Homogeneous relativistic cosmologies, Princeton University P 1975.
4 The standard Big-Bang model This chapter is devoted to the description of the historical pillars on which the standard Big-Bang model stands. Starting from the cosmological solutions described in Chapter 3, we present the observations proving the expansion of the Universe and the various consequences of this expansion. We start , in section 4.1, by describing the observational evidences of the expansion of the Universe from Hubble's observations to the most recent ones. This will lead us to discuss the value of the Hubble constant and its relation with the age of the Universe. Elements of thermodynamics in an expanding space-time are sumarised in section 4.2. We will deduce that the Universe has a thermal history. In particular , we will show how the departures from thermodynamic equilibrium playa central role. A remarkable feature of the Big-Bang model comes from the proof by Alpher , Bethe and Gamow in 1948 [1] tha t light nuclei can be formed during primordial nuc1eosynthesis. T his is discussed in section 4.3. Gamow [2] also deduced the existence of a microwave background of photons, the temperature of which was calculated by Alpher and Herman [3] in 1948. Relying on an estimate of the helium abundance and of the baryon density, they concluded that this temperature should be around 5 K. Section 4.4 describes the properties of the isotropic component of this microwave background radiation. These physical ingredients transform the (mathematical) cosmological solutions of F'riedmann- Lemaitre into the Big-Bang model. We will argue in section 4.5 that this model is an excellent standard model for cosmology, even if it suffers some problems, on which we will focus our attention in the following chapters of this book.
4.1
The Hubble diagram and the age of the Universe
If the Universe is expanding, galaxies must move away from one another (see Section 3.l.3; Chapter 3). The measurement of the systematic achromatic shift to longer wavelengths of the absorption or emission spectral lines gives access to the redshift z. For small redshifts , the recession velocity is given by vi c ~ z. As for distances , (3.99) shows that the luminosity and angular distances reduce to D A
~ D ~~z~ ~ L Ho Ho '
as long as z « l. In this regime, the slope of the curve of v as a function of D directly gives the Hubble constant Ho. For larger redshifts, this is no longer the case. First, the
178
The standard Big-Bang model
angular distance and the luminosity distance are no longer identical, and in addit other cosmological paramet ers also interfere in the determination of these distan (see Chapter 3). We describe the measurements for small redshifts in Section 4.1.1 and then th for supernOVffi that extend up to redshifts of order 1 in Section 4.1.2. We finish summarizing the methods allowing the age of the Universe to be determined in S tion 4.1.3 . 4 .1.1
The Hubble constant
The Hubble diagram is the most direct proof of the expansion of the Universe. T underlying idea is very simple and comes from the independent measurements of redshift and distance. These measurements must be performed at distances large enough for the pro velocity of the galaxies to be negligible. Typically, one should measure velocities lar than v rv 10 4 km· s-I, which corresponds to a redshift of z rv 0.03. The original measurements of Hubble, lie respectively, up to velocities of order 1000 km · S- l and then up to velocities of order v rv 20000 km · S-l (see Fig. 4.1). T analysis provided a value for Ho that was 10 times larger than the one adopted tod For a galaxy with recession velocity v rv 10000 km· s-l, the effect of its proper velo v rv 300 km . S- l leads to an uncertainty of approximatively 3%. This uncertainty be reduced by observing a large number of objects well distributed over the whole s Hubble (1929)
Hubble & Humason (1931)
~ ~
~
I
I
en
0
~ 1000 C 0
>
0
500
~
" en
0
0
0
0
10000
0 .;;;
0
en
<J) <J)
~
Jl,
"
0 .;;; 0
o
CIS 000 '0
'0 Q)
~ 20000
6
<J)
5000
0
'8
<J)
~
1 Distance (Mpc)
30
2 Distance (Mpc)
Fig. 4 .1 Observations by Hubble in 1929 (left) a nd by Hubble and Humason in 1931 (ri proving the expansion of the Universe. The first measurement of Ho included 18 gala for which both t he distance and the red shift were determined . Hubble concluded that H 500km · S - 1 . Mpc- 1 [4] .
4.1.1.1
Astronomical m ethods
The difficulty in constructing a reliable scale of distances [5] has limited progress in measurement of the Hubble constant. Thanks to the Hubble Space Telescope (HST is now possible to measure Ho with a precision of about 10%. In astronomy, distan cannot be measured directly. The parallax method , which is purely geometrical,
The Hubble diagram and the age of the Universe
179
only be used to determine the distances of the closest stars in the Milky Way. The general method is then to measure the intrinsic luminosity of a class of objects for which it is either constant , or correlated to another physical parameter , whose value is thought to be independent of the distance, as a way to calibrate the relative distances. Various classes of objects have been used to this purpose (see Ref. [6] for a review). 1. Cepheids: cepheids are variable stars the external atmosphere of which pulses
with a period between 2 and 100 days. These young and bright stars are very abundant in the close spiral and irregular galaxies. The pulsation mechanism is well understood theoretically and it has been established empirically that the pulsation period, which is independent of the distance , is correlated to the intrinsic luminosity of the star. The dispersion of the period- luminosity relation is of the order of 20% in luminosity. Since the luminosity scales as the inverse of the square of the distance, this implies a precision of 10% on the determination of the distance. For a sample of 25 cepheids, the statistic uncertainty drops to 2%. The Hubble diagram obtained from cepheids by the HST Key Project [7] is presented in Fig. 4.2. The most precise calibration of the period- luminosity relation is performed on the Large Magellanic Cloud (LMC) and its distance has been determined with a precision of 10%. At the moment, the det ection of cepheids is limited to about 20 Mpc. To extend the scale, obj ects brighter than simple stars should be found . Their luminosity can be calibrated by identifying such objects in the nearby galaxies that also contains cepheids. 2. Type Ia supernova;: type Ia supernOVffi (SN Ia) are the most promising distance indicators for modern observational cosmology. The origin of SN Ia is the explosion of a white dwarf. Their luminosity is comparable to that of an entire galaxy, making them observable at distances of several hundreds Mpc. As will be shown later in Section 4.l.2 , their luminosity curve shows a strong correlation between the characteristic evolution t ime and the maximal luminosity. The uncertainty of this relation is of the order of 12% on the luminosity, which translates into an uncertainty of around 6% in t he determination of their distance. 3. The Tully- Fisher relation: the total luminosity of spiral galaxies is strongly correlated to the m aximal rotation velocity of the galaxy. This relation, detailed in Chapter 7, reflects the fact that a brighter galaxy, and thus a more massive one, must rotate faster to compensate the gravitational attraction. The rotation velocity can be measured by spectroscopic observations from the Doppler effect. The Tully- F isher relation has been measured for around a hundred galaxies and it can be shown empirically that it has a dispersion of the order of 30% in luminosity, which implies a precision of the order of 15% in the determination of the distance. 4. Fundamental plane: the intrinsic luminosity of elliptical galaxies is correlated to the velocity dispersion of their stars. This correlation is analogous to the TullyFisher relation. Elliptical galaxies occupy a 'fundamental plane' where their luminosity L is correlated to t he surface brightness 10 and to the velocity dispersion 0 7 (J, by a relation of the form L ex 10 . (J3 . This relation can be understood by the fact that elliptical galaxies are roughly
The standard Big-Bang model
180
2000 ::;'
1 500
I
OJ
-5 00L-~--~
o
• '" • • []
~ ~
I
en
E
__~__~__~__~~__~__~__~__~__~__L-~__-" 20 30 10 Distance (Mpc)
2· 104
6
~~
Tull y-Ficher (Bande I) Fundamental plane Surface brightness Supernovre Ia Supe rnovre II
'8 0
~
~
I
"0.
::E I'
'(l
E
6
10 4
0 100 80 60 40 0
100
200 Distance (Mpc)
300
400
Fig. 4.2 (top): Hubble diagram for cepheids obtained by the HST Key Project collab tion [7]. Data are spread up to distances of around 20 Mpc, just as the measurement of Hu and Humason (1931). (bottom): Different objects are used and thanks to type Ia supernO the diagram can spread up to around 300 Mpc.
self-gravitating syst ems with a mass-to-luminosity ratio more or less const The virial theorem (see Section 7.2.2) then imposes (J ex M I ro. Moreover, surface brightness is well described by I (r) = 10 exp [- (r I ro) 1/ 4 ], called the Vaucouleur law. This can be integrated as L ex Ior5. Assuming that the mass luminosity ratio is of the form MIL ex M X , we obtain that £l +x ex (J4 -4x Ig The empirical law is recovered for x rv 0.25 . The precision of this relation is of the order of 10- 20%, which implies a preci of the order of 5- 10% on the det ermination of the distance.
5. Fluctuation of the surface brightness: the resolution of stars with a CCD cam depends on the distance. The luminosity of each pixel is due to a given num of stars. One can show t hat the Poisson fluctuations from one pixel to ano
The Hubble diagram and the age of the Universe
181
depend on the distance of the galaxy. This method has a precision of the order of 8% and is applied to redshifts up to z '" 0.02 . 4.1.1.2 S ummary of the m easurements These different methods have been put into application, mainly thanks to the HST, to construct a reliable astronomical distance scale. In particular , t his distance scale has been calibrated up to a level of 5% by cepheids. The results concerning the Hubble diagram are presented in Fig. 4.2. T he different values of the Hubble constant obtained by these various methods [7] are summarized in Table 4.1. Table 4.1 Value of the Hubble constant as measured by t he various methods d iscussed in t he text. Ho (km . s - 1 IMpc)
Method
71 ±2 ±6
SN Ia
71 ±3 ± 7
'TUlly- Fisher
70 ±5 ±6
surface brightness
72 ±9 ±7
SN II
82 ±2 ± 6
fundamental plane
72±8
combined methods
4.1.1.3 Physical methods In parallel with these astronomical methods , there exist at least three other methods to determine the Hubble constant, avoiding t he need to construct a scale of distances. 1. Expansion of the photosphere of SN II: the front of the SN II propagates at a velocity of order v / c '" 0.01 after t he explosion. This velocity can be measured
by the Doppler shift in t he supernova spectrum. The radius of the photosphere is then rphotosphere = v(t - texp losion) . If the angular diameter B under which the supernova is observed can be measured, then its distance can be deduced , D = v(t-texplosion )/B. Unfortunately, the angular diameter is difficult to measure directly for extragalactic supernOVffi. 2. Lensing effect: the arrival times of two gravitationally lensed images of the same point-like source depend on the path followed by light and on t he gravitational potentials along t hese paths (Chapter 7). The time delay between two light signals together with the angular separation between the two images of a variable quasar allow the Hubble constant to be deduced. However, this method has some problems. The gravitational lens is usually a galaxy, the mass distribution of which is not known independently. Thus, there is a degeneracy between the distribution of the lens mass and the Hubble constant. Ideally, we should need a measurement of the velocity dispersion as a function of the position. Only a dozen known systems, having both a favourable geometry and at least one variable source, is currently known . 3. Sunyaev- Zel 'dovich effect: the inverse Compton scattering of a microwave background photon (see Section 4.4) by t he ionized gas of a cluster , induces a distortion
182
The standard Big-Bang model
of the blackbody spectrum, known as the (thermal) Sunyaev- Zel'dovich effec The scattering probability P can roughly be expressed as a function of the cl diameter Dctuster and its average electronic density ne by P rv n e O"TDcluster being the Thomson scattering cross-section. The temperature and density of the hot gas can be obtained from an X emi map. Using the fact that the X flux , unlike the spectral distortion of the micro background, depends on the distance to the cluster, the latter can thus be dedu The main uncertainties of this method come from the heterogeneities of the distribution (which tend to lower the measured value of H o), from the proj ec effects, and the hydrostatic equilibrium hypothesis.
To conclude, it is reassuring to notice that methods based on very different phy principles lead to consistent values of the Hubble constant (see table 4.2), thems in agreement with those obtained by astronomical methods. Table 4.2 Measurement of the Hubble constant by physical methods. Ho (km . S - 1 / Mpc)
4.1.2
Method
73 ± 15
photosphere
72 ±7 ±15
lensing effects
60 ±20
lensi ng effects
60 ±4
Sunyaev- Zel'dovich effect
72 ±5
WMAP
Reference Schmidt et al. (1994) Tonry and Franx (1998) Fassnacht et al. (2 000 ) Reese et al. (2002) Spergel et al. (2003 )
The contribution of supernovre
As illustrated in section 4.1.1, SN Ia represent to date the most competitive candid to extend the Hubble diagram to large redshifts , for which the luminosity dista redshift relation is no longer linear. This offers the opportunity to measure cosmolo parameters other than the Hubble constant.
4.1.2.1 Distant supernovre
There exist two families of supernovre. Those for which the optical spectrum incl hydrogen traces are called type II, whereas those lacking hydrogen are of type 1. M over, type I supernovre are subdivided into three very different sub-families (a, SN Ib and Ic, just as SN II , are created by the explosion of massive stars (c Wolfe-Rayet) and lead to a residual black hole or neutron star. The mass lost by progenitor is more significant for the Ic than for the Ib so that the latter still ha layer of helium. SN la, on the other hand, appear in binary systems where one o two stars is a white dwarf that accretes the mass of its companion. When it rea the critical Chandrasekhar mass, 1.4 M G , the white dwarf collapses and explodes supernova. It has been established , thanks to the study of close supernovre, that luminosity curve could be calibrated using an empirical relation between the peak the width of the luminosity curve. Thus, SN Ia seem to be good standard candles can be observed up to large redshifts.
The Hubble diagram and the age of the Universe
183
Two teams have worked in parallel on the search for dist ant supernovre: the Supernova Cosmology P roj ect (SCP ) [9] and the High-Z Supernova Search Team (HZT) [10]. In 1998, they independently published t he Hubble diagram of type Ia supernovre (see Fig. 4.3 ). The SCP group used a catalogue of 42 SN Ia composed of 18 SN Ia with z < 0.101 and 24 SN Ia with 0.1 80 < z < 0. 830. T he analysis of t he HZT group relies on 34 close SN Ia and 16 SN Ia with 0.16 < z < 0. 62. Figure 4.3 summarizes t he observations of both t eams. Since then , the number and the maximal redshift of the observed supernovre have increased greatly. 3
44 • I-Iigh-Z SN Sea rch team ~I o Supernova Cosmology Proj ect o 42
i
..2'" 40 ::l
2
"0 0
- I-Iigh-Z SN Search team - - Supernova Cosmology .. Proj e~~.... ·····
E 38
t::
lS
is'"
= 0 .3 , QAO = 0. 7 = 0.3, QAO = 0.0 QmO = 1.3, Q AO = 0 .0
-
36
QmO
.. ... QmO - -
34
\.\0 <:> ..••••.•
o
6
J" l. n
~ O. o
.
~E O O~~~~~~~~tt~~~~Ir.~~ 1 fI -o. ~
f ~
0 .01
0 .10
1.00
- lLLww~~~~~~~~~~WWW
0.0
z
0.5
1.0
1.5
2.0
2 .5
Q mO
Fig. 4.3 The Hubble d iagram for the supernovre of b oth HZT and SCP t eams compared with the predictions of t hree ACDM models. The bot tom panel shows the differ ence between the data and a Universe with (Dmo, DAD ) = (0.3 , 0). (right): T he constraints on the cosmological parameters (Dmo , Dl\o) obtained by b oth exp eriments, compared with t hose obtained from the cosmological background (see Chapter 6) based on t he observat ions of MAXIMA and BOOMERanG balloons. Adapted from Ref. [11].
4.1.2.2 Implication for the cosm ological p aram eters As soon as t he redshift is not too small, t he relat ion between t he distance luminosity and the redshift becomes non-linear and t he next-t o-leading order term is proportional to ~( 1 - qO )z2 [see (3.99) ]. The analysis of both t eams proves t hat the deceleration parameter satisfies (4.1 ) qo < 0
> O. T his conclusion follows from the parameterizat ion (3 .92) t hat does not make any hypothesis on the material composition
at 2.80- with no other hypot hesis than Dmo
184
The standard Big-Bang model
of the Universe and relies only on the hypothesis that the Universe can be describ by a Friedmann- Lemaitre space-time. We can thus consider the fact that the Unive is accelerating to be a robust conclusion of the analysis of SN Ia data. l In a ACDM Universe, this relation mainly gives information on the two cosmol ical parameters n mo and nAO, as indicated by (3.94), and the constraint (4.1) imp that nAO > ~nmo. Figure 4.3 summarizes the supernovre constraints. These data ply t hat the cosmological constant must be nonvanishing and positive. The ellipses constraints of Fig. 4.3 can be roughly summarized as
(4
For tighter constraints, one should combine the data of other observations such as th from the cosmological microwave background. The latter indicates that the Unive is almost fiat , n mo + nAO c:::: 1.02 ± 0.02 according to the analysis of WMAP ( Chapter 6). Thus, for a ACDM Universe,
nmO
rv
0.3.
(4
The order of magnitude of this analysis is confirmed by precise statistical analysis
4.1.2.3
Robustness of the conclusion
The greatest uncertainty lies in the hypothesis that distant supernovre are stand candles and can be calibrated in the same way as close supernovre. Three effect s sho be quantified with precision in order to know if this effect is real or if it is an artefa
- The behaviour of supernovre could vary with cosmic time, mainly due to change of metallicity, i. e. of their chemical composition beyond helium-4. If dist supernovre have a luminosity peak systematically weaker than closer ones, conclusion nAO > 0 would be softened. If the explosions of SN Ia are strong then the conclusion would be reinforced. It is difficult to obtain information the intrinsic luminosity and one can more easily study other parameters of light curve. If two families (close and distant) are in all aspects identical, it however , highly probable that their luminosity is also the same. - The absorption by the interstellar medium could give the illusion that dist supernovre are systematically dimmer. The analysis of the light curve in seve colours can give an estimate of this dimming. The effects of gravitational lens could also soften the luminosity. - The possibility of an observational bias should also be considered.
The increasing number of observed supernovre provides a better understanding of th physics and current studies tend to prove that the acceleration of the Universe can be explained by anyone of these effects [ll]. Figure 4.4 compares the astrophysi and cosmological effects .
lWe emphasize that this conclusion can now b e reached by the combination of other da ta ( eMB a nd weak lensing) witho ut taking the SN Ia into account.
The Hubble diagram and the age of the Universe
185
0.5
~
5" ¥I
0.0 ... '.
r
-0 .5
Q = O
_. - .....
Absorption by interstellar medium or evolution QmO = 0.35, Qi\O = 0.65 QmO = 0.35, Qi\O = 0.0 QmO = 1.0, Qi\O = 0.0
.... "
.
...... ..... .
z
Fig. 4.4 Hubble diagram for type Ia supernovce compared to the one for an empty Universe (OmO,OAO) = (0,0). The measurement of the SN Ia at z = 1.7 is incompatible with some astrophysical effects that could mimick an acceleration of the Universe at lower redshifts . Adapted from Ref. [11] .
4.1.2.4
Cosmological constant or dark energy
The previous results assume that only a cosmological constant and ordinary matter dominate. However , any kind of matter with equation of state W < -1/3 can lead to qo < 0 [see (3.94)]. It is legitimate to wonder whether it can be checked from the data that the matter responsible for the acceleration of the Universe is indeed a cosmological constant, i.e. a component with an equation of state w = - l. Figure 4.5 presents the constraints on the pair of parameters (w, D l11o ) for a flat Universe if w is assumed to be constant. For Dmo ~ 0. 3 we obtain w < - 0.6. (4.4) The constraints on the equation of state depend on which data have been combined and on the parameterization of the equation of state. Figure 4.5 compares the constraints obtained by combining the observations from the microwave backgroud (WMAP) , galaxy catalogues (SDSS), the supernovre, and from the Lyman-a forest in various ways. It is currently conventional to call dark energy the component responsible for the acceleration of the Universe. This d ark energy could be a cosmological constant but other candidates also exist (see Chapter 12 for a more detailed discussion). The majority of these candidates have a dynamical equation of state. The constraints on the equation of state of the dark energy change very rapidly with the flow of new observations. Let us cite, for instance, a constraint [12] on a parameterization of t he form W= Wo + waz/(l + z); it leads to +0.193 Wo = - O.981 -0. 193'
wa
=
0.83 O. 05 +-0.6 5'
(4.5)
186
The standard Big-Bang model - 0.2,-------- - - - - - - , _
D D - 0.4
WMAP + gal + SN WMAP + gal + bias + Lyman - a \VMAP + gal + SN + bias + Lyman - a
- 0.8
-1.0
- 1.2
- 1.4
-1.6 ~~--~--~~--~--~--L-~--~
0.6
0.20
0.24
0.28
0.32
0.36
Fig. 4.5 (left): Constraints on a const ant equation of state of dark energy as a functio 12mo for a fiat Universe. Adapted from Ref. [9J. (right) : Same thing by combining various of observa tion (contours at 68% and 95%) , cosmological microwave background (WMA galaxies (SDSS) , supernOV ffi and Lyman-a forest , in different ways. Adapted from Ref.
at 1 a . The determination of the nature of this dark energy is one of the impor challenges of modern cosmology and will be discussed in the final chapters of book. 4.1.3
The age of the Universe
A constraint on the cosmological parameters can also be obtained from the meas ments of the age of t he Universe. A lower bound on this age can then be set that sh be compatible with t he dynamical age of the Universe computed from the Friedm equations. Dat a from supernovce imply that the dynamical age of t he Universe f flat ACDM model is I
t = 14.9+ 14 ( - ' o - l.l 0.63
)-1
X
10 9 years
h
'
to = 14.2! 6g ( 0.63
)-1 X
109 years ( '
using data, respectively, from SCP [9] and HZT [10].
4.1.3.1
A ges in the Milky Way
The oldest obj ects in the Milky Way are low metallicity 2 stars and are locate the halo of the Milky Way. T here are three methods to determine the age of t stars [13].
1. Nuc1eochronology: conceptually, this is t he simplest method , analogous to radi t ive dating methods, and it represents the most direct way to date the Unive
2By m etallicity, we mean here t he content in chemica l elements beyond helium-4 . The ra tio [F traces t his met a liicity.
The Hubble diagram and the age of the Universe
187
tu . The age of the stars is estimated from the abundance of long-lived radioactive
nuclei for which a half-life is known (for details , see Ref. [14]). The radioisotope 23 2Th (thorium) with half-life of 14 million years has been used to date t he age of the stars of the Galaxy (in particular t he star CS 22892) but its abundance is only halved on a period equal to the age of the Universe. In principle, 238U is a better indicator since its half-life is 4.5 million years . The abundance of this isotope was derived by observing the star CS31082-001 8 [15], 10g(U /H) = - 13. 7 ± 0.14, which corresponds to an age of (12.5 ±3) x 10 9 years . 2. Cooling of white dwarfs: white dwarfs are t he terminal stage of the evolution of stars with mass smaller t han around 8M0 . As t hey get old, white dwarfs get cooler and dimmer. Using a model of their cooling, their cooling rate, and thus the age of t he Universe, can be calculated. Unfortunately, since they are not very bright , white dwarfs are very difficult to observe so that the studies are reduced to white dwarfs in the neighbourhood of the Solar System and enable us to estimate the age of the local disk to approximatively 9.8 ::: 6:~ x 109 years. 3. Main sequence: theoretical models of stellar evolution make it possible to understand the posit ion of a star in the temperat ure~lumino sity plane. During the main sequence, which corresponds to the longest part of their life, stars burn their hydrogen to produce helium. At the end of t his period, the luminosity of the star increases and its surface temperature decreases. It can be shown that t he luminosity of the stars at this transit ion of t he main sequence is the quant ity least sensitive to errors in the t heoretical models and can be used to determine the age of globular clusters up to precision of 7~ 10 % . One should also determine the distance of the cluster that limits the precision of the method (an error of 10% on the distance translates into an error of 20% on the age!). Thanks to the satellite Hipparcos a better calibration of the local distance scale was possible. The age of globular clusters is roughly is the r ange (11 - 14) x 109 years.
Table 4 .3 Summary of the main constraints on t he age of the Universe. Age (109years)
Method
Reference
15.2 ±3 .7
nucleochronology (star CS 22892)
Sneden et al. ( 1996)
12.5 ±3
nucleochronology (star CS 31082-0018)
Cayrel et al. (2001)
11. 5±1.3
5 methods (glob ular clusters)
1l.8 ± 1.2
main sequence (globula r clusters)
14 ± 1.2 13.7 ± 0.2
+ binary) + la rge-scale structure
main sequence (globu lar clusters WMAP
C h ab oyer et al. ( 1998) Gratton et al. (1997) Pont et al. (1998) Spergel et al. (2003)
4.1 .3.2 Age of the Universe To conclude, the minimum age of our Universe can be estimated by specifying the ages of the oldest object s in our Galaxy. To obtain the age of t he Universe, the t ime taken to form these objects should also be estimated. Unfortunately there is no robust theory
188
The standard Big-Bang model
for the beginning of stellar-formation processes. The beginning of star formati estimated at z rv 5 - 20, which corresponds to a period of (0.1 - 2) x 10 9 years. allows us to conclude (see Table 4.3) that 9.6 < 9 tu S; 15.4. - 10 years
4.1.3.3
Compatibility between the Hubble constant and the age of the Univers
Having measured the Hubble constant, the dynamical age of the Universe ca obtained from (3.66) which only depends on the cosmological parameters Dmo DAO (the duration of the radiation era being negligible). To understand if the constraints on the Hubble constant and on the measurem of the age of the Universe are compatible, we first present a comparison between from supernovre and the theoretical computation of the dynamical age of the Uni (Fig. 4.6). We t hen compare Hotu with Hoto(Dmo, DAO ), assuming that Ho and t known up to 10%. These measurements are consistent if the cosmological constant not vanish , as independently indicated by the results from supernovre. An E inste Sitter Universe is marginally excluded (up to 1.50-) by these measurements.
4.2
Thermodynamics in an expanding Universe
The previous sections have convinced us t hat the Universe is expanding. Since radi scales as a- 4 while pressureless matter scales as a- 3 , the Universe was dominate radiation in the past for redshifts larger than
obtained by equating the matter and radiation energy densities and where 8 TCMB/2.725K [see (4.137) below]. Since t he temperature scales as (l+ z ), the tem ature at which the matter and radiation densities were equal is Teq = TCMB(l + which is of order
Above this energy, the matter content of the Universe is in a very different form that of today. In particular, when the temperature T becomes larger t han twic rest mass m of a charged particle, the energy of a photon is large enough to pro particle- antiparticle pairs. Thus, when T » m e, both electrons and positrons present in the Universe, so that the particle content of the Universe changes d its evolution, while it cools down. Here, we describe this phase and first fo cus on the particle distribution in librium and then on some out-of-equilibrium processes leading, for instance, t production of thermal relics. 4.2.1
Equilibrium thermodynamics
For a description of thermodynamics in the relativistic context and in cosmology can refer to Refs. [16- 18] .
Thermodynamics in an expanding Universe
h
=
0.7 , tu
=
189
12.10 9 yrs
2.0
1.5 + 20 - + 10
1.0
- -10 . - -20
0.5 +--+------~~------~~--rQ~h-=~O---+--(h = 0.5 , tu = 12.10 9 yrs) 0.0
l..---'1..._ _- ' - -_ _L..----'-_ _-'------"'-----'-_ _...L..----'_ _....IL----'
o
1
Fig. 4.6 Comparison bet ween t he constraints on the product of the Hubble const ant and the age of the Universe, H ot u , and the product of the Hubble const ant and the dynamical age of the Universe, H oto. The latt er depends on the cosmological paramet ers. We have fixed h = 0.7 and tu = 12 X 109 years. T he dark curve represents a flat Universe (D mo + DM = 1) in which case the abscissa is D hO . T he light curve is for a Universe wit h no cosmological const ant (OAO = 0), for which the abscissa is then D ruo. T he grey regions indicat e t he areas at 1 and 20" for Hotu and the plain horizontal line is the case where h = 0. 5. The circle corresponds to an Einst ein- de Sitter Universe for which H otu = 2/3. From Ref. [6]. 4.2.1.1
Dis tribu tion fun ctions and thermody namical quantities
To describe physical processes during t he radiation-dom inated era, the distribution function f i (P , T ) of the p a rticles present in the Universe should be d etermined.
Distribution functions P art icle interactions are mainly cha r acterized by a r eaction rate f . If t his rea ction rate is much lar ger tha n the Hubble expa nsion r a t e, then it can m a intain these pa rticles in thermodyn amic equilibrium a t a temper ature T. P articles can thus b e treated as perfect Fermi- Dirac a nd B ose- E instein gases with distribut ion 3
(4.10)
3The distri but ion function dep ends a priori on (x , t) and (p , E ) b ut the homogeneity hypothesis implies t hat it does not dep end on x a nd isotropy implies that it is a function of p2 = p 2. Thus , it follows from t he cosm ological principle tha t f (x , t, p , E) = f eE , t) = f [E, T (t )] .
190
The standard Big-Bang model
where 9i is the degeneracy factor, /-Li is the chemical potential and E2 = p2 + m 2 . T normalization of Ii is such that Ii = 1 for the maximum phase-space density allow by the Pauli principle for a fermion. Ti is the temperature associated with the giv species and, by symmetry, it is a function of t alone, Ti (t). Interacting species have same temperature. Among these particles, the Universe contains an electrodynam radiation with blackbody spectrum [see Section 4.4] at a temperature of 2.725 K tod Any species interacting with photons will hence have the same temperature as th photons as long as r i » H. The photon t emperature T-y = T will thus be called temperature of the Universe.
If the cross-section behaves as (J rv En rv Tn (for instance, 11, = 2 for electrowe interactions) then the reaction rate behaves as r rv n(J rv T n +3 ; the Hubble const behaves as H rv T2 in the radiation period. Thus, if 11, + 1 > 0, there will always a temperature below which the interaction decouples while the Universe cools dow The interaction is then no longer efficient; it is said to be frozen , and can no lon keep the equilibrium of the given species with the other components. This prope is at the origin of the thermal history of the Universe and of the existence of rel This mechanism, during which an interaction can no longer maintain the equilibri between various particles because of the cosmic expansion , is called decoupling. particle will be said to be coupled if r ~ H and decoupled if r H. This crite of comparing the reaction rate and the rate H is simple; it often gives a corr order of magnitude , but a more detailed description of the decoupling should be ba on a microscopic study of the evolution of the distribution function. As a concr example, consider Compton scattering. Its reaction rate, rCompton = ne(JTC, is of or 3 rCompton rv 1.4 X 10- Ho today, which means that, statistically, only one pho in 700 interacts with an electron in a Hubble time today. However, at a redsh z rv 10 3 , the electron density is 10 3 times larger and the Hubble expansion rate is order H rv HoJOmo(l + z)3 rv 2 x 104Ho so that r Compton rv 80H. This means th statistically at a redshift z rv 103 a photon interacts with an electron about 80 tim in a Hubble time. This illustrates that backward in time densities and temperatu increase and inter actions become more and more important. Note that if the thermal equilibrium can be maintained by interactions withi cosmic plasma, by causality, these interactions cannot establish such an equilibri between two causally disconnected regions that would have different initial tempe tures. The hypothesis of the homogeneity of space implies that the Universe is assum to be initially in thermal equilibrium. The justification of this state will be discus in Chapter 8.
:s
Thermodynamical quantities
The distribution function (4.10) can be used to define macroscopic quantities su as the particle number density, 11" energy density, p and pressure, P, for each spec 'i' , as 4
4Since E2 - m 2 41TV E2 - m 2 EdE.
p2 implies that pdp
EdE a nd, b ecause of isotropy, d 3 p = 41Tp 2 dp
Thermodynamics in an expanding Universe
191
(4.11) (4.12) (4.13) These three quantities can be expressed in terms of the integrals I (m,n) ( i
.
.)
=
X t , Yt -
1
00
Xi
m( U 2
2)n/2 d U exp [U(- )Yi] ± 1 '
U
-
Xi
(4.14)
with Yi
!-Li ==, T
(4.15)
as (4.1 6) Table 4.4 summarizes the expression of these quantities within various limits for fermions (F) and bosons (B), where ((3) ~ 1.202 is the value of the Riemann zeta function at 3.
Table 4.4 Summary of the main limits of the thermodynamical quantities (4.11)- (4. 13) for fermions (F) and bosons (B). particles
Limit
p
n
P
B
g«((3) /1r 2 )T3
(1r2/3 0 )gT4
p/3
F
(711"2/8
p/3
F
g(3(3) /41r 2 )T3 gJ.L3 / 61r 2
T»m,J.L < - T
B,F
(g /1r 2 )el"/TT 3
gJ.L3/81r 2 (3g/1r 2 )el"/TT4
T«m
B ,F
g(mT / 21r)3 /2 eCI"-=) /T
(m +3T/2)n
T »m, J.L J.L » T »m
X
30)gT 4
p/3 p/3 nT
4.2.1.2 Number of relativistic degrees of freedom The radiation density at a given temperature T can be computed from these expressions (see Table 4.4) (4.17) 9. represents the effective number of relativistic degrees of freedom at this t emperature ,
g.(T)
=
L i=bosons
Ti ) 4 7 gi ( T +8
L i=fermions
gi
(Ti)4 T
(4.18)
192
The standard Big-Bang model
The factor 718 arises from the difference between the Fermi and Bose distributio In this regime, the matter content is dominated by radiation and the curvatur negligible, so that the Friedmann equation is of the form 2
(7r30 )
H2 = 87rG N 3
g.
T4.
(4.
Numerically, this amounts to H(T)
1/2
rv
T2
= 1.66g. lVI'
(4.
p
or equivalently,
t(T)
~ 0 .g. 3 -1/2 Mp T2
-
obtained by integrating H = 4.2.1 .3
rv
2 42 .
- 1/2
g.
(_T_) 1 MeV
-2
s,
(4 .
- TIT.
Chemical potential and particle- antiparticle aBymmetry
Inelastic scattering processes such as Brehmsstrahlung e + p f----+ e + p + 'Y im that the number of photons is not conserved and thus t heir chemical potential sho vanish (4. /L, = O.
This implies that photons must have a Planck distribution. Any particle, A, kept in chemical equilibrium with its antiparticle by a reaction the form A + A f----+ 'Y + 'Y must then satisfy the constraint /LA
+ /LA = O.
(4.
As long as T» mA , using (4.11) , this implies an asymmetry nA _ nA
~ g;~3 [7r 2(~) + (~) 3]
between particles and antiparticles. At lower temperatures (T is exponentially suppressed nA - nA
~ 2g A (m;:) 3/2 e-
mA
/
T
sinh
«
(4.
mA) this asymme
C"J; ).
(4.
When T becomes smaller than mA , A and A annihilate and only this small exc survives. This is, for instance, the case for the electrons. Note t hat the electrical neutrality of the Universe implies that the number protons np is equal to n e -ne . Using the constraint npln, rv 5 X 10- 10 [see Section (4.142)]' we obtain n 9 7r 2 I/. --..E. rv ---'=---~ rv 10- 8 (4. n, g, 6((3) T ' so that 1.34/Le1T rv 6.5 can thus be neglected .
X
10-
10
.
The chemical potential of the electrons and positr
Thermodynamics in an expanding Universe
4.2.1.4
193
Entropy
In order to follow t he evolut ion of the processes, it is convenient to have conserved quantities such as the entropy. Combining t he conservation equation (3.27) , rewritten in the form d(pa 3) = - Pda 3 , and the derivative of the pressure (4.13) with respect to the t emperature,
we get that t he quantity 5 p + P - np, (4.27) T satisfies 6 the equation d (sa 3 ) = -(p, / T)d(na 3). Thus, the product sa 3 is constant (i) as long as matter is neit her destroyed nor created , since t hen na 3 is constant , or (ii) for non-degenerate relativistic matter , p,/ T « 1. We thus deduce that in the cases relevant for cosmology, (4.28)
s ==
'------=-----''---
To interpret this quantity, let us consider t he expression (4.27) which implies t hat Td( sa 3) c::: d (pa 3 ) + Pda 3 , in t he limit p,/ T « 1. We recognise S = sa 3 as the entropy and thus s is the entropy density. Hence, (4.28) implies that s ex: a- 3 as long as p,fT « 1. T hen, using (4.27), the entropy can be expressed as (4.29) where q* is defined by q. (T) =
L
L
Ti ) 3 7 gi ( T + "8
i = bosons
gi
(Ti T
)3
(4.30)
i= ferm io ns
If all relativistic particles are at the same temperature, Ti = T, then q. = g•. Note also that s = q*7T 4 / 45((3)n, ~ 1.8q.n" so that the photon number density gives a measure of the entropy. SIn the case of a vanis hing chemical potential, th is expression can b e easily derived by starting from the general expression of t he entropy of a system as TdS = dE + Pdll. If we choose T and V as our variables to describ e the thermodynamical state of the system t hen, since they a re intensive quantit ies, P and p are functi ons of T alone; p(T) a nd P(T). It foll ows that dE = d(pll) = pdll + lI(dp/dT) dT. Thus,
+ ~ dp dT. T TdT We deduce that (8S/8V)T = (p + P) /T a nd (8S/8T)v = (1// T )(dp/ dT ). T he integra bility condit ion then implies that dp /dT = (p + P )/T. Plugging these expressions in dS we deduce that d S = P + P dll
dS = d
[ ~(p + P)]
,
from wh ich we get the expression of the entropy up to a n addit ive constant . 6S tarting from TdS = dE + Pdll - /-LdN and us ing t h at 11 ex a 3 , so t hat E ex pa3 , and Sex sa 3 we deduce that Td( sa 3 ) = d (pa 3 ) + Pda 3 - /-Ld (na 3 ). Tak ing into account that the first two terms of the r.h.s. cancel, we deduce t he evolutio n equation of sa 3 .
194
The standard Big-Bang model
The number of particles in a given volume, N ex na 3 , is constant if no matt created or annihilated. The relation (4.28) thus implies that
y= ~ s
(4
represents the number of particles per co moving volume. For relativistic particles, quantity remains constant during the evolution. It can be easily computed in limiting cases
(4
(4
4.2.1.5
Decoupled species
If at a given time , the interaction rate is no longer high enough to maintain thermodynamic equilibrium of the species i, i. e. if f i ;S H , then this species evo independently from the others . If this species was in thermodynamic equilibrium be decoupling, then its distribution function at the time of decoupling, tD , is 1
f i(P, tD)
=
exp [(E - f.L i)/Ti(tD)]
(4
±1
After decoupling, particles propagate freely and the form of the distribution functio conserved. Only the momentum is redshifted by the expansion as p(t) = p(tD)aD/ It follows that the distribution function at any t > tD is given by
f i(P, t)
a(t) ] fi [ aD p, tD .
=
(4
Thus, if a species was in thermodynamic equilibrium at a given time in the histor the Universe , one can determine its distribution at any time. If the decoupling occurred when the particle was relativistic, (T » m , f.L), then distribution function (4.35) takes the simple form 1
(4
fi(P, t > tD) = exp [E/Ti(t) ] ± l '
where TD = T(tD). The temperature of this decoupled species always decrease a - 1 and its entropy Si is conserved separately. If this particle becomes non-relativ after t NR » tD , we have E(t) c::::: m for any t > t NR but the distribution function k the form (4.36). If the decoupling occurred when the particle was non-relativistic (T « m) , E c::::: m + p2/2m and the distribution function is given by
f ,(p , t > tD) o
= e-(m - I-' )/T De -
p 2 /2mTi
(t) ,
T (t) i
=
T
D
[aD] 2 a(t)
(4
Thermodynamics in an expanding Universe
195
Thus, everything goes as if the particles had an effective chemical potential , J.-l( t) m+ [J.-lD - mJT(t) / To. We then obtain ni
4.2.1.6
~
gi
(
mTO -)
3/ 2
271"
(a--O)3e - m / To . a
(4.38)
Temperat ures and their evolution
Temperature of the Universe Since the total entropy is constant, q. (T)T3a 3 remains constant during the evolution of the Universe . If, moreover , a species i decouples from the cosmic plasma at temperature To then the entropy of this species is conserved on its own so that and
S,
=
271"2
3 3
-q·(T)T a 45 ' "
(4 .39)
are both constant. The entropy, S - Si = (271"2/45 )q, (T)T 3 a 3 , of the particles in thermal equilibrium with the photons, is t hus also constant. We thus conclude that the temperature of the Universe varies as (4.40) where q, is given by the relation (4.30) but only summing over the relativistic particles that are in thermal equilibrium with the photons. Temperature of a decoupled species As long as the number of relativistic species does not vary, the temperature decreases as a - I . When a species becomes non-relativistic, its entropy is transferred to the other relativistic pa rticles remaining in thermal equilibrium and therefore see their temperature suddenly increase (q. is a decreasing quantity) as
T(+) =
[q,(_)] q, (+)
1/3
T( - ),
(4.41 )
where + and - refer, respectively, to quantities evaluated after and before the decoupIing. At the time of decoupling T = Ti = To so that comparing the quantities (4.39) at temperatures To and T < To, we get that the temperature of the decoupled species is related to that of the Universe by (4.42) This expression generalizes (4.36) to the case where qi(T) is not constant. In particular, qi(T) can vary if the species i has a new interaction allowing it to annihilate to produce another particle that does not interact with the cosmic plasma.
196
The standard Big-Bang model
4.2.1.7 Neutrinos as an example
Neutrinos are in equilibrium with the cosmic plasma as long as the reactions v + D + e + e and v + e +----> v + e can keep them coupled. Since neutrinos are not charg they do not interact directly with photons. The cross-section of weak interactions is given by u rv E2 ex: T2 as long the energy of the neutrinos is in the range m e « E « mw. The interaction rate thus of the order of r = n(uv) c:::: G;T5 . We obtain that
G;
r
c::::
G;
(_T_) H
(4.
3
1 MeV
'
where we have used (4.20). Thus, close to To rv 1 MeV, neutrinos decouple from cosmic plasma. For T < To , the neutrino temperature decreases as Tv ex: a- 1 a remains equal to the photon temperature. Slightly after decoupling, the temperature becomes smaller than me . Between and T = m e there are 4 fermionic states (e -, e+, each having ge = 2) and 2 boso states (photons with g-y = 2) in thermal equilibrium with the photons. We thus h that 7 11 (4. q-y (T > me) = 8 X 2 X 2 + 2 = 2· For T < m e only the photons contribute to q-y and hence
q-y (T < me) The conservation of entropy implies that after the neutrinos and the photons are related by
(4.
= 2.
e-
e annihilation, the temperatures
_ (11) 4 1/3 Tv.
(4.
T-y -
Thus, the temperature of the Universe is increased by about 40% compared to neutrino temperature during the annihilation. Since nv = (3/11)n-y, there must ex a cosmic background of neutrinos with a density of 112 neutrinos per cubic centime and per family, with a temperature of around 1.95 K. The energy density of th neutrinos, as long as they are relativistic, is Ov = (7/ 8) (4/11 )4/ 30-y per family neutrino. Using the value (4.141) of O-yo, we conclude that today r. 2 HvO h =
1.68
X
10 -5
(Nv) :3 84 -
(4.
2.7'
if their mass is smaller than 10- 4 eV.
4.2.1.8
Number of relativistic degrees of freedom
The e - e annihilation is the last annihilation to happen , so that knowing Tv / T-y gi the values of g. and q. today 7
g.o = 2 + 3 x - x 2 x
8
( 4 ) 4/ 3 -
11
= 3.36,
if there are 3 families of massless neutrinos.
q.o = 2+3 x
~ x 2x
C
4 1) = 3.91 (4.
Thermodynamics in an expanding Universe
197
The functions q* (T) and g* (T) depend on the particle content of our Universe. As an example, Table 4.5 and Fig. 4.7 summarize their evolution in the context of the standard model of particle physics, discussed in Chapter 2.
Table 4.5 Particle content of the Universe as a function of the t emperature in the context of the standard model of particle physics. ~ 150-400 MeV characterizes the quark- hadron phase transition, which is assumed to be adiabatic. The thermodynamic history of this transition is not yet well understood. This table also assumes that the Higgs boson has a mass larger than the mass of the W ±, ZO bosons and of the top quark (t). The last three lines of this table are not certain since the Higgs sector and the electroweak transition are not well understood. At larger temperatures, 9* depends greatly on the theoretical model for the fundam ental interactions. For instance, for a minimal SU(5) grand unification model, 9* (T > TG UT ) = 647/4 and in the case of the minimal supersymmetric model, 9*(T < Tsusy) = 915/4 since the number of degrees of freedom is almost doubled.
T:r
Threshold (GeV)
T
< me
0 .511
X
10 -
Relativistic particles
"( (+
3
g*
2
+ e±
11/2
TD(V) - ml-'
0.106
v start to interact
43/4
mJ-L - mW
0.135
+ J.L ±
57/4
+ rr±, rro
69/4
m e - TD(V)
(2 - 4)
X
mrr - T:!h
T2h -
ms
0.194
10-
3
,,(, 3v's, e±, J.L± U, d ,
U,
ms - me
1.27 ± 0.05
+ s,
d, 8
S
205/4
g 's
247/4
mC-mT
1.78
+ c, c
289/4
m T - mb
4.25 ±0.1O
303/4
mb-mW
80.3 ±0.3
mW-mt
180 ± 12
+ T± + b, b + W±, ZO +
t, t
423/4
+ HO
427/4
mt - mHO mHO -
4.2.2
3 decoupled v)
T;:w
300 (7)
345/4 381/4
Out-of-equilibrium thermodynamics
The evolution of a decoupled species can thus easily be described . However, the description of the decoupling , or of a freeze-out of an interaction, is a more complex problem that requires us to go beyond the equilibrium description that has been developed previously.
198
The standard Big-Bang model
q
1
10-5
10--4
10- 3
10- 2
10-1
1
T (GeV)
Fig. 4 .7 Evolution of the functions 9. and q. as a function of the temperature in the s dard model of p article physics SU(3)c x SU(2)L x U(l)y with two hypotheses concerning critical temperature of the quark~hadron transition, Tcqh .
4.2.2.1
Boltzmann equation
Evolution of the distribution function
The evolution of the distribution function is obtained from the Boltzmann equat
L[jl = C[f]'
( 4.
where C describes the collisions and L = d/ds is the Liouville operator, with s length along a worldline. The operator L is a function of eight variables taking explicit form
a - r'"{3-yP{3P-y ap"" a L[fl -- P'" ax'" In a homogeneous and isotropic space-time, time, f(E , t) , so that
f
L [f l = E af _ H
at
( 4.
is only a function of the energy 2
af . aE
(4.
p Using the definition (4.11) of the particle density, and integrating this equation w respect to the momentum p , we obtain 7 71n order to evaluate the second term of (4.5 1), we use that
Thermodynamics in an expanding Universe
199
( 4.52) The difficult part lies in the modelling and the evaluation of the collision term . Here, we restrain ourselves to the simple case of an interaction of the form i+j ~ k+l,
for which the collision term can be decomposed as Ci
(4.53)
= Ckl ->ij
- Cij -> kl
with
(4.54) with a + sign for bosons and a - sign for fermions. IMlij->kl are the matrix elements describing the interaction. The Dirac delta function imposes the conservation of momentum and of energy. This form also shows that the probability for i to disappear is proportional to fdj , i.e. roughly to the density of the interacting species. 8 The factors (1 ± fk) arise from quantum mechanics and are related to the Pauli exclusion principle for fermions a nd to stimulated emission for bosons. If CP invariance holds, as we assume here , then Ckl->ij and Cij->kl involve a unique matrix element, 1M 12 , determined by the physical process. Indeed , this invariance implies t hat the process we consider is reversible and thus that i + j -+ k + I a nd k +1-4 i + j have the same matrix elements. It follows that
(4.55) Equation (4.52) thus involves three sources of evolution for the number density ni, namely dilution (3H ni), creation (C i = Ckl ->ij ) and destruction (C i = Cij->kl). Evolution of the particle density In cosmologically interesting situations, E - p, » T. Quantum effects can thus be neglected and 1 ± f c::: 1. Equation (4 .52) then takes the form
J
afd 3 p = -3 -p2 E aE
J
fd 3 p
by integrating by parts.
8For this reason , we may decompose C ij -> kl as Cij->kl = n i njCij->kl. Since Cij_kl has dimensions of a volume divided by a time , we can interpret it as the average cross-section multiplied by the relative speed of the interacting particles, that is C ij->k l = (aij_kIVij) . If we define the reaction rate by rij~kl = nj ( a ij _ kl Vij) then (4.52) takes the form
200
The standard Big-Bang model
(4
In this limit, the distribution functions are of the form f ex exp[(/1 - E)IT ] so the particle density (4. 11) can be expressed as a function of that at /1 = 0 as
(4 Furthermore, t he conservation of energy implies t hat Ek term !kfl - fdj takes the form
+ El = Ei + E j
such t hat
The Boltzmann equation (4.56) can t hus be written as
nn ) n + 3Hn = -(uv ) ( n·n· t J - ~nknl _ _ nknl 1.
1.
,
(4
where (uv) is defined as
(4
This equation is very general, but to go further we should know the form of the ma element IM I. We simply note t hat when nj (uv) » H , we recover t he fact that (4 can only be satisfied if the term in brackets cancels, i.e. if
(4
The system is then in chemical equilibrium. This equality is broken as soon as system is out of equilibrium.
4.2.2.2 Frozen interactions and relic particles
A massive particle X is in thermodynamic equilibrium with its antiparticle X temperatures larger than its mass. Assuming this particle is stable, then its den can only be modified by annihilation or inverse annihilation
If this particle had remained in t hermodynamic equilibrium until today, its density, n ex (mIT)3/2 exp( - miT) would be completely negligible. The relic den of t his massive particle, i. e. the residual density once the annihilation is no lo efficient , will actually be more important since, in an expanding space, annihila cannot keep particles in equilibrium during the whole history of the Universe. T particle is usually called a relic.
Thermodynamics in an expanding Universe
201
To evaluate the factor (Jxfx - fLfr) , note that land [ are in thermal equilibrium so that , after neglecting their chemical potentials, fL = exp( - EI I T) and fr = exp ( -ErIT). Conservation of energy imposes Ex + Ex = El + Er, so that
fdr = exp ( -
- E x+ Ex) T = fxix·
The Boltzmann (4.58) now t akes the simplified (integrated) form
+ 3Hnx =
nx
- (O"v) (n~ - n~) ,
(4.61 )
where (O"v ) is defined by (4.59). The conservation of entropy implies that nx + 3Hnx = sYx , where Yx = nx I sis the number of X particles per co moving volume, so that (4.61 ) can be written as (4.62 ) Y.x = -(O"V)s (Y x2 - Y-2) x . Using the variable x [see (4.15) for its definition] and the fact that dx /dt equation can be rewritten as
x = - (O"V)s (2 Y - Y-x2) . dx Hx x
-dY
= Hx, this (4.63)
During the radiation era, H ex: x - 2 , so tha t introducing ~ == Yx - Yx , and expressing H and s, respectively, by (4.19 ) and (4.29) , it t akes the final form
d~ dx
-
dYx
=- - - - AX
dx
with A = (O"V)
- 2
~
(
- ) ~+2Yx ,
rrr ~*/ 2mMp. V45 g*
To go further , we need the form of (O"v) and the value of mechanism in the next section. 4.2.3
4.2.3.1
(4.64)
(4.65)
Yx- We illustrate this
Two limiting cases
Cold relics
For a cold relic, the decoupling occurs when the particle is non relativistic. Using our previous results to express nx [(4 .11)] and s [(4.29)]' we get that
Y- x -_ -45ji9X - - x 3/ 2 e - x . 27r4 8 q*
(4.66)
The quantit ies (O"v ) and A depend a priori on the velocity. In general, (O"v) ex: v 2 n with n = 0 (annihilation of s waves) or n = 1 (annihilation of p waves). This quantity can be parameterized in a gener al way as
(O"V) = O"of( x) ,
(4.67)
where f (x) is a function of m i T only since (v 2 ) ex: T. Equation (4.64) can be integrated numerically, but the main properties of its solutions can be obtained analytically.
202
The standard Big-Bang model
Before decoupling, (1 < x « xd , Y x rv Yx and .6. and its derivative remain v small. Thus, (4.64) has the approximate solution .6.
2
rv _
Since d In Yx / dx
-
1_ _ x_ d In Yx 2>'0 f(x) dx '
_
= - 1 + 3/ 2x
(4.6
»
1, we get that before decou pling
1
x2
-1 for x
rv
.6.
rv
(4.6
__
- 2>'0 f(x)·
After decoupling (x » xd, Yx decreases exponentially and becomes negligible comparison to Y rv .6.. Equation (4.64) then takes the form .6. 2
d.6.
-
= - >'0---2
dx
Integrating this equation between .6. -1
f(x)x
and x =
Xf
_
00
.6. f
>. 0
1 =
00,
1
f(x) dx 2·
>'0
Xf
f(x)
(4.7
X
[ 1 7 ]-1 00
Yoo c:::
we get
00
Xf
It follows that
(4.7 ·
(4.7
dx
Xf should now be determined. The decoupling happens when Y x differs significan from Y x , i.e. when .6.(xd = cYx(xd, c being a number of order unity that can determined by comparison with a numerical integration . By estimating .6.(xd with value (4.69), Xf is the solution of
(4.7 A similar expression is obtained using the criteria f(xd
= H(xr), which gives
(4.7
From today's entropy density [see (4.140)], we can deduce the relic density of t particle X, nxo = sOYoo
(4.7
Thermodynamics in an ex panding Universe
203
To give an estimate of t his relic density, let us assume that f(x) = x- n and that q*(xf) ~ g*( xt) . Since px o = mnxo and Dx = Px / Pcrit , we get
D
xo
h 2 ~ 0.31 g* [
(
-1 / 2
(n
Xf )]
100
1
+ 1) xn+1 f
( q*o ) 8 3 (
3.91
2. 7
ao
10 - 38 cm2
)-
(4.76)
Since X f depends only weakly on the mass , these expressions give a good estimate of the relic density as a function of the particle 's mass and cross-section. Figure 4.8 depicts the numerical integration of t he Boltzmann equation (4.64). We recover both asymptotic regimes discussed in this section. 1 10- 2 ~
.~ I: '"
10-4
Q)
"Cl
....
Q)
10- 6 increasing
.D
E ;:l to:
'OJ)
~
10- 8
to:
.;;:
~
0
E 0
y"
- - --
U
~
---
10-1 2
10- 14 10 1
1
103
10 2 Time
miT
-+
Fig. 4.8 Numerical integra tion of (4.64). We recover b oth asymptotic regimes a nd the d ependence in (cyv ) of the relic density Yx.
4.2.3.2 Ho t relics
A hot relic decouples from the plasma while it is relativistic, so t hat
(4.77) where cFB = 1 (boson) or 3/4 (fermion). The interaction is no longer efficient when Yx is constant , so t hat the relic density is simply given by its value at equilibrium
Ycxo =
-
Yx(xr)
45((3)
gx
= -2 4 CFB-(-). 7f
q*
Xf
(4. 78)
204
The standard Big-Bang model
If the evolution of the Universe remains adiabatic after this transition then nXO
=
SOYoo
q.o ) CFBgx 8 - - (-) 3.9 1 q. Xf
= 776.8 ( - -
3 2.7
cm
-3
.
(4
If this particle is still relativistic today, then it behaves as the massless neutrino cussed in Section 4.2.1.6. If its mass is greater than the temperature of the cur cosmic background, i.e. m > 1.7 x 10- 4 eV, then the associated energy densi pxo = mxsoYoo, so that using (4.140) , its density is 2 ( q.o ) ( mx ) cFBgx nxoh = mxsoYoo = 0.739 3.91 10 eV q.(xd·
(4
The constraint n x h 2 < 1 implies that m < 13.5q.(xd/cFBgx eV. The precise v of this constraint depends on the value of q. at decoupling. For instance, for slightly massive neutrinos, or for particles with a small mass decouple at around 1 MeV, q. (xd = 10.75, and we get
n
xo
h2
_ mx -91.5 eV '
(4
which gives a constraint on the particle mass. The relic density is inversely proporti to q. (xd. A relic will thus be less abundant if it decouples early. For instance, if particle decouples at around 300 GeV then q.(xd cv 107, so that
n
xo
h2 =
mx 910eV'
(4
So, a particle decoupling while q.(xd » 1 will have a very small relic abundance t emperature compared to the photons of the cosmic background. These are usu called warm relics.
4.3
Primordial nucleosynthesis
For temperatures above or of the order of 100 MeV, the Universe is dominated relativistic particles in equilibrium: electrons, positrons, neutrinos and photons , the nuclear reactions can be activated. The contribution from neutrons and prot which are then non-relativistic, is negligible in the mass budget. The weak interact between neutrons, protons and leptons, Ve
+n
f---+
p
+ e,
(4
and the interactions between electrons and positrons,
(4
maintain all these particles , together with non-relativistic baryons, in thermodyna cal equilibrium.
Primordial nucleosynthesis
4.3.1
205
Main stages o f the mechanism
The synopsis of primordial nucleosynthesis (usually referred to as Big-Bang nucleosynthesis, BBN) can be decomposed into three main stages:
(1) T» 1 MeV, the components of the Universe are in thermodynamic equilibrium. Radiation dominates the Universe and 9* = 10.75. Thus, the neutron to proton ratio is approximatively given by its value at equilibrium
Q=
mn - mp = 1.293 MeV. The fraction of each atomic nucleus can then be obtained from purely thermodynamical considerations.
(2) T rv 1 - 0.7 Me V , weak interactions can no longer maintain the equilibrium. Neutrinos decouple and the nip ratio deviates from its equilibrium value. At the freeze-out temperature, Tf rv 0.8 MeV , it is of the order of 1 rv
-
5 Free neutrons then decay into protons while atomic nuclei remain in thermodynamic equilibrium. In terms of time, this stage starts roughly round t rv 1 s [see
(4.21)J. (3) T
rv 0.7 0.05 MeV, the nuclear thermodynamic equilibrium can no longer be maintained. Electrons and positrons have annihilated each other and reheated the photon bath. 9* is then 3.36 . The atomic nuclei are then formed by a series of two-body reactions. For that , deuterium needs to be synthesized (p+n --7 D +,,(), which is possible only when the radiation density is low enough so that the inverse reaction , the photodissociation of deuterium , is negligible. This determines the temperature TNuc obtained by no/n"{ rv 1, i.e.
T) 2
exp
(ED) ---
rv
1.
TNuc
Around T Nuc , (n/p)Nuc = (n/p)f exp( - t Nuc/Tn) rv 1/ 7. All neutrons are then bound in helium-4 nuclei, with abundance (in mass with respect to the total number of nucleons, nb = n + p) of the order of
2(~)NUC
1+ -n) ( P
rv
0.25,
Nuc
because the number density of helium-4 is n/2 so that it accounts for a mass 4 x n/2 = 2n in nuclear units. The increase of the Coulomb barrier for nuclei of larger atomic mass together with the drop in the temperature and density (and hence in the probability of two- body reactions) associated with the absence of stable atomic
206
The standard Big-Bang model
nuclei of m ass A = 5 and A = 8 explain why primordial nucleosynthesis does p roduce any nuclei b eyond helium in a ny significant amount . BBN roughly st when T ~ 0.05 MeV , which corresponds to t ~ 530 s [see (4.21) ]. Key parameters
In this mechanism , departure from thermodyn amic equilibrium is crucial, since nucleons would otherwise b e in t he form of iron if the equilibrium was maintai during the entire nucleosynthesis. The a bundances predicted by primordial nucleos thesis dep end m ain ly on a few pa ra meters .
- 9. : the number of relativistic degrees of freedom. 9. determines the radiation d sity a nd thus the value of the freeze-out t emperature Tf. This paramet er depe on the number , N v , of neutrino fa milies. The tempera ture Tf is roughly obtai from the Friedma nn equation
O;Tl ~ VON9.Tl. -
T n: t he neutron lifetime det ermines the ratio (n / p) Nuc a t the beginning of nu osynthesis from its freeze-out value. Tn dep ends on various fundamental const a
- 1
Tn
636 2 2 ) 5 )- 1 = 1.27r . 3 0F (1 + 39A m e = (887 ± 2 s
- T/: t he number of b ar yons p er photon T/ == nb/ n, ~ (5 - 6) x 10- 10 [see (4.14 This quantity is not directly measurable with a great accuracy as it is difficul determine which fr action of da rk matter is in baryonic form. BBN thus offe way to m easure this paramet er. - The fundamental constants: ON and OF are involved in the determination t he freeze-out temperature. The fine structure const a nt 0: det ermines Q and reaction r at es , since it enters the value of the Coulomb ba rriers.
This chapter will detail these main lines. To obtain precise results , one sho integra t e numerically the network of nuclea r reactions in an expanding sp ace-ti We refer to the original papers [1- 3, 19, 20] a nd reviews [21- 25] for more deta expla nations and to t he numerical code [26] tha t is freely availa ble [27].
4.3.2 4. 3.2.1
Initial state Thermody namic nuclear equilibrium
x1
At t emperatures high compared to 1 MeV, all nuclei are kept in thermodyna equilibrium by nuclear interactions. The density of these non-rela tivistic nuclei is t given by nA = 9A (
mAT) 3/ 2
~
e
- (mA- /LA )/T
.
(4.
As long as the reaction ra te is greater than the expa nsion rate, the chemical equilibr imposes the value of the chemical potential
(4.
Primordial nucleosynthesis
207
where I-£p and I-£n are the chemical potentials of protons and neutrons. Neglecting the mass difference Q = mn - mp c:::: 1.293 MeV in the pre-factors, the neutron and proton densities are (4.87) where m N = ~(mp + mn). Using (4.86), the exponential factor in the expression (4.85) can be expressed as exp
( - mA T- I-£A)
=
[exp (1-£'; )]Z [exp (1-£;: )] A- Zexp (mA) - T .
Expressing the first two factors as a function of the proton and neutron densities (4.87) , we get (4.88) where B A is the nucleus binding energy (4.89) The mass fraction of a nucleus with atomic number A with respect to the total nucleon mass is defined by (4.90) In terms of the parameter TJ defined by (4.91 ) we finally get that the abundances of the atomic nuclei in t hermodynamic equilibrium, using that n'Y = 2((3)T3 11f2, are given by
(4.92) (4.93) This illustrates the influence of the entropy on the a bundance of the different nuclei. If 1] ~ 1, X ~ is stable as soon as T '" B A since at this temperature the formation of the nuclei, controlled by the factor exp ( BAIT), dominat es over its destruction by photodissociation, controlled by t he factor TJA - l. Now, if TJ « I , the balance between formation and photodissociation is reached only when exp(BAIT) '" TJ A- 1 , that is at lower temperatures.
208
The standard Big-Bang model
The temperature at which the a bundance of the nucleus A becomes of order un which we denoted by TA, can thus be estimated from (4.92) . Neglecting the numeri factor F(A) and with Xp '" Xn '" 1, we have
(4. For the lightest elements , we obtain, with Tf '" 5
X
10- 10 ,
D
T
BA (MeV)
2.22
6.92
7.72
28.3
TA (MeV)
0.066
0.1
0.11
0.28
So, even at thermodynamic equilibrium, nucleosynthesis does not start before arou
0.3 MeV. This is due to the very small value of Tf. Even if nuclei are energetica
favoured , entropy acts in the opposite way and favours free neutrons and protons.
4.3.2.2
Neutron abundance
Since the equilibrium between protons and neutrons is maintained by weak interacti (4.83) implies that (4. f..in + f..iv = f..ip + f..ie· We conclude from (4.87) that
n nn P == np
= exp
(Q + T
f..ie - f..iv )
T
.
(4.
As shown by (4.26) , the electrical neutrality of the Universe implies that f..ieiT 10- 10 - 10- 8 . Neglecting f..iv , we get that
(4.
For temperatures larger than 0.1 MeV, the study of the previous section teaches that XA « 1 so that nb = nn+n p . Thus , the abundance offree neutrons in equilibri is (4. Xn ,eq = ( 1 + e Q/T ) - l
At large temperatures, we therefore have Xn = Xp = ~ and XA c::: O. For instance, T = 1 MeV, deuterium has an abundance X 2 '" 6 X 10- 12 . 4.3.3
Freeze-out of the weak interaction and neutron to proton ratio
Weak interactions maintain neutrons and protons in equilibrium. These interacti freeze-out at a temperature Tf determined by H = r w(Tf ). For temperatures T < and T > m e the reaction rate of the weak interactions is given by
rw =
77r(1 60
+ 3g A2 )C2 T5 F
'
(4.
Primordial nucleosynthesis
for the reaction p
+ e ----+ V e + n.
209
We obtain from (4.19) with g* = 10.75 that
r
(
H"::'
T 0.8 MeV
)3 '
(4.100)
so that the freeze-out temperature of the weak interactions is of the order of Tf
~
(4.101)
0.8 MeV.
Thus , the freeze-out occurs around tf = 1.15 s. Applied to neutrons, the Boltzmann (4.56) can be re-expressed in terms of Xn as (4.102) The reaction rate n ----+ p , Anp , is related to that of the reactions p ----+ n, Apn, by Anp = Apn exp( Q /T) simply because the reactions (4.83) are reversible . In equilibrium, i.e. when Xn = 0, the solution of this equation is given by (4.98). We can first note that the evolution equation (4.102) has an integral solution
Xn(t)
=
it
J(t, t')Apn(t')dt'
+ J(t, ti)Xn(ti),
(4.103)
t,
where the function J is defined by J(t, t') = exp { -
Jt~ [Apn(U) + Anp(U)] du} . Since
at high temperatures the reaction rates are very large, J(t, t i ) will be negligible if the time ti is chosen to be far enough in the past, so that Xn (td plays no role and one can choose ti = 0 (see, e.g., Ref. [20]). It follows that
Xn(t) =
lot J(t, t')Apn(t')dt' "::' Xn ,eq(t) - lot J(t, t') d~' [Xn,eq(t')] dt'.
(4.104)
Since the reaction rates are large compared to their relative variation , it follows that (4.105) Thus, the neutron abundance follows its equilibrium value until T f . When the weak interactions freeze-out , the neutron abundance is therefore
Xn(Tr) "::' Xn,eq(Tr)
=
[1
+ exp(1.293/0 .8) r 1 ~ 0.165 ~ ~,
(4.106)
that is (n/p)rf ~ 0.198 ~ 1/5. The numerical coincidence Q/Tf = 0(1) implies that Xn is neither equal to Xp nor very small. As a consequence, the quantity of helium synthesized will be of the same order of magnitude as the quantity of hydrogen. We stress here that this coincidence is unexpected since Q is determined by the strong and electromagnetic interactions , whereas Tf is fixed by the weak and gravitational interactions.
Th e standard Big-Bang model
210
To evaluate the remaining abundances after the freeze-out, Xn(t use the explicit form of the reaction rates (see Ref. [22])
A(nVe
---t
pe) = A
A(ne+
---t
pve ) = A
A(n
pe- ve ) = A
100 dqvq~qe(Ev +
100 1
»
tr) one sho
Q)(l - f e)fv,
dqeq;qv(Ee + Q)(l - fv)fe,
v Q2- m~
---t
dqeq; qv(Ee + Q)(1 - fv) (l - f e).
(4.1
One can show (see Ref. [20] for details of this computation) that the leading part these integrals comes from particles with energy large compared to the t emper at during BBN. The Fermi- Dirac distributions can thus be approximated by Maxw distributions, f e = [1 + exp(Ee/Te)] - l '::::' exp( - E e/Te). Since the Boltzmann fact are small in a diluted gas , quantum effects are negligible and the Pauli factors can neglected, 1 - f '::::' 1. The last approximation assumes m e = 0 in the computation the two first int egrals, which are dominated by Ee,v » m e, so t hat
This approximation is precise at a level of 15% as long as T > m e, when the react rates have become very weak. The last reaction r at e gives a relation between const ant A and the neutron lifetime
-1= AQ5[V1 - in (1- - -3_m 2- -m 2 _4 ) + -in4cosh-1(1)] '::::' 0.0158AQ 5 in ' 2
T
n
with in
5643
4
== m e/ Q. Finally, the total reaction rate Anp '::::'
253
- - 5 (12 Tn X
Anp '::::' 2A(nve
---t
pe) takes the form
2
+ 6x + x ),
(4 .1
with x == Q / T , if we neglect the decay of neutrons during the freeze-out. Actually, t term only becomes important when T < 0.13 MeV , i. e. for x > 10. We recover th for x « 1, the reaction rate is given by Anp ~ 12 X 253 / Tn X 5 rv G;T5, which justif a posteriori our approximation in the determination of Tf. The expression (4 .108) allows us to compute the function J(t, t') as
J( x, x')
exp[K(x) - K( x') ]'
=
with
/00 1 + e- u
K( x ) = b J
u4
x
= b
(12
+ 6u + u 2 )
du
[( -4+-3+ -1) + (4- + -1) x3
x2
X
x3
x2
e
-x] ,
Primordial nucleosynthesis
211
where the parameter b is given by b = 253J45/47f 3g.Mp /Tn Q2 = 0.252. From the expression (4.104) , we can derive the evolution of the neutron abundance
Xn( X)
=
Xn,eq(x)
+ fox[X n,eq(X,)]2 exp [K(x)
- K(x')] eX'dxl
(4.109)
Finally, we find that the neutron abundance freezes out at X n (=)
This value is reached at T 4.3.4
rv
0.25 MeV (x
= 0.150. rv
(4.110)
5), i.e. at t c::: 20 s.
Abundances of the light elements
The previous computation does not take into account the change in g* and the production of deuterium, two processes happening below 1 MeV. Light elements will be formed by a series of nuclear reactions, summarized in Fig. (4.9). No significant abundance can be produced before the deuterium is formed since the densities are too low to allow for more than two-body reactions, such as 2n + 2p -> 4 He, to play any important role. Furthermore, the production of deuterium from two protons (p + p --+ D + e+ + v e ) involves the weak interaction, and is thus negligible. Light isotopes cannot be synthesized before the temperature drops below 0.1 MeV. This is due to the fact that deuterium has a very weak binding energy and is easily photodissociated. Its synthesis only starts when the radiation density is low enough. Moreover, the reaction chain cannot continue before the deuterium is synthesized, this is often called the 'deuterium bottleneck'. 4.3.4.1
Deuterium
The synthesis of deuterium is thus a key stage of primordial nucleosynthesis. It must be produced in sufficient quantity so that the heavy elements can then be synthesized. As already mentioned, the synthesis of light isotopes only starts at around T = Tnue = 0.066MeV. At this temperature, the electrons and positrons are no longer relativistic and g* = 3.36, so that (4.21) implies that t nue = 303s. Since n(Tnue) = n(Tr) exp( -tnue/Tn) and p(Tnue) = p(Tr) + n(Tr)[I- exp( - tnue/Tn)], which imply that (n /p)(Tnue ) rv 0.133, we have (4.111) The reaction rate per neutron of the deuterium-forming reaction (n
+ p -> D + ,)
is (4.112) It can be shown that this reaction freezes out well after the end of nucleosynthesis so that the deuterium abundance (go = 3) is given by its equilibrium value (4.85)
(4.113)
212
The standard Big-Bang model
l.p H n 2. p (n, r)d 3.d(p, r)JHe
4. d (d, n)JHe 5. d (d, p)t 6. d (d, nlHe 7.( (ex, rYLi 8. JI-Ie (n, p)t 9. J I-Ie (d, pj4He 10. J]-[e (ex, rYBe 11. 7Li (p, a/He 12. 7Be (n, p YLi
Fig. 4.9 Simplified network of nuclear reactions involved in the synthesis of light nu during BBN.
4.3.4.2
Helium- 4
Since the binding energy of helium-4 is larger than that of deuterium, helium favoured by the Boltzmann factor exp(B /T) in comparison to deuterium. As c be seen from Fig. 4.10 , the growth in the deuterium abundance is followed by th of helium-4. Heavier nuclei are not synthesized in significant quantities mainly two reasons: (1) the absence of stable nuclei with A = 5 or A = 8, so that production chains for the reactions n + 4He, p+4 He, 4He+ 4He are blocked (the trip a process, that produces carbon-12 in stars, cannot be efficient in the primord Universe); (2) the important Coulomb barrier for reactions as 3H+ 4He--7 7Li+')' a 3He+ 4He--7 7Be+,),. At T nuc , all free neutrons are bound in helium-4 nuclei. Since helium-4 has t neutrons, its primordial abundance is therefore
(4.1
A more precise estimate [20] of Tnuc gives Tnuc = 0.086 MeV, which implies t nuc 178.5 s and hence Yp rv 0.238. This analytical estimate gives the correct orders of magnitude and predictions w a precision of t he order of 1% as compared with what can be obtained from numeri integrations. Let us recall that our estimate assumed f1v = 0 to determine the ratio (4.97) this is not the case, the constant parameter ~ == f1v/T can be introduced. When t parameter is small, its main effect is to modify the reaction rate n + Ve --7 P + e,
Primordial nucleosynthesis
213
p
n
10- 5
10- 15
Tn =
10- 20
887 s
Nv= 3 Q bOh 2 =
10-2 5 LU~~L-L-~
1
0.01
__L-~~LLLL~L-L-~__L-__~ 0.1 0.01 T (MeV)
Fig. 4.10 Evolution of the abundances of light elements during primordial nucleosynthesis. Dashed lines represent abundances in thermodynamic equilibrium. that Anp = A pn exp (Q / T + ~ ) . This changes the freeze-out neutron abundance (4.110) to Xn(OO ) = Xn (oo , ~ = O)exp (- O . The results from numerical simulations can be summarized , in the interval 3 x 10- 10 < TJ < 10- 9 , by
Yp
= 0. 245+ 0.01 4(Nv -
3)+ 0.0002 (Tn- 887 s)+ 0.009 ln (
TJ
5 x 10-
10)
- 0.25~,
(4. 115)
which gives t he sensitivity t o t he most critical parameters (see Ref. [28] for a more detailed study).
4.3.4. 3
Oth er elem en ts
At the end of nucleosynt hesis , at around t "-' 1000 s, traces of deut erium (XD "-' 10- 4) , helium-3 (X 3He "-' 10- 4) and lit hium (X 7Li "-' 10 - 9 ) are still present. T he abundance of helium-3 includes some of the trit ium that disappears by (3 decay, and similarly, the one of lithium-7 includes beryllium-7. The production of elements beyond A = 7, such as boron, is negligible. Unlike helium-4, t he abundances of t hese nuclei are very sensitive to t he value of 'f/ (see F ig. 4.11 ) . T he abundances of deut erium and helium-3 decrease with TJ . The lithium-7 can be produced by two channels: (1) 4He+ 3 H--> 7Li +,), that competes with 7Li+p-->4He+ 4He for TJ < 3 X 10 - 10 and (2) 4He+ 3 He--> 7Be+)', followed by a (3 decay for'f/ > 3 X 10- 10 . The t ransit ion between both channels explains t he shap e of the lithium- 7 abundance curve (F ig. 4.11 ). We can show [23] that in the band 10- 10 < TJ < 10 - 9 , the primordial abundances of the light nuclei are given by
214
Th e standard Big-Bang model
~) = ' 36 (H
X
1O- 5 ±o .06 71 -
(4.11 6)
He ) = 1 2 H'
X
1O- 5 ±o. 06 71 -
(4.117)
1.6 '/ 5 .5 ,
p
-3 (
0 . 63 '/5.5 ,
P
L 1') H
= 1 ' 2 X 1O- 11 ±o.2
_7
(
(71 - 2 .38 '/ 5 .5
+ 21 ' 7712 .38 ) '/5.5
,
(4.118)
P
wit h rJ5 5 = rJ / 5.5
X
10- 10 10- 2
Q boh 2
0.26 0.25
4He
~
ci" 0
0.24
'0 ()
'"
J::
'" ::;;'"'"
0.23 0.22
:t: 0
10-4
£
--:t: OJ
'"
10-5
10- 6
10- 9 :t:
---
r~
Fig. 4.11 Helium-4, deuterium , helium-3 and lit hium-7 abundances as a fun ct ion of t he baryon to photon rat io, 'rj , or equivalent ly of the baryon density, n bo h2. The hatched areas correspond t o the spectroscopic observations discussed in the text: Refs. [29-31] for helium-4, Ref. [32] for deuterium, and Ref. [33] for lit hium-7. T he vert ical band represents t he constraint on 'rj obt a ined from t he W MAP satellit e [34] . From Ref. [35].
Primordial nucleosynthesis
4.3.5
215
Observational status
The comparison between the observed and predicted abundances is difficult , mainly because the primordial abundances can be modified by various nuclear processes during the evolution of the Universe. Moreover, these modifications are different for each nucleus. For instance, the abundance of helium-4, which is very stable, increases due to the stellar production, whereas the deuterium is much less stable and can only be destroyed but never be synthesized in stars. Helium-3 and lithium-7 have more complex evolutions as they can be both synthesized and destroyed. Astronomers therefore try to measure the abundances in environments as primordial as possible.
4.3.5.1 Observations We briefly review the status of the light-nuclei abundance observations. For a more detailed review on these topics, see Ref. [24].
- Helium-4: is detected through its emission lines in various astrophysical environments (planetary atmospheres, young stars, planetary nebulae), which could be galactic, extragalactic or intergalactic. In order to eliminate the production of stellar helium, its abundance is correlated to that of other elements such as nitrogen and oxygen. The best system to carry out these studies seems to be the HII regions of compact blue galaxies . Recent measurements from various groups give Yp = 0.2443±O .0015 [29], Yp = 0.2391±O.0020 [30] and Yp = 0.249±O.004 [36]. The recent analysis of 82 HII regions performed in 76 compact galaxies has given [31] Yp = 0.2421 ± 0.0021. - Deuterium : is easily destroyed in stars as soon as the t emperature increases above 6 x 105 K. Any measurement of D / H is therefore a lower bound, giving an upper bound on TJ. Its abundance could have been reduced by a factor estimated from 2 to 10. Its spectral lines have actua lly never been detected in any star, which implies that D / H < 10- 6 in stellar atmospheres . They can be detected in giant planets, in local interstellar environments and in the absorption spectrum of low-metallicity clouds on the line of sight of quasars . The finest measurements in absorption systems lead [32] to D / H= 2. 78:+:g : ~~ x 10- 5 . Conservatively, local measurements within the interstellar regions give the lower bound [37] D / H= (1.5 ±0.1) x 10- 5 , while the absence of any detection in systems with high redshift fixes the upper bound [32] D/H< 6.7 x 10- 5 . - Lithium-7: the system that seems to enable the best measurements of lithium7 is the stellar halo of our Galaxy. This halo is mainly composed of stars from populations I and II and thus has a very low metallicity. Extrapolating to zero metallicity, we get a primordial abundance Li/H=1.23 :+: g~~ x 10- 10 [33]. - Helium-3: has only been observed in HII regions of the galactic disk (thus in a relatively high metallicity environment). In these regions, its abundance is 3He/H = (1 - 2) x 10- 5 [38] although it is difficult to track back to its primordial value since this isotope has a complex stellar history and can be both produced and destroyed during the galactic evolution. Helium-3 is thus unlikely to provide a precise cosmological test today. At the moment, it can be mainly
216
The standard Big-Bang model
used to constrain the astrophysical mechanisms in which it is produced or stroyed. However , by comparing the primordial abundance of 3He with the abundance, we can conclude that it has not evolved much in 14 billion years
4.3.5.2
Constraints on cosmological models
Primordial nucleosynthesis allows us first to measure t he ratio T/. If only spectrosc data are considered and assuming N " = 3, then T/
= (5.15 ± 1.75) x 10- 1°,
(4.1
which corresponds to D bO h 2 = 0.01 8 ±0.006. Current da t a allow us to impose constraints on parameters such as N ", the num of neut rino families with mass lower than 1 MeV , and ~ , but also to test the valu some fundamental constants such as the gravitational const ant , or the fine struc constant (see Ref. [39]). Primordial nucleosynthesis is thus a test not only of the Bang model, but also of nuclear physics and of general astrophysics. The agreem between theory and observations makes it one of the pillars of the Big-Bang mod
4.3.5. 3 A recent uisis?
For a long time, primordial nucleosynthesis and the spectroscopic determinatio light-element abundances were t he only reliable ways to quantify the baryon con of the Universe. Recently, D bO h 2 has been determined with an extreme precision t hanks to the analysis of t he cosmic microwave background anisotropies (see Chapte D bOh 2 = 0.0224 ± 0.0009 ,
(4.1
°.
which corresponds to T/ = (6.14±0.25) X 10- 1 Using t his value for the baryon den we can then deduce the expected abundances [35]: Yp
D/ H
0. 2479 ± 00004 2.60:t: g~~ x 10- 5
7Li/H 4. 15:t:g!; 1O -
3He/H 10
(104 ± 0.04) x 10-
6Li/H 5
(1.2 ± 0.08) x 10-
This has opened up a reanalysis of the nuclear reaction rates involved during cleosynthesis (Fig. 4.9 ). The situation is summarized in Figs. 4.11.
4.4
The cosmic microwave background radiation
As long as the temperature of t he Universe remains large compared to the hydro ionization energy, matter is ionized and photons are then strongly coupled to elect through Compton scattering. At lower t emperatures, the formation of neutral at om thermodynamically favoured for matter. Compton scattering is then no longer effic and radiation decouples from matter to give rise to a fossil radiation: the cos microwave background (CMB). The temperature of t his cosmic background was predicted by Alpher and H man [3] in 1948 , following the arguments developed in the same year by Gamow This prediction was confirmed in 1964 by Penzias and Wilson [40] who discov
The cosmic microwave background radiation
217
an unexplained noise signal, identical in every direction. This signal was immediat ely interpreted by Dicke et al. [41] as being the cosmic microwave background radiation at a temperature of 3.5 ± 1 K. In this section, we first describe the recombination and decoupling mechanisms before describing the properties of the isotropic component of the cosmic background as revealed by the COBE satellite in 1987; in particular , we describe the constraints on the possible distortions of the Planck spectrum. For more details on this subj ect, we recommend the reviews [42- 44]. 4.4.1
Recombination
4.4.1 .1 Prediction of the existen ce of the cosmic microwave background To st art with, let us briefly review the analysis of Alpher and Herman who predicted the temperature of t he cosmic background in a surprising way, based uniquely on an estimate of the helium abundance and on t he baryon density. To explain a helium-4 abundance of approximatively 25 %, primordial nucleosynthesis must have happened at around 109 K. At this time , the baryon density is of the order of n b rv 10 18 cm - 3 . Today, it is of order nbO rv 10- 7 cm - 3 , from which we deduce that , at the time of nucleosynthesis, the redshift is 1+ Since T scales as (1
+ z) , the
=
ZBBN
(
-n b )
1/3
n bO
rv
8 2 x 10 .
current t emperature of the photon bath is thus around
ToCM B
=
T BBN
1 + ZBBN
rv
5 K.
4.4.1. 2 R ecombination and decoupling As long as the photoionization reaction (4.121) is able to maintain the equilibrium , the relative abundances of the electrons, protons and hydrogen will be fixed by (4 .60). Since the Universe is electrically neutral, one has ne = np . For simplicity, we introduce the ionization fraction
x e -_
ne n p + nH
,
(4.122 )
where the denominator represents the t ot al number of hydrogen nuclei, n b = np + nH [ne = np = X en b, nH = (1 - Xe )n b]. In this particular case, (4.60) is known as the Saha equation. Once t he equilibrium quantities are computed , it implies that (4 .123) where Er = m e + mp - mH = 13.6 eV is the hydrogen ionization energy. The photon temperature is T = 2.725(1 + z) K = 2.348 x 10- 4 (1 + z) eV and the baryon density nb = 7]n"o(1 + z)3 cm - 3 so that the previous equation becomes
218
The standard Big-Bang model
log
X;- ) (1- Xe
2 (1 + z)3/2] - 25163 = 20.98 - log [ DbOh -. 1+Z
(4.12
Notice first that when T """' E r, the right-hand side of (4.123) is of order 10 15 , that X e(T """' Er) """' 1. Recombination only happens for T « E r· To see that , w deduce from (4.124) that if recombination is defined by Xe = 0.5 , 0.1 , 0.01 then Zrec 1376,1258, 1138, respectively. Thus, X e varies abruptly between Z = 1400 and Z = 12 and the recombination can be estimated to occur at a temperature between 3100 an 3800 K. Note that (4.123) implies that
3
[2
X ; = -In (meT ) -Er - In Tj-((3)T I n -1 - Xe 2 271" T 71"2
3] ,
from which we deduce that
(me) 271"T
-Er = -3 In T
2
[
2 )X;- ] . In Tj - In -((3 71"2 1 - Xe
This gives a rough estimate of the recombination temperature since the last term ca be neglected. It gives T """' 3500 K. The electron density varies rapidly at the time of recombination, t hus the reactio rate fT = neaT , with aT the Thompson scattering cross-section, drops off rapidly that the reaction (4.121) freezes out and the photons decouple soon after. An estima of the decoupling time , td ec, can be obtained by the requirement fT(td ec) = H(t dec The reaction rate fT takes the form
(4.12 Only matter and radiation contribute significantly to the Hubble expansion rate that
H 2 = Dmo H
5(1 +z )3 (1 +~). 1 + Ze q
(4.12
Thus, we conclude that the decoupling redshift is solution of (1
+
)3/2 = 280.01
Z
dec
X e(OO )
(n 2)-1(n ~ 0.02
~2 )1 / 2 0.15
1 + 1 + Zdec 1 + Zeq
(4.12
Xe( 00) is the residual electron fraction, once recombination has ended. We can estima (see the next section) that Xe(oo) """' 7 X 10- 3 . 4.4. 1. 3
Recombination dy namics
To describe the recombination, and to determine Xe( oo ), one should solve the Bolt mann equation. A complete treatment , taking the hydrogen and helium contribution into account , is described in Refs. [45,46]. Actua lly, the helium recombination ends lon before that of hydrogen begins. For simplicity, we will neglect the helium contributio
The cosmic microwave background radiation
219
Consider the Boltzmann equation (4.58) for the electron density. Using ne = nbXe and the fact that nba3 is constant , the term ne + 3Hne can be written as nbXe. Then , using that nH = nb(1- X e) and l1e l1 p /l1H = (m eT /27r)3 / 2 exp( -Er/T) , it reduces to (4.128)
with
~ = (~~) 3/2 cyCZ) exp ( -
i ),
(4.129)
These parameters are given in terms of the temperature by ¢2(T ) ::::' 0.448 In
(i) .
(4.130)
The expression for ¢2 is a good approximation , better than 1% for T < 6000 K. The coefficient Cr should in principle be equal to unity. This description is actually too naive [45] . When an electron is captured by a proton, a hydrogen atom is formed in an excited state. This a tom returns to a lower excited state by emitting a series of resonant Lyman- a photons. To reach the ground state, one of the photons must have at least the transition energy 2s ----; Is. These photons have an important cross-section and thus a high probability to be absorbed by surrounding hydrogen atoms, exciting them to a st ate from which they can easily be ionized. Thus, this process does not lead to any net change in the hydrogen density. It has been shown by Peebles [45] that only transitions from the level n = 2 can lead to significant changes in the hydrogen abundance. The fine structure of this level therefore plays a central role during recombination. Two leading effects should be taken into account . - The transition 2s ----; I s is forbidden to first order. From conservation of energy and angular momentum , two electrons should be emitted during this t ransition. This process is slow and its reaction rate is A 2s = 8.227 S- I. This is the leading mechanism and the reaction r at e of recombination is very different from the one predicted by the Saha equation. - The transition 2p ----; I s produces some Lyman-a photons. The frequencies of these photons are redshifted by the cosmic expansion. Thus, their energy gets lower and they are no longer resonant , this lowers the probability to reionize a hydrogen atom. These two effects are equivalent to fixing the value of the reduction coefficient , Cr, to
C _ r -
Aa.
Aa. + A 2s + A 2s + ~(2 )
(4.131)
with
~(2) = ~ exp (~) ,
8H
Aa.=~, A a. n
ls
Aa.
4 3E[
= --
=
1.216
X
10- 5 cm -
l
For T < 10 5 K t he hydrogen abundance in the state I s can be replaced by nls (1- Xe)nb. The ~(2) coefficient describes the ionization , whereas An + A2s is the total
220
The standard Big-Bang model
reaction rate of both dominant processes . A more complete study, with details o the processes is presented in Ref. [46]. Table 4.6 compares the approximation from Saha equation with the integration of (4.128) for a fiat Universe with DbOh2 = 1.
Table 4.6 Comparison between the approximation from the Saha equation and the in gration of (4. 128) for a flat Universe with DbOh2 = 1. z
1667
1480
10- 1
Sah a
9.43 x
(4.128)
9.2 x 10-
Ref. [46]
9.14 x 10-
1 1
1296
1111
3.8
X
10 -
1
2.6
X
10 -
2
4.0
X
10-
1
7.2
X
10-
2
4 .0
X
10 -
1
7.15
X
10-
2
8.8
X
9.8
X
9.68
X
926
10 -
4
5.28
10 -
3
9.2
10-
3
A detailed analysis of (4.128) shows [46] that for 1500 > _dX_e
dz
= - 61 72
.
D bO h
9.01
Z
741
X 10- 6 X X
10 10-
4
2.4 l. 23
4
l.2
X X X
10
10
10
> 800, it reduces t
2
J( z) [ 1 + 2 26 x 104 ze- 14486/(z827) ] - 1 X2 (Dmo h2 ) l / 2 · e'
(4
with J( z ) = (1 + 1.45z/zeq )-1/2 , if we only keep the dominant contribution. A approximation of the solution of this equation is
(4. as long Dba and Dmo are both between 0.05 and 1.
4.4.1.4
Last scattering surface
The optica l depth can be computed using the ionization fr action as a function o redshift around decoupling, T
=
j .n X e
(7
e T
dX::::::'
- 1
0.42e1 4 .486(1 -827)
(
Z )14.25 1000 '
--
(4.
where the Dmo dependence disappears in an obvious way. More interestingly, also independent of Dba. The optical width varies rapidly around z '" 1000 so the visibility function g(z) = exp( - T)dT/dz, which determines the probability a photon to be scattered between z and z + dz, is a highly peaked function Fig. 4.12). Its maximum defines the decoupling time, Zdec ::::::' 1057.3. This red defines t he time at which the CMB photons last scatter; the Universe then rap becomes transparent and these photons can propagate freely in all directions. Universe is then embedded in this homogeneous and isotropic radiation. The ins when the photons last interact is a space-like hypersurface , called t he last scatte surface. Some of these relic photons can be observed. They come from the intersec of the last scattering surface with our past light cone. It is thus a 2-dimens sphere, centred around us , with comoving radius X(Zdec). This sphere is not infin thin. If we define its width as the zone where the visibility function is halved get .0. z dec = 185 .7. As can be seen in (4 .133) and (4.134), the value of Zdec is
The cosmic microwave background radiation
221
very sensitive to nboh2 and n mO h 2 [it is only independent of these parameters in the approximation of (4.132) ]. From the analysis [34] of the WMAP satellite, we can conclude that (4.135) Zdec = 1089 ± 1, .0.Zdec = 195 ±2, which is in good agreement with our analytical approximation. p
,, 5
-tl~
_ QOh2 = 1
--·QOh 2 =0 .1 ...... Q Oh 2 = 0.01
4
\'
Equality
a
Today
0
3
,...;
,£
.S
0 2 t:
2 0
~
:0 '<;j
\.
/
>=
.............. 0 0
500
1000 Redshift
Fig. 4.12 (left): During the expansion of the Universe, the density, p, decreases with time. Slightly after recombination, the free-electron d ensity rapidly decreases and the Universe becomes transparent. (right): The v isibility function is highly p eaked , which allows us to define the last scattering surface.
4.4.2
Properties of the cosmic microwave background
In Big-Bang models, the energy inj ected into radiation between the electron- positron annihilation at a redshift of order Z rv 109 and Z = 10 - 20 is very small. The fossil radiation emitted around Z = 10 3 must hence have a spectrum very close to a Planck distribution. We had to wait until the COBE satellite [42] to check this prediction with accuracy. In this section, we detail the observed properties of this radiation. 4.4.2.1
Observed temperature and spectrum
The temperature of the cosmic microwave background, defined as the average of the temperature of the whole sky
To == T-yo =
~ 47T
jT(e , 'P) sineded'P,
has been measured with precision by the FIRAS experiment on board the COBE satellite [47]
222
The standard Big-Bang m odel
= 2.725 ±0.001
To
(4 .
K
at 20". We define the dimensionless parameter
e 2 .7 -= ~K 2.725 .
(4.1
T his temperature corresponds to an energy of E-yo = 2.345 X 10- 4 eY. The observed spectrum is compatible with a blackbody spectrum
I(v) ex
v3
e"
IT
- 1
(4.1
.
F igs. 4.13 and 4. 14 compare the measurements of the spectrum of the cosmic microw background to that of a blackbody at 2.725 K. The fact that this spectrum is so close to a blackbody proves that the fossil radia could have been thermalized , mainly thanks to interactions wit h electrons. Howe for redshifts lower than z ~ 10 6 , the fossil radiation do not have time to be thermali Any energy injection at lower redshift would induce distortions in the Planck spect and can thus be constrained from these observations (see Fig. 4.13). Wavelength (em)
10- 17
~
l.0
10
Freq uency (GHz)
0.1
l.0
10- 18
0.1
I
to
::r:: ~
...
I
10-19
N
I
E 10-20
t
~ ~
u
2.726 K
~
• • o
10- 21
o
•
Cl
U.
Blackbody FIRAS DMR UBC LBL-Italy Prin ce ton Cyanogen
"'1
"-
0.01
~
0.001
0.0001
10- 22 1
10
100
Frequency (GHz)
1000
10 5
10 7
(J + z)
Fig. 4.13 (left): Comparison between the measurements of the cosmic microwave backgro spectrum to a blackb ody at 2.725 K. (right ): Upp er bound on the inj ect ed energy compa wit h the constraints from FIRAS on t he distortions of t he cosmic background spectrum . F Ref. [48] .
CMB photons have a Planck distribution at a temperature To. From the express (4. 11 ) and (4.12), we can conclude (9"1 = 2) that
(4.
This implies that P-yo = 0.26032 eY . cm - 3 = 4.96 x 10- 13 ey4. We then deduce f (4.29) that the current entropy density is
The cosmic microwave background radiation
30
300 3.5
0.03
I.
400
I
~
g3.0
1 300
\
"-
:>,
~
.3
-; 200
~
~
\
0"------'----'----'---""'o 5 10 15 20
/"
\
/"
/"
\
0-
100
/"
\
2.5
/"
\
E ~ 2.0
'"
~
Wavelength (em) 3 0.3
223
/ -Planck / ... Compton distorsion , y / - - Che mical potential, J1 _ . Free-free, YU·
\ \ \
l.5 0.1
1
Frequency (cm-1 )
100 10 Frequency (GHz)
1000
Fig. 4.14 (left): Spectrum of the fossil radiation measured by the C OBE satellite, the error bars are multiplied by 200. (right): Different kinds of distortions of the blackbody spectrum . None of these distortions has b een observed. From Ref. [48]. q* ) 83 So = 2889.2 ( 3.91 - 2.7 cm -
3
q* ) = 7.039 ( 3.91 n, o,
(4.140)
and that the photon density paramet er is (4.141)
Using n bO = DbOPcrit / m N "-' 0.11(DbO / 0.02) m - 3 we deduce that
7]
= 5.514
X
10 -
10
h2) 8 -
D (~ 0.02
3 . 2.7
(4.142)
4.4.2.2 Distortions of the Planck spectrum Three kinds of distortions can affect the sp ectrum of the cosmic background. 1. At redshifts lower than
Z de c , the cosmic plasma can be heated by the injection of energy with t emperatures higher than the one of the CMB photons. The Compton scattering of hot electrons then t ends to shift the spectrum towards higher energies , while m aintaining the number of photons constant, so that the low frequencies (Rayleigh- J eans part) get deplet ed in favour of the higher frequencies (Wien part ). This distortion is called Compton distortion and is characterized by a unique paramet er
(4.143)
which represents the integrat ed Compton optical depth. 2. For redshifts between Z "-' 2 X 10 6 and Z "-' 10 5 , the cosmic microwave background can be thermalized through Compton scattering , but the thermalization of t he energy between the plasma and the photons happens too late for the spectrum to
224
The standard Big- Bang model
have time to relax into a Planck spectrum. The spectrum is then deformed Bose- Einstein spectrum with non-vanishing chemical potential 1I
3
J(lI) ex e (V-IJ, )IT
-
(4.1
1.
This difference can be characterized by the dimensionless parameter p, T"{
(4.1
p, = - .
3. Finally, there can be some so-called free-free distortions , mainly coming from
emission of photons by electrons that scatter on charged particles. The effec the cosmic background is then proportional to the parameter Yff defined as
(4.1
2)3_-----,::-n-"~==
with ri,
where 9
rv
==
8ftg ( e 3v6 47fco
meT~JmeTe '
2 is called the Gaunt factor.
The current constraints on these three kinds of distortions [48] are mainly obtai from the analysis of the FIRAS spectrum, and give
iyi < l.9
x 10- 5 ,
(4.1
These strongly constrain any processes capable of inj ecting energy into the Univ from z rv 106 onwards, and even from primordial nucleosynthesis (see Fig. 4.13). 4.4.2.3
Monopole and dipole
The observations by FIRAS [49] depend on the position and frequency. They have b expanded in spherical harmonics into monopole, dipole and residual anisotropies. Even though each point in the sky has a Planck spectrum, the spectrum temp ture is slightly higher in one half of the sky and lower in the other half. This dip distortion was measured by FIRAS
bT(e) = (3.346 ±0 .017) x 10- 3 K cos e,
and is compatible with a local kinematic origin associated with the motion of the S System and of our Galaxy with respect to the CMB rest frame. The dipole would t arise from a Doppler effect , i. e. from a t emperature fluctuation of the form -T =
To
e
[ 1 - v 2 / c2 (1 - V - cos e)] -
J
c
1
rv
v2
1 + -Vcos e + -1 - 2 cos 2e c 2c
+0
(v3- ) , c3
where is the angle between the line of sight and the dipole, i.e. the directio motion. We thus conclude that the centre of gravity of the Solar System moves in
The cosmic microwave background radiation
225
direction (l, b) = (263. 85° ± 0.1O°, 48.25° ± 0.04°) in galactic coordinat es, with velocity v '" 368 ± 2 km . S - l. The corresponding velocity of our Galaxy and the local group is then v rv 627 ± 3 km· S- l rv 0.0022c in the direction (l , b) = (276° ± 3°, 30° ±3°). From this dipole, we can determine the absolute motion of the satellite with respect to the CMB rest frame, i. e. the reference frame for which no dipole is present . Note that this dipole could have a cosmological origin (its effect would not be distinguishable from that of a kinematic dipole) . However , since the value of the quadrupole is around 1% of the dipole, the kinematic interpret ation is commonly accepted. 4.4.2.4
R esidual fluctu ations
Residual fluctuations
oT(B , cp) = T(B , cp) - To ,
(4.148)
with charact eristic amplitude of ((oT/ T) 2) 1/2 rv 1.1 X 10- 5 , have also been detected by the COBE satellite. To reach these fluctuations, various foreground effects had to be eliminat ed first (dust , galactic emission , synchrotron , free-free, ... ). The spectrum of these emissions is very different from a Planck spectrum . Fig. 4.15 represents their relative amplitudes as a function of the frequency. These contributions can thus be substracted (more or less easily). After having subtracted these effects, some t emperature anisotropies remain , with relative amplitude rv 10- 5 , which correspond to t emperature fluctuations of the order of 30 ~K. These anisotropies correspond to anisotropies in the cosmic microwave background at the time of recombination and redshifted by the cosmic expansion. Chapter 6 is dedicated to the study of these temperature anisotropies. Wavele ngth (cm) 3
30
0.3
2.7 K CMB
Synchrotron Free-free I I
CMB /',T
0.01 1
10 Frequency (GHz)
100
Fig. 4 .15 Different foreground effects spoiling the cosmic microwave background. None of these effects have a Planck spectrum . From Ref. [48] .
226
Th e standard Big-Bang model
To conclude , t he cosmic microwave background confirms that t he Universe eme from a state in t hermodynamic equilibrium and that its matter was ionized in the Its t emper ature allows us to determine the photons density. T his contribution re sents around 93 % of the electromagnetic radiation , integrated over all wavelengths can thus deduce the radiation density, the best measured cosmological parameter. cosmic microwave background can be used to strongly constrain any energy inj ec since z rv 106 . 4.4.3
Another proof of the expansion of the Universe
The evolution of the t emperature of the cosmic microwave background gives an a tional proof of the expansion of the Universe. Indeed , it must scale as the inver the scale factor, so t hat T( z ) = 2.725( 1 + z ) K. (4.
These photons can excite the fine structure levels of some atoms . The t empera of the CMB photons at a redshift of z = 2.33771 has recently been measured from t he observation of absorbing clouds. The measurement is based on the excita of the two first hyperfine levels of neutral carbon. T hese excitations are induce collisions and by the tail of the CMB photon distribution. After subtracting t he contribution, which can be evaluated from other transitions, one can determine distribution temperature of the relic photons. It leads to the constraint
6.0 K < T < 14 K ,
(4.
while the theoretical prediction is T = 9.1 K. F ig. 4.16 summarizes the variou tempts to check the relation (4 .149) and proves that it seems to be satisfied . provides an additional proof of t he expansion of the Universe.
4.5 4.5.1
Status of the Big-Bang model A good standard model.. .
The Friedman-Lemaitre solutions derived in Chapter 3 describe homogeneous and tropic expanding Universes. This expansion is now established by various observa (see Section 4.1.1 ) and t he Hubble constant has been measured with a good preci Questioning the reality of this expansion would require the reinterpretation of var observations, and this has not yet been achieved by any alternative model (see example, Ref. [51] for an original and instructive attempt) . The expansion of Universe and the redshift represent t he first observational pillar of t he Big-Bang mo The cosmological principle, i. e. the homogeneity and isotropic hypothesis, imp t he symmetries of cosmological space-times and is at the origin of their mathema simplicity. This hypothesis is supported by the observations of the cosmic microw background that prove t he high level of isotropy of the Universe around us. The Bang model does not address the origin of this homogeneity and isotropy. It is an important issue to fur ther test the Copernican principle observationally. The study of nuclear and electromagnetic processes in cosmological space-t has led to two main conclusions: (1) the exist ence and abundances of light nucle
Status of the Big-Bang model
30
0 GE et al . (1997)
• o • ... --
g ~
227
Lu et al. (1997) Roth and Baue r (1999) Songaila et al. (1994) Srianand anel Petitjean (2000) T(z) = 2.72 6 X (1+z )
20
3
......'"
Q)
0.
E
~
co ::E u 10
Reelshift
Fig. 4.16 Measurements of the temperature of the cosmic microwave background at different redshifts. The dashed line represents the prediction from the Big-Bang model. From Ref. [50]. naturally explained to within ten orders of magnitude, Section 4.3; and (2) there must exist a cosmic microwave background of photons of cosmological origin, Section 4.4. Both these predictions have been confirmed by observations , providing two other observational pillars of the model. As already seen, out-of-equilibrium mechanisms playa central role in these predictions. These conclusions rely on sensible, even conservative, hypotheses, namely the validity of general relativity and nuclear and electromagnetic physics at temperatures lower than 100 MeV, providing a robust description of the history of the Universe from 10- 4 s after the Big Bang. This Big-Bang model is therefore an excellent standard model for the description of the Universe. We will use it as a starting point in the rest of t his book and it is nowadays adopted unanimously. A number of conclusions are generic to this model: - the Universe was dominated by radiation in the past. Thus , a radiation-dominated era and matter-dominated era succeed each other. Then , there can possibly be an era dominated by a cosmological constant or by the curvature of the Universe. - the Universe has a thermal history. During its expansion , the Universe cools down. The interactions that are efficient at high temperatures tend to decouple as soon as their reaction rate becomes smaller than the expansion rate. - the Universe emerges from a state where matter at high temperature is ionized and is in thermodynamic equilibrium. The dynamics of the Big-Bang models depend on their matter content. As seen in
228
The standard Big-Bang model
Chapter 3, all background observables depend on the function E(z) only. A detai assessment of this content can be made [5 2]. As seen in this chapter, Big-Bang mo els are characterized by 5 independent parameters (h,OK,Om,Ob,Or,OA) with constraint that the sum over the Os should be 1. The values of these parameters, deduced from observations, are summarized in Table 4.7. From these parameters few quantities can be deduced that characterize the thermal history of the Univer They are summarized in Table 4.8.
Table 4.7 Cosmological pa ra meters of Big-Bang models and constraints discussed in t chapter. 0.72 ± 0.05
Section 4.1.1
curvature parameter
- 0.0049 ± 0.0l3
Chapter 6
dark-ma tter density
0.113 ± 0.009
Chapter7
baryon density (BBN)
0.018 ± 0.06
(4.119)
Hubble const a nt
baryon density (CMB)
0.0224 ± 0.009
fl -yoh2
photon density
2.4696 x 1O - 5e~.7
(4.141 )
flvO h2
(m assless) neutrino density (3 famili es)
(4.47)
fl ro h 2
radiation density
1.68 x 1O- 5et7 4. 148 x 1O - 5e 7
cosmological const a nt
0 .72 ± 0 .03
Section 4.1. 2
dark- energy equation of state
w < - 0.6
(4 .4)
fl AO w
t
Table 4.8 Characteristic quantities of the thermal history of the Universe . 'r/
baryon/ photon r atio (CMB)
(6.14 ±0.25) x 10-
10
(4.120)
baryon/photon ratio (BBN)
(5.15 ± 1.75) x 10-
10
(4.119, 4.1 42)
To
present CMB temperature
Zeq
red shift at equa lity
T eq
temperature at equa lity
Zd ec
Ll. zdec Tdec
(4.137) (4.8)
5.65e2.~(flmo h2) eV
(4.8) Section 4.4
red shift at decoup ling
1089 ± 1
width
195 ± 2
Section 4.4
tempera ture a t decoupling
2970 K
Section 4.4
~ 10 10
Section 4.3
ZB BN
redshift at nucleosynthesis
TBBN
temperature at nucl eosynthesis
4.5.2
2.725 ± 0.001 K 3612(fl m oh 2 /0.1 5) e 2.~
~1
MeV
Section 4.3
... but an incomplete model
The Big-Bang model offers a convincing picture of the evolution of the Universe. higher temperatures the model is less reliable since the laws of physics have to extrapolated to regimes where they have not been tested otherwise. For example, still do not have any model explaining the origin of baryons. But it is sensible to ho that we will be able to extrapolate, at least qualitatively, up to the grand unificat scale, i.e. to around 10 16 GeV. Beyond this value, probably at energies close to t
S tatus of the Big-Bang m odel
229
Planck scale at 10 19 GeV , we expect t o have t o consider quantum gravity effect s. We also get into more and more speculative zones but that gives us the possibility t o test the phenomenology of theoretical models at these energies. Putting this aside, t he Big-B ang model has a list of problems inherent to its formulation. 4.5.2. 1 Flatness problem The equation for the evolution of the curvature (3.53) is of the form dD K
dina =(3w+ 1)(1 - DK) DK '
(4.151)
if we neglect the recent contribution from t he cosmological const ant. If w is const ant , this equation can be integr at ed easily DK
=
(1 - DKO )( l
D KO
+ z )1+3w + DKO·
So that if t he present curvature is small, i.e. IDK ol = IDo - 11 < 0.1 , as observations tend to prove, then we need
(4. 152) at equality and
(4.153) at the Planck t ime. The Big-Bang model does not give any explanation for such a small curvature at the beginning of the Universe. We could have ended with the same conclusion from t he dynamical analysis of Sect ion 3. 2.2 that concluded that , for any equation of st ate w :::: 0, the fixed point DK = 0 was unst able. We should therefore det ermine a mechanism leading to such a small curvature or . explain why we strictly have DK = O. One solution is already suggested in Fig. 3.6 that illustrates that DK = 0 is an at t r actor if the Universe is dominat ed by a mat t er component with equation of stat e w < - 1/ 3. In the primordial era , as long as the Universe is domina ted by ordinary matter , it is always very fl at , so that the fl atness problem can actually be seen as an age problem: why does our Universe look so young? 4.5.2.2 Horizon problem The cosmological principle, which imposes space to b e homogeneous and isot ropic, is at the heart of the Friedmann and Lemaitre solutions. By const ruction, they cannot explain the origin of this homogeneity and isotropy. So it would be more satisfying t o find a justificat ion of t his principle. From an observational p oint of view, the CMB radiat ion supports this hypothesis. Its isotropy could be easily explained if the photons were emitted once the causal contact was est ablished b etween them. For t hat , the photons should be in causal contact since the t ime of decoupling. To compute t he size of the causally connected regions at decoupling, we work in conformal time (see Fig. 3.12) .
230
The standard Big-Bang model Infinite redshift / /
zdec
/
E
~
/
o.,
/
/ / /
/ /
,,
,,
,,
/
I
I
,,
I
,,
\
I
\
I
,,
/ /
I
,,
/ /
,,
/ /
I I I
,,
/
/,,"
,,
Decoupling
/ /
CJ
Las t sca ttering surface
o
,,
,,
,
",
\
A' "
\
,,
,,
",I" A" /
/ /
A
B"",
,,
\ \
, Space
B
,, ,
/
" ,, ,
/ / /
... _-
-_ ...
Fig. 4.17 (left) : The past light cone of an observer 0 intersects the surface of last scatt on a sphere of diameter A' B' that determines the size of the observable Universe. Th of the causally connected regions (A' A" , ...) is determined by the intersection with the f light cone from the Big-Bang hypersurface. (right): The horizon at decoupling and the su of last scattering.
As can be seen from Fig. 4.17, the regions in causal contact at the time of decou are smaller than the size of the observable Universe. Thus, the surface of last scatt includes around N
~
(TJO - TJdec ) TJd ec
3
~ 8(1 + Zdec)3 j 2 ~ 105 _
106
»
1
regions in causal contact. Such a causal region at decoupling is now observed u an angle ~ 2 TJdec ~ 10. (4 TJo - TJdec It is difficult to explain why the temperature of t he cosmic microwave backgr is the same up to 10 - 3% in the entire sky, while the latter is composed of a 10 6 causally independent regions. The standard cosmological model predicts that small regions of the surface of last scattering should be correlated, whereas l scales should necessarily be uncorrelated. Another way to formulate the horizon problem is to give an estimate of the nu of initial cells, with an initial characteristic Planck size length , present today i observable Universe. This number is typically of order
e
N
~
(1Cp+HoZPl)3~
1087.
Here again , the study of galaxies tends to show that their distribution is homogen on larger scales , so it is difficult to understand how init ial conditions fixed on causally independent regions can appear so identical (at a 10- 5 level!).
Status of the Big-Bang model
231
The horizon problem is related to the stat e of thermodynamic equilibrium in which the Universe is, discussed in Section 4.2 . The cosmological principle imposes a noncausal initial condition on spatial sections of t he Universe and in particular that the temperature of the thermal bath is the same at every point. The horizon problem is thus closely related to the cosmological principle and is therefore deeply rooted in the Friedmann- Lemaitre solutions. 4.5.2.3 Problem of the origin of struct ures Friedmann- Lemaitre solutions describe strictly homogeneous and isotropic space-times. However, the Universe is filled with numerous structures, whose origin and properties cannot be addressed and explained within t his framework. As formulated , t he cosmological model is thus incomplete. Since gravity is purely attractive, any density perturbation will tend to collapse to create structure. We can thus think that small fluctuations, of thermal origin, for example, could explain t he large-scale structure. But this mechanism is actually not very efficient in an expanding Universe (see Chapter 5). As long as t he Universe is radiation dominated , radiation pressure acts against gravitational collapse, and hence delays the beginning of large-scale structure formation. T hus, density fluctuations can only grow significantly for Z > Zeq rv 10 4 . We should t herefore find a mechanism able to generate density fluctuations with non-negligible amplitude. Such fluctuations do indeed exist and have been detected by the COBE satellite. Thus, a new horizon problem appears. In a radiation- and matter-dominated Universe, the causal radius, roughly the Hubble radius, increases more rapidly than t he physical wavelength of perturbations. Thus, the wavelengths of all cosmological perturbations observed today must have been larger than the causal radius at the beginning of the expansion of the Universe. In particular, the sat ellite COBE has detected temperature fluctuations with wavelengths larger t han the horizon at decoupling. Moreover, these fluctuations seem to have a scale-invariant spectrum. Can these initial density perturbations be generated by a causal mechanism? How can we explain their amplitude and t heir spectrum? Another problem related to the existence of these perturbations is the transPlanckian problem. All physical wavelengths observed today were smaller than the Planck scale at some t ime in the past. Thus, they must have been described by some physics that we do not know. Until which point are cosmological observations robust enough against the hypothesis of this (unknown) physics? Can they be used to extract some information on this physics? Note that t he Hubble radius also becomes smaller than the P lanck length , it is t hus unclear whether the description of the Universe by a classical space-time is still acceptable. This problem is related to the existence of an initial singularity. This version of the trans-Planckian problem is slightly different from the one presented in Chapter 8 where only t he wavelength of perturbations becomes trans-Planckian, whereas the Hubble radius remains larger t han the Planck length . 4.5.2.4
Relic problem
While the Universe cools down , some symmetries can be spontaneously broken (see Chapter 11) . During these phase transitions, topological defects can be formed.
232
The standard Big-Bang model
I I
I I I
I I
a
I I I I I I I I I I
I I I I I I I I
I I
Physical distance
Fig. 4.18 Horizon problem for a perturbation with physical wavelength A and the Planckian problem related to the existence of a singularity. All physical lengt hs, in part wavelengths, become smaller than the Planck scale. The Hubble radius also becomes s than the Planck scale and the classical d escription of space-time can b e questioned i limit.
In particular, in the grand unified theory framework, non-gravitational intera are assumed to have a symmetry described by a group (} . This group must be b to give rise to the one observed at low-energy, namely SU(3) cx SU(2) LxU(1)y Chapter 2). If the group (} is a simple group, then during this symmetry bre monopoles are produced with a mass of order Mmonopole '" T CUT '" 10 16 GeV. the fields giving rise to these monopoles are not a priori correlated on distances than the causal horizon , we can assume that they are created with a unit densi volume during the phase transition, nmonopo le '" (2tCUT) -3 . We get that the mon density is Pmo nop o le '" Mmonopol e nm o n opole. At tCUT , t he Universe is dominated by radiation , so that tCUT '" T c0T (see C ter 11 for a description of phase transitions in the cosmological framework). It fo that 2 Dmonopole h
17 '"
10
TCUT ( 1016
Ge V )
3 (
Mmonopole ) 10 16 Ge V
.
(4
As soon as they are formed , such relics would therefore overdominate the m content of the Universe. The effects of this matter on the evolution of the Un would be catastrophic. This problem was initially referred to as t he monopole pro To conclude, the overproduction of massive relics in the primordial Unive expected in various theoretical frameworks, the grand unified theory in particula should explain why these relics do not affect the dynamics of the Universe.
Status of the Big-Bang model
233
4.5.2.5 Dark-sector problem According to the conclusions from cosmological observations, only around 5% of the cosmic matter, so-called baryonic, is in a form expected by the standard model of particle physics. We should postulate the existence of a dark sector representing 95% of the matter and having at least two components: - the dark matter , representing 23 ±3% of the matter. This non-relativistic matter cannot be in baryonic form . The production of relics in the primordial Universe provides a mechanism that could lead to such matter. We should determine the interactions involved and the nature of this particle. - the dark energy, representing 72 ± 3% of matter , is required to explain the recent acceleration of the Universe. The cosmological constant has been a natural candidate for a long t ime. A satisfying cosmological model cannot fail to have a deep understanding of this dark sector. In particular , we should understand why dark matter and baryonic matter have comparable abundances. We should also remember that baryogenisis is still not a well-understood process. This mechanism should explain why 'rf = nb /n-y rv 10 - 1 As for t he dark energy, two possibilities are to b e considered:
°.
- the cosmological constant is a manifestation of the vacuum. In this case, its energy density can be computed from the theory of particle physics. However, any vacuum energy coming from a theory compatible with the standard model of particle physics should be at least of order of the energy of the electroweak symmetry- breaking scale, p;(4 rv 1 TeV. But the cosmological value is of the order of p;(4 rv 10- 3 eV. The disagreement between the theoretical estimate and the cosmologically determined value is thus at least of the order of 1060 , and can go up to 10 120 if we extrapolate to the Planck scale. This problem is known as the cosmological constant problem. - The dark energy is a dynamical quantity. These models will be described in detail in Chapter 13. However, we could ask ourselves why this matter only starts to dominate the matter content of the Universe today. This is what is called the coincidence problem. The coincidence problem is actually more general and relies on the fact that the different matter components are in the following ratio today
(4.156) How can we understand that these components of matter , whose origin depends a priori on differing mechanisms , come with such a ratio? Why do we live at a time when the two components of the dark sector are in comparable quantities? Also, note that the existence of structure and the cosmological observables are very sensit ive to these ratios. To finish, note that any information obtained on this dark sector has been deduced from the observation of luminous matter. These conclusions rely on a central hypothesis: the validity of general relativity to describe gravity. This hypothesis is so central that it is mandatory to t est it on cosmological scales . This last point merges with
234
Th e standard Big-Bang model
the effective constancy of the fundamental constants of Nature on cosmological scales. Note once more that these conclusions have been obtained by imposing tha Universe is well described by a Friedmann- Lemaitre space. Most observations, in particular the supernovre data, from which we deduce t he recent acceleratio the Universe, are localized in our past light cone. This implies that dependenci space and time are in general degenerate. This degeneracy is lifted by the symm hypothesis in the framework of Friedmann- Lemaitre models . The conclusions c thus be different in a Universe without these symmetries. For instance, the supern data can be reproduced in a Universe with spherical symmetry, filled only with (Lemaitre-Tolman- Bondi space described in Chapter 3). The expansion of su Universe is not accelerated but the observation of supernovre would be interpreted a wrong symmetry hypothesis and would lead to the conclusion that the expan is accelerating, whereas it actually only reflects the spatial dependance of the me Once more , we should check the Copernican principle as much as possible and veri conclusions that can be deduced from the observations: they depend on the hypot made on the space-time structure. 4.5.3
Conclusion
The Big-Bang model is a good standard model, giving a satisfactory description o Universe on large scales , where it can be described by a homogeneous and isot space-time . It has, nonetheless, various problems that primordial cosmology aim resolve. At higher temperatures, physics is less well understood, and its cosmolo implications can offer solut ions to some of these problems, but can also constrain physics. It is thus important for primordial cosmology to have a standard model such a Big Bang . This starting point will allow us to formulat e some variations, to test t and to study their phenomenology. The rest of this book is dedicated to this prog
References [1] R.A. ALPHER, H. BETHE and G. GAMOW, 'The origin of chemical elements', Phys. Rev. Lett. 73 , 803 , 1948. [2] G. GAMOW, 'The evolution of the Universe', Nature 162, 680, 1948. [3] R.A. ALPHER and R. HERMAN, 'Evolution of the Universe', Nature 162, 774, 1948. [4] E. HUBBLE , 'A relation between distance and radial velocity among extra-galactic nebulae', Proc. Natl. Acad. Sci. USA 15, 168, 1929. [5] S.K. W EBB, Measuring the Universe, the cosmological distance ladder, SpringerVerlag, 1999. [6] W. FREEDMANN , 'The Hubble constant and the expansion age of the Universe' , Phys. Rep. 333 , 13, 2000. [7] W. FREEDMANN et al., 'Final results from the Hubble telescope key project to measure the Hubble constant ' , Astrophys. J. 553 , 47, 2001. [8] M. BIRKINSHAW, 'The Sunyaev-Zeldovich effect' , Phys. Rep. 310 , 97, 1999. [9] S. PERLMUTTER et al. , 'Measurement of n and A from 42 high -z supernovre' , Astrophys. J. 517, 565 , 1999. [10] A.G . RIESS et al. , 'Observational evidence from supernovre for an accelerating Universe and a cosmological constant ', Astron. J . 116, 1009, 1998. [11] A. FILIPPENKO, 'The accelerating Universe and dark energy: evidence from type Ia supernovre' , Lect. Notes Phys. 646 , 191 , 2004. [12] U. SELJAK et al., 'Cosmological paramet er analysis including SDSS Ly-a forest and galaxy bias: constraints on the primordial spectrum of fluctuations, neutrino mass, and dark energy ', Phys. Rev. D 71 , 103515, 2005. [13] B. CHABOYER, 'The age of t he Universe', Phys. Rep. 307, 23, 1998. [14] J.J. COWAN, F.-K. THIELEMAN N and J .W. TRURAN , 'R adioactive dating of the elements', Annu. Rev. Astron. Astrophys. 29 , 447, 1991. [15] R. CAYREL et al. , ' Measurements of stellar ages from uranium decay', Nature 409 , 691 , 2001. [16] E. KOLB and M. TURNER, The early Universe, Addison Wesley, 1990. [17] R. TOLMAN, R elativity, thermodynamics and cosmology, Cambridge University Press, 1934. [18] J . BERNSTEIN, Kinetic theory in the expanding Universe, Cambridge University Press , 1988. [19] P.J.E. PEEBLES, 'Primordial helium abundance and the primeval fireball ', Astrophys. J. 146, 542, 1966. [20] J . BERNSTEIN, L .S. BROWN and G . FEINBERG, 'Cosmological helium production simplified ', R ev. Mod. Phys. 61 , 25 , 1989. [21] P.J.E. PEEBLES , Physical cosmology, Princeton University Press, 1971.
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[22] S. WEINBERG , Gravitation and cosmology: principles and applications of the g eral theory of relativity, John Wiley and Sons, 1972. [23] S. SARKAR, 'Big-Bang nucleosynthesis and physics beyond the standard mod R ep. Prog . Phys. 59 , 1493, 1996. [24] K.A . OLIVE, G. STEIGMAN and T .P. WALKER, 'Primordial nucleosynthesis: t ory and observations ', Phys. R ep. 333 , 389, 2000. [25] V. MUKHANOV , 'Nucleosynthesis without a computer' , Int. J. Theor. Phys. 4 669 , 2004. [26] RV. WAGONER, 'On the synthesis of elements at very high temperatures ', A trophys. J. 148, 3, 1967. [27] http://www-thphys.physics.ox.ac.uk/users/SubirSarkar/bbn.html. [28] K.M . NOLLETH and S. B URLES, 'Estimating reaction rates and uncertainties fr primordial nucleosynthesis' , Phys. R ev. D 61 , 123505, 2000. [29] YI. I ZOTov et al. , 'Helium abundance in the most metal deficient blue comp galaxies' , Astrophys. J. 527, 757, 1999. [30] V. LURIDIANA et al. , 'The effect of collisional enhancement of Balmer lines the determination of the primordial helium abundance', A strophys. J. 592, 8 2003. [3 1] YI. IzoTov and T.X. THUAN, 'Systematic effects and a new determination the primordial abundance of 4He and DY / dZ from observations of blue comp galaxies ' , Astrophys. J. 602 , 200, 2004. [32] D. KIRK MAN et al., 'The cosmological baryon density from t he deuterium to h drogen t owards QSO absorption system: D/ H towards Q1243 +3047' , A stroph J. Supp. 149, 1, 2003 . [33] S.G. RYA N et al. , 'Primordial lithium and big Bang nucleosynthesis ' , Astroph J. 523 , 654, 1999. [34] D.N. SPERGEL et al. , 'First year Wilkinson microwave anisotropy probe (WMA observations: determination of cosmological parameters', Astrophys. J. Sup 148, 175, 2003. [35] A. Coe et al. , 'Updated big Bang nucleosynthesis confronted to WMAP obs vations and to the abundance of light elements ', Astrophys. J. 600, 544, 2004 [36] K. OLIVE and E. SKILLMAN , ' A realistic determination of the error of prim dial helium abundance: steps toward non-parametric nebular helium abundance A strophys. J. 617, 29, 2004. [37] J. LINSKY, ' Atomic deuterium/ Hydrogen in the Galaxy', Space Sci. Rev. 10 , 49 , 2003; H.W. Moos et al., 'Abundances of deuterium , oxygen, and nitrog in t he local interstellar medium: overview of first results from FUSE missio Astrophys. J. Suppl. 140, 3, 2002. [38] T .M. BANIA , RT. ROOD and D.S. B ALSER, 'The cosmological density of baryo fro m observations of 3He+ in the Milky Way', N ature 415 , 54, 2002. [39] R CYBURT et al. , 'New BBN limits on physics beyond the standard model fr He4' , A stropart. Phys. 23 , 313 , 2005 . [40] A. PENZIAS and R WILSON , 'A Measurement of excess antenna temperature 4080 Mc/s', A strophys. J. 142, L419 , 1965.
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[41] R DICKE, P. PEEBLES , P. ROLL and D. WILKINSON, 'Cosmic blackbody radiation ', Astrophys. J. 142, 383, 1965. [42] G. SMOOT, The CMB spectrum, in The cosmic Microwave Background, C.H. Lineweaver et al.(eds.) , NATO ASI series 502 , Kluwer Academic Publishers , 1997. [43] F.R BOUCHET, J.-L. PU GET and J.-M. LAMARRE, The cosmic microwave background, in l 'Univers primordial, Springer , 1999, pp . 103. [44] RB. PARTRIDGE, 3K: the cosmic microwave background, Cambridge University Press, 1999. [45] P.J .E. PEEBLES , 'R ecombination of the primeval plasma', Astrophys. J. 153, 1, 1968. [46] B.J. JO NES and RF. WISE, 'Ionization of the primeval plasma' , Astron. Astrophys. 149, 144, 1985. [47] J.C. MATHER, 'Callibrator design of the COBE far infrared absolute spectrophotometer ', Astrophys. J. 512 , 511 , 1999 . [48] G.F. SMOOT and D. S COTT, 'Cosmic background radiation ', in The review of particle physics, S. E idelman et al. , Phys. Lett. B 592, 1, 2004. [49] D. FIXSEN et al., 'The cosmic microwave backgound spectrum from full COBE FIRAS data set' , Astrophys. J. 473 , 576 , 1996. [50] R SRIANAND, P. P ETITJEAN and C. LEDOUX , 'The microwave background temperature at t he redshift of 2.3371', Nature 408 , 931 , 2000. [5 1] G.F.R ELLIS, R MAARTENS and S.D. NEL , 'The expansion of the Universe', Month. Not . R. Astron. Soc. 184, 439, 1978. [52] M. FUK UGITA and P .J.E. PEEBLES, 'The cosmic energy inventory', Astrophys. J. , 616 , 643 , 2004.
5
The inhomogeneous Universe: perturbations and their evolution
Friedmann- Lemaitre space-times are approximations t hat give a good description our Universe only on very large scales. In fact , our Universe is not strictly homogeneo and isotropic since it contains structures such as galaxies, galaxy clusters, etc. In t 1980s, it was understood that the large-scale structure can result from the gravitatio amplification of the density fluctuations of a pressureless fluid, the cold dark matt see R efs. [1- 3J. The aim of this chapter is to study Universes with a geometry 'close' to a Frie mann- Lemaitre space-time and to understand the evolution of the density perturb tions under the influence of gravity in an expanding space-time. To do so, we will st by recalling the Newtonian limit in Section 5. 1. Then , we will focus on the gene relativistic case in Section 5.2 to present the theory of cosmological perturbation This part, which is quite technical, is one of t he cornerstones of modern cosmolo and is essential in order to address the properties of t he large-scale structure of t Universe. In Section 5.3 , we will study some solutions of these perturbat ion equatio The charact eristic scales that should be t aken into account in the general case, and t form of the power spectrum are described in Section 5.4. We will finish in Section with a description of the observations of these large-scale structures. R eferences [4- 9J can complement this chapter.
5.1
Newtonian perturbations
Before developing t he general case of t he evolution of perturbations in the relativis framework, let us focus on the simplified case of the evolution of perturbations of initially homogeneous fluid , in the Newtonian regime. In fact , as we will see in Section 5.4, such a Newtonian approach is sufficient describe the late time evolution of the density perturbations during the matter era sub-Hubble scales. A general description of gravitational dynamics in the Newtoni framework can be found in R ef. [4J .
5.1.1
Case of a static space
In a static Euclidean space, the hydrodynamic equations take t he usual form [10J o continuity equation and the Euler equation
Newtonian perturbations
OtP + V. (pv) = 0 ,
239
(5.1) 1
Otv+ (v· V)v = -- Vp - VcI> ,
(5.2)
P
for a fluid with density p, pressure P and velocity field v , evolving in a gravitational field with potential cI> . 5.1 .1.1
Without gr avi ty
If we neglect gravity and decompose the density in terms of a homogeneous component and a perturbation as p = 15 + op and the pressure similarly as P = P + op, (5.1) and (5 .2 ) reduce to a unique wave equation
O;op - tJ.op = 0.
(5.3)
Let us introduce Xa == (oP/oP)a/P, called the isothermal or adiabatic compressibility coefficient , where a stands either for the t emperature (T) or the entropy (S). When the propagation time of heat is brief, the flow can be considered to be isothermal, whereas it will be adiabatic if there was insufficient t ime to reach equilibrium, which is the case for long wavelengths. In the latter case, we infer that t he wave equation (5.3) takes the form (5.4)
Thus , hydrodynamic perturbations propagate with constant amplitude and with velocity cs ' the speed of sound of the given fluid. 5.1.1.2
Effect of gr avity
Let us now turn on gravity. The gravitational potential is determined by the Poisson equation (5.5) so that the evolution equation for the density fluctuations now has an extra term and takes the form
(5.6) By decomposing the perturbations into plane waves as op ex exp[i(wt - k.x) ], we obtain the dispersion relation (5.7)
AJ is the Jeans length above which any perturbation is unst able and grows exponentially as
240
The inhomogeneous Universe
Qualitatively, the J eans length characterizes the transition from a regime for which t evolution of the perturbations is dominated , at long wavelengths , by gravity (A > AJ to a region (A < AJ) where gravity is negligible and in which we recover sound wav The value of the J eans length can actually be deduced by considering the tw dynamical t imes of the problem. The time associated with sound waves (i. e. the t im taken by this wave to propagate along a distance equal to its wavelength A) is given t sound cv AIcs ' The gravitational collapse t ime taken by a structure wit h density 15 collapse is t gr av cv (GNP) - 1/ 2. Hence , at small wavelengths, one has tso und « t grav a the pressure has time to compensate t he gravitational collapse of the fluid , which th remains quasi-homogeneous. When A becomes larger than the J eans length, t so und larger than t grav , the pressure can no longer balance gravity and large inhomogeneiti can develop.
5.1.1 .3 Energetic aspects
From an energetic point of view, each p oint of the fluid can be seen as a harmon oscillator whose displacement from its equilibrium position is denoted by 8 . The velocity of each element of the fluid is thus 8 and , using the conservation (5.1 t he density perturbation is op = - V· 8. The equation of propagation (5.7) can th be integrated to give, after averaging over a volume V , t he equation for conservati of energy (5.
where (EK ) = ~ (82 ) is the mean kinetic energy, (EE) = ~c; (V8 2 ), the mean elas energy, and (Ee) = -27rGN 15 (82 ) , the mean gravitational energy. The kinetic ener takes t he form -1
2
(8.2) -_
-1 W
2
2(8 2)
'
We deduce that the total energy per unit volume is
(EM)
= ~ (82) [w 2 + c; (k2 - k3) ] .
The dispersion rela tion teaches us that there is equipartition of t he energy betwe the kinetic energy a nd the potential energy (which has two contributions). The me energy is t hus given by (EM) = (82) (k2 - k3) , (5.
c;
and is, as expected , negative for wavelengths larger than the Jeans length. 5 .1. 2
Case of an expanding space
In an expanding space, one can introduce t he co moving coordinates x by
r(t)
=
a(t)x ,
(5. 1
where a(t) is the scale factor. The velocity field is then given by
v(t) = r = Hr where u =
ax is the proper velocity.
+ u,
(5.1
Newtonian perturbations
5.1.2.1
241
Eulerian formulati on
Equations (5 .1 ) and (5 .2) can be rewritten by noticing that , in the previous case of a static space-time, the time and space derivative operators defined from t and r were considered to be independent. This is no longer the case here, and it is convenient to use space derivatives defined with respect to the comoving coordinates x. Consequently, the gradient V r is replaced by V x/a, and the time derivative Otp(r, t), that previously assumed constant r , becomes Otp(x , t) - Hx . V xp(x , t). The two equations are thus re-expressed in co moving coordinates as .
p(x , t)
+ 3Hp(x , t) + -1 V x [p(x , t)u(x , t)] a
=
(5 .1 2)
0,
and
u(x , t)
1
+ Hu(x , t) + - (u· a
where we have used that V x
V x) u(x , t)
1
1
a
ap
= - - V x(x , t) - - V xP(x, t),
(5.13)
= 3. If t he density contrast, 6, is defined by
. X
p(t)[l
p =
+ 6(x , t) ],
(5.14)
then the first equation can be rewritten as the continuity equation .
1
6 + - V [(1 + 6)u] = 0,
(5. 15)
a
where from now on, unless otherwise stated , we omit the reference to co moving coordinates that is implicit in what follows. After some manipulations, an evolution equation for 6 can be deduced from these equations
.. . 1 1 1 [ . "] 6+2H6 = -211P +2V. [(1+6) V ]+2oiOj (1 + 6)u"uJ pa
a
a
.
(5. 16)
Note that this equation does not m ake any assumptions on 6 and in particular it does not assume 6 to be small compared to 1. The gravitational potential is determined by the Poisson equation, which then takes the form
(5.17) Solving (5.17) , one gets that the gravitational potential is given by
(x , t )
-
= - G NPa
2
J
3 I 6(XI , t) d x Ix _ xi i'
(5. 18)
These equations provide a Eulerian description of the dynamics. 5.1. 2.2
Lagrangian description
A Lagrangian approach can also be developed by following the trajectories of particles or fluid elements instead of considering the dynamics of the density and velocity fields.
242
The inhomogeneous Universe
The central quantity is then the displacement field X(q, t) , which maps the initia position of a particle labelled q into its Eulerian position at time t according to
x(t) = q
+ X (q , t).
The equation of motion for particle trajectories x(t) is then
or equivalently in terms of the conformal time TJ
x" + Hx' = - V",
Taking the divergence of this equation, one obtains an equation for the displacemen field
(5.19
where we have made use of the Poisson equation and of the fact that 15d 3 q b(x , t)]d 3 x, which implies that 1 + b(x, t)
=
1
det(bij
= 15[1 +
1 + Xi,j) - J(q, t)'
- --
where X i, j == 8Xd 8qj, and J(q , t) is the Jacobian of the transformation from Euleria to Lagrangian space. Note that the gradient in Eulerian coordinates can be transforme to a gradient in Lagrangian coordinates as
Such a description can be useful for some problems and more details can be foun in Refs. [11 , 12]. Note, however, that when there is shell crossing, that is when th fluid elements with different initial positions q end up at t he same Eulerian positio x, the Jacobian vanishes and the density field becomes singular. At these points, th description of the dynamics in terms of a mapping no longer holds . 5.1.2.3
Linearized systems
Let us now suppose that the fluid density is only weakly perturbed with respect t the homogeneous configuration. This translates into the two conditions b « 1,
(dut)2 «b,
where u is the characteristic fluid velocity and d the co~ere nce length of density per turbations. The second condition implies that the gradients must be small. Let u recall that in a flat Universe with no cosmological constant, the Friedmann equatio implies that 67fG N 15t 2 = 1 during the matter-dominated era (since a ex: t 2 / 3 ) .
Newtonian perturbations
243
By linearizing (5.16), we get the propagation equation (5.20) We recover the J eans length defined by (5.7). Unlike the case of a static Universe, the Jeans length now depends on time via 15. A perturbation with given wavelength will thus be able to pass from a sound-wave regime to a gravitational collapse regime , and this at a time depending on the value of the wavelength. 5.1.2.4
Growth of density p erturbations
For a fluid of matter with negligible pressure, (5.20) is a second order linear differential equation involving only time. One can therefore look for solutions of the form
where the functions dx) correspond to the initial density field. We will refer to D+ as the growing mode and to D _ as the decaying mode. The functions D are solutions of the equation (5.21 ) where we have expressed the mean matter density as 41rG NPm (t) = ~ H2 flm(t). For a flat Universe with no cosmological constant (flrn = fl mo = 1, a ex t 2 / 3 ) , one can check that (5.22) In the general case, this equation needs be integrated numerically. A convenient way to solve it is to use the scale factor , a, as the time coordinate, so that (5.21) takes the form d D (~dH +~ ) dD _ ~flrno D = 0 (5.23) da 2 H da a da 2 a5 Ho '
2+
(li) -2
where we have used the fact that 41rG Np = 41rGNPrnoa- 3 and assumed that ao = 1. One can check that D _ = H is a solution. The second solution can then be obtained from the method of variation of parameters as
r
D (a) - ~ H(a) fl da' + - 2 Ho rnO o [a' E(a,) ]3 '
(5.24)
J
where E(a) = H(a) / Ho is given in (3 .39). The normalization constant has been chosen so that in a fl at matter-dominated Universe, we recover D+ = a. As an example, in a Universe with A = 0 and flrn < 1, it can be shown that the decaying and growing modes are given by D
-
=
J1+X x 3
'
3 D+ = 1 + - + 3D _(x) In x
[vT+x - Fx] ,
where x == fl;:;/ - 1. Note that when flrn --> 0, that is when x D_ '" l/ x so that the perturbations cease to grow.
»
1, D+
(5.25) -->
1 and
244
The inhomogeneous Universe
0.4 0.2 0.2
0.4
0.6 a/ao
0.8
1
Fig. 5.1 Evolution of t he growing mode of the density contrast, D + (a) , as a fun ctio of the scale factor , a , for n mo = 1 [solid line], (n mo , n AO ) = (0.3 , 0.7) [dashed line] an (n mo , nAO) = (0.5,0) [dotted line].
In more general cases, (5.24) for D + has to be integrated numerically, as illustrate in Fig. 5.1. It has been shown [13] that for a ACDM model it is well approximated b
(5.26
Actually, when K = 0, it can be checked [14], using the solution (3.46) of the Friedman equations, that
where a = HoVDi\o = 5. 1.2. 5
J A/3 and 2Fl
is a hypergeometric function.
The Hubble parameter from the growth fun ction
Equation (5.23) can also be seen as a function for E(a) once rewritten as 2
2
dE+ 2 [ 3 + d D (dD)-l] - l =0. E 2 - 3-Dmo D (dD) da a da 2 da a5 da
This first-order equation for E2 is easily integrated to give
E 2(z)
=
dD) - 3D mo (1 + z )2 ( -d z
-2
J OO - D- -dD d z, z
1 + z dz
(5 .27
in terms of the redshift 1 + z = ao / a. It follows that for the class of ACDM model there exists a rigidity between the background dynamics and the growth of large-sca structure. Such a feature can be useful to test this model (see Chapter 12) .
Newtonian perturbations
245
5.1 .2.6 Density - velocity relation Since 6 ex D(t) , (5. 15), implies that 1
e(x, t) == aH V . u(x, t) = - f(Omo, OAO)6,
(5.28)
_ dlnD(a) f(O mO, OAO) = d In a .
(5.29)
with
There exist many analytical approximations for this function. For instance, Ref. [13] suggests
f+(OmO, OAO)
~ o~8 + ~~O
(1 +
~Omo) .
e
Note that represents the local fluctuations of the Hubble constant. This is an observable quantity, independent of the value of the Hubble constant. Using (5.13), it follows that the velocity field has the general form
u
=
-~ f+(OmO , OAO) V
aHO m
x U '
(5.30)
where U is an arbitrary vector function. Considering again the example of a flat Universe with no cosmological constant, the solution (5 .22) implies that (5.3 1) Thus, the growing mode corresponds to a velocity field falling towards the potential well, which does indeed increase 6. The decaying mode corresponds to a configuration where the fluid escapes from the potential wells and tends to erase the initial fluctuations. Since the velocity field can now be observed [15], it opens the possibility of testing the gravitational dynamics and provides a way to measure the cosmological pm' ameters. 5.1.2.7 Gravitational potential In the linear regime, and in the Newt onian approach we have followed in this section, the gravitational potential can be obtained from the Poisson equation (5.5) so that it behaves as
=
~Omo ( 8H~~C)
20"8.
It follows that O"ip ~ 10- 5 if 0"8 ~ 1. This estimate has two consequences. First , it implies that
246
The inhomogeneous Universe
contrast has entered the non-linear regime. Second , if the power spectrum is (almo scale invariant, as predicted by inflation (see Chapter 8), then this amplitude of t gravitational potential is its typical amplitude on all scales. 5.1.3
Predictions and observables
We have characterized the density and velocity fields of t he cosmic matter fluid . T density field results from the gravitational collapse of primordial density perturbatio In this book, we will try to present t he models capable of generating these primord fluctuations. It is, however, important to discuss some aspects of the comparison the predictions we will make with the observed Universe.
5.1.3.1
The need for a statistical approach
Note first of a ll that t he primordial fluctuations are not directly observable. On different obj ect s, observed at different epochs of their evolut ion, are accessible. F instance, although E± (x), the field of primordial fluctuations, is not directly accessib we can observe the large structures of the Universe (galaxies, clusters ... ), and yet can only measure a limited set of properties of these obj ects spatial distribution. As will be seen later , models of t he primordial Universe allow us to predict t statistical properties of the primordial fluctuations of which we will be able to dedu t he distribution of some observables such as O. The density field , as well as all the oth fields, are t hus now to be considered as stochastic variables. Every statistical predicti will hence depend on the statistical properties of the initial pert urbations and on th dynamical evolution. The Universe is thus modelled as a stochastic realization o statistical set of possibilities.
5. 1.3.2 Statistical description
Given t he above, we will thus be dealing with classical random fields, such as t density contrast , for instance, and compute some of t heir statistical properties such the correlation function ~(x, r) = (o(x) o(x + r) ) . (5.3
Let us draw t he implication of t he Copernican principle when applied statistically the fields living in a Friedmann- Lemaitre space-time. Statistical homogeneity means t hat all t he moments of a random field or t he jo distribut ion probabilities p[o(rd, o(r2) , .. .J remain unchanged under the action of a space translation. So t hese probabilities only depend on the relative separations , su as rl - r 2. Statistical isotropy, on the other hand , demands t hat these quantities be a invariant under t he action of rotations. Therefore, the probabilities eventually depe only on the modulus of t he relative separations , namely e.g. r12 = Irl - r 21. Th hypothesis can be considered as the statistical version of t he cosmological princip but it is only an hypothesis and needs to be tested (as should the Copenican princi itself). Assuming statistical isotropy and homogeneity, the correlation function (5.32) th only depends on r := Irl, i. e. ~(x, r) = ~(r).
Newtonian perturbations
247
It is convenient to expand b(r) into Fourier modes as
(5.33) and since we consider a real-valued field, the coefficients bk satisfy the conjugation relation (5.34) The random field is thus completely characterized by the statistical properties of the random variables bk . For instance
(5.35) which, after integrating over x , gives
(5.36) 6(3) being the Dirac distribution , not to b e confused with the density perturbation. The power spectrum P/i is related to the correlation function by
(5.37) a relation that can be inverted to yield an expression for the correlation function , ~(r) , in terms of the power spectrum, namely
() J
~T=
( ) e-d3k - /P ik (27f)3
i k·r
=
1= 0
dk k2p/(k)Sinkr -i -27f2
kr
(5.38)
5.1.3. 3 Link with observable quantities and smoothing
Quantities such as ~(r) cannot actua lly be measured as we only observe a finite part of a single realization of the stochastic process. So , operationally, only the spatial average is accessible
(5.39) where 60 bs is the observed field density. For any statistical observable, let us say 0 , one must construct an estimator , O. The probability of measuring the value 0 in a cosmological survey (e.g. galaxy catalogue) given a theory is called the cosmic distribution function , P (0), and the ensemble average of 0 is simply
(0)
=
JOP(O)dO.
Most estimators are non-linear and are thus biased, in the sense that their ensemble average differs from the real value O. The cosmic bias bo, defined as
248
The inhomogeneous Universe
bo
==
(0) - 0 0
does not vanish in general, except when the size of the survey t ends to infinity, sin in that case one measures the actual value of the observable. A good estimator mu have a minimum cosmic bias and also minimize t he cosmic error, which is obtained b computing the variance of P. This description is still idealized and simplistic. Indeed, what does it mean measure 6obs(r) ? For inst ance , a catalogue of galaxies can give us access to the me surement of the mean number of ga laxies per volume V so that an estimate of continuous field should be inferred from the observation of the number of objects in given volume. This gives an estimate of the average density in this volume, namely
(5.4
there by defining the (0 bervational) window function W R- This function must satis a normalization condition
(5 .4
as well as the condition
(5.4
which guarantees that its mean width is finite and of characteristic size R. Th archetypes of such 'window ' functions are the 'top-hat ' and Gaussian filters. No that if the Fourier components of this filter are denoted by WR (k) , then the Fouri coefficients of the filtered field are given by (5.4
since (5.40) is a convolution. The cosmic error and t he cosmic bias have three main contributions (for a discu sion of these effect s, see Refs . [12, 16]) :
(1) Finite-volume effects, due to t he fact that there are approximatively N rv VIL independent volumes of size L in a survey of volume V. In particular , the mea value of the density is not always well det ermined . These effects are proportion to the average of the 2-point correlation function over the survey and are usual referred to under the name of cosmic variance. (2) Edge effects, related to the geometry of the catalogue and to the fact that th estimators in general give less weight to the data on the edge of the survey. (3) Discreteness effects, related to the fact that the theoretical field is assumed co tinuous while the observed one (e. g. galaxies) is discret e. These effect s scale the inverse of the number of obj ects in the survey to some power.
Thus, for any physical quantity Q( r) we would like to predict the probabili distribut ion function (PDF) of the filtered quantity QR(r ) (we restrict ourselves the one-point function for the sake of simplicity). This PDF, p( Q, R)dQ , represents th
Newtonian perturbations
249
probability to lie between Q and Q + dQ. For example, if Q is the number of galaxies and the filter a top-hat , then the PDF of Q R will give the probability of finding N galaxies within a volume of radius R. Models such as inflation predict that these initial fluctuations are Gaussian , i.e. P(Qi, R)
Q~R ] , = V2ii a1i(R) exp [ - 2a;(R)
(5 .44)
where a;(R ) = \ Q~) is the initial variance. The second moment completely characterizes this distribution , which we would like to compare with that of QR today. There are thus two effects to consider: - the evolution of the modes can modify the initial spectrum by a transfer function that may depend on k , - non-linear evolution can modify the initial statistics. There is a third effect that will not be discussed in this book, which comes from the fact that only luminous objects are observed. It should therefore be checked that luminous matter is a good tracker of the total matter density. This is at the origin of the notion of biased measure [12, 17- 19]. The rest of this chapter is dedicated to the first aspect that requires a det ailed analysis of the evolution of perturbations in the relativistic framework. We briefly illustrate the second aspect in the next section. 5.1.4
Towards the non-linear regime
The Newtonian description allows us to address the behaviour of perturbations in the non-linear regime. This aspect cannot be developed further in this book, but we refer the reader to the review [12] for a very detailed study. Whatever its initial value, 5 grows and , given enough time, eventually becomes of order unity. The time at which this happens depends on the initial amplitude of the fluctuation and on the growth factor , that is on the cosmological parameters. As we will see, the normalization of the initia l power spectrum is such that the (top-hat) variance of the density field in a sphere of radius R s = 8h- 1 Mpc is of order unity (5.45) The length scale Rs therefore characterizes the scale below which non-linearities cannot be negl cted, i. e. the scale below which the density contrast is too large for the linear approximation to be applicable. To take the non-linear evolution into account, the density field can be expanded perturbatively as 5
=
5(1 )
+ 5(2 ) + ... ,
where the term 5(n ) is of the order of en in the initial density field [e(X) first order corresponds to the solutions previously found 5(1 ) =
D(t)e(x) ,
«
1]. The
(5.46)
250
Th e inhomogeneous Un iverse
whereas, to second order, t he evolut ion equation (5 .1 6) reduces to J (2)
+ 2H 5 (2) = 4nG N p [0(1)] 2 + ~ a 2 V 0(1) . V + ~OO a2 2 J
[U1)U j(1) ] 2 ('
.
(5.47
Using t he 1st -order solut ions, one can solve this equation to obtain [20]
0(2) = D2(t)
{ ~[1 + I\:(t) ]E2 (x) + V E' V +
with
D- I\:(t) ] cy2} ,
(5 .48
1
cy 2 = CYij cy ij ,
CYij = OiOj - 30ij l:..
T he function I\:(t) is slowly varying, and to a goo d approximation is given by [21 ] I\:
~ ~D - 2/63
- 14
m
,
when the cosmological constant vanishes. 5.1.4.1
Application: Sk ewness
In order to illustrate the effect of non-linear dynamics, let us consider the t hird-orde moment of t he density field
where t he first t erm in t his expansion vanishes fo r Gaussian init ial conditions. We se t hat t he non-linear evolut ion induces a departure from Gaussianity of t he density field As discussed previously, one should consider the smoothed density field, and at t h lowest order, we obtain
with
, 21+
J(k , k ) =
I\:
k' (kkk' .k') + (12 - ) (k.k ')2 kk'
+ k
I\:
Neglecting t he filter and assuming t hat t he spectrum behaves as a power law P (k) e k n , we obtain [21] 3
S3(0)
= (0(02)2 =4+41\:-(n + 3)-:::= 3417 )
(
-2 / 63
Dm
)
- 1 -(n+3) .
(5.50
For a t op-hat filt er , t his result is modified [22] to
S (R) = S (0) 3
3
+
d lnCYJ(R) d ln R '
(5.5 1
where CYs( R) is the variance of t he density field , filtered at t he scale R. Such a relatio can be compared wit h observations (see, for inst ance, Ref. [23]) , e.g. to show tha large-scale structures are compatible with Gaussian initial condit ions.
Gauge invariant cosmological perturbation theory
5.2
251
Gauge invariant cosmological perturbation theory
The Newtonian approach is valid for sub-Hubble perturbations and is relevant for the description of t he evolution of p er turbations in t he late Universe. However , the gravitational dynamics must be described in a relativistic framework , particularly for long-wavelength modes. T he following section is one of the cornerstones of contemporary theoretical cosmology. In particular, we will present a derivation of the equations generalizing (5.1) , (5.2) and (5.5). There exist two for malisms for cosmological perturbation theory. We choose to present an approach in which we introduce arbitrary parameterizations of the perturbations from which we will then construct gauge invariant variables. This method , introduced by Bardeen [24, 25J , is the most common one and various aspects are detailed in many reviews [26- 29J . For the approach t hat we do not develop , Refs. [30- 32J can be consulted. Both approaches lead to equivalent physical results. 5.2.1
5.2. 1.1
Perturbed space-time
Metric of the perturbed space-time
At linear order, we decompose the space-time metric according to ds2
=
a 2 ('T)) [-( 1 + 2A)d'T)2
+ 2Bi d x i d'T) +
hij
+ h ij )dxi d x j ] ,
(5.52)
where the small quantities A , Bi and hij are unknown functions of space and time to be determined from the E instein equations. Writing this metric as (5 .53) where !hw is the Friedmann- Lemaitre metric, the inverse metric is then given by (5.54) to first order in the perturbations . In what follows, h will denote the t race of the spatial perturbations h
==
j hij,i ,
and we recall that V IJ. is the covariant derivative associated wit h glJ.l/ ' T he Christoffel symbols, Riemann, Ricci and Einstein tensors and the scalar curvature associated wit h this metric are given explicitly in Appendix C.
5.2.1 .2 SVT decomposition In what follows, we will perform a so-called scalar-vector-tensor decomposition [24J (denoted by SVT in the following) of t he per turbation variables. This is , in a cosmological background, the same decomposition as that introduced in Chapter l. This decomposition is a generalization of the well-known fact t hat any vector field can be decomposed as the sum of t he gradient of a scalar and a divergenceless vector as B i = D i B + IF with Di Bi = O. (5.55) For a velocity field, B will b e t he potential and Bi the vorticity. One sees t hat the 3 components of the vector have been split into 1 scalar (B) and 2 vector (Bi) components.
252
The inhomogeneous Universe
In an analogous way, any rank 2 symmetric t ensor can be decomposed as with
--=ij
DiE
-
i
= 0, E i = O.
(5.56
Equation (5.54) implies that the indices of these quantities are raised and lowered wit --==i .. respect to the unperturbed spatial metric , lij , for example, B = , t) B j. We adop the convention in which any ' barred ' quantity is divergenceless (and traceless if it ha two indices). The 6 components of h ij have thus been split into 2 scalar (C and E), vector (E j ) and 2 tensor E ij components . Note that the quantities B and E are not unique. In fact , B is a solution 6.B = Di B i (the Laplacian being defined by 6. == DiD i ), whose solution is uniqu only if some boundary conditions are imposed. The 10 degrees of freedom of the metric have thus been decomposed as - 4 scalars: A , B , C and E , corresponding to 4 degrees of freedom , - i
-i
- 2 vectors: B and E corresponding to 2 x (3 - 1) = 4 degrees of freedom, - 1 tensor:
It j , corresponding to 3 x 2 -
1- 3
= 2 degrees of freedom.
The advantages of this decomposition lie in the fact that, to leading order , the thre types of perturbations are decoupled and can thus be studied separately.
5.2.1 .3
The ga uge problem
To introduce the perturbed space-time metric, we have made t he hypothesis (5. 53 that the metric 9,J.V was 'close' to that of the Friedmann- Lemaitre g/-Lv . In field theory, space-time is usually fixed and once the system of coordinates chosen , one can define the perturbation of any quantity Q(x , t) as
oQ(x , t) = Q(x , t) - Q(x , t),
where Q(x , t ) represents the unpert urbed configuration. At any point of space-tim the quantity Q(x , t ) can be compared with Q(x, t) at the same point. In general relativity, there exist an important difference with this standard ap proach. Indeed , we would like to compare two different space-times and make th hypothesis that (M , 9/-Lv ) is 'close ' to a Friedmann- Lemaitre Universe (M , g/-Lv). I making this comparison, there is some arbitrary freedom related to the way of iden tifying the points of these two space-times. Indeed, an isomorphism 7jJ should be in troduced to relate the points of the two space-times (see Fig. 5.2). There is actuall no ' natural' identification between the two spaces , and there is therefore a freedom o choice in the identification of the points in each spaces. This implies that there exis some unphysical degrees of freedom related to the choice of the coordinate systems o t he two manifolds. The decomposition (5.52) implicitly assumes that a system of coordinat es has bee chosen in the perturbed space. Any change of coordinates will modify the form o the metric coefficients. To illustrat e the effect of such a change of coordinates, le us consider an unperturbed Friedmann- Lemaitre space-time, as in Fig.5.2, and le
Gauge invariant cosmological perturbation theory
M
253
M
Fig. 5.2 Any perturbed quantity is defined via a mapping between the Friedmann- Lemaitre space-time, M , and the perturbed space-time M.
us perform an arbitrary space and time-dependent change of coordinates xi - ~i (xj , fJ). We then get ds 2
Xi --; yi
=
= a 2 (fJ) [- dfJ2 + 2~~dyidfJ + (rij + 2D(i~j))dyidyjl.
There appear two artificial terms Bi = ~~ and Ei = ';i that do not correspond to metric fluctuations, since this space is still a Friedman- Lemaitre space, but to a choice of the system of coordinates. The dependence of the quantities that are intrinsic to t he manifold in which the perturbations evolve should thus be separated from the artificial ones, related to the arbitrary choice of a particular coordinate system. 5.2.1.4
Gauge invariant variables
In order to extract t he physical degrees of freedom , let us consider an active transformation of the coordinate system defined by a vector field ~. The coordinates of any point change according to (5.57) where the displacement ~J.L is decomposed into 2 scalar degrees of freedom (T and L) and 2 vector degrees of freedom (D, which is divergenceless DiD = 0) as ~o
= T,
(5 .58)
Under this change of coordinates, the metric transforms as [see (l.57)]
To first order in the perturbations, this is (5.59)
254
The inhomogeneous Universe
which in turn implies that the metric perturbation variables (5.52) transform as 1
A - t A + T' + HT Bi - t Bi ~ DiT + L~ hij
-t
hij
(5.6
+ DiLj + Dj L i + 2HT'Yij '
(5 .6
Using the scalar-vector-tensor decomposition (5.55) and (5 .56) we get
+ T ' + HT,
A
-t
A
B
-t
B - T
C
-t
C + HT,
E
-t
E
-i
+ L' ,
-i
-,
E -t E + V ,
(5.6
+ L,
respectively, for the S, V and T modes. These transformations are similar to the gaug transforma tions encountered , for inst ance , in electromagnetism. Therefore , one ma reduce the arbitrariness by imposing some additional conditions on the variables: other words, and again as in electromagnetism, one needs to fix the gauge before doin any actually relevant calculation . As stressed in the previous section , it is interesting to define gauge-independe quantities. To do so, notice that some combinations of the previous quantities do n depend on L i and T , for instance,
== - C - H(B - E') , == A + H(B - E') + (B <]?i = E i' _ B i , IlT
1>
(5.6 ~
E')',
(5.6
(5 .65
~
ij
E .
(5. 6
ij
We initially had 10 perturbation variables (A , B ,C, E , E \ B i, E ) and 4 gaug degrees of freedom (T, L , Ii ) that can be a bsorbed into the definition of the gaug invariant variables, then described by 10 ~ 4 = 6 degrees of freedom parameterize These quantities can be considered as the 'rea for instance, by 1lT , 1>, <]?i and space-time p ert urbations in the sense that they cannot be removed by any change coordinat es.
Itj .
1 For complet eness, we give the expressions of t he Lie d eri vatives. They are easily obtained fro the express ions of the C hrist offel symbols of the Friedmann- Lemaitre space-time and from ~i" a 2 ( - T , Li ). They are given by
£e 900 = - 2a 2 (T'
+ 7tT) ,
£e 90i = a 2 (- B;r + L ;J , £e 9ij = a 2 (Di L j
+ D j L i + 27tTiij ).
Gauge invariant cosmological perturbation theory
255
Other gauge invariant quantities could have been constructed, such as, for instance,
X=A -C- (~)' ,
(5.67)
but they can always be expressed in terms of the two functions, and W. For instance, in the case of X defined just a bove, (5.68) as can straightforwardly be checked .
5.2. 1. 5 Stewart- Walker lemma Just as the metric transforms as (5.59) under a change of coordinates of the form (5.57), any scalar quantity oQ transforms as
to first order in perturbations . Since each vector field ~ generates a gauge transformation, we can conclude that the only gauge invariant quantities are the ones for which This result is known under the name of the Stewart-Walker lemma [33]. Note an important corollary of this result. Since all relativistic equations are covariant by definition , they can always be written in the form Q = 0, where Q is a tensor field . It is thus always possible, t o first order in perturbations, to write all the relevant equations in terms of gauge invariant quantities only. Such a treatment would ensure t hat no gauge, i.e. unphysical, degree of freedom, is misleadingly used. 5.2.2
Description of matter
In this section, we introduce various quantities that come into the description of a perfect fluid for which the energy-momentum t ensor in an unperturbed space is of the form as was seen in Chapter 3.
5.2.2. 1 The energy-momentum tensor of a perturbed fluid The energy-moment um tensor of t his perturbed fluid takes the general form (5.69 ) where u!-' = vP+ou!-' is the four-velocity of a comoving observer, satisfying u !-'u!-' The normalization condition of vI' (to zeroth order) provides t he solut ion
=
- l.
256
The inhomogeneous Universe
and since the norm of uP + buJJ. should also be equal to - 1, we infer that 2uPbuJJ bgJJ.vupu v = 0, and thus that buo = - Aa. We then write bui == vi/a, so t hat
(5.7 and we decompose Vi into a scalar and a t ensor part according to
Vi = D iv +Vi ·
(5 .7
In the decomposition (5.69) , 'TrJJ.v is the anisotropic stress tensor and characteri the difference between the perturbed fluid and a perfect fluid. It can be chosen be traceless (gJJ.v'TrJJ.v = 0) , since its trace can be absorbed into a redefinition of t isotropic pressure, bP. It is also symmetric, since the energy-momentum tensor is, a can be chosen orthogonal to uJJ. , i. e. uJJ.'Tr JJ.V = O. This implies that, without lack generality, one can set 'Troo = 'TrOi = O. The anisotropic stress t ensor then consists only of a spatial part, which can decomposed into scalar , vector and tensor parts as
(5 .7 where the operator
t'::..i j
is defined as
(5.7
We infer from (5 .70) , (5 .71) and (5.72) that the components of the perturbation the energy-momentum tensor (5.69) are
bToo
=
pa 2 (b
bToi = _pa 2
+ 2A) ,
[(1 + w) (Di v + Vi ) + Di B + B i]
(5 .7
,
(5 .7
(5.7
where we recall that b == bp/ p is the density contrast. It will be useful to introduce the entropy perturbation, f, defined by the relatio
(5.7 which takes the form wf
1
= - (bP - c;bp) . p
(5 .7
For an adiabatic perturbation , bP/ bp = c; and hence f = 0 and we introduce t notation used in the literature bPnad = Pf , (5 .7 where the index ' !lad ' stands for non- adiabatic.
Gauge invariant cosmological perturbation theory
5.2.2.2
257
Gauge invariant quantities
Upon performing a change of coordinates of the form (5.57) , scalar and vectors transform as and so that it follows that , after SVT decomposition , OP , op, v and Vi transform as
op OP
----+ ----+
op + piT,
v
----+
v - L',
oP + PiT,
(5.80)
The quantities 71", Hi and Hij are gauge invariant from the Stewart- Walker lemma, since the unperturbed part of the anisotropic stress tensor must vanish to be compatible with the perfect fluid hypothesis imposed by the symmetries of the Friedmann- Lemaitre space-time. One can also check t hat , as expected again from the Steward- Walker lemma, r is gauge invariant since r ----+ r + (Pi - c; p')T = r. As in the case of t he metric perturbations, one can likewise define the gauge invariant quantities associated with the quantities (5.80). Different choices are possible, and we define I
ON
= 0 + ~(B - E'), p
OF = 0 _ pi C
(5 .81) (5.82)
P 7-{ ' C pi o = o+ -(v + B) ,
(5 .83)
V
(5. 84)
p
=
v+E' ,
(5 .85)
Vi =Vi+ Bi , -
-I
W i=Vi+ E i ·
(5.86)
The pressure perturbations op N , OpF and ope are defined in an identical way. The different gauge invariant variables are related to each other and one can check that C
o In all these relations , we recall that 0 5.2.3
pi
= ON + - V,
(5.87)
P
= opl p is the density contrast.
Choosing a gauge
From the Stewart- Walker lemma , we know that any perturbation equation can be written in terms of gauge invariant variables only. Indeed , such an equation always takes the form Q = 0 and is, by construction , linear in the first-order perturbation variables . One method for obtaining these gauge invariant equations is thus to write
258
The inhomogeneous Universe
the equations in an arbitrary gauge and to regroup the terms to make the gaug invariant variables appear . A more subtle way is to work in a given gauge from the beginning so that th variables in that gauge reduce to gauge invariant variables. For instance, the longitu dinal gauge is fixed by the condition E = Bi = 0, so that C = - \fJ , A = <1> .... Th equations will then be derived directly in terms of gauge invariant variables. We ca then change gauge or re-express the equations in t erms of the initial ga uge dependen perturbations by using, e.g. , (5.63)- (5.66) . Let us stress an important, and often confusing, point. Although the calculation can be made in any gauge, one in general needs to perform a gauge transformation t relate the results of these calculations with observable quantities . In fact , the gaug invariant quantities do not have a priori any direct physical interpretation and on needs to find a gauge where the perturbation variables reduce to the gauge invarian quantities . For instance, ON can only be interpreted as the density contrast if it measured by an observer using this gauge. Note also that locally, all systems of coordinat es are equivalent. Therefore, w only expect to find differences between the different gauges on large scales, or, mor precisely, above some yet-unknown characteristic scale. We will see shortly that th is indeed the case and that the scale in question is fixed by the Hubble radius. A final cautionary point at this level: if, for some reason, a supposedly small (pe t urbative) quantity, calculated in a fixed gauge , becomes large, while all quantitie calculated in a different gauge remain small , this does not necessarily mean that t h first-order perturbation theory breaks down , but neither does it mean the opposite In fact , it merely implies that the coordinate change to get to this gauge is no longe infinitesimal. One then needs to go on to second-order calculations, at least: if th second-order correction happens to be smaller than the first order in all gauges , t he it is likely that t he particular gauge used to begin with was pathological at t his poin This, however, really needs be checked before any physical conclusion be reached . We now turn to the extremely useful task of presenting the various gauges fre quently used , before deriving the equa tions of evolution for the cosmological pertur bations.
5.2.3.1
Newtonian or longitudinal gauge
This gauge is the one where the scalar part of the perturbed metric is diagonal , namel B = 0
and
E = O.
(5.88
Given these conditions , the vector part cannot be diagonal, and one can arbitrari choose to impose the condition B i = 0, (5 .89 which fixes the gauge completely since one can pass from an arbitrary gauge to th one with the tra nsformation (5 .58) with parameters defined by
T = B - E',
L= -E
(5 .90
In this gauge, the expansion is 'seen ' as isotropic. The quantity \fJ represents the cu vature perturbation of constant t ime hypersurfaces [see (C .31)] and it is this potentia
Gauge invariant cosmological perturbation theory
259
that arises in the Poisson equation, i.e. the one that reduces to the Newtonian potential on small scales, hence the gauge name. In this gauge, one finds the remaining perturbed quantities (5.91) in terms of the gauge invariant variables.
5.2.3.2 Flat-slicing gauge The fiat-slicing gauge is defined by imposing that the scalar part of the curvature perturbation of the spatial sections vanishes. Using (C.31) , this imposes that C
= 0,
E =0
and
Ei = O.
(5.92)
As the longitudinal one, the fiat-slicing gauge is also completely fixed since one can pass from an arbitrary gauge to this one with the transformation (5.58) defined by (5.93) In this gauge , the remammg perturbed quantities are given in terms of the gauge invariant variables by
E' = - \Ii
A=X,
v = V
7-{ '
and
Vi = Wi·
(5.94)
The main advantage of this gauge is , as we shall see, that the conservation equations take the same form as in the Newtonian version of the previous sections.
5.2.3.3
Synchronous or Gauss gauges
Synchronous gauges are defined by the constraints
A =0
and
Bi = o.
(5.95)
In such a system of coordinates, only the spatial sections are perturbed, the worldlines with equation Xi = const. are geodesics orthogonal to constant-time hypersurfaces and the proper time of a co moving observer corresponds to the cosmic time. This is thus a system of coordinates in which every point corresponds to a free-falling observer. This intuitive physical interpretation had made this gauge very popular. However, it suffers from pathologies, and in particular it is not completely fixed. Indeed , one can still perform coordinate transformations of the form t ---+ t' = f(t) or xi ---+ yi = r(x j ), involving only time or space separately and that are compatible with t he constraints (5.95). The transformation to pass from an arbitrary gauge to a synchronous gauge is given by
T = - -1 a
J
T
Aa dry + -C a
,
and
r
=-
J
Bi dry + ~, (5.96)
Ci
where CT and = (CL'~) are four integration functions that can depend on the spatial coordinates xi. These functions have simple physical interpretations, namely
260
The inhomogeneous Universe
CT represents the residual freedom in the choice of the origin of the proper time of eac observer , while C L corresponds to an arbitrary choice of the system of coordinates o each const ant-time hypersurface. The existence of this arbitrary choice in the definitio of the gauge leads to the presence of gauge modes when solving the evolution equation for the perturbations. T hese gauge modes are not physical and correspond to spuriou solutions that may lead (and have led) to confusion. Furthermore, since the gauge i not completely fixed , a variable computed in this gauge cannot always be expresse in terms of gauge invariant quantities. It is conventional to define the residual metric perturbations in this gauge by h == 6C + 2tJ.E 5.2.3.4
and
'rJ = -C.
(5.97
Comoving gauge
The comoving gauge is a gauge that, as its name indicates, tracks the overall motio of the matter. In this gauge, the velocity of the matter fluid is thus imposed t o vanish i.e. (5.98 1ST? = o.
When several fluids are present, a co moving gauge can be defined with respect to an given component (baryons, photons, etc .. .) or to t he total matter. In this text , unles stated otherwise, by 'comoving gauge ' we really mean 'comoving gauge with respec to t he total matter ' . T he condition (5.98) above cannot always be satisfied, as it implies that Vi+Bi = O It is always possible to find a system of coordinates for which the scalar part of th velocity vanishes (fo r instance, in a potential flow , Stokes theorem ensures that th flow lines do not intersect and one can thus choose a system of coordinates that follow these lines). This is not the case for the vector part that corresponds to the vorticit and t here are current vortices t hat cannot be suppressed by a change of coordinates Due to t his fact, the following two conditions must be added
E = 0
and
Ei = O.
(5.99
T his gauge has the special feat ure, reminiscent of one of the flat-slicing gauge property as we shall see, that the Poisson equation it leads to is identical to t hat obtained i the Newtonian case of the first sections. Having shown a few frequently used gauges, let us turn t o the actual dynamica description of the p erturbation evolution, namely the E instein equations expanded t first order. 5.2.4
Einstein equations: derivation
E instein equat ions for the perturbations take the simple form ISG~
= ",IST/:.
All required perturbed quantities are gathered in Appendix C. We will treat t he tenso modes, followed by the vector ones, which are mathematically simpler , before finishin with the scalar modes .
Ga uge invariant cosmological perturbation t heory
5.2.4.1
261
Decomposition of 3-dimensional vector and tensors
Vectors In Fourier space, we have the splitting with
(5.100)
so that ,i lives in the subspace perpendicular to ki . This is a 2-dimensional subspace so that Vi has been split into 1 scalar (V) and two vector modes (Vi ) that correspond to transverse modes. Let us now consider the base {e 1 , e 2 } of the subspace perpendicular to ki. By construction , it satisfies the orthonormalization conditions
with a, bE {I, 2}. Such a basis is defined up to a rotation about the axis ki . Now, the vector modes can be decomposed in this basis as
L
V i (k i , 7]) =
Va(k i , 7]) ef(ki ) .
(5 .101 )
a = 1,2
This defines the two degrees of freedom , Va, which depend on ki since the decomposition differs for each wave number. The two basis vectors allow us to define a proj ection operator onto the subspace p erpendicular to k i as
(5.102) where ki = kdk. It trivially satisfies P]Pk = Pk, P]kj = 0 and pij,ij = 2. It is also the projector on vector modes so that we can always write Vi in the form
(5.103) thereby making a scalar-vector decomposition, similar to the SVT one for tensors. Tensors Analogously, any (3-dimensional) symmetric tensor field, Vij , can be SVT-decomposed as (5.104) Vij = T,ij + D.ijS + 2D(iVj) + 2Vij , where we recall that D.ij
== DiDj - 1 D.' ij and DNi = O,
j V:= O=DiV .
(5 .105)
The symmetric tensor V ij is, according to our convention , transverse and trace-free. Hence, it has only two independent components and can be decomposed as
Vij(k i , 7] ) =
L )., =+ , x
V).,(k i ,7])C:;j(kd,
(5.106)
262
The inhomogeneous Universe
where the polarization t ensors have been defined as
(5.10
It can be checked that they are indeed traceless (E;/yij = 0) , transverse (E;j ki = and that the two polarizations are perpendicular (E;jE ~ = o~). This defines the tw t ensor degrees of freedom. Introducing the proj ector operator on tensor modes by
Aab 'J
=
pa pb _ ~p .p ab '
J
2
'J
,
and the 'trace-ext racting' operator
the SVT t erms are extracted as follows
(5.10
In this expression , Vi j has been split into 2 scalars (T and S) , two vector (Vi ) and t t ensor (V ij ) modes. Thus, we can always split any vectorial equation such as Vi = by projecting it along ki (scalar) and P] (vector) , while any t ensorial equation of t
kind Vij = 0 can be proj ected along ,ij (scalar) , Ti j (scalar), P/kj (vector) and A (tensor). Having these decompositions , let us move on to their actual dynamics. 5.2.5
5.2.5.1
Einstein equations: SVT decomposition
Ten sor modes
The tensor modes are explicitly gauge invariant by construction. Their evolution equ tion is obtained by extracting the tensor part of the Einst ein equation as
and they reduce to a single evolution equation
(5.10
Up to the curvature t erm and the damping t erm due to the expansion , this equati is analogous to (1.132 ) obtained in Chapter 1 for gravitational waves in a Minkow space-time.
Gauge invariant cosmological perturbation theory
263
It can be interesting to rewrite this equation, by introducing the reduced quantity
n defined in (3 .36), in the form
(5.110) It is also convenient to rewrite the equation for gravitational waves for its two polarizations using the reduced quantity
(5.111) where cZj is the polarisation tensor (see Fig . 1.9) . Equation (5 .109) then takes the form II
UT
+
(
all)
2K - .6. - --;:
UT
=
IW
3
P1f>..
(5.112)
In the case of a Universe with Euclidean spatial sections (K = 0) the left-hand side term of this equation is analogous to that of a field evolving in a potential, here time dependent , given by a" / a. Note that this equation also has the same mathematical form as a time-independent one-dimensional Schrodinger equation for a particle in a potential, the latter being d2 ?jJ(x)/dx + [E - V(x)] ?jJ(x) = O. This formulation will come in handy when studying gravitational waves during inflation (see Chapter 8).
5.2.5.2
Vector modes
There are two E instein equations for the vector modes that follow , respectively, from the components (Oi) and (ij). Using the results of Appendix C and the expressions (5 .75) and (5.76) for the perturbed energy-momentum tensor, we readily obtain the two equations (5.113) (5.114) As for the case of (5 .110), it can also b e interesting to rewrite these two equations, by introducing the reduced quantity D, defined in (3.36) , in the form
+ 2K)
- /
(5 .115) (5.116)
5.2.5.3 Scalar modes To derive the Einstein equations for the scalar modes , we will consider the following four combinations:
- bCg
+ 3'HD;lbC?,
- the traceless part of <SCj,
- bC? ,
264
The inhomogeneous Universe
- oGi
+ 3c~oGg.
F irst, we obtain two constraint equations,
(6.
+ 3K) w=
K,
2
C
(5 .11
"2 a pO ,
(5 .11
The first equation, as announced above , t akes t he classical form of the Poisson equati when expressed in terms of oC, using (5.87). The second equation tells us that the tw gravitational potentials are equal if the scalar component of t he anisotropic stre tensor vanishes (most frequent case) . The other two equations are equations of evolution given by
w' + }{cI> = - ~a2p( 1 + w )V, w" + 3Ji(1 + c;) w' + - (Ji
2
(5 .119
[2 Ji' + (Ji 2 - K)( l
+ 2Ji' + K)
+ 3c; )] w -
c; 6.w =
[~r + (3Ji 2 + 2Ji')7f + Ji7f' + ~6.7f]
- 9c;Ji 2 (Ji 2 + K) 7f,
(5.120
where we have used, in the second equation, (5.118) to replace by its value in term of w and 7f. 5.2.5.4
Equivalent forms
It can be useful to give some equivalent forms of the Einstein equations for the sca p erturbations, in particular using other gauge invariant density contrasts . The Poisson equation (5.117) takes the following two forms depending on wheth ON or OF is used
(6.
+ 3K) w = ~a2p [ON
6. w- 3Ji 2X = ~a2poF 2
- 3Ji(1
+ w)V]
,
(5 .12
(5.12
'
while (5. 120) can be expressed in terms of the variable X alone as
JiX' 5.2.6
+ (Ji 2 + 2Ji') X
=
~a2 p ( wr + c;OF + ~w6.7f)
.
(5 .12
Perturbed conservat ion equation for a fluid
T he fluid conservation equations take the simple form
Again, all the required perturbed quantities are gathered in Appendix C. We treat t vector modes first , and then the scalar ones. Note t hat , since fluids are described scalar and vector quantities only, there is no tensor equation.
Gauge invariant cosmological perturbation theory
265
5.2.6.1 Vector modes There is a unique conservation equation for the vector modes and it takes the form
-V , + H (1 ,
3c
2) -V = -1- -w - (t. + 2K) 'if_ ' . 21+w '
(5.124)
s'
This form is sufficiently simple that one can solve it explicitly in most of the interesting cases.
5.2.6.2 Scalar modes For the scalar modes , the 4-dimensional conservation equations will give us two equations analogous to the continuity equation and to the Euler equation, obtained, respectively, from the t ime-like and space-like component. Working in the Newtonian gauge, one obtains
1
5N '
+ 3H(c~
- w)5 N = -(1
V , + H (1 - 3c2) V s
+ w) (t.V -
31lJ') - 3Hwr,
c;
- - 5 N - -W-
= - <1> -
l +w
l +w
[
(5.125)
I
r + -2 ( t. + 3K )_] 'if 3
(5.126)
.
The conservation equation (5 .125) can be rewritten in the form
5N - )' (l +w
= - (t.V - 31lJ') - 3H~r ,
(5.127)
l +w
if we use t he expression (C.19) to re-express Wi. It is also useful to give the expression c for these two equations in terms of the density contrasts 5F and 5 , namely
5F '
+ 3H(c~
- w)5 F = -(1
+ w)t.V -
3Hwr ,
c;
(5.128)
V, + H ( 1 -2 3c) V = - <1> - 3c21lJ - - - 5 F - S
S
l +w
W
-
l +w
[
r + -2 ( t. + 3K )_'if] 3
,
(5.129)
V 5.2.7
, + HV =
c;
- <1> - - - 5 c - -Wl +w l +w
[
r + -2 ( t. + 3K )_] 'if 3
.
(5.130)
Interpretation of the perturbation equations
At first glance , the conservation equations are analogous to those obtained in the Newtonian approximation. In particular, as announced before, we recover (5.1) and (5.2) if we place ourselves in a static space (H = 0) with non-relativistic matter (w = c; = 0 and 'if = 0) for adiabatic perturbations (r = 0) if J is identified as JF . However, there are some important differences we ought to stress. 1. There are vector and tensor perturbations in addition to the scalar perturbations.
266
The inhomogeneous Universe
2. There are two potentials,
5.3
Evolution
The perturbation equations derived in the previous sections can be solved in variou simple cases. In general , it is useful to expand the perturbations into Fourier mod (see Appendix B) and to study the evolution of each mode as a function of tim Indeed , since the equations of evolution involve spatial derivatives only through th Laplacian, they will reduce to linear ordinary differential equations in Fourier space In many cases, we will distinguish two regimes that we introduce now
k k
« »
71 71
super-Hubble mode sub-Hubble mode
wavelength larger than Hubble radius , wavelength smaller than Hubble radius .
Note that in the literature, the super- and sub-Hubble modes are often, wrongly (an confusingly), called sub- and superhorizon. This (historically based) misuse of languag can be understood as in the st andard Big-Bang model presented in Chapter 4 t h Hubble radius coincides, up to an irrelevant numerical factor , with the particle horizo radius. However , this is not always the case, in particular if there has been a perio of inflation, which precisely has, among its aims, the purpose of solving the horizo problem (see Chapter 8)! The fact that the spatial gradients are negligible in th evolution equations of the perturbations is not at all related to causality but to th comparison between the damping t erm due to the expansion of the Universe, 71 - 1 an the characteristic time of oscillation of mode k, given by ~ k - 1 Note that the behaviour of the growing mode on super-Hubble scales cannot b inferred intuitively from causality arguments, contrary to what is sometimes state
Evolution
267
when claiming that on superhorizon scales the perturbations are 'frozen ' (see, e.g. , Ref. [34] for a det ailed discussion). 5.3.1
5.3.1.1
Vector and tensor modes
Vector modes
As long as ni vanishes or is negligible, (5.114) implies that -
iPi
IX
IX
a-
a
- 2
(5.131)
,
and the Euler equation implies that
Vi
(1 - 3c 2
s) .
(5.132)
So that the vector metric perturbations are washed out by the cosmological evolution. The same is true for the vorticity of the cosmic fluid if the latter is non-relativistic. Thus, the vector modes are of litt le interest in the formation of the large-scale structures, even if they can play a role , for example, if magnetic fields or topological defects are present . 5.3.1.2
Tensor modes
Let us assume that n i j vanishes or is negligible and let us write (5.109) in Fourier space for a Universe with Euclidean spatial sections. Assuming that the scale factor behaves as a IX r( then 2E _ d E ij 2v dE i j (5.133) -d 2 + --d - + i j - O, x X x with x == k'rj . The solution of this equation can be expressed in t erms of Bessel functions as (5.134) where A ij and B ij are two transverse and traceless constant tensors. Keeping only the mode that does not diverge when x --> 0 (the divergent mode corresponding to a decaying mode when 'rj increases indeed becomes negligible) we obtain
- .1 / 2 - v J v E ij -x
I
2
( X ) A ij -- x i -v J·v- i
(X )
A ij·
(5.135)
In particular, we obtain, respectively, in the matter- and radiation-dominated epochs Matter
Radiation
is const ant as long as k'rj < 1, i.e. as long as the wavelength of the considered mode is larger than the Hubble radius. When the wavelength becomes shorter , the mode develops a damped oscillatory behaviour (see Fig. 5.3).
E ij
268
Th e inhomogeneous Universe 1.0
1.0
0.8
0 .8
0.6
0 .6
Eij
Ei,i
0.4
0.4
0.2
0.2
0.0
0.0
- 0.2 -3-2 -1
0 1 lnx
2
- 0.2 -3-2-1
3
0 1 ln x
2
3
Fig. 5.3 Evolu t ion of gr avitational waves in a radiation-domin ated Universe, i. e. w = v = 1, (left) and pressureless matter , W = 0, v = 2 (right) in t erms of In x = In kT/. T mode is const ant as long as t he wavelength is sup er-Hubble (x < 1) and develops a damp oscillatory b eh av iour as soon as it b ecomes sub-Hubble (x > 1).
5.3.1.3
Tensor modes in a Universe domin ated by a mixture of matter and radiati
Let us now assume that the Universe is dominated by a mixture of pressureless matt (w = 0, 1/ = 2) and radiation (w = ~,1/ = 1). Introducing the quantity a
y= - ,
(5.13
ae q
where a e q is the value of the scale factor when r adiation and matter have the sam density [Pm(a eq ) = Pr (a eq )], we deduce t hat y = Pm / Pr and that the equation of sta of the mixture of the two fluids is 1
1
w= - - - . 31+y
(5.13
The evolution (5.109) of the gravitational waves can then be rewritten as 2-
d E ij dy2
+
-
4 + 5y dE ij 2y( 1 +y) dy
+ (~) k eq
2
_2- E i = 0 l +y J '
(5.13
where we have switched to t he variable y : to obtain this equation, we have used t fact that t he derivatives with respect to y are given by E~j = y'dEij / dy , and that t Hubble parameter is given by H = y' / y. The Friedmann equation, with this variab takes the form H2 = H2 1 + y (5.13 e q 2y2 '
where H eq is the value of H at a eq . We have also introduced k eq , the value of the wa number corresponding to t he mode entering t he Hubble radius at equality, defined
(5.14
Evolution
269
Equation (5.138) can be integrated numerically and the solutions for different values of k/ keq are represented in Fig. 5.4. Note that the earlier the mode enters the Hubble radius, the more damped it is. 1.0
I--_=--_=--_~
0.8
0.6
0.2
0.0
y
Fig. 5.4 Evolution of gravitational waves in a Universe dominated by a mixture of matter and radiation in terms of y = aj a eq (y = 1 at equality) for the modes of value kj keq = 10, 1 and 0.1.
5.3.1.4
General solutions for long-wavelength modes
In a flat Universe (K = 0) , the form (5.112) of the gravitational waves evolution equation admits the (formal) integral solution
uT(T/ , k) = a(7]) Adk) + A2(k)
[
/
2 '7 a2(~) dd - /ii 1 + k 2 /'7 a2(~) di} a (i})u T (i}, k) . (5.141)
For long wavelength modes, that is satisfying k
«
a/l / a, it reduces to
(5.142) where the two integration constants Al and A2 maya priori depend on k. Note, in the analogy with the Schrodinger equation, that this is simply the first term of a Born expansion for describing the deflection of a particle in a potential.
5.3.2
Evolution of the gravitational potential
We are interested in the evolution of the gravitational potential, so let us rewrite its evolution equation in the form <1>/1
+ 37-{ (1 + c~)
<{>I
+ [27-{1 + (7-{2 - K)
(1
+ 3c~)l
<{> -
c~~<{>
when we neglect the contributions from the anisotropic stress (7f
=
~a2 pr,
= 0).
(5.143)
270
The inhomogeneous Universe
5.3.2.1
Another formulation of the evolu tion equation
A reduced form of t his evolution equation can be obtained. First, we introduce t auxiliary function
~
8=
a
(_p _)1/2 (1 - ~)1 /2, +P
(5.14
Kpa 2
p
which can be rewritten as
H 8= ~
[23 (H2 - HI + K) ]-1/2
(5.14
We then define the auxiliary variable
(5 .14
thanks to which, and after a tedious, but straightforward, computation, one can sho that (5.120) eventually t akes the form U
/I
s
-
(81/ + 2) ~ 8 s -
C
4
U
s
8 pr = -K3H -a '
(5.14
which shares many features with the form (5. 112) and can thus be analysed usi similar tools.
5.3.2.2 Integral solu tion for long-wavelength modes
T he homogeneous part of this equation can be int egrated for any long-waveleng mode by noticing t hat u = 8 is a solution. In Fourier space, we obtain
(5 .14
This solution is valid for all modes in t he matter era as long as t he entropy perturbati is negligible. As discussed above , it can be compared with (5.142) by replacing a 8.
r
5.3.2. 3
First integral
Equation (5 .143) admits a first integral for the adiabatic evolution of super-Hubb modes [35]. Indeed , by introducing the quantity [36] I'
=
., -
2
+ 3H
I + H 1 +w '
(5.14
(5. 143 ) takes the form 1'1
.,
=
~~ {~ 3 1 + w H2
[1
2_w )
+ 3w I 3 ( s 2H + C
]
pr _ + ~~ 2 H2
2(~) 2 H
Cs
}
'
(5.15
where we have used (C. 19) and (C .22) to evaluate Wi and HI . We infer t hat as lo as the curvature is negligible , i.e. in the regime for which K / H 2 « 1, the adiabat
Evolution
271
evolution (f = 0) of the super-Hubble modes (k / 7-{ « 1) has (as first integral, namely that the approximation (' = 0 (5.151) is valid under these assumptions. Not e that if the evolution is not adiabatic, then (5.150) t akes the form ,
( =
7-{
p
+ p5Pnad ,
(5.152)
so the conservation of ( is not ensured if entropy perturbations are produced. The quantity ( turns out to have a very simple geometrical interpretation: considering the expression (C.3 1) for the spatial curvature, we notice that ( is exactly this curvature in the comoving gauge . Let us also introduce the curvature perturbation in the flat-slicing gauge as 1
5p
1
5
( BST == - c+ - - = - c+ -3 -. 3p +P 1+W
(5.153)
The variable (B S T' being gauge invariant, can be expressed in terms of the gravitational potential as (5.154) We infer that
(BST
therefore satisfies (5.155)
Note t hat unlike (5. 150) , the right-hand side term of this equation does not involve the spatial curvature K. Thus, (B S T remains constant in time for the adia batic (f = 0) and super-Hubble (k / 7-{ « 1) modes provided that <1>' + 7-{ does not diverge when k/7t ~ 0, which is the case if the decaying mode is negligible. It is easy to see that bot h quantities differ only on sub-Hubble scales and that they are related by 1 ~ (5.156) ( = (B S T - :3 7-{' - 7-{2 . It should be noted at this point that the conservation of (B S T' although derived in the Einstein equations framework , does not in fact depend on the underlying gravity theory, but only on \7 /1.T/1.V = 0, i.e. the fact that the stress tensor is conserved [37J. This assumption is much weaker and renders this variable useful in wider fr ameworks. 5.3.2.4
EHect of a variation in the cosmic fluid equation of state
For illustrative purposes, let us consider the evolution of the gravitational potential during a change in the equation of state. We thus assume that w varies from WI for 'rJ < 'rJ- (era 1) to W2 for fI > fI- (era 2) and we assume that K = 0 and f = O.
272
The inhomogeneous Universe
In each phase, the scale factor evolves as a ex t Pi ex 7) n i , with ni = Pi/ (l - Pi ) 1 + Wi = 2/ (3Pi). The general solution (5.148) for long-wavelength modes implies =
~
[A _
+ A+
J
2
a (1
+ W)d7)]
(5 .
,
where A+ and A _ are two const ants that char acterize the growing and decaying mo If we asssume that the decaying mode in the era 1 has had enough time to bec negligible, then , before the t ransition, the only mode that remains is the constant namely <1>(7) < 7)*) rv A+ (1 + wd Pl/ (Pl + 1) = 2A+ /[3(1 +Pl)]' while at a large eno time after the transition, when the decaying term is also negligible, one similarly up with <1> (7) » 7)*) rv 2A+/[3(1 + P2)]. We then have 1 + Pl 1 + P2
(5.
In the special case of a transition between a radiation-dominated era (Pl = ~ ) a matter-dominated era (P2 = ~ ), this leads to 9 10 '
(5.
for long-wavelength modes. Note that one could have reached the same conclusion from the fact that ( is stant during the evolution , since from (5.149) , ( rv {1 + 2/ [3(1 + w)]} for the grow mode, during each era. As it turns out , in this particular case, the relation (5.159) also be obtained by using the analytical solution for the scale factor [27]. 5.3.3
Scalar modes in the adiabatic regime
We assume in the following examples that the influence of the anisotropic stress t e and of the entropy p erturba tion are negligible (7f = 0 and r = 0). In particular , implies that the two gravitational potentials are equal, = w. Thus, this chapter mainly focus on the study of the evolution equation (5.143) with r = O. Note t hese assumptions are valid in most of the evolution of the Universe.
5.3. 3.1
Universe dominated by a fluid with equation of state
W
=I
0
Let us consider a flat Universe (K = 0) dominated by a fluid with constant equa of stat e w =I O. In particular , this implies that c; = w. The scale factor of su Universe evolves as a ex 7)v with 1/ = 2/ (1+3w), so that in Fourier space, the evolu equation (5.143 ) t akes the form 2
df
d X 2+
~df X
dX +
[ _ I/(I/ + 1)]f=0 W
X
2
'
(5.
after introducing the function f = x V and x = k7) . The general solution of equation is known and yields the gravitational potential. The latter can be expre in t erms of Bessel functions as
273
Evolution
(5.161) where jl/ and nl/ are spherical Bessel functions defined in Appendix B , and where the second equality defines ZI/ ' The Poisson equation (5 .117) allows us to deduce that the density contrast is given by (5.162 ) and (5.119) then implies that kV
3 1= - -x
1/ [ ZI/ ( cs x )
4
)] . - - Cs-x ZI/ - l (csx 1I + 1
(5.163)
Using the asymptotic behaviour of the Bessel functions (see Appendix B), we obtain the following behaviour in the two limiting regimes cs X « 1
_J
c
-kV
+
2 (1) + x2 _
2 [
+ 11- 1) x - 2v ] /[3 11(1 + w)]
+ x 1 - v cos (cs x - a v ) / (3112) 2
The phase (XI/ is given by (X I/ = 7r(1I + 1) / 2. For wavelengths much shorter than the acoustic radius (cs x » 1) , oCbehaves as a sound wave that is damped if -~ < w < ~ . Radiation (w = ~ ) is the limiting case where the amplitude of the density perturbation remains constant (Fig. 5.5). Outside the acoustic radius (cs x « 1) , the growing mode of the gravitational potential remains constant.
(f
1.0
0.00
0.5
-0.05
0.0
-0.5
0.2 0.1 kV 0.0 -0.1 -0 .2
- 0.1 5
-1.0 - 2 -1 0
1
2
In (cs x )
3
4
-0.20 -2 -1 0 1 2 3 I n (cs x )
4
-2 -1
0
1
2
3
4
In (cs x)
Fig. 5.5 Evolution of the density contrast (left) , of the gravitational p ot ential (middle) and of the velocity p er t urbation (right) for t he scalar modes of a radiation fluid (w = c~ = ~) in a radiation-dominated Universe (v = 1) in t erms of cs x = cs k'T). The mode is constant as long as the wavelength is larger that the sound radius (cs x < 1) and it starts a damped oscillatory behaviour as soon as it is smaller (cs x > 1).
We also deduce from these calculations that during the radiation era, the superHubble density fluctuations of the r adiation fluid grow as the square of the scale factor , (5.164)
274
The inhomogeneous Universe
1.5 1.0 Flat slicing 0.5
Newtonian
-- - --- - - - --
8 0.0
Comoving
-0.5 - 1.0 -l.5
L........~~~~~~~~~~~~~~~~~~---'
-2
o
-1
1 In (cs x)
2
3
4
Fig. 5.6 Evolution of the density contrasts be (solid line), bN (dashed) and bF (dotted) terms of Cs x = Cs k7J. As long as the wavelength is larger than the sound radius (c s x < there is a difference between these three quantities that converge rapidly as soon as th become smaller than the sound radius (c s x > 1). This illustrates the effect of the ga choice.
5.3.3.2
Matter fluid in a matter-dominated Universe
In the case of a dust-dominated Universe (w = c; = 0 and v = 2), and using the sa notations as in the previous section, (5.160) takes the form , d2 / dx
2 d/ x dx
6 x
-+ - - - - 2 /=0 2
'
(5.1
whose general solution provides the potential, density contrast and velocity, namel
(5.16 We conclude (since a ex rt 2 ex x 2 ) that the density contrast of matter grows as
(5.16
independently of the wavelength, which is the conclusion we had reached from Newtonian analysis for sub-Hubble modes [see (5.22)]. The constancy of the gravi tional potential in a matter-dominated Universe and the fact that the density contr cannot grow faster than the scale factor led Landau in 1946 to the conclusion that gravitational instability in an expanding space-time could not explain the large-sc structures observed in the Universe [38]. Indeed , if the initial fluctuations are sim thermal fluctuations, they can only grow, according to (5.167), by a mere factor 10 4 during the matter-dominated era, which is far too small to explain the existen of the present fluctuations. We thus need to find a mechanism capable of generat perturbations of the order of 10- 4 at the beginning of the matter-dominated era order for those to have enough time to grow non-linear by the current epoch.
Evolution
275
5.3.3.3 Effect of curvature Curvature, behaving as a- 2, can only dominate at late times, i.e. at a time at which radiation is negligible , i.e. during the matter-dominated epoch we are considering here. The solution for the evolution of the scale factor in such a matter and curvature Universe was obtained before, in Section 3.2.1. If we only consider the matter- curvature transition (i.e. the case a = ~) , we can integrate (5.143) analytically to obtain (5.168) where we have assumed K = - 1. Note that as soon as the curvature dominates (T) > 1) then
Je m
5.3.3.4
rv
_ 2k267rB(k) - A(k) sin 3 (T)/ 2)
11- no l no
(5. 171 )
Effect of the cosmological constant
A cosmological constant can also only dominate at late times , i.e. at a time where radiation is negligible. Introducing YA = a/aA = (A//'£p)1/3, one readily finds that the transition between the matter-dominated era and the cosmological-constant dominated era occurs at redshift 1 + ZA =
(g:~r/3,
and the Friedmann equation takes the form
H
2=}(7-.(1 2) - -+YA .
Equation (5.143) then takes the form
2
YA
276
The inhomogeneous Universe
8~1
1. 2
12
1.0
10
c 8
0.8
8 111
0.6
6
0.4
4
0.2
2 2
3
6
4
2
I) / 1)0
4
8
6 I) /1)0
Fig. 5.7 Evolut ion of t he d ensity contrast 8~ in a Universe with spherical spatial se tion (left) for no = 1.2, 1.3, 1.4 and a Universe with hyperbolic spatial sections (right) f no = 0.2, 0.3, 0.4.
(5. 17
The density contrast is then given by
(5. 17
As soon as YA » 1, the potential decays as l /YA and the density contrast freezes a constant value (see F ig. 5.8). In such a Universe t he large-scale structure formatio stops at a redshift of the order of ZA. 10 , - - - - - - - - - - - - - - - - - - - - - - ,
10,---- - - -- - - -- - - - - -- -- -- .
8
8
6
6
4
4
2
2
- 1.5 -1 - O.S
0 0.5 log (y)
1
1.5
2
- 1.5 - 1 -0.5
0 0.5 log (y)
1
1 .5
2
Fig. 5.8 Evolut ion of t he gravitational potential (left) and of t he density contrast (righ in a U niverse dominated by a cosmological constant . On the right , we have shown t he thr modes k / 'H A = 1, 2, 3.
Evolution
5.3.3.5
277
Case of a Universe dominated by a mixture of m atter and radiation
Let us now consider the case of a fiat Universe (K = 0) dominat ed by a mixture of matter (w = 0) and radiation (w = ~). We again introduce the variable y , the scale factor (5 .136) normalized at equality and we have W
= :31 ( 1 + y )- 1 and
c; =:31 ( 1 + 43)-1 Y
(5 .174)
Using iP l = JtydiP / dy , <1>/1 = Jt 2y 2d 2<1>/dy2 + y/ldiP / dy and the form (5.139) of the Friedmann equation, we can rewrite the evolution (5.143) as 2
d <1> 1 ( 1 8 ) d 1 dy2 + 2y 7 - 1 + Y + 4 + 3y dy + y(1 + y)(4 + 3y)
8
1
1
+:3 4 + 3y 1 + Y
(k)2 3 keq iP = - 2y2 wf ,
(5 .175)
where keq is defined by (5 .140) . In the case f = 0, we have integrated this equation to obtain the solutions depicted in Fig. 5.9 . For non-vanishing entropy perturbation, one needs to know the actual behaviour of r in order to solve (5.140) . As we shall show later , for large wavelengths , f remains essentially constant, so that the results of Fig. 5.9 are then a good approximation for these modes in the adiabatic case.
1.0 0.8
r~-~~~~====d
0.6
cp 0.4
0.2 0 .0
- 0.2 -2
-1
o
1
2
3
log (y) Fig. 5.9 Evolution of the gravitational potential in a Universe dominated by a mixture of matter and radiation, assuming that r = 0 for k / k eq = 10- 3 , 0.1, 1, 2, 10, 100. This figure should be compared with Fig. 5.1 2 where the time variation of the entropy is taken into account . It is interesting to note that in the regime k/ keq « 1, this equation can be solved analytically. Indeed , setting rjJ = y 3 iP / VI + y , we easily obtain diP / dy ex y2( 4+ 3y) /( 1+ y)3 /2 which can be integrated to give
278
The inhomogeneous Universe
<1> =
1~~3
(16Jl+Y + 9 y 3
For a super-Hubble mode, we recover <1>(y of the result (5 .159). 5.3.4
»
+ 2y2 -
1)/<1>0
8y - 16) .
(5.17
= 9/ 10, which is a particular ca
Mixture of several fluids
In the actual cosmological context, we should consider several fluids , for instanc baryons, dark matter , radiation , etc. These different fluids interact through gravi but can also be coupled by non-gravitational forces. Figure 5. 10 summarizes the ma components that should be considered and their potential interactions. Matter
! Dark
Fig. 5.10 The different cosmological fluids and their interactions . Electrons and baryo are coupled via the Coulomb force and photons interact with charged matter via Compt scattering. Dark matter only interacts gravitationally with ordinary matter. It is not exclud that dark energy interacts with ordinary matter or with dark matter in an as yet unkno way.
5.3.4.1
Description of the interactions
In the most general coupled case, the conservation of the energy-momentum tensor only valid for the total energy-momentum, whereas the individual energy-momentu tensors satisfy conservation equations of the form
(5.17
where Q~ represents the forces acting on the component a of the fluid . These term must satisfy the constraint
(5.17 which is simply the act ion-reaction law. It can be derived from the Bianchi identit that imply the conservation of the total energy-momentum tensor.
Evolution
279
To lowest order, space-time symmetries imply that (5.179) so that Q~Q ~gf.tV = - Q;. Note t hat (gf.tV + uf.tuv )Q~ = 0, so that the symmetries of the background space-time imply that there cannot exist any force between the fluids, at the background level. Taking the couplings into account, the conservation equations generalize to (5.180) and we find (5.181) where Wa and c~ are , respectively, the equation of state and the sound speed of t he component a. At linear order in perturbations, Q~ can be decomposed as Q~ = Qau~
+ FI:,
with
Fl:u~ =
O.
(5.182)
We define af!: = (8t + uf.tuv )F::, so that f2 = 0 and f~ can, as usual , be decomposed . -i . into a scalar and a vector as f~ = fa + D' fa. Under a gauge transformation of the form (5.57), these quantities transform as (5. 183) and f~ is gauge invariant , since it vanishes at the background level. T hus, we can define the quantity (5 .1 84) 8QN a = 8Qa + Q~(B - E'), which is gauge invariant. With t hese notations, the continuity equation (5.125) and the Euler equation (5 .126) for t he component a generalize as
)' = - (IlVa (~ 1 + Wa
3W') -
3H~ra + a [8Q~ + Qa (<1> - 8~)] , 1 + Wa Pa + Fa (5.185)
and
5.3.4.2 System of fluids interacting gravitationally We consider the simplified case in which a set of fluids interact only through gravitation. In this case, Qf.t = 0 for each fluid and the energy-momentum tensor is thus the sum of t he energy-moment um tensors
280
The inhomogeneous Universe
Each energy-momentum tensor can be decomposed as in (5.69) so t hat t he total den and pressure can be defined as
(5.1 a
a
a
We have
(5.1 a
and
(5 .1
For the sake of simplicity, let us consider a system of two fluids. It will be descri by two density contrast s, rSa and rS b , and two velocity fields , v~ and vt . It is conven to transform these variables into a set of variables, rS and v, describing t he proper of t he total fluid and a set of variables S ab and V;b' describing their relative propert T he definitions (5. 187) imply that
(5.1 a
a
for the scalar modes. These quantit ies satisfy the cont inuity equation (5.125) the Euler equation (5 .1 26) . T he entropy perturbation, r , is defined as before by relation (5.78), so that , for such a mixture of fluids, it takes the form
(5. 1 a
a
T he first term corresponds to the entropy contribut ion from the components of mixture and the second term , which does not vanish even if r a = 0 for each compone represents the entropy of mixing. One can check that t his second contribution is ga invariant. The E instein equations take the general form obtained previously but fo r the qu tities describing the total fluid . In particular, both (5.117) and (5.118) can be rewrit in the form
(ll + 3K ) \[I =
~ 7-{2 L narS~,
(5.1
a
\[I -
L na
W a7Ta ·
(5.1
a
Now,
Sab
and
Vab
are defined , respectively, by
Sab
-_
~ 1 + Wa
_ _
rS_b_
1 + Wb
and
Vab = Va - Vb,
(5. 1
which are , by construction , gauge invariant . Using the relations (5 .1 90), we ea obtain the inverse relation
Evolution
281
(5.195) and the analogous expression for bb is obtained by interchanging a and b. The evolution equations for the quantities Sab and Vab are obtained by combining the continuity equation in the form (5.127) and the Euler equation (5.130) for each fluid. We get (5.196)
V~b = - 7-{Vab -
(c; -
- [~(6. +
c~)
1:
C
3K)1fab
W - [C;(l
+ rab]
+ Wb) ~ + c~ (1 + w a) ~a]
1
:b
w
(5.197)
,
with the definitions (5.198) (5.199) To completely solve t he evolution of a system of two interacting fluids , we thus need to solve the evolution (5.143) for the gravitational potential, where r is now given by the relation (5.191) , together with the pair of equations (5.196) and (5.197): the relation C (5. 192) then gives b . 5.3.4.3
Adiabatic and isocurvature modes
Two different types of initial conditions can be distinguished for a system of two fluids: we can decide that t he entropy perturbation vanishes initially (adiabatic initial conditions) , or, on the contrary, arrange that t he gravitational potential initially vanishes, C i.e. 'balance' the density perturbations so that b = 0 (isocurvature initial conditions). We thus define the two limiting regimes ~
adiabatic: it is assumed that
r
=
Sab
=
0, from which we deduce that (5.200)
~
isocurvature: in this case, we impose that \IT
= 0 and thus we have (5 .201)
Since the evolution equations are linear, any system can always be decomposed into a linear combination of an initially adiabatic system and an initially isocurvature system.
282
The inhomogeneous Universe
5.3.4.4
Mixture of two perfect fluids
The discussion in the previous section is very general. In practice, we are interested simpler special cases. Let us consider that of a mixture of two perfect fluids (r a = 1fa = 0) with purely gravitational interactions. Using (5. 188) and (5.189) , the re tion (5.191) reduces to W
+ wa) (1 + Wb) ( 2 _ 2) S r -- _ DaDb (1D2(1+w) Ca Cb ab, .
c
so that a closed system can be wntten for 0 and Sab, namely <1'>/1
+ 3H (1 + c;)
<1'>'
+ [2H' + (H2 - K) (1 + 3c; )]
~H2 DaDb (1 + w a ) (1 2 D (1 + w)
+ Wb)
(
<1'> -
2 _ 2) S
ca
cb
c; ~<1'>
=
(5.202
ab,
(~+ 3K) <1'> = ~H2DOc , S/I
ab+
HS' _ Da (1
ab
(5.203
+ w a) c~ + Db (1 + Wb) c~ ~S _ ( 2 _ 2) ~Oc ab- Ca
D(l + w)
Cb l+w'
(5.204 This system reduces to a system of two coupled second-order differential equation the gravitational potential is eliminated from (5.202) by differentiating (5.203).
5.4
Power spectrum of density fluctuations
We will now be interested in t he power spectrum of dark-matter density fluctuatio For this , and for the sake of simplicity, we assume that all m atter is dark (i.e. DbO Dco ~ Dmo ~ 1). Moreover, we restrict ourselves to a flat Universe (K = 0 and D = with no cosmological constant. The Universe thus contains, by construction , only da matter and radiation. We choose the normalization ao = l. As before, we introduce t he scale factor y normalized at equality [see (5 .136)] that the Hubble law takes the form (5.139). The value of y today is aO
Yo = -
a eq
4
= 1 + Ze q ~ 2.4 x 10 h
2
[see (4.8) ]. The Friedmann equation evaluated today makes it possible to compute t numerical value of the mode entering the Hubble radius at the time of matter- radiati equality (5. 140) , namely k eq
Dmoao 2 - 1 2 - - - = 0.072 Dmo h Mpc ,
= Ho
a eq
(5.20
corresponding t o a wavelength -1
- 1
14
k eq = H eq = ~h2
mD
Mpc.
(5.20
Power spectrum of density fluctuations
283
In general, a given mode k enters the Hubble radius when k = H, and the Friedmann equation (5.139) allows us to determine that this occurs at a redshift
y. (k) = 1 + v'1+8k - 2'
-) 1 + z. ( k
4k
Yo = -(_) ,
y. k
(5.207)
where we have introduced the reduced wave number
k k =-
(5.208)
k eq
for fur ther notational convenience. 5.4.1
Two equivalent approaches
We can follow two approaches to study this physical system . The first is to consider the 4 equations of evolution for the density contrasts and the velocity perturbations of each fluid , and the Poisson equation. The second is to use the results of the previous section for t he total density contrast and the entropy perturbations. We will use one or the other formulation depending on the sit uation. 5.4.1.1
Evolution of the den sity contras ts
Our system is composed of a mixture of radiation and matter that only interacts gravitationally, so that the system of equations is deduced from (5.125) and (5.126). In Fourier modes, they reduce to u rn =
,N'
k 211,m
N'
4 2
+ 3,*,If... ' ,
(5.209)
,
6 r = "3k Vr+4<1> ,
(5.210)
V;~1
(5.211)
= - HVm - <1>,
,I N V; = - <1> - :i6r,
(5 .212)
where we have used t hat W = <1> since 1f = O. Another equation should be added to determine the gravitational potential and we choose to use the Poisson equation (5.213) or (5. 119) in the form (5.214) These two equations can also be used in a combined way to express <1> in terms of 6% and 6~ only.
284
The inhomogeneous Universe
5.4.1.2
Evolution of the gravitational potential and of the entropy
Using the expressions (5.174) to express c; and w , the system (5.202)- (5 .204) reduc to
. (
1
8 ) 2y + y(1 + y)(4+3y)
c
/j
.. s+
2 (
<1.'>+ 7 - 1 +y + 4+3y
4 ( k)2
=-3
3y + 2 · 2 S = -2y(1+y) 4+3y
keq
(
k - ) keq
= (4
+ 3y)y2
/j
y2 1 + y ,
2 (
/j
e
C
-
YS)
1 +y (5.215
(5.216
y ---S) 1+y '
(5.217
with S = /jm - ~/jr ' We have also assumed that for a Universe with Euclidean spat sections and vanishing cosmological constant (K = A = 0) , the matter and radiati density paramet ers can be rewritten simply in terms of t he rescaled scale factor y
y
and
Om= - 1 +y
1 Or = - - . 1+y
(5.21
This system generalizes (5.175) to the case where the entropy perturbation is no long neglected . The density contrast /jN is then given by
(5.21 and the density contrasts of each fluid are /j
_ 3(1 + y)/j /4+S 1 + 3y/4 '
/j =
and
m -
r
(1
+ y)/j -
yS
1 + 3y/4
.
(5.22
5.4.1 .3 Initial conditions
To solve the systems (5.209) - (5.212) or (5.215) - (5 .217), initial conditions for the p t urbations need to be imposed (see, for example, Ref. [39]). These initial conditions a fixed deep within the radiation era (Yi « 1) and for super-Hubble modes (k/ Hi « As seen earlier , we can distinguish two types of initial conditions:
- adiabatic: the super-Hubble modes are assumed constant (in other words t decaying mode has had enough time to decay) and, by definition , we have,
<1.'>'
=
0,
S = 0,
S' = 0,
which fixes everything for the system (5. 215)- (5.217) . The Poisson equation i plies that C
2
2
/j t =- -x 3 t , where
Xi
C
C
/j r,i = /j i,
== k/ H i . (5.214) then implies that
(5.22
285
Power spectrum of density fluctuations
1
v."i =
kV,;t = --x
V;n ,i
= Vi,
(5.222)
and (5.219) that
O~,i = O~,
(5.223)
which completely fixes the initial conditions for the system (5.209)-(5.2 12) . - isocurvature: we still assume the super-Hubble modes to be constant and the definition yields
S' = o.
The Poisson equation implies t hat (5.224) so that (5 .214) t hen implies V;· ,i
=
Vm ,i
= Vi = 0,
(5.225)
while (5.219) provides (5.226) which again completely fixes the initial conditions for the system (5.215)- (5.217). 5.4.2
Different regimes
The behaviour of the solutions of t hese systems mainly depends on t he value of the wave number k that determines the time at which the mode becomes sub-Hubble. In order to have an idea of the orders of magnitudes , the following Table gives the values of y and of the redshift at the time a given mode becomes sub-Hubble for several selected values of k. k (hY*
1
Mpc)
10 5
X
4.75
10- 5 X
5
X 10- 4
108 4.75
X
5
X
10 7 4.74
10 X
106
5.1
X
4.63
10X
10 -
0.1
1 3
2
105
6 .4 3.7
X X
10 104
1
2
26 9.05 x 10 2
10- 3 2.5
X
10 3
8.4
We t herefore have four kinds of behaviour to take into account, depending on whether the mode is (i) super-Hubble today, (ii) has become sub-Hubble during the matter-dominated era, or (iii) during the radiation-dominated era , or (iv) whether it has always been sub-Hubble. 5.4.2.1 Super-Hubble m odes
Let us first consider long-wavelength modes. In practice, one can neglect all the t erms that involve a Laplacian, which are proportional to k 2 . Equation (5.196) then tells us that the entropy remains constant. Actually, we can also reach this conclusion by
286
The inhomogeneous Universe c
noting that (5. 216) implies that 5 ex. k 2 iP . In the set of equations (5.215)-(5.217) , t t erms k 2 iP and k 2 S are negligible in comparison to , respectively, it! and S. We th infer that in (5.217) , k 4 iP / k;q « 5, which implies that S is const ant. c
Note then t hat using the transformation (5 .87) to introduce 5r which is of the ord c of 5 ex. k 2 iP in the radiation era, the two equations (5 .211) and (5.212) imply that
and therefore that v;.m ex. a- 1 during the radiation era. In the case of adiabat ic initial conditions, S = 0 and the solution can be obtain ana lytically in the form (5. 176) . Deep enough into the radiation era, both the gra tational potential and the total density contrast are constant . In particular ,
whatever y. In the radiation era, we obtain 5~ = - 2iP(y
and in the m atter era (y
»
»
1),
1)
9 iP(y » 1) = 10 iP(y
5~(y
«
«
12 1) = - 5 iP(y
1),
«
(5.22
1) ,
To illustrat e the effect of the initial conditions, let us now consider some isocurv ture initial conditions and let us assume t hat initially S = S i 1= 0 and iP = o. As have seen S remains const ant. Deep in the radiation era, (y « 1) , 5~ '" 5N = 2iP = and thus 5~ = S i . Note then that for super-Hubble modes , since iP and S are consta iP = - 2S is a particular solution of (5.21 5). The general solution satisfies t he init condition iPi = 0 and is thus obtained by adding this particular solution to the gene solution (5.176) where the constant iPo should be taken equal to iPo = 2Si , iP
= 2Si3 lOy
(16Jl+Y + 9y3 + 2y2 -
In the matter-dominated era (y conclude that , for y » 1
»
1) , 5 ~ '" 5N
8y - 16) - 2Si .
= - 2iP and iP(y » 1) = - S;/5. W
1 iP = -5 S t,
5.4.2.2
(5.22
(5.22
Modes entering the Hubble radius in the m atter-dominated era
As can be seen in Fig. 5. 11 , even though the mode is sub-Hubble from a redshift the order of z* '" 8.5 , it is always constant. As~ually, to est ablish this result , we h
Power spectrum of density fluctuations
287
to assume that the Laplacian term was negligible in the evolution equation for the gravitational potential. This is legitimate as long as c~ k 2 « 7-{2 , i.e. if
2y2
-2
---,-----:--:'-------,--,- k « l. 3(1 + y)(l + 3y/4)
(5.230)
For the modes represented in Fig. 5.11 , k = 10- 3 a nd this ratio is still equal to 1. 7 X 10- 4 today. One can check that this ratio is of order unity for k rv keq so that for all the modes entering the Hubble radius in the matter-dominated era, the 1.0 -
1.00
-
0.75
0.5 0.01'--------------1
0.25
-0.5
0.00
-1.0 I------~
-2.0
~---...-::_:_-=--I -2
\
\ \
'"'----~~~~~~~~~......::.=
-4
-~~~~""----------
0.25
-1.5
- 2.5
0.50
o
2
4
\
0.50
\ \ \ \
0.75 -4
-2
o
2
4
y
y
Fig. 5.11 Evolution of the total density contrast bN (thick solid line), the entropy S (thick dotted line) and gravitational potential (thick dashed line) as well as the two contrasts b;' (thin dashed line) and b);, (thin solid line) for a mode k = 10- 3 assuming either adiabatic initial conditions (
gravitational potential remains constant, put aside the small variation due to the jump of the equation of state from ~ to 0 (see Fig. 5.12). The Poisson equation then tells us that c 6rn ex: a. (5.231)
5.4.2.3
Modes entering the Hubble radius in the radiation-dominated era
When the Universe is dominated by radiation, the gravitational potential is determined by t he density fluctuations of radiation. It is therefore given by the general solution (5 .161) and its evolution is represented in Fig. 5.5. Thus, as soon as the modes becomes sub-Hubble during the radiation-dominated era, the gravitational potential is subject to damped oscillations and t ends rapidly to O. The dark-matter fluid density perturbations are then obtained by solving their evolution equations with the gravitational potential determined by the solution (5.161), (5.232) where the Hubble parameter is given by 7-{ = Tf The solution of this equ ation is of the form
1
deep enough into the radiation era.
288
The inhomogeneous Universe 0.00 r== : : : : : : - - -----::-====::-1
-0.05
e
e
-0.10
-0.15
-3
-2
1
o
2
3
-0.20_~3--_-'::-2~~--70-~-..:::>2:-.--:;J3
y
y
Fig. 5.12 Evolution of the gravitational potential for k = 10- 2 , 0.1, 0.5,1 Mpc- 1 for ad batic (left) and isocurvature (right) initial conditions . These figures should be compared wi Fig. 5.9 for which the entropy was neglected.
(5.23 where the particular solution is obtained in the integral form
(5.23
When y« 1, 5N ~ 5~ ~ 2i. Assuming adiabatic initial conditions (5% = 35~/4) the 5~n ~ 3d2. It turns out that the contribution of the particular solution is negligib so that we are left with A = 3d2 and B = O. As shown by the solution (5.161) , the gravitational potential decays rapidly soon as the mode becomes sub-Hubble and is constant before. The integrand vari significantly only around r/ = 1/ k , so that the upper bound of the integral can extended to infinity in the particular solution. This means that the above integral c be performed to give (5 .23 where the constants
(5.23 depend on the function j, itself given explicitly by
obtail).ed by using the solution (5.161). Numerically, we obtain A "':' that the general solution takes the form
~6
and B "':' 9
(5.23
Power spectrum of density fluctuations
289
Obtaining the exact value of these coefficients would of course require a more careful calculation. To conclude, let us recall that
O%~o~cx lna
(k'T/ » 1);
(5.238)
during the radiation era (see Fig. 5.13 for a numerical integration).
1.4
25
1.3
20 zE
;zoE
'<> 15
'<> 1.2
10
1.1
5
1.0 r--~-----150 - 1.25 - 1.00 - 0.75 -0.50 -0.25
y
Y
Fig. 5.13 Evolution of the matter density contrast 0;:' for k = Ih - 1 Mpc- 1 (solid line), k = 5h- 1 Mpc- 1 (dashed line) and k = lOh- 1 Mpc- 1 for adiabatic (left) and isocurvat ure (right) initial conditions. These modes become sub-Hubble during the radiation-dominated era.
5.4.2.4
Sub-Hubble modes
Let us now consider the modes that were sub-Hubble during the radiation era and let us try to evaluate how the perturbations of the matter fluid are affected by the radiation- matter transition. For these modes , we have seen (Fig. 5.6) that t he density contrasts were the same independent of the gauge. For any sub-Hubble mode during the radiation era, we have determined that Or is roughly constant , whereas from the previous section we know that Om ~ In a . Even though Pm is small compared to Pr, the contribution of the matter density fluctuations can dominate over those of radia tion in the Poisson equation, since Pmom - ~ ylny. Pror We will thus focus on the regime where Or « Om· The total density contrast is then given by y
0 = 1 + yOm, and the PoiSson equation takes the form k 2 if>
= _ ~k2 0 4y eq m,
(5 .239)
as long as the velocity contributions can be neglected. We now use the two equations (5 .209)- (5.210) rewritten in terms of y. After differentiating (5.209) and eliminating
290
The inhomogeneous Universe
Vm using the equation (5.211) and 11m again with (5.209) , we obtain a second-or equation for b involving
bm
+
2 + 3y
2y(y
3
.
+ 1)
bm
-
2y(y
+ 1)
bm
=
(5.2
O.
This equation has an affine solution and the second solution can be obtained using method of variation of the parameter. One can check that two independent soluti are and
D _ (y)
In the matter-dominated era (y a decaying mode since
»
1 = D +(y) In ( vr+Y+ vr+Y ) - 2J1+Y. (5.2 1+ y - 1
1) , they correspond, respectively, to a growing a
This solution can be matched to the solution (5.233)- (5.235) at the time where mode enters the Hubble radius at y.(k) (see Fig. 5.14 for a numerical integration) 1.50 r - - - - - - - --
2.5
--"""T---.
2.0 1.00
~ 15 ~.
zE ~
c
c
.,.',
.';'
- 0.5
0
0.5
1.0 Y
1.5
2.0
- 0.5
0
0.5
1.0
1.S
2.0
Y
Fig. 5.14 Evolution of the matter-density contrast b;;; for k = Ih- 1 Mpc- 1 (solid lin k = 5h- 1 Mpc- 1 (dashed) and k = lOh- 1 Mpc- 1 for adiabatic (left) and isocurvature (rig initial conditions.
5.4 .2.5
SummaIY
In the simplified example of a Universe containing only dark matter and radiat interacting gravitationally, the previous study shows that the behaviour of
Power spectrum of density fluctuations
291
depends on the value of t he mode k and on the era during which the mode becomes sub-Hubble. However, this analysis neglects four important effects: 1. Matter contains baryons that are coupled to radiation by Compton scattering.
We describe this coupling and its effects on the perturbations in Chapter 6. 2. We have described radiation using a fluid approximation. This description is very rough and should actually be extended using a kinetic approach and radiation should be described by a Boltzmann equation. Such a description implies a hierarchy of equations, of which the fluid equations are the two first moments . This will be described in det ail in Chapter 6 that is dedicat ed to the study of the cosmic microwave background. 3. The recent evolution of pert urbations can be affected by the existence of curvature or of a cosmological constant. These effects were illustrated in Figs. 5. 7 and 5.8. 4. Due to the presence of neutrinos, 1> i- 1¥, which changes, for instance, the ratio 9 j 10 for the variation of the gravitational potential on large scales. 5.4.2.6
Transfer fun ction
This simplified study, nevertheless, allows us to understand the general shape of the large-scale structure power spectrum. To see this, define the transfer function (5 .242) where ai is an initial time and D +(a) the growth function (5.24). Note that ai is arbitrary as long as it corresponds to a time before any scale of interest has become sub-Hubble. This transfer function characterizes the modification of the initial matter power spectrum during the evolution a nd depends on the cosmologica l parameters and on the matter content of the Universe. The term D + has been added by hand because during the matter era 61> = ~H2n D +(a) _0_ 2 0 rnO a D +(a) and the factor D +(a) j a is approximately equa l to unity during the matter era (Fig. 5.1). The definition (5 .242) implies that 1>(k, a)
= T(k, a)1>(k, ai).
The observable modes today were initia lly super-Hubble and the theoretical models of the primordial Universe allow us to predict the matter power spectrum of these modes. This implies that the gauge in which the initial density contrast o(k , a = ai) is considered should be clearly specified. Actually, since for these modes 1> is constant, we will have the same transfer function if we use OC in the definition (5.242) since the two density contrasts are related to the gravitational potential by the same Poisson equation and we t hus have (5.243)
292
The inhomogeneous Universe
and
(5.2
since P8(k , ai) = k 4 pif>(k , ai). The approximate shape of the transfer function can be deduced from our previ analysis (see Fig. 5.15). Indeed , neglecting the logarithmic growth during the radiat era, the modes that became sub-Hubble during this era are frozen and have not gro between the time they entered the Hubble radius and equality. Their relative amplitu compared to the long-wavelength modes (that is k < k eq ) is thus lowered by a fac (k eq /k)2. We thus expect the limiting behaviour to be T(k , ao)
~
(k « keq) ,
1
T(k , ao) ~
2
keq
(
k
(5.2
)
This result should be compared with Fig. 5.16 obtained from a numerical computat that includes in particular baryons and their coupling to the radiation. Note also t the transfer function will depend on the kind of initial conditions in the primord Universe (adiabatic or isocurvature). Different approximations for this function ex in the literature [19]. As an example, for a cold dark matter model , one has
T(q)
=
2 q In(\+3 .34 ) [1 + 3.89q + (16.1q)2 + (5.46q) 3 + (6.71q)4j - l /4 , . 4q
T(q) = (5.6q)2
124} - 1/1.24
{
with q = k/(fMpc initial conditions. 5.4.3
(5.2
1 + [15q + (O.9q)3 /2 + (5.6q)2 ] . 1
)
,
(5 .2
and f = DmOh2 , respectively, for adiabatic and isocurvat
Some refinements
To close this chapter, we give the complete set of equations, in the fluid approximati for a Universe containing photons, neutrinos , dark matter and baryons. We encour the reader to write a program for this system and to try to reproduce the curves of t chapter and to understand the different effects neglected in the previous presentati
5.4.3.1
Dark matter
Dark matter is governed by the equations used all through this chapter ,
(5.2
v; =
-JiVe -
where we have maintained the distinction between
5.4.3.2
(5.2
<1> , <1>
and
iji.
Baryons
Electrons and nuclei have the same density contrast since the electromagnetic for requires the charge density to be extremely close to zero. Their conservation equatio
293
Power spectrum of density fluctuations
~
h
T-l
~--~~l'-------~------~--~ Y
7J=k +--
y=l
Yo
k
Radiation era - - - + + - Matter era
Fig. 5.15 (left): The modes that become sub-Hubble during the radiation era (k > keg) remains almost constant (neglecting the logarithmic growth) from 'f/ rv l / k to 'f/ rv l / keg. (right): The amplitude of the d ensity contrast on these scales suffers from a lack of growth of the order of (k eg / k) 2 For k > keg we thus expect the transfer fun ction to behave as T rx (k eg /k)2, while it remains of order unity for k < keg .
Fig. 5.16 Transfer function for different scenarios . The general form of this transfer function can be understood from the analytical study we have carried out, d . (5 .245). From Ref. [8J.
are the same as for dark matter but their Euler equation will be different because the Thomson scattering couples them to photons. The momentum exchange between the baryon and the photon fluids should thus be taken into account. It can be shown [41 ] that the force that enters the Euler equation is
294
The inhomogeneous Universe
(5.250
where O"T is the Thomson scattering cross-section and ne the free-electron density. The speed of sound does not vanish and can be computed assuming that we have monoatomic gas of mean molecular mass J.l (it is actually the mean mass of electrons protons and helium atoms) at a temperature Tb
c~
=
kBTb (1 _ ~ dlnTb ) J.l 3dlna '
(5.251
neglecting the time variation of J.l (actually j.t is important only during the recom bination period for which the baryons contribute very little to the pressure of th photon- baryon fluid , so this turns out to be a good approximation). To obtain th temperature evolution equation, let us start from the first law of thermodynamic (see Ref. [42]) 5Q = ~5(Pb/ Pb) + Pb5(1/ Pb) with the heat variation rate given b Q' = 4(p, / Pb)aneO" T (T, - Tb). We conclude that the evolution of the baryon temper ature is governed by
(5.252
Since Pb « Pb one can neglect Wb and c~ in the evolution equations apart from th term c~ 6,5~ . We eventually obtain
5~'
V~ 5.4.3.3
=
k2Vb
= -
+ 31l"
,
[{Vb - 1> -
(5.253
c~5~ + ~ P, aneO"T (T, 3 Pb
- Tb) .
(5.254
Photons
As will be seen in the next chapter, radiation should be described using a Boltzman equation, which will have the effect of replacing the photon and neutrino propagatio equations by two hierarchies of equations. Here, we only refine our fluid descriptio by adding an anisotropic pressure term , Jr" and the coupling to the baryon fluid
(5.255
(5.256
We now have a force t erm stemming from the effect of the baryon fluid and 0", == Jr,/6 We must accept without proof at this stage that the evolution equation for 0", is
(5.257
Actually, (5.256) contains the contribution from other multipoles of the Boltzman hierarchy and t he fluid version is thus only a truncation of this hierarchy.
Power spectrum of density fluctuations
5.4.3.4
295
Neutrinos
As stated above , neutrinos should also be described by a Boltzmann equation. They follow equations identical to those describing photons apart from t he absence of coupling with the baryons. We thus have ON' v
= i3 k 2 v:v + 4w' ,
,
I N
Vv = - 4"0 v - with O"v == of 0" v is
7r v
+ k 2 0"
(5.258) V ·
(5.259)
/6. As for the photon, we accept without proof that t he evolution equation '=-~V: 15
O"v
V·
(5.260)
The more general case of massive neutrinos is described in detail in Ref. [44] .
5.4.3.5 Einstein equations The Einstein equations reduce to (5.261) and (5.262)
5.4.3.6 Initial conditions Just as in the previous sections, we must fix the initial conditions for the super-Hubble modes in the radiation era. For adiabatic initial conditions, we obtain (see, for instance, Ref. [39])
As before, we have (5.264) from which we deduce that (5.265) and (5.266)
296
Th e inhomogeneous Universe
5.4.3.7 Tight-coupling approximation
Note t ha t befo re decoupling, Vb = V')' as the Compton scattering strongly couple bot h species . Aft er decoupling, the Universe becomes almost neutral and both fl uid are decoupled. This system can t hus be solved in the 'tight-coupling' limit where on goes from one regime to the other almost instantaneously (see, for instance, R ef. [43] Befo re decoupling we t hus have Vb= VI'= V,
5.4. 3. 8
(5.267
Conclusions
The syst em of equations given in detail in t his section leads to a more realistic pictur of the evolution of cosmological perturbations. While remaining in t he framework a fluid description, it now takes into account t he departures from a perfect fluid fo photons and neutrinos via t he introduction of t heir anisotropic pressure . T he tw gravit ational potentials are t hus no longer equal. T he coupling between baryons an photons has been included , but its derivation requires a kinetic approach. T his kinetic description of rad iation will be discussed in t he next chapter in whic we will not only justify t he unproved equations but also provide a more precise versio of these equations. One should a lso describe the dynamics of recombinat ion t o hav access to the t ime variation of the electronic density and go beyond the strong- couplin hypot hesis. T he resolution of such a system allows us to obtain t he transfer fu nction of F ig. 5.1 to a good precision for a la rge class of cosmological models .
5.5
The large-scale structure of the Universe
Up t o now t his chapter was essentially focused on the linear regime. In Section 5.1 we have alluded to the slightly non-linear regime. We now briefly describe what ob servations tell us on t he matter power spectrum . 5. 5.1
5.5.1.1
Observing the large-scale struct ure
Galaxy catalogues
The measure of t he matter power spectrum is mainly based on t he construct ion o large galaxy cat alogues. R ecently two large catalogues , SDS S (Sloan Digital Sky Su vey) [45] and 2d fGRS (2-degree field Galaxy Redshift Survey) [46], have drast icall increased both the number of observed obj ects and the sky coverage. SDSS cove around a quarter of t he sky and measures the position and absolut e brightness o around 100 million celestial object s and the distance of more than one million galax ies; 2dfGRS determines the position and redshift of more t hat 250 t housand galaxie These catalogues probe the struct ure of t he Universe up to a redshift of around 0.3 A sample of the cata logue construct ed by 2dfGRS is presented in F ig. 5.1 7. The link between these observations and the three-dimensional power spectrum o dark matter is not direct. Indeed , only luminous matter , i.e. baryonic, can be observed In order t o derive any relevant conclusion as t o t he total matter distribution , w
The large-sca le structure of the Universe
297
2dF Galaxy Redshift Survey
~
.....
.c~
~
~
4° slice of 82821 galaxies
Fig. 5.17 The distribution of galaxies in a 4 wide range of t he catalogue 2dfGRS. These observations of the large-scale structure of the Universe are at the basis of the measurements of the matter power spectrum. 0
should convince ourselves that this luminous matter is a good tracker of the darkmatter distribution. This implies t hat the density contrast of luminous matter should be proportional to that of dark matter. This coefficient of proportionality, b, is called the bias parameter [12, 18], Oluminous
= boc '
(5.268)
The validity of this hypothesis, the possibility that b m ay depend on scale and numerous other problems have been studied numerically and observationally [12]. Another problem lies in the fact that observations determine with a great precision the position of objects in the sky and their redshift. In redshift space, t he radial separation includes information not only on the position of the galaxy but also on its velocity. Thus, at small scales, a spherical structure in gravitational collapse would appear, in the redshift space, as an ellipse flattened in the radial direction. This distortion in redshift space due to proper motion, must be corrected.
5.5.1 .2 Constraints on the power spectrum The large galaxy catalogues allow for the reconstruction of t he matter power spectrum over around two orders of magnitude. The linear spectrum can be obtained from the anisotropies of the cosmic microwave background (Chapter 6). On smaller scales, new techniques, such as weak gravitational lensing (Chapter 7) and the study of Lyman-a forests that allow for the reconstruction of t he distribution of extragalactic gas, enable us to extend the measure by almost two orders of magnitudes in wavelength. Note that the use of gravitational lensing has the advantage of avoiding the bias problem. Indeed, t his method directly measures the distribution of the gravitational potential (see Chapter 7 for details).
298
The inhomogeneous Universe Wavelength (h - I Mpc)
10 4 ~
"
~ ~
~
1000
>.
'"
"0
8
E
g ""~
100
T
~
"~ 0
0-
10
• CMB • SOSS Galaxies a bundance • Gravita ti o nal lensing .. Lyman - a forest
~ C lu ste r
0.0 1
0.00 1
0.1
10
Wave numbe r k (h f Mpc)
Fig. 5.18 T he matter power sp ectrum compared with various observations from linear scal (cosmic microwave background) to non-linear scales (galaxy cat alogues, here SDSS), gravit tionallensing and Lyman-a forests . From Ref. [47].
To conclude, the matter power spectrum can be reconstruct ed over around fi orders of magnitude. Figure 5.18 summarizes the state of observations and the r construction of the power spectrum performed using these different methods. No however, that , alt hough impressive at first sight , this figure cannot be directly d duced from observations. Indeed , for small scales (k 2:, O.lh/Mpc), only t he pow spectrum in the non-linear regime is observed. The linear reconstruction thus depen on a 'mapping ' between the linear and non-linear regimes (see Fig. 5.19). Moreove the reconstruction of the power spectrum from the Lyman-a forests is not direct eith and , until further data is gathered , should be t aken with caution.
5.5.1 .3 Baryon aco ustic oscillations
As we have already explained, during the radiation era photons and baryons are co pled. This implies that the photon- baryon plasma undergoes acoustic oscillations. Th radiation pressure competes with gravitation and sets up oscillations in t he photo fluid (see Fig. 5.5). The tight coupling between electrons, baryons and photons du to the Compton scattering then causes the baryons to oscillate in phase with th radiation. Thus, the whole plasma oscillates due to these sound waves.
The large-scale structure of the Universe
299
.11 =-2 "n; -1.5 b. n :;: ; - 1 on ;O
1000
Virgo ACDM
100
;;: ("\.<J
1000
100 10 0.1 0.1 IO
10
~
100
Mpc - 1
h
Fig. 5.19 (left): Comparison between the analytic mapping proposed by Ref. [51] and numerical simulations [53] . For each model, five epochs are represented. From bottom to top, 1/( 1 + z) = 0.6, 0.7, O.S, 0.9 , l. (right): The power sp ectrum in t he non-linear regime at fi ve epochs, z = 0, 0.5, l.0 , 2.0, 3.0 from bottom to top. T he solid line corresponds to t he mapping of Ref. [53] and the dotted line to that of Ref. [52] . From Ref. [53].
As we shall see in t he next chapter , t he imprint of these sound waves on the cosmic microwave background has been detected. These waves are related to a characteristic scale that corresponds to t he sound horizon. These oscillations are also imprinted in the power spectrum of galaxies. Moreover , their amplitude is suppressed by a factor of order Db rv 0.05 since they are imprinted only in the baryon fluid and not in t he dark matter. It was recently detect ed [48] in t he correlation function of luminous red galaxies from the SDSS survey. This correlation exhibits a peak at a scale of order 100 h - 1 Mpc at a redshift of about z rv 0.35. 5.5.2
Structures in the non-linear regime
The matter equations of evolution are non-linear and we have only solved them in t he linear regime. To end this chapter , we present some approaches to the study of this regime that completes the analysis of Section 5. 1.4. 5.5.2.1
Hierarchical m odel of structure form ation
To understand the non-linear regime , it is crucial to determine when a scale enters t his regime. If t he density contrast was complet ely frozen during the radiation era, then all modes would become non-linear at the same time since the growth of perturbations in the matter era is independent of the wavelength . However, there is a small growth of pert urbations during t he radiation era (Figs. 5.13 and 5.14). The earlier a wavelength becomes sub-Hubble, the more it is amplified , so that small scales will be the first ones to become non-linear. The smallest structures are therefore t he oldest. This is what is called a hierarchic m echanism.
300
The inhomogeneous Universe
At matter- radiation equality, the amplitude of perturbations is typically of th order of 10- 5 . The small wavelengths enter the non-linear regime when 6c (k , t) ~ and st art forming structures. If, at this epoch , the Universe had been observed wit a resolution larger than these scales, it would seem very homogeneous, so the densit field can always be described by a continuous field in the linear regime. The process continues and larger and larger wavelengths enter the non-linea regime. So, the small over dense regions (protogalaxies) form first before merging t form larger and larger galaxies and then clust ers. The scale of non-linearity today i of the order of 20 to 40 Mpc , which is only one order of magnitude larger than the siz of galaxy clusters (around 2 Mpc). So clusters are not distributed in a homogeneou way and form large structures , such as filaments , separated by voids the characteristi size of which is of the order of the non-linearity scale. At this scale , the structure o the Universe has the morphology of a sponge, but smoothed over a larger scale, it i homogeneous again. 5.5.2.2
Spherical collapse model
A simple analytical model of the non-linear evolution of a structure can be constructe by considering a spherical overdense region. To study its global evolution , it is no necessary to know the exact density profile since thanks to the Gauss theorem , onl the mean density contained in a sphere of given r adius is necessary to det ermine th evolution of this sphere. In a matter-dominated Universe, the time evolution of the sphere radius t akes th parametric form r = A(l - cos B) , and t = B(B - sin B) , (5.269
and the Newtoni an equation of motion, r = - G N M/r 2 implies that A3 = G N MB2 The sphere collapses to a point (r --+ 0) after a time tc = tI O=27r = 27rB. The mea density in the sphere is Pint = 3M/ 47rr 3, whereas that in the exterior space-time i Pext = (67rG N t 2)- 1 for an Einstein- de Sitter Universe. The ratio between these tw densities gives the density contrast 6 = Pint! Pext - 1. At the beginning of the collapse, B « 1 so that
r (t)
~
/3 [1 - -1 (6t)2 /3] . -A (6t)2 2
B
20
B
Initially, the sphere density contrast grows as 6 ex t 2 / 3 , in agreement with our previou analysis (5.22) in the linear regime. This model shows the following stages of the gravitational collapse. If we compar the exterior evolution to the linear evolution in a flat Universe with Dm = 1, thre epochs can be distinguished.
1. Th e decoupling of the expansion. The overdensity is a system gravitationall bound. Its expansion reaches its maximum at B = 7r and t hen decouples from
the global expansion of the Universe. The density contrast then reaches the valu 6 = [A(6t/ B) 2/3/ 2]/r 3 = 97r 2 / 16 ~ 5.55 whereas the linear regime predicts 6 = (3 / 20)(67r) 2/3 ~ 1.06.
The large-scale structure of the Universe
301
2. Gravitational collapse. If no other mechanism comes into play, the sphere collapses into a singularity at = 27f and b --> 00. The density contrast deduced from the linear analysis extrapolated until t c is equal to b = 1.69. 3. Virialization. Some dissipative mechanisms convert the kinetic energy into thermal motion. The sphere then reaches a stationary regime and stabilizes its radius.
e
This basic analysis shows that t he linear analysis is no longer valid as soon as b > 1. A more realistic approach to gravitational collapse in an expanding space-time is obtained from the solutions of the Einstein equations for a distribution of matter with spherical symmetry, known under the name of Lemaitre- Tolman- Bondi [49] (see Chapter 3). 5.5.2.3
Th e non-linear power spectrum
A more precise analysis can be obtained by N-body numerical simulations. These simulations try to solve the Newtonian equations (5.15) and (5.16) and the Poisson equation (5.5) . A detailed description of the various methods , problems and the stateof-the-art of these simulations can be obta ined in Ref. [12]. Various codes [50] of N-body simulations are freely available. From these simulations , various mappings have been established [51- 53] between the matter power spectrum in the linear regime and the power spectrum in a non-linear regime. These mappings make it possible to discuss qualitatively the deformation of the power spectrum in the non-linear regime. A fruitful hypothesis is that of stable clustering that postulates [54] that , in the non-linear regime, the regions of very high densities virialize, while keeping the mean density fixed. The correlation function of the density of these systems would then evolve as ~(r, t) ex. 1/ 15 ex. a 3 . It can then be shown [54] that , in a fiat Universe, if the initial spectrum is of the form P ex. k n , then this hypothesis implies that
(5.270) The spectrum in the non-linear regime would then keep a memory of the initial spectrum .
We can thus postulate that the non-linear effects can be described by a universal mapping inl. Indeed, the correlation function, averaged on a large volume,
- 31
~nl( X ) = 3" X
X
Y2 ~nl(y)dy ,
0
measured by numerical simulations can be parameterized by a function of the linear regime correlation function [51]. The scale transformation between both regimes can be deduced by considering a collapsing spherical halo. This halo is constituted of spherical shells each of them reaching a maximal expansion before collapsing. If there is no shellcrossing, the mass in the interior of each one of them is conserved, so that
m«
r) =
347f r 3 p«
47f
3
r) = 3 £ p«
£) = mo«
C),
where mo( < £) is the initial mass distribution in a sphere of radius £ (in the linear regime) . If we identify 1 + ~nl with the density amplification factor, then
302
The inhomogeneous Universe
X3 [ 1 +
~nl( X,
t) ]
= £3 ,
where x represents the non-linear scale and £ the associated one in the linear regim After this scale transformation , we suppose that (nl(X, t) = inl [(lin(£, t) ] . This procedure transposes easily into Fourier space. If we define
(5 .27
then the function inl is given by with
(5.27
Under the hypot hesis of stable clust ering, one can show [51J that this function mu behave asymptotically, for x » 1, as
g(D) = D+(a) , a
(5.27
where D+ is the growing mode (5 .22) of the linear regime. An analytical form inl has been proposed ; it is calibrated on N-bod y numerical simulations for ACD models [51- 53J. The existence of such a universal function can also be justified the retically [55J. Figure 5.20 represents the result of numerical simulations, while Fig. 5.19 compar this analytical mapping to the results from numerical simula tions and its effect on th power spectrum. These mappings will be useful when discussing the gravitation lensing by the large structures in Chapter 7.
The large-scale structure of the Universe
303
Fig. 5.20 Results of numerical simulations (top) for n = - 2 to l / (l+ z) = 0.2 , 0.45 and 0. 55 from right to left and (bottom) for n = 0 to 1/( 1 + z) = 0.25 , 0.63 and 0.83 . The influence of the spectrum on the morphology of the large structures (more filaments for n = - 2 and smaller structures for n = 0) can be seen with the eye. From R ef. [53] .
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[44] J. LESGOURGUES and S. PASTOR 'Massive neutrinos and cosmology' , Phys. Rep 49 , 307, 2006. [45] SDSS: http://www . sdss. ~rg. [46] 2dfGRS: http ://www.aao.gov.au/2df / . [47] M. TEGMARK , A. HAMILTON and Y. Hu , 'The power spectrum of galaxies i 2dF lOOk redshift survey' , Month. Not. R. Astron. Soc. 335 , 887, 2002. [48] D. EISENSTEIN et ai. , ' Detection of baryon acoustic p eak in the large-scale corre lation function of SDSS luminous red galaxies' , Astrophys. J. 633 , 560, 2005. [49] R .C. TOLMAN, Proc. Nati. Acad. Sci. 20, 169, 1934. [50] GADGET: Code in C described in V. SPRINGEL et al., 'GADGET: A code fo collisionless and gas dynamical cosmological simulations' , New. Astron. 6, 79 2001 , available at http://www.mpa-garching.mpg.de/gadget/right.html ; HYDRA: described in H. COUCHMAN et ai., 'Hydra Code R elease', Astrophys. J 452 , 797, 1995, available at http://coho.mcmaster.ca/hydra/hydra/hydra.html; TPM: described in P. BODE et ai. , 'The Tree-Particle-Mesh N-body Gravity Solver Astrophys. J. Suppi. 128, 561 , 2000, available at http://www.astro.princeton.edu/~bode/TPM/.
[51] A. HAMILTON et ai., 'Reconstructing the primordial spectrum of fluctuations o the Universe from the observed nonlinear clustering of galaxies', Astrophys. J Lett. 374, L1 , 1991. [52] J.A . PEACOCK and S.J. DODDS, 'Non-linear evolution of cosmological power spec tra', Month. Not. R. Astron. Soc. 280, L19 , 1996. [53] R.E. SMITH et ai., 'Stable clustering , the halo model and non-linear cosmologic power spectra', Month. Not. R. Astron. Soc. 341, 1311,2003. [54] P.J.E. PEEBLES, 'The gravita tional instability picture and the nature of the dis tribution of galaxies', Astrophys. J. 189, L51 , 1974. [55] J.S . BAGLA and T. PADMANABHAN , 'Critical index and fixed point in the transfe of power in nonlinear gravitational clustering', Month. Not. R. Astron. Soc., 286 1023, 1997.
6 The cosmic microwave background The aim of this chapter is to provide the basis for the current understanding of the physics of the cosmic microwave background (CMB) and to detail the computation of the expected signatures from various scenarios. We start by presenting in Section 6.1 a simplified version of the computation of the angular power spectrum of the temperature anisotropies in order to discuss the different contributions to this spectrum. The origin of the temperature anisotropies of the cosmic microwave background as well as the relationship between the temperature fluctuations observed today and those at decoupling are given in detail. This will allow us to understand and discuss in Section 6.2 the shape of the angular power spectrum. We then present the modern description of the microwave background theory in Section 6.3. This description relies on a kinetic approach, which will require the study of the perturbed version of the Boltzmann equation in an expanding space-time. Finally, we discuss the effects of various parameters on the angular power spectrum in Section 6.5, for both cosmological parameters (e.g. the different density contrasts, the Hubble constant, etc.) and parameters describing primordial physics. For general reviews on this topic, see Refs. [1-4].
6.1
Origin of the cosmic microwave background anisotropies
After decoupling, photons no longer interact with matter and one can assume, to a first approximation, that they follow null geodesics. A simplified approach to the cosmic microwave background physics can then be obtained by computing the total redshift (between decoupling and today) of a photon propagating in a perturbed cosmological space-time. In this simplified approach, we adopt a fluid description of matter and assume that the surface of last scattering is infinitely thin , i.e. we assume that decoupling happened instantaneously, (see Section 4.4.1, Chapter 4). This simplified presentation has the advantage of keeping the physics transparent and a more rigorous formalism, based on the kinetic approach, will be presented later in Section 6.3.
6.1.1
Sachs- Wolfe formula
The Sachs- Wolfe formula relates the present energy of a photon to its energy at decoupling [5,6] .
308
The cosmic microwave background
6.1.1.1
Propagation of a photon in a p ert urbed space-time
Let us st art by studying the propagation of a photon in a space-time with a met 9IJ.// of the form (5.52). The geodesic of this photon is a worldline XIJ.(A) where A is affine parameter. T he tangent vector, kIJ. , to this wordline satisfies the two equatio (see Chapter 1)
Notice that two conformal space-times have the same causal structure so that th null geodesics will be identical (since these geodesics define the light cones). Defini the metric gIJ. // through 9IJ.// = a 2 gIJ.//' it can be easily checked that if kIJ. is the tange vector of a null geodesic of the metric 9 , then kP = a 2 kIJ. will be the t angent vector a null geodesics of t he metric g, i.e.
where VIJ. is the covariant derivative associated with gIJ.// . Let us decompose the vector kIJ. as
(6
E is a constant 1 and the four components of bkIJ. reduce to (M, be i ). Only three these four components are independent, since kIJ.kIJ. = 0 implies that
(6
To first order in perturbations, the geodesic equation for kIJ. takes t he simplified fo
(6
Using the decomposition (6.1) , its time-like component (v
dM ds
= -
where we have set dMj ds 6.1 .1.2
AI
-
2 Ai >:l A e
Vi
= 0)
gives
- "21 hiij eAi eAj + Di B jeAi eAj ,
(6
== eioiM + ooM.
Sachs- Wolfe formula
As seen in Chapter I, see (1.122), t he frequency, or equivalently the energy, of a phot measured by an observer comoving with the velocity field uIJ. is related to its frequen at the t ime of emission by
Eo(e)
(6
1 As a simple check, remember t hat in a Friedmann- Lem aitre space-time, Uo ex a so that t he ene of a photon scales as k!'u!, = k Ouo = (/';;o /a 2 )a, that is as l /a since E = /';;0 is constant .
Origin of the cosmic microwave background anisotropies
where by
ul"
309
is the baryon velocity field. The emission and reception points are related
XE = Xo + e(7]O - 7]E),
(6.6)
where e = -e stands for the direction of observation. This trajectory is a geodesic in the unperturbed space-time. In what follows, we will work in the Born approximation in which all perturbed quantities are evaluated along an unperturbed trajectory. We infer from (6 .5) that the temperature observed in a direction e is related to the temperature at emission by (6.7) where we have used ul" = (- 1 - A, Vk + Ed and Vbi is the velocity perturbation of the baryon fluid (see Chapter 5). Since recombination is dictated by the photon- electron interaction, which is itself governed by the coefficient o-Tn e (see Chapter 4, Section 4.4.1), we can model the surface of last scattering as a surface of constant electron density, ne = const. Because of the density fluctuations , recombination does not occur at the same time at each point in the Universe (see Fig. 6.1). The time of emission is then decomposed as
Both temperatures can be decomposed , respectively, as
where TE is the spatial average of the temperature field at the time 7]E, and
To(e) = To (7]0) [1
+ 8 0 (e) ] .
8 o(e) represents the density fluctuations in a direction e compared to the average of the temperature on the sky, TO(7]o), Inserting these two decompositions into (6.7), we obtain that at lowest order in the perturbations
To a(fJE) TE - a(7]o) ,
(6.8)
which is simply the scaling of the temperature of a blackbody in terms of the redshift in a homogeneous and isotropic expanding space-time [see Chapter 4]. Using that, to first order in the perturbations, T E(7]E)a( 7]E) = TE(fJE) [1 +(T' /T)57]E] a(i)E) [1 + (a' /a)57]E] = TE(fJE)a(fJE), we infer that the temperature contrast observed in the direction e is given by
(6 .9) Again, in the Born approximation, the term in brackets being first order in the perturbations, it can be evaluated either in fJE or fJR.
310
The cosmic microwave background
Time
,,
,< ,, , , , ,
;' XE ,"
Last ',scattering " , surface
"4 ryE
~
~
Fig. 6.1 The last-scattering surface is a surface of constant electron number dens n e = const . T hus, it is not a constant-time hypersurface. The time of emission is rela to the mean t ime by T/E = fiE + cST/E.
To compute the first term on the right-hand side of the previous equation, notice that t he electron density, n e , is a function of the photon and baryon ene densities, P"'( and Pb· As can be seen from Saha's equation [6], the effect of the bary can be neglected , so that the last-scattering surface is well approximated by the surf P"'( = const. (see Chapter 4) . Stefan- Boltzma nn 's law then implies that
GE(XE, fiE) =
1
40",((XE, fiE) .
(6.
To evaluate t he contribut ion [M]~, we use the geodesic equation (6.4) that can rewritten in the integral form
[M]~ = -2[A]~ +
lO(A' - ~h~jeiej +
DiB jeiej ) ds ,
(6.1
the integral being performed along the unperturbed geodesic. We infer fro m(6 .9), (6. 10) and (6 .11) t hat
Go(e ) =
[~6"'( + A -
i
e (Vbi
+ Bi)] (XE, fiE) +
lO
[A' - G'
j -eie (DiD jE' + DiE~ - DiBj + E: j ) ] ds + f(O) ,
(6.
where f(O) is a function of the perturbation variables evaluated today. In terms . / . - 1gauge invariant quantities , after expressing eJ D j (Ei - B i ) = eJ D/P i as -i + di/ this expression takes the form
G(Xo , 170 , e) =
[~6~ + +
lO
-lO
('
i
e (DNb
+ V bi + ~i)] (XE' fiE)
+ Wi) [X(17) ,17]d17
eie j E :j [x( 17) , 17]d17,
-lo ei~:[x(17) , 17]d17
(6.13
Origin of the cosmic microwave background anisotropies
311
where we integrate along t he unperturbed geodesic that we chose to parameterize as x = Xo
+ e(1]0 -
(6 .14)
1] ) ,
so that the integral now runs over the conformal time 1]. We have also voluntarily omitted the quantity 1(0) that is a function of the observation point and cannot be measured. T his relation is known as the Sachs- Wolfe equation.
6.1.1.3
Interpretation
The relation (6.13) can be decomposed as t he sum of three t erms
(~ b~ + <J) )
8 sw
=
8 dop
= _e i (Vbi + q>i ) (X E, ryE),
8rsw =
kO [(
<J) '
(XE ' ryE) =
(~bF, + <J) + W)
+ Wi) - eiq>: - eie j E: j ]
(XE' ryE) ,
(6.1 5) (6 .16)
d1].
(6.17)
The first term only has a scalar contribution and is called the ordinary Sachs-Wolfe term, the second has a scalar and vector contributions and represents the Doppler term, while the third one contains all three kinds of perturbations and is called the integrated Sachs- Wolfe t erm. The ordinary Sachs- Wolfe t erm, 8 s w , is evaluated at the decoupling t ime, i. e. in (XE, ryE) , and contains two t erms: - the density contrast of the radiation fluid translat es t he fact that a denser region is hotter, by virtue of Stefan- Boltzmann 's law, - the gravitational potential indicates that a photon emitted in a pot ential well has an addit ional gravitational redshift (E inst ein effect ) . The t erm 8 dop is simply a Doppler effect term , coming from the fact that the emitter and observer do not have the same velocity. The integral t erm , 8rsw , depends on the photon 's history between its emission and reception and contains t he time derivative of the two gravitational potentials (in SVT decomposition) . These t erms will have a contribution coming from among other things, - structures in formation t raversed by the photon. Indeed , the gravitational potential of a collapsing structure, even spherical, is not 'seen' symmetrically by a photon that t hen acquires an additional redshift , - in a Universe with a cosmological const ant or non-vanishing curvature, we have seen t hat t he gravitational potential is not const ant in our recent past. 6.1.2
Angular power sp e ctrum
The Sachs- Wolfe equation rela tes the observed temper ature fluctuations to the cosmological pert urbations. So 8 (x o, 1]0 , e) is a stochastic variable that can be characterized by its correlation function
(6.18)
312
The cosmic microwave background
The statistical isotropy, which follows from the cosmological principle, implies that t correlat ion function only depends on the relative angle between the two directions observation, el and e2; cos 19 = el . e2. It is thus convenient to expand this correlati function in a basis of Legendre polynomials as
,,2£ + 1 C(19) = L --Ce Fe( el . e2) , e 41r
(6 .1
which defines the angular power spectrum Ce. A multipole £ corresponds appro mately to an angular scale 1r /19 (see Appendix B) so that C e is a measure of t variance of the t emperature fluctuations at this scale. If the temperature fluctuatio have a Gaussian st atistics, this function entirely charact erizes the temperature dist bution. Expanding 0(xo , 7)0 , e) in spherical harmonics as
0(xo , 7)0, e) =
L
aem (XO , 7)O )l'£m(e),
(6.2
fm
the expansion coefficients aRm (xo, 7)0 ) are then given by
aem(Xo, 7)0)
=
J
2 d e 0(xo , 7)0 , e)Yc~(e).
(6.2
We can then show, using (B.17) to express the Legendre polynomial in (6.19) and t h inverting t his relation using (B.15) twice, t hat
(6.2 m
We shall now compute this angular power spectrum as a function of the cosmolo ical perturbations in the SVT decomposition. For each mode, we can expand the te 0(xo , 7)0 , e) in Fourier modes as
0(xo, 7)0 , e) =
J
d3k
'
(21r)3/2 0(7)0, k , e),
(6.2
where t he phase exp(ik.xo) has been absorbed into the definition of 8. We infer th
(6.2 From now on, we omit the dependence of aem on (xo , 7)0)·
6. 1.2.1
Scalar modes
After expressing
XE
with (6.6) , the scalar part of (6.13) , tells us t hat
8(7)0, k , e) = [8sw(k) - ikp,Vb(k)] eik /.l6.""E +
J(fiJ'
+ ~1)eik/.l6.""d7),
(6 .2
with f:,.7) = 7)0 - 7) and where p, is defined by t he relation k . e = kp, . The co ficient exp( ikp,f:,.7)) appears in the integral since the Fourier transform contains t
Origin of the cosmic microwave background anisotropies
term (I
313
+ WI) exp[ik
. x('1])]. The relation (6 .14) then implies that exp[ik . x('1]) ] = exp(i k . xo) and we have assumed that the constant phase was absorbed into the definit ion of the Fourier coefficients [see (6 .23)] . T he same argument explains the coefficient exp( ikfJj~.'1]E) for the two first terms. Each term of the expression (6.25) is a random variable that can be decomposed as X(k , '1]) = X(k , '1])a(k) where a(k) is a random variable satisfying exp (ikfJ'!~'1])
(a( k )a*(k l) )
=
6(3)(k - k l) ,
(6.26)
so that 8('1]0 , k , e) = 8('1]0 , k, f..l )a(k) with (6.27) Using the relation (B .21) to expand the exponential factor in t erms of spherical harmonics, we obtain (6.28) with
(6.29)
Inserting these expressions into (6 .22) and then using (6 .26), we obtain
J = ~J
(2£ + 1)C[ =
L
2 2 d e 1d e2
(~:~3 8('1]0 , k, f..l d8*('1]o , k, f..l2)YRm(edYe~(e2) '
m
2 k dk
L f,m1 f2m2m
8~;) (k)8~;)*(k)
J
2 2 2 d k d e 1 d e2
The integration over k = k / k leads to a term 6f1f26m1m2' and the integrations over el and e2 to 6f1f 6m1m and 6U2 6mm2' respectively, so t hat
(6.30)
Fig. 6.2 decomposes the power spectrum of t he scalar modes into its ordinary SachsWolfe, Doppler and integrated Sachs- Wolfe cont ributions.
314
The cosmic microwave background
<0
~,
/
/ /
1000
10
\ \
~ 102
,
~
1\ 1\
__ ><'~~';>:::""" '/'<"" :\"'"
......
100
\
....
~ 10 1
~
"-
c.J 100
..........
,....,
~
+ 10- 1
-Total \ - - - Ordinary Sachs-Wolfe \ - . - Integrated Sachs-Wolfe " /', ....... Doppler
~ ~
~10-2
To tal - - - Ordinary Sachs-Wo lfe - . - Integrated Sachs-Wolfe ........ Doppler
10-3 ~~~~~~~~L-~~~~
10
t
100
10
1000
100
100
Fig. 6.2 The a ngular power spectrum a nd its t hree contributions, ordinary Sachs- Wo Doppler and integrated Sachs-Wolfe for adiabatic (left) and isocurvature (right) initial c ditions and a scale invariant primordial sp ectrum (ns = 1) , as predicted by inflation ( C hapter 8).
6.1 .2.2
Vector modes
We now perform the same exercise for the vector contributions. The decompo tion (6.23) now gives
8(1]0, k, e)
=
_ ei [Vbi( k ) + <1?i(k )]
eiklJ.~1)E
-
Jei<1?~eiklJ.~7) d1],
(6.3
The vector variables are transverse, so that their Fourier coefficients do not have a components parallel to the wavevector. They can thus be expanded in the basis (e1, e of t he space perpendicular to k as
X i (k,1]) =
L
X:>.(k,1])e?(k)a:>.(k),
(6.3
:>.= ±
with t he complex quantities e~ == el ± i e2 so that e±ei = sinB exp(±i. (k) now satisfy the relation
(6.3
so that after using the relation (B.21) to expand the exponential factor in terms spherical harmonics, we obtain
8(1]0 , k , e) = 41f
LL
(sin 7gei :>''P) ei~} (k)Yem (k )Yc~ (e)a:>. (k) ,
(6 .3
:>.= ± em
with
(6.3
Inserting these expressions in the relat ion (6.22) , using the relation (6.33) and th p erforming the integral over d 2 k, we obtain
315
Origin of the cosmic microwave background anisotropies
(2£ + 1)C[
~1
=
k 2 dk
L le~~), (k) 12 L .e l
,)..
2 11 d e sin
1geiA~YEm(e)Yc:m, (e) 12
mml
In order to compute the integral over e, first notice that sin 1geiA~ = - A2J27r /3Y1A (e) before using the integral (B.23). We can then perform the sum over m and m1 using the relation (B.25) of the 3j-Wigner symbols. This sum only keeps the terms with £1 = £ ± 1 so that the Bessel functions jE, that appeared in the expression for e~~), recombine to lead to the function [see (B.46)] .(11)( ) _
Je
x -
f1!fSi(£ + 1) je(x) . 2
x
(6.36)
We finally obtain
C[ = ~
1L le~~)(k) 12
2
A
e~~)(k) = 6.1.2.3
(6 .37)
k dk,
[V A(k) +
1
(6.38)
Tensor modes
The computation for the tensor modes is noticeably the same as that for the vector modes. The expansion (6.23) now gives (6.39) The tensor Eij can be expanded as
E ij (k,1])
=
L EA(k, 1])s7J aA(k),
(6.40)
A
where the random variable aA(k) satisfies the condition (6.33) and where the polarization tensor is defined as (6.41) In particular, this implies that stjeie j = sin 2 eexp(2iA
8(1]0, k, e)
=
47r
LL
(sin21ge2iA~) e~~)(k)Yem(k)Ye~(e)aA(k),
(6.42)
A= ± Em
with (6.43)
UNlVE!5!DAD -
~
M.-\DmO UiK':;'")TcCA C~'t,)C:A.S
I
"'WONOIY~ DE· ,
~
316
The cosmic microwave background
~
10000
~cJ
r---------------------
--------
,,
1000
, \ \ \
100 - - Scala r modes - - - - Gravitational waves (1· = 1)
10
'''' '',
.\
t
Fig. 6.3 Angu la r power sp ectrum for the tensor modes, compared to that of scalars fo ratio r = 1 between the p rimordial spectra of the two mo des.
Inserting these expressions in (6.22) , using (6.33) and then performing the integ over d 2 k we find that
(2C+ l)CJ =
~
1
2
k dk
L le~~],(k) 12 L e, ,'\
mm,
1
2 2i 2 d esin 1g e ,\
To compute the integral over e , the procedure is t he same as for the vector mod noticing t hat now sin 2 1ge±2i
(C + 2)! je(x) (C - 2)! x 2
.
(6.4
We finally obtain
1L e~~] (k) 1E~(k)jk22)(k!:::.T))dT). ceT =
2
le(T\k)1 k2dk , '\ ,e
2-. 7f
(6.4
,\
= -
6.2
(6.4
Properties of the angular power spectrum
The previous description allows us to understand the general shape of t he pow spectra of the scalar and t ensor modes. The vector modes will be neglected in t rest of t his chapter. Complementary references on t he study of t he properties of t temperature power spectrum can be found in Refs. [2 , 7] .
Properties of the angular power spectrum
6.2.1
317
Large angular scales
For large angular scales, i.e. scales corresponding to super-Hubble modes at the time of decoupling, the gravitational potential remains constant (Chapter 5). For these modes, we can estimate the angular power spectrum at small £ as a function of the characteristics of the power spectrum of the gravitational potential. This part of the angular spectrum is usually called the Sachs- Wolfe plateau.
6.2.1.1
Adiabatic perturbations
The initial conditions (5.223) imply that initially J~(O) = - 2w(0) , i.e. that the photon over dense regions correspond to potential wells, and V, = O. On super-Hubble scales at the time of decoupling, V, did not have time to grow significantly. Equation (5.255) then implies that J~ -4W remains constant. In the matter era, J ~ /4 = J~/3 = -21>/3 [(5.220) and (5.227)] so that 1
8s w = -1>
(6.4 7)
3 is the main contribution to the temperature fluctuations, in agreement with Fig. 6.2. Equation (6 .30) then implies that the angular power spectrum is the convolution of the primordial spectrum with Bessel functions (6.48)
since only the first term of (6.29) contributes significantly and D.rJE = rJo - rJLSS' where 'fiE = rJ LSS is the time of last scattering. In the matter era, the gravitational potential is constant and is related to the curvature perturbation by 1> = 3(/5 for super-Hubble modes (5.149). It follows that Pip = 9Pc,l25. Using the definition Px = k 3 Px /27r 2 , we infer that
cl
= 7r
J
A§(k)j;[k(rJo - rJLss )]dkk ,
(6.49)
using a spectrum of the form (8.227) as predicted by inflation, with the definition (8 .234) giving A§ = 4Pc,l 25. This integral is dominated by modes such as krJo ~ £ so that one expects to have Ce ex A§(k ~ £/ rJo), More precisely, for a power-law spectrum (see Appendix B),
CS ~
e-
7r A 2
ko
s( )
j
~ r (3 - ns)
[
2
23 - n s
r2
r(£+~) 2 ( n) ( 5- n) , 2 - ~ r £+ _ _ s 2
2
choosing the pivot k. = ko = 1/ rJo , and with the approximation rJ LSS we obtain
£(£ + 1)Cl
= ~A§(ko) ,
(6.50)
«
rJo. For ns = 1, (6.5 1)
which corresponds to the plateau of the angular power spectrum at low multipoles (Fig. 6.2). For large £, r(£ + a) ex £a so that
318
The cosmic microwave background
(6.5
The spectral index of the primordial spectrum influences directly the slope of t Sachs-Wolfe plateau. Also notice that (6.47) implies that Gsw = -b~/6. So the co regions correspond to over-dense regions at the time of decoupling.
6.2.1.2
Isocurvature perturbations
For isocurvature perturbations, we initially have b~(O) = (O) = 0 and VI' (O) = 0 the radiation era (Chapter 5). If S(O) is the initial entropy, then in the matter e = - S/5. For long-wavelength modes, b~ - 4\[1 remains constant so that b~/4 + + \[I ~ 2 . The leading term in the temperature fluctuation in the matter era is th Gs w = 2 = - 2S/5 ;
(6.5
see (5.229). Using the same argument as for the adiabatic modes , we obtain
(6.5
For ns = 1, we have again a plateau at large angular scales
J!(J! + l)Cf = 367TA§(ko). 6.2.2
(6.5
Intermediate scales
The origin and structure of the peaks of the angular power spectrum can be understo by studying the system of photons and baryons during the radiation era. The evolution of the density of photons coupled to baryons is dictated by t equations (5.255) and (5.256), as well as (5.253) and (5.254) N VI',=I-4 15 I' -
+ -61 k 2 7TI' + 7 '( Vib
V~ = - 7-{Vb -
+
7'
R (VI' - Vb),
-
VI' ) ,
(6.5
(6.5
with 7' = an e(JT. The residual anisotropic pressure term is related to the radiati velocity by k2 8 k2 -7T = - - -V (6.5 12 I' 45 7' 1"
This relation reflects the fact that locally, the quadrupole can be generated fro the gradient of a velocity field. It will be justified when the effect of polarization Thomson scattering will be taken into account [see (6.231) below]. Notice that if t polarization of radiation is neglected, the factor 8/45 is replaced by a factor 2/15.
Properties of the angular power spectrum
6.2.2.1
319
Dynamics of the photon-baryon fluid
Let us start by considering wavelengths that are large compared to 1 IT' . The Euler equations for the photons and the baryons can be expanded to leading order in kiT' to obtain, if the anisotropic pressure term is neglected,
Vb
=
V"
The second equation implies that the entropy remains constant during the evolution. In this tight coupling regime , we can combine both Euler equations to obtain
[(1
+ R)V,j'
=
1
-46~
-
(1
+ R).
(6.59)
The continuity equation for photons can then be used to deduce that the evolution of the density contrast is dictated by (6.60) with
R = ~Pb 4 p, '
2 1 1 c = ---
s
31
+R
(6 .61)
This equation describes a forced damped oscillator. The forcing term is mostly determined by the Bardeen potentials, which tend to be mostly dominated by the density contrast of the CDM as R increases. It induces a shift mean value of the oscillations. Equation (6.60) can take the more compact form (6.62) This equation is at the basis of the understanding of the acoustic oscillations of the photon- baryon plasma. The density contrast of the photons oscillates due to the two opposite forces: the radiation pressure and gravity. The frequency of these oscillations is Ws = kc s . In a WKB regime where the period of the oscillations is small compared to the characteristic time of the variation of the amplitude [that is when 6~' « 6~ws and 6~/I « (6~ws )' but taking into account that w~/ws = -R'/2(1 + R) which is of the same order as the damping term], the homogeneous equation has two solutions
(6.63) where
rs
=
loTI cs dr/ = loTI J3(~7]~ R)
is the sound horizon at time
7].
The general solution of (6.60) is thus of the form
(6 .64)
320
The cosmic microwave background
with the Green function
, _ 6a(r/)6b(1]) - 6a(1])6b(r!,)
9(1],1]) = 6a(1]')6£(1]') _ 6~(1]')6b(1]')' =
V3 [1 + R(1]'W/ 4 . I: [1 + R(1])]1 /4 sm [krs (1]) -
krs (1]
so that the gener al solution is [1
+ R(1])]1/46~
=
6,(0) cos [krs(1])] +
V3 +41: 6.2.2.2
r
10
V; [6~ (0) + ~R'(0)6,(0)]
sin [krs(1])
'
. " kr s(1])- kr s(1])]F(1])d1]'. [l +R(1]) ]3/4 sm[
Sachs- Wolfe term
Let us assume to start with that the Bardeen potentials are almost const ant corresponds to a matter-dominated Universe. Furthermore, we shall neglect the variation of R. The solution (6.66) then implies 8 sw
=
[8s w (0) + R<.p] cos krs (1]) +
8'
~w
(0)
Cs
sin krs (1]) - R<.p ,
and rs reduces to rs = Cs 1]· The Sachs- Wolfe term undergoes oscillations in tim are the signature of the density waves, i.e. sound waves, propagating in the ph baryon plasma with a velocity cs ' Notice that the minimum of the oscillations, is shifted with respect to zero because of the baryons [compared with the study pure radiation fluid in Chapter 5]. The amplit ude of the oscillations increases w and thus with the amount of baryons. In reality, the variation of R and of the Bardeen potentials cannot be neg over several periods and the evolution of 8 sw is obtained from (6.66) . Its amp thus decreases as (1 + R)1/4 during the cosmic expansion. Figure 6.4 illustrat time evolution of 8s w for several modes before decoupling. 6.2.2.3
Doppler term
Neglecting the time dependence of the Bardeen potentials and of R, the cont equation for the photons implies that k 2 V, /3 = 8 + (\[I - <.P)" so that i approximation ,
sw
kV --f" =
-Cs
[8s w (0) + R<.p] sin krs (1]) +
8'
s~
(0)
cos krs (1]).
In the tight coupling limit , this velocity can be identified with that of the ba that is at the origin of the Doppler contribution. Since the direction of the ba is random , we can estimate that 8 dop ~ k V, / V3. This t erm oscillates symmet around zero, in quadrature with 8 s w , with an amplitude V3c s = (1 + R) -1/2 sm The relation between the phases of the density and the velocity implies that the associated with the Doppler term fill in the troughs ofthe spectrum of the Sachsterm. When R --7 0, the Doppler term has the same amplitude as the Sachsterm, so that the resulting oscillations of the spectrum are very damped. Wh becomes large enough the Doppler term is smaller and the final spectrum t he oscillations, but less marked than that of the spectrum of 8 s w (Fig. 6.2).
Properties of th e angular power spectrum - k = 0.001 Mpc- 1 --· k = 0.1 Mpc- 1 2 ",--,~~~---Y~--l .. .... k = 0.2 Mpc- 1
- k = 0.001 Mpc- 1 --· k = 0.1 Mpc- 1 .. .... k = 0.2 Mpc- 1 O ~~~~~~~~~~~"".~~ . _-_~-~
1
~ O ~,~tj~~~~~~~~~~
-1 -2 ~-+~~4-~+-+-~~-+~~~ 2
,g
321
rff -0.1 -0.2 0 .05
1 0 jd~~--I'-f'T+--I'--':'-¥---!'''"""'~'''''~~
-1 -2 ~~~~~~~~~~~~~~-"
100
200
300
100
T/ I Mpc
200
300
T/ I Mpc
0.5 0.1
O ~m.b. .. ~.. ~..~.. ~ .. --~+++H~--~~~~~
......
- 0.1 -0.5 - 0.2 1-8_S_\\_' _ _- 10- 3
0.Q1
0 .1
k l Mpc
10-3
0.Q1
0.1
k I Mpc
Fig. 6.4 Evolution of 8 sw = 6~ / 4 + and of 8 dop ~ kV-.Jv'3 as a function of time for three modes with wave number k = 10- 3 (solid line), 10- 1 (dashed) and 2 x 10- 1 Mpc - 1 (dotted). The contribution at the time of decoupling as a funct ion of k is presented at the bottom. We compare the adiabatic (left) to the isocurvature (right) initial conditions .
6.2.2.4
Positions of the aco ustic p eaks
The position of the acoustic peaks depends on t he initial conditions of the perturbations. Below the sound horizon , the Bardeen potentials are negligible. As will be seen in the next section, as soon as the scales become of the same order as the scattering scale (6 .80), the phot on density contrast is exponentially damped. We then obtain (6.69) with D(k , 7) ) == exp (- k 2/k~) . C A and C J are two integration constants, the values of which depend on the initial conditions.
- Adiabatic initial conditions: in this case, initially o ~ (O) = -2 iI>(0) . If the photon velocity V, is negligible, the conservation equation reduces to J~' = 4iI>', so that
322
The cosmic microwave background
c5~(1]) = - 2cI>(0) + 4cI>(1]) - 4cI>(0). After the Hubble crossing, t he gravitation potential tends to zero so that c5~(1]) c::: - 6cI>(0). So, at t he time where t he mo enters the sound horizon c5~/4 = - 3cI>(0) /2 and c5~' = 0, so that CA = -3cI>(0) and C J = 0, 3 V(k , 1]) (6.7 8 s w = - 2cI>(0) (1 + R)1/4 coskrs '
For adiabatic perturbations, t he gravitational potential is constant so that F constant in the (6.60). In the limit krs --+ 0, this constant excites t he cosine mod - Isocurvature initial conditions: initially c5~(0) = cI>(0) = O. As can be seen fro (5.220) , c5 ~ ex in the radiation era and S = S(O) remains constant. T gravitational potential then grows since cI> ex (5.228). During t his era ex so that F ex 1] in (6 .60) . In the limit krs --+ 0, this term excites the sine mod Solving the evolution equations, we then obtain
- as
- as
CJ =
a
V6 k e q S(O) . 4 k
The factor keq / k can be understood as c5~ ex Jeans length, i. e. until a '" ke q / k.
- as
(6. 7
until t he mode reaches t
The term in cos( kr s ) or sin( kT s ) is excited, depending on the nature of the init conditions. We infer that at the time of decoupling (1] = 1]LSS )' 8 s w (k, 1]LSS ) is maxim for p7r , (adiabatic) (6.7 _~ ) 7r , (isocurvature) k(p)rS(1]LSS ) = {
(p
with p = 1, 2, .. " The physical scale corresponding to these wave numbers, A(p) a(1]Lss )7r/ k (p), is observed with an angular scale of 19 (p) '" A(p)/ D A (1]LSS )' where D A the angular distance (3.74) , i. e.
(6.7
An angular scale 19 corresponds approximatively to a mult ipole g '" 7r /19 (see t properties of the Legendre polynomials, Appendix A). The angular power spectru therefore has a set of peaks located at t he multi poles (adiabatic)
p, { p -
~.
(6.7 (isocurvature) .
The position of the peaks thus mainly depends on the cosmological parameters v the values of ZLSS and r S(ZLSS )' and very little on the primordial spectrum of t perturbations. Adiabatic models have a set of peaks from the series (1:2:3: ... ), wh
Properties of the angular power spectrum
323
the peaks series of the isocurvature models is (1:3:5: . . . ). The peak p = 1 in the isocurvature models is usually too weak so that the first peak is localized at
£(1)
rv
220, (adiabatic) { 330, (isocurvature)
(6.75)
for a fiat Universe with no cosmological constant. This difference in the peaks position makes it possible to constrain the relative importance of the two types of perturbations. The angular spacing between the peaks is more or less constant and fixed by the ratio of the co moving radius at decoupling to the co moving angular distance. The acoustic peaks therefore encode information on the cosmological parameters. In particular , for a Universe with nA = 0, f K( zLss ) rv 2/(Hono) and r S(zLSS ) IX n~ 1 /2, so that for an adiabatic model, the position of the first peak gives almost directly an estimate of the curvature of the Universe, 220 £(1)
rv
yn;;- .
To finish, let us stress tha t the existence of these acoustic peaks was noted by Sakharov as early as 1965 [8], well before the observation of the temperature anisotropies. The acoustic peaks are therefore often called Sakharov oscillations. 6.2.3
Small scales
On small scales, ki T' is no longer negligible. One should go beyond the tight coupling approximation as soon as k rv T' , and the mean free path of photons becomes of the same order of magnitude as the wavelength of the perturbations.
6.2.3.1
Silk damping
Let us consider modes that have become sub-Hubble during the radiation era. Unlike the ones that come into play at intermediate scales, these modes have become sub-Hubble while t he photons were dominating the matter content of the Universe. This implies that the gravitational potential has been greatly reduced during the oscillations. For a qualitative analysis , we can thus neglect the Bardeen potentials and forget about the gauge issue. Moreover, the characteristic time of acoustic oscillations is very small compared to the Hubble time so that one can neglect the effects of the expansion, i. e. assume R is constant. The Euler equations (5.254) and (5.191) for the baryons and radiation then take the simplified form
V,,
1 1 2 = - -0 4 ' + -k 6 7r'
-
T
'(
v:, -
Vlb) .
behaves as l / T' and in order to go beyond the tight coupling limit we must determine the contribution in l / T' of (Vb - V, ). Taking t he difference of the two Euler equations t he terms of order (T')O t each us that 7r"(
(6 .76)
324
The cosmic microwave background
Both
(R +
Euler l)V~
equations
+ R(V~ -
can
V~). Using
then b e combined to form V~ + (6.76) to express (V~ - V~), we obtain
R2
, 1 1 2 1 6~ ( R + 1)V =-- 6 +- k7f - - - - - . , 4' 6 ' 4 1 + R T'
RV~
(6.7
Including the effect of polarization, 7f, is related to V, by (6.58) [see (6.231)]. Co bining this equation with the equation of continuity for the radiation, we then obta
(6.7
When neglecting the effect of polarization, the factor 16/15 should be replaced by factor 4/5. The WKB solution of this equation is 2
k 6, ex exp ( - k~
)
exp (±ikrs ) '
(6.7
with - 2
kD =
1
t [
6" 10
1 (16 1 + R 15
R2) ]---;;d1]' .
+ 1+ R
(6.8
The term in brackets varies very little during the cosmic history since it is 16/15 f R = 0 and 1 for R -+ 00. The mean free path for Compton scattering is AC ~ l/T l So the photons under a random walk that drifts in a time 1] of a length AD ~ VN AC, the number of ste being of the order of N ~ 1]/ AC. The scattering length is thus of the order of AD )1]/T'. This is what (6.80) indicates: AD ~ k;;l ~ )1]/T'. In a flat Universe with cosmological constant, assuming that Zeq ~ Z LSS' it is of the order of
(6.8
This damping [9] leads to an exponential decrease of the angular power spectru that starts to appear at around f! ~ 140)D bO h 2 /O.019h, i. e. b efore the first Dopp peak. The multipoles beyond f! ~ 2000 are almost completely damped, which fixes t angular resolution that can be reached by observations. This effect is illustrated Fig. 6.5.
6.2.3.2
Effect of the thickness of the last-scattering surface
As has been detailed in Chapter 4, recombination is not instantaneous and the surfa of last scattering h as a given thickness. The visibility function g(1]) or g(z), chara terizes the probability that a photon underwent its last scattering at a time 1], or
Properties of the angular power spectrum
- - 7 .. .. . 7
-
= =
0.0 0.1
325
<0
:3
7=0.2
(;? C\J
.......
cJ '...... """"' +
~
~ 103 h
0~~~10~~~~lWOLO~~~~~
10
£
£
100
1000
Fig. 6.5 (left): Effect of the reionization on the temperature angular power spectrum. (right): The dashes represent the anisotropies of a standard cold dark matter model without taking into account the Silk damping and the thickness of the surface of last-scattering surface (top) or the thickness of the surface of last scattering alone (middle) or the Silk effect alone (bottom). The solid line represents the exact computation and the dotted line corresponds to a computation including t he effect of the thickness of the last scattering surface and the Silk effect by an analyt ical correction deduced from our estimates. From Ref. [3J.
redshift z . The visibility function can be deduced from the optical depth of an object that emits its light at 7], itself related to the differential opacity T' by
T(7]) =
(')O
J7)
T'(u)du.
Notice the choice of sign in this expression that implies that dT/d7] bility function is then g(7]) == T' exp( - T)
(6.82)
= - T'! The visi(6.83)
the integral of which is 1 by construction. The temperature observed in a direction e is then an average weighted by the visibility function , of the form
(6.84) Moreover, the purely gravitational interactions to which the photons are subjected, at a given redshift, only affect the fraction of photons that underwent their last scattering. This fraction is expressed in terms of the visibility function as
(6.85) which varies from 0 to 1 between 7]in and 7]0 . The integrated Sachs-Wolfe term is then of the form
326
The cosmic microwave background
8 rsw =
J
g,(rJ) [
+ Wi] (rJ, l'rlo -
rJ ]e) dr).
(6.8
Assuming for simplicity t hat the temperature fluctuations are t ime independent th the expression (6.84) is simply the convolution, in the radial direction, of the tempe ature fluctuation with the visibility function. In Fourier space,
8(Xo,rJo,e)rv
J (
A
A
A
)
g( k) 8sw+8dop + 8rsw (rJLss,k)e t
k e
dk
'70)2n.
As can be seen in Fig. 4. 12, the visibility function can be described by a Gaussi function centred around rJ Lss ,
so that
g(k) ex exp (
-~k20"~ss ) .
(6.8
The thickness of the last-scattering surface has the effect of exponentially washi out the temperature anisotropies on scales smaller than the thickness of the las scattering surface. This effect is of the same order of magnitude as the Silk dampin
6.2.4
Other effects
After decoupling , the photons propagate almost freely so that only the gravitation effect can significantly change their energy. At lower redshifts, the Universe can reionized , and the interaction with the plasma can also modify the photon temperatur
6.2.4.1
Integrated Sachs- Wolfe
The variation of the Bardeen potentials and primordial gravitational waves give contribution to the temperature fluctuation, 8rsw, whose expression is given by (6.17 This component possesses two contributions:
- Early Sachs- Wolfe effect: the variation of the Bardeen potentials at the matte radiation transition leaves a small contribution to the integrated term. For ad batic models , this effect is dominant for the modes with wavelength entering t sound radius between equality and decoupling. Its contribution is thus domina for multipoles close to the first peak. For these modes , 8sw rv -
Properties of the angular power spectrum
327
close to zero for the sub-Hubble modes at the time of decoupling. Since
6.2.4.2 R eionization As soon as the first sources of ultraviolet photons appear, the Universe can be reionized. The free electrons can then (re)scatter t he photons of the cosmic microwave background after decoupling. This scattering tends to isotropize the anisotropies of the cosmic microwave background by averaging the temperature anisotropies from many lines of sight that meet at the scattering event. If we assume that reionization is homogeneous, we can approximate the visibility function by a sum of two Dirac distributions localized at TJ LSS and TJre, respectively, for the decoupling and the reionization, then
8(xo , 170, e)
c:::'
(1 - e-r,e) (8 + e.Vb) [e(TJO -
TJre) , TJre]
+e- r,e (8 + e.Vb) [e(170 - TJLSS) , TJLSS ],
(6.88)
where Tre is the optical depth at reionization and where we have neglect ed the integrated Sachs- Wolfe effect. The first t erm describes t he averaging from different lines of sight as well as the generation of new anisotropies due to the scattering on moving electrons at reionization. The second t erm represents the temperature that would be observed if there were no reionization and has been weighted by the fraction of photons that have not been scattered . 8 [e(TJO- 17re) , TJre], which is evaluat ed at the time of reionization, is the average of e - e'. Vb on the electron last-scattering surface. For t he long-wavelength modes, i.e. k(77re - 170) « 1, it gives 8 [e( TJo - TJLSS) ' TJLSS ], and it vanishes at small scales. Thus, for the super-Hubble scales at the t ime of reionization , the observed temp erature anisotropy becomes
8(xo , TJo, e)
--7
8(xo , TJo , e) +
(1 - e- r , . )
e . ~ Vb ,
where ~ Vb is the difference between the electron velocities at reionization and at the previous scattering along the line of sight. At these scales , the Doppler t erm is subdominant so that 8(e) is hardly affected . For sub-Hubble scales at the time of reionization 8( Xo , 17o , e) --7 e- r ,e8( Xo , TJo , e) + (1 - e- r , . ) e. Vb (17re). This effect is illustrat ed in Fig. 6.5. The analysis of the cross-correlation between temperature and polarization of the WMAP observations suggests that T re rv 0.17 from a redshift of Zre rv 15.
328
The cosmic microwave background
6.2.4.3
Sunyaev- Zel 'dovich effect
The interaction of the photons of the cosmic microwave background with a hot plas can lead to a modification of their t emperature. In particular , the hot gas of clust has a t emperature of the order of 10 7 - 108 K, which radiat es, via Brehmsstrahlung the X-ray region. The inverse Compton scattering with the free electrons in the clus can modify the energy distribution of photons by shifting the low-frequency phot to higher frequencies. This is the thermal Suny aev- Zel 'dovich effect , which represe the dominant contribution to the secondary anisotropies. In a nutshell, the modification of the temperature of the cosmic microwave ba ground photons is of the order of 6.Tsz rv L( neTe), where the bar stands for an aver along the line of sight. L is the characteristic length of this line of sight in t he clus Te and n e are the temperature and the density of t he free electrons. Not ice that clust er surface brightness in X is of the order of
where V = Di 82L is the volume of the cluster , assumed to be observed under angular diameter 8. To illustrate this effect, let us consider a cluster with electronic density profile the form
ne (r)
=
no [1 +
(:J 2j -3f312'
for 0 < T < Rctuster and zero otherwise, where the temperature of the photon is given by
Tc
(6.
is the core radius. The variation
(6 . assuming that the temperature of the hot gas is independent of r. In the limit 00, we infer that
R c lust e r
(6.
Unlike Thomson scattering , the thermal Sunyaev-Zel'dovich effect induces a sp tral distortion with a temperature decrease at long wavelengths and an increase short ones. It allows us to probe galaxy clusters and represents a powerful tool detect clusters at large redshifts (see Ref. [10] for a review) . If the cluster has veloc v with respect to the cosmic microwave background , the Doppler effect induces a s ond effect , called the kinetic Sunyaev-Zel'dovich effect , which depends on the rad component of the velocity.
Kinetic description
329
6.2.4. 4 Gravitational waves It is difficult to obtain a relation equivalent t o (6 .50) for gravitational waves since (6.45)
involves t he time derivative of EA' In the matter era, the solution of the gravitational wave equation is given by (5.135) so that (6 .92)
The relation (6.45) t hen implies that , for a primordial spectrum of the form P T ex k nT as predicted by inflation (8 .234) ,
C T ex (£ + 2)! £ (£ - 2)!
J~ [l Yo1 -
]2
YO
j £(yo - y ) j 2(Y) d ( Yo -)2 Y -Y- Y
nT
Y07)LSS/ 1)O
(6.93 )
where we have set Y = k7) and chosen ko = 1/ 7)0' This integral is non-negligible only when Y rv 2 and Yo - Y rv £ simultaneously, since otherwise at least one of the two Bessel functions is close to zero . The integration domain implies that Yo < Y7)O/7)L SS ' so that both conditions can be satisfied simultaneously only if
£ < 2~ rv
rv
7) LSS
2) z
LSS
+1
rv
60
(6.94)
.
Figure 6. 3 indicat es that C'{ decreases rapidly from £ rv 100 onwards, which is compatible with t his estimate. This implies that the spectrum C'{ is weakly sensitive to the cosmological parameters. To evaluate the integral (6 .93) , notice that the second Bessel function is peaked around y rv 2. It can be approximat ed by j2(Y) / Y rv 2- 1 / 2<5(y - 2) , which implies, after integrating over y , that C T ex (£+ 2)! £ (£ - 2)!
when we assume that Yo
»
J~
Y6 -
[j£(YO)]2
nT
Y6
'
(6.95)
2. We infer that CT £
(£+ 2)!r(£- 2 + ¥) ex (£ _ 2)! r (£ + 4 + ~T)"
(6.96)
Using the same expansion as for (6.50 ) , we obtain
£(£ + l)C'{ ex £n T
(6.97)
for small multipoles, £ 2: 2.
6.3
Kinetic description
To obtain a more precise and realistic description of the photon propagation and of their decoupling from matter , we must develop a kinetic approach. This approach is mainly necessary for relativistic fluids , which have a large mean free path. The kinetic theory in the relativistic framework is described in detail in R efs. [11- 13] and complementary mat erial on distribution functions can be found in Ref. [14]. The Boltzmann equation in the cosmological framework is detailed in Refs. [15- 19] .
330
The cosmic microwave background
6.3.1
6.3.1.1
Perturbed Boltzmann equation
Distribution function
Just as in Chapter 5, we introduce t he distribution function f(xl-', PI-') that depen on the position and the momentum, which are considered to be independent. For particle of mass m, the distribution function lives on the mass shell
(6.9 where the tangent space is defined as
M being the four-dimensional manifold describing space-time and Yx its tangent spa at the point x. To decompose this distribution function , it is convenient to introduce the vect field normal to the constant t ime hypersurfaces , {ry = const. }. With a metric of t form (5.52) , t his vector field can be expressed as NI-' = - a(l
+ A, 0) ,
(6.9
to first order in perturbations. One can then decompose the t angent vector kl-' to t geodesics of t he photons (we shall use pI-' for the general notation and kl-' only for nu vectors) as
(6. 10
with l i j nin j = 1. This decomposition ensures firstly that kl-'kl-' = 0 and secondly allows us to identify E with the energy of the photon that would be measured by observer comoving with the four- velocity NI-' since NI-'kl-' = - E. In perturbation t heory, the distribution function can then be decomposed into part that , because of isot ropy and homogeneity, depends only on ry and E and in a perturbation, that will depend on t he space-time position and the 4 components the momentum, as
(6. 10 The evolution of this distribution function is dictated by the Boltzmann equation
L[J] = C[J],
(6.10
where L == d j dry is the Liouville operator and C[J] the collision term. This equatio implies that the photon density in phase space is only affected by collisions.
Kinetic description
6.3.1.2
331
Collisionless Boltzmann eq uation
To start with, let us consider the Liouville term, (6.103) where A is an affine parameter along the geodesic with tangent vector k"' , so that dT]/ dA = kG. The geodesic equation (1.37) can be rewritten as
dk'" dA
= _ r'" kO< k{3 0<{3
(6. 104)
,
so that the Liouville operator takes the general form (6.105) From the decomposition (6.101) it can be rewritten as (6 .106) Now, we must evaluate t his function to first order in t he perturbations. The last term is easy to obtain since of / oni is of order 1 in perturbations so that dn i / d1] should only be evaluated at the background leveL To this order, (6.104) implies t hat
E' = - HE ,
(6. 107)
(3)qk being the Christoffel symbols of the spatial metric last term of (6.106) is thus
lij
(see Appendix A). The
i dn of __ (3)r i j k of n n oni' d1] oni jk The second term is also very simple to obtain since dx i j d1] = k i j kO. Since first order in perturbations, we only need dxijd1] = ni so that
od is of
i dx o f _ i<')' f d1] oxi - n u, . To evaluate the third term of the expression (6.106), one should compute dE/dT]. For this it is sufficient to consider the time component of (6. 104) ,
which directly gives i ) - r °{kO< k{3- ( 1+2A. ) -d ( -E ) = -E (A' +noiA 3-
d1]
a
a
0<
Eja
332
The cosmic microwave background
Using the expressions (C.6) and (C.26) for the Christoffel symbols, we obtain
(1
dE [ . A + - h' . -D(B) d7]= - E H + n'o' 2 'J 'J
) ..J n'n J
.
Gathering all these terms together , we then obtain the final form of the Liouv operator to first order in the perturbations
i j oj ij n n on k .
(3) r k
-
[] = J ,+ n kOkJ- [ H+n kOkA+ LJ
(6.108
(1 ,
-h - D(B) 2 'J 'J
)
i J-J nn EoJ -oE
(3) r
k-nnJ i - o on (6.109
'J
With the decomposition (6.101), we thus obtain the two evolution equations L [J] =
l' -
HE
oJ
(6.110
oE
(6 .111 To lowest order, the distribution function of the photons is a Bose- Einstein dis bution function (Chapter 4),
J(E , 7)) =
{ exp
[T~7)) J - 1 } -
1
=
J (~) .
(6.11
(6.110) then implies that T ex l / a , as was already given for the energy of each phot by the geodesic equation. To linear order, if we consider the distribution function to be of the form
and expand the temperature as T(7) , Xi, n j ) = T(7)) [1 i
j
_
-
oj
+ B(7) , Xi, n j )] i
j
J(7) , X , E , n ) - J(7) , E) - E oEB(7) , x , n ).
then
(6.11
The temperature field is thus inhomogeneous (it depends on xi) and anisotropic depends on n j ). We stress that this ansatz implicitly assumes that B does not depe
Kinetic description
333
on E. This hypothesis can only be justified after the collision term has been studied. As we shall see, this term does not depend on the energy, so the Thomson scattering, in this regime , does not induce any spectral distortion. 6.3.1.3 Macroscopic quantities A series of macroscopic quantities can be defined from the distribution function , the first of which are fluid quantities. For this we introduce a space-time tetrad, or vierbein, (e~)a=03, to be locally in a Minkowski space-time. We choose this t etrad such that
The vector kJ-L can then be decomposed as k a = kJ-Le~ . The energy-momentum tensor is then obtained, as in Minkowski space-time, by integrating over the momentum as
r
TJ-LV(x) =
3
kJ-Lk V f (x CX, k(3 )
d k
,
(21T) 3Ek
}Pm(x)
(6.114)
with k 2 = 6abkakb and Ek = .Jk2 + m 2 = k for a massless particle. To simplify the notations , the factor (21T)3 is included in the distribution function. The energymomentum tensor of radiation is then given by
r
TJ-LV( x) =
kJ-Lk Vf (XCX, k(3) EdEd 2 n.
}Pm(X)
(6.115)
.
This energy-momentum tensor can be expanded as (5.69) . We then infer that
P= P = IIJ-LV =
J ~J J
(kJ-L U J-L)2 f EdEd 2n ,
(6.116)
V 2 (kJ-Lk l..J-Lv)f EdEd n,
k cx k(3
(l..~l..~ - ~ l..CX(3 l..J-Lv)
(6.117) f EdEd 2n,
(6.118)
where we recall that l..J-Lv = gJ-LV +uJ-Lu v , uJ-L is defined by (5 .70) and thus uJ-L = a(-l A,Vk + Bk)' kJ-LkJ-L = 0 implies that P = p/3 and we recover the equation of state for radiation. We first note that kJ-LuJ-L = - E and then notice that the integral of
Jn i
6p =
(p + P)(v i + Bi) = IIij
[1 -
ni(vi
+ B i )]
,
over angles vanishes, so that we infer that
J J
6f E 3 dEd 2n ,
(6.119)
ni 6f E3 dEd 2 n ,
(6.120)
= P1T ij =
J
n ij 6f E 3dEd 2 n,
(6.121)
with
(6.122)
334
The cosmic microwave background
6.3.1.4
Brightness and temperature
The brightness, 1, is defined by integrating the distribution function over the energ
(6.12
It therefore represents the energy density per unit solid angle that propagat es alo a direction ni at the space-time point (1] , x j) . This quantity can be decomposed 1 = I + c5I with
Notice that from (6 .116) , the photon energy density, P'Y' is simply the monopole this brightness, i. e. 2 n . . (6 .12 P'Y (1] , xJ) = 47f 1(1], x" nJ),
. Jd
and (6.110) then implies t hat
l' + 4HI = 0,
P~
+ 4HP'Y = 0
(6.12
and also that I = 15"1' which is obvious since I does not depend on ni because of t background space-time symmetries. Similarly, for the collision t erm entering (6.102) , we define
(6.12 Integr ating t he (6.111) over the energy, we then obtain
c5I'
+ n k[Ac5I + 4Hc5I + 4 [n kOkA +
(~h~j -
j DCi Bj ») nin ] I
_(3)r k nin j oc5I = e [c5I] 'J onk .
(6.12
The t emperature contrast is defined from the brightness by ..
e(1] ,x" n' ) =
1 c5I
4I '
(6.12
and corresponds to the definition introduced by the (6.113) . The monopole of th quantity is directly related to the contrast of the temperature of the photons
(6 .13 The evolution equation for using (6 .126),
e can be easily inferred from that of the brightness (6 .12
Kinetic description
e' + nkokB -
(3) r tninj ::k
+
[nkOkA +
(~h:j -
j D(i B j) ) nin ]
=
335
C[B], (6 .131 )
where we have defined C[B] = C[Of]/41. This equation can be obtained directly by inserting the decomposition (6.113) in (6. 111 ). However , a difference exists since here represents the rela tive flu ctuation of the brightness and the integration over the energy implies that this quantity does not depend on the temperature , an hypothesis implicit in the definition (6. 113) . Both quantities are identical and satisfy the same Boltzmann equation as long as t he collision term does not depend on the energy.
e
6.3. 1. 5 Multipole expansion Just as in (6 .125) , t he different fluid quantities appear as the successive moments of brightness
(p
. . J
+ P)(v' + B' ) =
2
Of n·d' -n ,
(6. 132)
41T
We can thus expand the Boltzmann (6.128) by expanding the brightness in a series as
'I( T/, Xi ,n j) -- " . (T/ , X.i) . ~( n il . . . n ie ) T.' l ... 'e
U
(6.133)
£
The only metric perturbations that are not gauge invariant are scalar and vector modes so that only Io and I iI are modified under a gauge t ransformation. All higherorder components are gauge invariant. Such a decomposit ion is not unique but this problem can be fixed by imposing t he quantit ies I il . il to be symmetric and t raceless. Under t hese conditions I il . ie possesses only (2£ + 1) independent components [only (€ + 1)(£ + 2)/2 out of t he 3£ components of I ilie are independent if the tensor is symmetric and the condition on the trace imposes e(£ - 1)/ 2 constraints among these components]. As seen previously, only the first three t erms of this expansion are fluid quantities, and as such will enter the Einstein equations. A similar series expansion can be performed for t he temperature field, leading to the defini tion of the coefficients
Bil .. il(T/ , Xi). Notice that the expansion (6.133) allows us to simplify the for mulation of the Boltzmann equation since in nkok6I , the derivative is only applied on I il ... il , so that this term can be combined with t he Christoffel symbols to lead to a (spatial) covariant derivative. Besides, this is the only proper way to deal with terms such as oB / k . Equation (6.13 1) then takes the form
an
e' + 2: n k (ni l ... n il ) DkBil ie + [nkOkA +
(~h:j ~ D(i Bj)) nin j ] = C[B]. (6.134)
£
The metric terms do not depend on n i so that they only come into play in the equations for the monopole and the dipole, i.e. at the level of the fluid description.
336
The cosmic microwave background
The SVT decomposition, introduced in Chapter 5, can be generalized to all or We recall that a trace-free symmetric tensor of rank 2, such as t he anisotropic s tensor (5.72) , can be decomposed as 7rij =
6. ij 7r
+ D(i1fj) + 1fij ,
where we recall that 6. ij is defined by (5 .73). We thus obtain the scalar, vector t ensor parts as
where 'STF' means t hat we extract t he symmetric and trace-free part. For the de position to be unique , it is clear that we should impose 7rB' ) and 7r;J) to be trac and transverse. This decomposition can be generalized to any tensor of rank £ and the compo of order (m) can be decomposed as
(6
Iml = 0, 1,2 correspond, respectively, to the terms S, V and T. Decomposing Fourier coefficients, {j , and choosing a basis such that e3 = k /k, then the scalar of (jil .. .ie will be proportional to k i 1 ... kie (j~O). Contracting it with t he SFT pa nil ... n ie , it is clear t hat we obtain a Legendre polynomial of order £ so that n i1 ... nie{j(O) . ex Yeo{j~O) 'll··
.ze
,{.
The vector modes have two polarizations that are vectors orthogonal to k [see decomposition (6 .32)] so we can choose e± = el ±e2 as the basis for the vector m . ' . . ± ' The vector part of Bi, .. ie IS then proportlOnal to the STF part of k i' ... k ie _, e ie B In this case, the contraction with ni , ... nie gives terms of the form n
i,
... n
ie '(±1) '(±1) Bi, ... ie ex YC±IBe .
T his construction can be generalized for all m. Thus, the Fourier component ( can be decomposed as
(6 m=-e
with
±
±]
'(.m). ex [ B ., ... 2e ki,,,.kie _ meie_m+, Q9 ,,· Qg ei e
STF '(m)
Be '
(
6
choosing the + sign when m 2: 0 and the - sign otherwise. One can then show Chapter 5 of Ref. [3] or Ref. [20], which contains a description of the constructi the STF tensors) that ni1 ... n ie {j(m). ex Yem{j (m) . (6
2, .. " e
e
Each type of perturbation possesses 2 degrees of freedom apart from the scalar m that only has one, so that for a mode £, t here are 2£ + 1 degreeS of freedom.
Kinetic description
337
This construction allows us to conclude that the temperature field B(1] , x, n) of radiation at the point x that propagates in t he direction n can be expanded in the basis Gem (x , n) defined by
.) e{;fh7r -n--e 2{. + 1
G em ( x , n ) = ( -z
i k. x
Yem (n) ,
(6 .1 39)
as
-J
d k ~ '(m) (27r)3/ 2 ~ Be (1], k)Gem(x, n).
B(1], x , n) -
3
(6.140)
em Only the modes m = 0, ± 1, ± 2 are excited by t he matter and space-time geometry perturbations.
6.3.1.6 R elation with the SVT decomposition Expanding the scalar modes as
=_
(O) Q tJ
(00 _~o3 .ll) t
J
tJ
Q (O)
,
then
n i Q i(0) = - kG lO ,
Q (0) = Goo,
(6. 141)
As for the vector modes, they can be expanded as Q t(±l)
= _~ v'2 e t±Q(O) ,
Q (± l ) tJ
=
0
<X
- kG2±I'
' Q (± l ) j) ,
(t
so that i
(± l )
n Qi
=
G l± l ,
i
j
(± l )
n n Qij
(6.142)
This decomposition is analogous to that of (6.32) which we have used initially. The tensor modes can b e expanded as
Q
(±2) tJ
= _ ~e±
V8
t
0 e ± Q (O) J
'
so that (6 .143) Notice that this expansion differs from t hat initially used (6.40) by a factor
fi78.
338
The cosmic microwave background
6.3.1.7
Collision term
In general, the collision t erm can be decomposed as
C [I ] = dl+ _ dld1] d1]
(6.1
where 1+ and I _ represent the incoming and outgoing distribution functions dur the scattering. In the rest frame of the baryons and electrons, the two terms invol in (6. 144) are given by
dl; = J1 T'
where
T'
w(n , n') d~;' ,
(6. 1
is defined as previously as
(6.1
is the cross-section of the Thomson scattering and n e the density of free electro The function w( n , n') contains the angular dependence of the Thomson scattering is given by
O"T
(6.1
P2 being the Legendre polynomial of order 2. Notice an important property: wand t the collision term does not depend on the energy so that the Thomson scattering d not induce any spectral distortion. This is no longer the case when the temperat of the photons is very different from that of the electrons, as, for instance, for Sunyaev- Zel 'dovich effect clusters where the temperature of the electrons is of order of a few keY, while that of the photons is only a few Kelvin. This property implies that the brightness fluctuation (6.129) does indeed correspond to a temperat fluctuation , as introduced by the ansatz (6.113). At the background level, the expression for w(n, n') implies that
C [f] =0,
(6.1
which is to be expected since at this order the symmetries of the Friedmann-Lema space-time impose that all fluids have the same velocity, proportional to ot:. To first order in the perturbations , we get
C[ol] =
T'
[}01(1] , xi, E , n ') d
2
n' -01(1],
~
xi, E , n)
+ ~nij }01(1], Xi, E, n')n~j d 4
2
n
~
(6.1 where E is the energy in the rest frame of the electrons and the baryons. Using fact that E = - kl-'u~ = E [l - (Bi + v~)ni], we obtain that
The integral over the energy then gives
Kinetic description
e[M]
=
T'
[/
d2n' _ . . OJ ~ - OJ + 4I(B" + vb)ni
3 .. /
+ =tn")
339
. _ d 2n'] OJ(T) , x", E, n')n~j ~ .
(6 .150) The first term reduces to the monopole and the last one to 3nij rr ij = (4I)ninj7rij/16 , so that (6.151) where eo is the monopole of the temperature (6.130). Notice again that O[e] vanishes in the unperturbed space-time. From the Stewart- Walker lemma (Chapter 5), O[e] is thus gauge invariant. We also stress t hat in the expansion scheme (6.133) , the only term that contributes beyond the multipole £ = 2 is O[e] = - T'e . So, for £ > 2 no coupling term to the baryons or to the metric will contribute to the Boltzmann hierarchy. 6.3.2
Gauge invariant expressions
Some complementary material on the discussion of the gauge dependence of the distribution function can be obtained in Refs. [19,21]. 6. 3.2.1
Gauge invariant distribution function
The decomposition of the distribution function 1 as 1 = /+01 is not unique. Just as in Chapter 5, the definition of the perturbation 01 implies the choice of an isomorphism 1/J between the spaces where / and 1 are defined , i.e. of a map
which allows us to decompose
1 as 1 =/01/;+0].
(6.152)
Any change of coordinates , xC> ----> yC> = xC> - ~C> , on M induces a transformation on the tangent space in which t he momentum transforms as (6.153) This transformation is thus gener ated by t he vector with coordinates (6.154) in the natural basis (0J.L , oP'"). To linear order, the distribution function transforms as
1 - t 1 + £Td·
(6.155)
To obtain the trahsformation law of 01, one should first notice that / is not defined on TM but only on the subspace Pm. But in general, T~ is not tangent to P m, so that £T~/ is not well defined . / should therefore b e extended on an open set of T M
340
The cosmic microwave background
containing P m. P erforming a gauge transformation and keeping only the first ord terms, we obtain 11 0 'l/Jl + 6h ----> 11 0 'l/Jl + 6h + £TE}I ,
since £Tf,6 h is of second order. However , one should notice that there can exist seve decompositions of the same function, 1 = 11 0 'l/Jl + 6h = 12 0 'l/J2 + 612. In this ca there exists a vector field ( such that 12 = II - £ re}1 so that
The variation of
61
under a change of coordinates is thus given by
1
One can show that T~ II is para llel to Pm so that £Tf, 1I does not depend on t extension of 1. The computation of this t erm (see Ref. [21]) allows us to show th under a gauge transformation , the perturbation of the distribution function transfor as 2
(6.15
where we have used the notations of Chapter 5. A more intuitive way to obtain th result is to remind ourselves, (6.101), that 61 is a function of (T), Xi, E, n j ), where fro (6.99) , E is given by E = - NI-'kl-' so that E
= a(l + A)ko.
Under a coordinate transformation of the form x'" We infer that E --> E - En i 8i T so that
--> x"' - ~"' ,
kl-' transforms as (6.15
The unperturbed Boltzmann (6.110), i. e. L[ll = 0, can be used to express HE8f/8E so that we recover the previous result (6 .156). We can therefore define a gauge invariant distribution function as
l'
(6.15
where we stress here that E refers to the metric perturbation in the term (E' - B) a otherwise to the energy. The density contrast deduced from the expression (6 .157)
2Be careful here since T does not correspond to t h e temperature but to the v = 0 component the vector field ~I' , as defined in Chapter 5.
Kinetic description
341
Thus, F corresponds to the perturbation of the distribut ion function in the flat-slicing gauge. We could have defined another gauge invariant distribution function by (6.158) F corresponds to the distribution function in the Newtonian gauge. After integrating over the energy, we obtain that (jJF = (jJ)'" - 4\]i p"(" The associated temperature contrasts, defined in (6.129) , will be denoted by M (flat-slicing gauge and obtained from F) and 8 (Newtonian gauge and obtained from F). They are related by the relation
M = 8 - \]i , and satisfy the equations
= C[M]
(6.159)
=C[8] .
(6.160)
Since the metric t erms do not dep end on n , these two equations are identical, apart from the monopole and the dipole.
6. 3.2.2 Boltzm ann equation for the temperature We restrict ourselves temporarily t o a space-time with flat spatial sections and fo cus on the scalar modes. In Fourier space, the Boltzmann (6.160) for the t emperature takes the form
8 1 + i kp,(8 + 1'»
=
\]il + TI
(8 8+ ikp,Vb + 0 -
116 n injIIij) ,
(6.161)
with p, = k.n/ k. We can decompose the radial part of 8(k , 7) , p,) into Legendre polynomials as 8(k , 7) , p,) = -i)e8 e(k , 7))Pe(p,)e+ik .x . (6.162)
'2) e
Using the property of the Legendre polynomials
(£ + 1)PH 1(p,) = (2£ + l)p,Pe - £Pe- 1, the term ikp,8 is rewritten as
k"L (:£: 13 8 e+l - 2£
e
Then noticing that \]il
~ 1 8 e- 1) (-i/ Pe(p,).
+ TI8 0 = (\]il + TI8 0 )Po, iTlkp,Vb = - kirb( -i) lp1 and, from
(6.132) , IIijninj / 16 = (_ i)2 P2 8 2 / 10 , the Boltzmann equa tion can be decomposed into the hierarchy
342
The cosmic microwave background
(6 .16
(6.16
(6.16
(6.16
As expected, the perturbations of the metric and of the baryon energy-momen are only involved in the equations for the monopole and the dipole. Moreover, £ > 2, the collision term reduces to - T'8 e, as expected. With these conventions, the first multi poles of 8 are related to the fluid quanti by 5
82 = 12k
2
7r . -y
(6.1
The factor - k between 8 1 and V-y is a consequence of the relation (6.141). The two equations of the hierarchy are therefore only the continuity and Euler equati for radiation , as always in kinetic theory. Now, for the tensor modes, the Boltzmann equation reduces to - , . .J 8' + ik ~ll 8 = -E ·n tn ~
T'
(
1 tJ.. ) . 8 - 16n t·nll J
(6.1
Decomposing Eij as Eij = H (±2)Q~;2) [see (6.141)]' the hierarchy of the tensor p starts at £ = 2, namely
8' v'5 9, . - = - k - 8- 2 - -T 8 2 - H 2 7 10 ' 8'
e
6.3.3
=
k (
V£2 -
48 _ 2£ _ 1 e- 1
J (£2£+ +1)23 -
(6.1
48 £+ 1
) - T'8 e.
(6.1
Thomson scattering and polarization
In the previous section, we have assumed that radiation was not polarized. Howe Thomson scattering tends to polarize radiation in the direction perpendicular to scattering plane, in particular if the radiation is anisotropic before its scattering. should therefore take this polarization into account and determine its influence on collision term. For complementary studies on polarization, we recommend Refs. 26]. The multipole approach is detailed in Refs . [3,27].
Kinetic description
6.3.3.1
343
Stokes param eters
The state of polarization of an electromagnetic wave is described in t erms of the Stokes parameters [28] . The electric field of any monochromatic electromagnetic plane wave that propagates along the direction e3 can be decomposed as
E(x , t )
= £ exp [i(wt
- k.x) ],
(6 .171)
e
If (h = 2, E l and E2 have the same phase and the wave is polarized linearly. Its polarization vector then forms an angle such that tan a = E2 / El with el. If El and E2 have different phases, the wave is polarized elliptically. Instead of the two amplitudes, al and a2, and the two phases e l and e2, we can describe this wave with the four Stokes parameters,
== Q == U == V == I
(ail + (a~) = h + h (ai ) - (a~ ) = h - h
(6.172)
e2)), e2)) .
(6 .174)
2(ala2 cos (e l
-
2(al a2 sin(e l -
(6.173) (6.175)
Here the averages are taken over several periods , assuming that the amplitudes and phases could vary in time, but slowly compared to the p eriod 27r /w of the wave. Only three of these parameters are independent since they clearly satisfy the relation (6.176) I is the total intensity of the wave and the parameter Q measures the prevalence of the linear polarization along el compared to that along e2 . U and V give information on the phases. Expanding the wave (6.171) in the basis e (±) = (el ± ie2)/V2, then V = (a~) - (a~) . Thus, V represents the difference between the positive and negative helicity intensities. Under the action of a rotation of angle 'ljJ of the plane (el ' e2) around the axis e 3, e~
= - sin'ljJel +cos 'ljJe2,
I and V remain invariant while Q and U transform as
) (Q' u'
= (
cos 2'ljJ sin 2'ljJ ) sin 2'ljJ cos 2'ljJ
-
(Q)
(6 .177)
U·
In an equivalent way (h , h , u, V) transforms as I~ U'
(V'
2 sin 2 'ljJ (sin2'ljJ)/ 2 0) cos 'ljJ -(sin2'ljJ)/20 - sin 2'ljJ sin 2'ljJ cos 2'ljJ 0 0 0 0 1 2
I{)
( COS 'ljJ sin 2 'ljJ -
(II~2l)
.
(6.178)
344
The cosmic microwave background
From Q and U , we can thus construct two quantities that transform as a obj ect under the action of rotation
(
Since Q and U vanish in the unperturbed space-time, they are first -order ( invariant) quantites and they evolve without coupling to the metric perturb (since such t erms would be of order 2). Their Boltzmann equations then take the
(
Here, we have neglected the curvature t erm that could easily be added. By rede the Stokes par ameters as
( and using the evolution (6.126 ) for the brightness for the polarization
I , we obtain the Liouville equ
( wit h
c 6.3.3.2
[(B)] ~ 41 J4~E2C [(B)] dE. j
(
Sph erical harmonics of spin s
For any direction e 3 defined by the two angles (&, ¢ ), one can introduce a tangent on t he sphere, (e l' e2), defined up t o a rotation. A function s1 (&, ¢) defined o sphere is t hen said to be of spin s if it transforms as sf' (&, ¢) = e -is'lj; s1(&, ¢) un rotation of angle 1j; around the axis e 3 . Any function of spin s on t he sphere can be expanded in the basis of sph harmonics of spin s [29], sYem , which satisfies the complet eness and orthogo relations
2". /1
1 o
L
d¢
d cos & sYf.~m' (& ,¢)sYem (& ,¢) =
beebmm"
(
-1
sYf.~ (&, ¢)sYem (e' , ¢') = b(¢ - ¢' )b( cos & - cos &' ).
(
em
A cent ral property in the construction of these spherical harmonics is t he exi of opera tors t hat can be used to increase (8+) or decrease (8- ) the spin of any fun
Kinetic description
345
in the sense where 8±sf transforms as (8±sf)' = e- i (s±1)"-'8±sf under a rotation of angle 'ijJ . The explicit expression for these operators is 8± sf(8, ¢) = - sin±s 8 (8e ± Si: 881»
sin~s 8 sf(8 , ¢).
(6.186)
In particular , for functions of spin 2 that satisfy 81> [±2f(J-L, ¢)] = im ±2f(J-L, ¢), these operators reduce to (6. 187) with J-L = cos8. We can then define the spherical harmonics of spin s from the action of these operators on the (usual) spherical harmonics
sYem ==
(£ - s)! + s (£ + s)! (8 ) Ycm ,
sYcm ==
(£ + s) ! (_ l) S(8 - )- Sy; (£ _ s)! em,
(6.188)
respectively, for 0 < s ::; £ and -£ ::; s < O. These functions satisfy the following conditions s Ye~ =
(_ l) S-sYe-m)
8± sYcm = ±.j(£ =+= s)(£ + 1 ± s) s±lYcm)
8- 8+ sYcm 6.3.3.3
=
- (£ - s)(£ + 1 + s) sYem'
(6.189) (6.190) (6. 191 )
E and B modes
Just as the temperature is expanded as
8(n) = LaemYem (n), em
(6.192)
the polarization field can be expanded as
(Q ± iU)(n) = L a~!2) ±2Yem(n) em
(6.193)
with a~~2) * = a~~;l. The relation (6.189) can be used to express this last expansion as ± 2
(8 ) (Q ± iU)(n) =
(£ + 2)!" (±2) (£ _ 2)! ~ aim Yem (n). em
(6. 194)
So we have been able to construct rotational invariant quantities from the Stokes parameters at the price of the action of a non-local operator . The coefficients of these expansions are given by (6 .195)
346
The cosmic microwave background
and
One of the difficulties introduced by this expansion comes from the fact that the Sto parameters are not invariant under rotation. So if t he latter are easily computed fo given mode k , they must be recomputed again in a fixed frame when integrating o k . To avoid this problem, it is convenient to decompose the p olarization as
(Q ± iU)(1J, x, n) = -
L
(Eem ± iBem) ±2Y'£m(n) ,
(6. 1
[ (+2) _ (-2) ] B em -- _ ~ 2i aCm aCm .
(6 .1
em so that
Eem =
( + 2) (- 2)] - "21 [aCm + aCm
,
The choice of the designation E and B is related to the properties of parity of th quantities. Under a reflection with respect to the plane perpendicular to k , Q rema unchanged and U t r ansforms into - U. The spherical harmonics of spin s transform i (-ll s YCm under this operation, which implies that ECm transforms into (_ l)e EC while B em t ransforms into (_ 1)£+1Bem. Thus , E represents t he scalar (or electric) p of the polarization and B the pseudo-scalar (or magnetic) part. Figure 6.6 illustra configurations of fields with pure E and B modes.
6.3.3.4
Relation with the Stokes parameters
To better understand the relation between the E and B modes and the Stokes para eters, it can be useful to work in the flat-sky limit in which the coordinates (ii ,
8(n) =
J~:f.8(f.)eiR.n, J~:f.
(Q ± iU)(n) = -
[E(f.) ± iB(f.)] f12 (a±)2 eiR n .
(6 .1
(6.20
In t he small angle limit, (6 .186) implies that
The previous expressions have been derived in the natural spherical basis (e r , eo , e but in the small angle limit, one can define a fixed basis in the plane perpendicu
Kinetic description
U =0
Q >0
~
E
/
E>O
•
~
•
~
/
U=O
Q <0
/
347
~
/
/
•
/
•
/ Fig. 6.6 The polarization figures correspond to pure E or B modes at a given point , chosen as the centre . The bars give the orientation of the polarization at the centre of the segment and thus give a simple representation of the Stokes parameters. These configurations repeat themselves identically to infinity. Since the components E and B are non-local, one should consider the configuration of the polarization field all around the point to determine these two components. We recover the properties of parity of both components, the top two configurati~ns (E modes) remain unchanged under such a transformation, while the bottom ones (B modes) are exchanged.
to the local vertical, z . The Stokes parameters in the two systems of coordinates are related by (Q ± iU)' = ef'2i> (Q ± iU) so that, in this frame, Q(n) = U(n) = Since
d2f
J J
.
-
[E(f)cos2¢e - B(f)sin2¢e]e tRn ,
-
[E(f)sin2 ¢e - B(f) cos2¢e] etR . n
27r d2f 27r
.
.
(6.201) (6.202)
348
The cosmic microwave background
cos 2>e = (C; - C~) / C2
(6.
and
,
the previous expressions take the form and
in real space. The variables E and B are therefore non-local. This non-locality, in ited from (6.194) , arises from the fact that the relation between the spherical harmo of spin 0 and of spin 2 is also non-local. 6.3.3.5
Temperat ure and polarization expansion
The analysis of the previous sections shows that the radiation temperature 8(ry, x at a point x propagating in the direction n can be expanded on the basis Gem (x defined by
Gem(x , n ) = (-i/
J
4 2C : 1 eik"'Yem(n),
(6.
and that the polarization field can be expanded on the basis
(6.
We t herefore expand the spatia l part of 8 and (Q ± iU) on a basis of com exponentials and their angular part on a basis of spherical harmonics of approp spin
8(ry , x , n) (Q±iU)(ry,x, n)
J f;;: = J(2~~~/2 L d3k " (27r)3/ 2
=
(mJ 8 e (ry , k)Gem (x , n) , [EimJ ±iBi
mJ
(6.
] (ry,k)±2Gem(x,n). (6.
em
The Stewart- Walker lemma implies that the variables E~mJ and B~mJ are gauge variant.
6.3.3.6 Scattering cross-section
The Thomson scattering cross-section for an incident wave with linear polarization scattered to an outgoing wave with linear polarization [ out is da dD
=~
87r aT
I[in* . [out l2
(6.
,
(see Ref. [30]). It is convenient to introduce the partial intensities _
2
1+ Q
h =a 1 = - 2 - ' An incoming wave scattered at an angle intensities
1-Q -2-'
(6.
e in the plane (n, E 2 )
(Fig. 6.7) has outg
_
2
12 = a2
=
Kinetic description
e
lout _ 3crT lin 1 87r 1 ,
2 lin l 2out -_ -3crT - COS 2 , 87r which in t erms of t he Stokes paramet ers translates into
349
(6.211)
E' 1
n
J... . ......
Incident light
.. !!t .~ .............................. 8CClttered l ' 19ht
()
F ig. 6 .7 Geometry of the Thomson scattering in the electron prop er fr ame. An incident photon in the d irection n is deflected by an angle () and comes out in t he direct ion n' .
lout ) = 3crT ( 1 + cos ( Qout 167r sin 2
2
e
e
2
sin e 1 + cos 2
) e
( l in ) Qll1 '
(6.212)
Under a rotation, (Q , U ) transforms as (6. 177). To determine the cross-sect ion of an init ial st at e (l in, Qin, Uin) propagating in t he direction n scattered into a final state (lout , Qout , uout) that propagates in the direction n' , we perform the following operations , using the plane (y , z) as reference plane, (see Fig. 6.8 for a definition of the angles and the vectors and Ref. [30] for details). 1. We perform a rota tion around n to bring t he plane (n , n') into the plane (n , z ), which t ransla tes into a rot ation (6.177) with ¢ = a for t he Stokes parameters.
2. We perform a rotation around t he axis z to bring the new plane (n , n') into the reference plane (y , z) . This operation does not influence the Stokes parameters. 3. We can then use t he relations (6.212 ). 4. We perform a rotation around the axis z to bring the scattering plane back to the old plane (z , n') . Again, this operation does not affect the St okes parameters . 5. Finally, we perform a rot ation around n' to recover the initial position , which translates into a rota tion (6.177) with ¢ = a' for t he Stokes parameters. Omitting t he parameter V t hat decouples from the ot her polarizations and remains zero if it 1Vas initially, the previous procedure allows us to relate the outgoing Stokes parameters to their incoming values (6.21 3)
350
The cosmic microwave background
z
y
x Fig. 6.8 Definition of the angles and vectors involved in Thomson scattering.
The matrix P can be decomposed as
(6.2 with
p rO)
p (2) =
2(1 - J.L2)(1 - J.L'2) J.L'2 4 ( 0
=~
+ J.L2J.L'2 J.L2
0) 1 0 0 0 '
(6.2
3 (4J.LJ.L' cos 5 0 2J.L sin 5) p el) = _ 0 0 0 4 ' -4J.L' sin 50 2 cos 5
(6.2
J.L2 J.L'2 cos 25 _ J.L2 cos 25 J.L2 J.L' sin 25) - J.L'2 cos 25 cos 25 -J.L' sin 25 , 4 ( _ 2J.LJ.L'2 sin 25 2J.L sin 25 2J.LJ.L' cos 25
(6.2
~
with 5 = cp' - cpo We then have access to the Stokes parameters by performing rotation
(h) ~
1(110)(1)
=2
~ ~1 ~
~.
(6.2
For each polarization, the collision t erm is given, in the rest-frame of the baryons
erA) = (JTn e
[J
f(A')(E' , n')PNn ,n,)d:;' -f(A)(E" n)] ,
(6.2
Kinetic description
351
where P;,(n , n') corresponds to the 2 x 2 block of P corresponding to (h ,h). T his expression generalizes (6.144). This collision term can be expanded on t he harmonics basis (6 .205) and (6.206). For that , it is convenient to introduce the vector T == (e, Q + iU, Q - iU). It can then be shown (which we do not demonstrate here, see Ref. [3 1]) that the collision term takes the form (6.220) The term
p (m)
is deduced from (6.214)- (6.2 18) and takes the form Y2rn(n') Y2rn (n)
p (m)
=
(
- V6 Y 2rn (n') 2 Y 2rn (n)
-Ii
2Y2rn (n ') Y2m (n )
32 Y 2m (n') 2 Y 2rn ( n)
- V6Y2rn (n ') 2 Y2rn(n) 3 2Y2m(n') - 2Y2m (n)
-Ii
-2 Y 2m(n') Y2rn(n) )
3 - 2 Y 2rn (n
') 2 Y 2rn (n)
.
3 - 2 Y 2m (n ') - 2Y2rn (n)
(6 .221)
Note that if one restricts attention to t he temperature, this expression reduces to (6.151). 6.3.3.7
Origin of th e cosmic microwa ve background polarization
The properties of the Thomson scattering that we have just described allow us to understand from a heuristic point of view the origin of the cosmic microwave background polarization (Fig. 6.9 ). A mode propagating along the direction x has a polarization in the plane (y , z) . If an initially unpolarized wave is scattered in the direction z , the outgoing wave ends up being linearly polarized in the direction y. However , if we consider incident waves coming from every direction, then statistical isotropy implies that t he outgoing wave is not polarized (on average) , see Fig. 6. 9a. If initially radiation has a dipolar anisotropy, the polarization along y of the waves propagating in the directions x and - x has on average the same amplitude as t he polarization along x of the waves propagating in t he directions y so that t he outgoing wave along z is, again, not polarized (on average), see Fig. 6.9b. To produce a polarized outgoing wave, t he radiation must have a non-vanishing quadrupole (see F ig. 6.9c). The hot and cold zones are then in orthogonal direct ions compared t o the scattering point. This implies t hat the polarization of t he cosmic microwave background will be weaker than the t emper ature anisotropies. Indeed , the T homson scattering, which sources this polarization , is only efficient when the Universe is ionized . At this epoch, photons an,d baryons are strongly coupled so that the quadrupole of the radiation is very weak. For scalar modes , t he polarization amplitude on super-Hubble scales (i. e. for multipoles lower than the position of t he first Doppler peak) must t herefore be strongly damped. The polarization field will in addition exhibit the same acoustic peaks as those
352
The cosmic microwa ve background
z
z
y
y
x
x
x
(b)
(a)
Fig. 6 .9 (a) Isotropy implies t hat t he Thomson scattering does not generate any polariz The p olarization amplitude of the waves propagating in t he d irections x and yare equa polarization of the outgoing wave carries away the y-component of t he incoming wav propagates along x and the x-component of the incoming wave propagating along y, have the same amplit ude. (b) Case of d ipolar anisotropy : the polarization of t he ou wave is the sum of the y-components of the incoming waves propagating along x (c from a hot zone) and -x (coming from a cold zone) that, on average, has the same amp as the x-compon ents of the incoming wave propagating along y. It is thus also unpola (c) Case of a quadrupole anisotropy: the outgoing wave has a polarization along y , inh from t he incoming wave along x that comes from a hot region, larger than that alo inherited from the incoming wave along y t hat comes from a cold region.
for the temperature sp ectrum but, due t o the quadrupole origin, the position o peaks will be out-of-phase. On the other hand , the primordial gravit ational waves intrinsically have a quadrupolar anisotropy, even on super-Hubble scales, can gen polarization fields , even at large angular scales. 6.3.3. 8
Temperature and polarization hierarchy
Choosing the direction as
e3
along the direction k allows us to express the gradient
.
n'oi
.
--+
in' ki
=
iy(h 3 kYlO( n).
This term will multiply the angular dependence of the temperature and t he p ization. This introduces terms of the form YlOY£m and YlO±2Yfm that are expr as
{hy
Y3
las
y; sKe y; _ ms y; fm= )(2£+1)(2£- 1)S f-1m £(£+ l) S fm
+
m s Ki+l
)(2£ + 1)(2£ + 3) s
y; i+ lm,
(6
wit h
(6
Kinetic description
353
The explicit form of the Boltzmann equation follows directly from this decomposition to give
e(m) ' =
k [
E (m) ' £
OKp
(2f!- 1)
£
=k[
2Kp
(2f! -1) - T'
B (m )' = £
k [
£- 1
E (m) _ £- 1
[E~m) +
2K p
(2f! - 1) - T
elm) _
B (m) £- 1
'B (m)
e
OKU1
(2f!+3)
elm)] _ T'e( m) + S(m) £+ 1
2m B (m) f!(f!+ 1) £
£
£,
K 2 U 1 E( m) ] £+ 1
_
(2f! +3)
(6.225)
V6P(m)6£ ,2] ,
+
(6 .224)
2m E (m) f!(f! + 1) £
K 2 U 1 B (m ) ] £+1
_
(2f! + 3)
(6 .226)
.
The source t erms are given by S~O) = T' p rO) , s (1) = T' p ( l ) 2
S~2 )
= T' p
+ ~
(6 .227)
(2) _ H (2) ' ,
if the Fourier coefficients of the vector and t ensor perturbations are expanded as (6.228) T he t erm by
p(m )
that reflects the anisotropic na ture of the Compton scattering is given p(m)
= ~ [elm) _ V6E( m) ] 10
2
2
. ,
(6.229)
and only includes the quadrupole of the temperature and of t he polarization E. Notice that the polarization source, p (m), only enters into the equation of evolution for the quadrupole of E. The equations for the temper ature are to be compared with those [(6.163) and (6. 166 )J obtained by neglecting the polarization. It is not necessary to solve these equations for Iml and - Iml . By symmetry, B~O) = 0, for any source. This set of equations replaces the fluid equations of evolution for the radiation. The other equat ions remain unchanged. For neutrinos , we can use similar equat ions with T ' = O. Notice that in the short-wavelength limit (k » T') , t he evolution equations for a scalar mode imply that p rO) =
~e (O) 4
'
and (6.230)
354
The cosmic microwave background
Using these relations (6.167) , we obtain
(6.2
which is the relation we have used in the study of the Silk damping [see Section 6.2
6.3.3.9 Power sp ectra With the previous expansions , the power spectrum is given by
(2£+ 1) 2C{Z
~
=
J t
m- 2
k 2dk
X im) *(1]o , k)Zi m)(1]o , k) ,
(6.2
m= - 2
where X and Z take the values 8 , E or B. Figure 6.10 illustrates the four non-vanish spectra that can be computed.
-.- . _.- .-'
1000
\ , .. , .. . .
-
~.~
-.-._._ ._._.-
,'"
100 I
10
....... - - -.......
"
"
0.1
"
......
... -...
E
\
I
\,
.................. .... .. ....... .... /
I.:
~
',IIJ' I
/.l···· ·
•
I' , I 't
h, "'I,
I
q:
'I
\ \ \
..
.1
h:~
'1
....... (
..'
\
\.'
hi,
.:, 0 ',
~
l:~~' I 'u~
~ !!I \
I
\,
:'
I, :
, ,
1
1
\,'t
'. ,
~:S"~I'ot.;j /:r.. 91~ :J:\
, / \'I: I
BE
" : ~.:.
\ \
.
."
\,
'. \.\
".,
,
0.01 10- 3
10
l
100
1 000
10
l
100
1 00
Fig. 6.10 The four sp ectra 8 - 8 , 8 - E, E - E and B - B induced by scalar (left) tensor (righ t) m od es for an inflationary model. As stressed earlier , the sp ectrum B only generated by t ensor m od es and the E - B sp ectrum vanishes by symmetry.
The numerical integration of the Boltzmann hierarchy can in principle be used determine the power spectra. Historically, this was t he first method used to solve problem and it amounts to evolving t he photon distribution function during t he en history of the Universe. Technically, this approach is not very good as one has to s
Kinetic description
355
a large number of coupled differential equations, typically of the order of 5£max for a large number of values k (typically of the order of a thousand in an interval krJo E [£min/ 10 , 2£max]) . One should also carefully control the way the hierarchy is truncated to avoid any (unphysical) reflection of the power towards the lower multipoles. 6.3.4
Numerical integration
6.3.4.1
R adial modes
It is useful to rewrite the expression for the functions sGem by expanding the complex exponentials into spherical harmonics thanks to the relation (B.20)
exp(ik· x)
2:) -i)\/47r(2£ + l)je(kr)Yeo(n),
=
e with e3 = k / k and x = - rn. The sign convention is such that n corresponds to the direction of propagation of the radiation toward the observer, i.e. opposite to the direction of observation. Using the recursion relations (B.46) between the spherical Bessel functions, we can rewrite the functions sGem as (6.233)
and
±2G2m = 1)- i)\/ 47r(2£+ 1) [f~m\kr) ±i,8~m\kr)] ±2Y£m(n).
(6 .234)
e
The functions j~e'm), f~m) and ,8~m) are then explicitly given by .(00).
1£
= Je,
.(10).,
Je
.(20)
= Je,
(l(i+l) je
.(ll) _
Je
Je
.(21 ) =
V~ x '
-
=
1£
+ Je . )/2 , / 3£(£ + 1) (je)'
(3 '/1 1£
V
x
'(6.235)
3 (£ + 2)! je "8 (£ - 2)! x 2
.(22) _
1£
2
-
and (0)
fe
=
.(22)
Je
(6.236)
,
r.-c--....,....,-.,.------,-
(1)
fe f
=
(2) -_
e
,8~0)
=
)(£ - 1)(£ + 2) je 2 ( x2 1 ( -Je . + I./1 n 4 ~
-
+ j~)
,
.'), + 2 -Je. + 4Je -
x2
(6.237)
X
X
0,
(6.238) (6.239)
r.-c--....,....,-.,.------,-
,8~1) . = )(£ - ;(£ + 2) ~, ,8(2) =
e
!2 (j~ + 2 jex )
.
Moreover, these functions satisfy f~-m) = f~m) and ,8~-m) = _ ,8~m).
(6 .240) (6.241)
356
The cosmic microwave background
6.3.4.2
Integral solu tion
The Boltzmann equations (6.224) - (6.226) have a formal solution in an integral form Using the expansion (6.233) and (6.234) for the functions sGem into radial and angula parts, we can show that the solution of the temperature evolution (6.224) is given b
(6.242 The solut ion for the pola rization takes t he similar form
[E~m) ± iB~m)] (T)o , k)
= - (2£
('0dT)T'e -
+ 1)V6 Jo
T
p (m)( T) )
[E~m) ± i,8~m)] [k(T)O - T))]
(6.243 This clever method [32,33] is actually a generalization of the Sachs-Wolfe formula sinc it is equivalent to looking at the temperature of the emitting zones and integratin along the line of sight. Its huge advantage lies in the fact that it is sufficient to comput precisely only the moments of order 2 of the perturbed distribution function by solvin the Boltzmann equation since the higher order moments do not enter into the sourc terms S~m). In practice, one can simply truncate the Boltzmann hierarchy at £ = even though it is preferable to go up to £ ~ 10. The other mult ipoles are deduce from the integral expressions. This step is quick since the Bessel functions can b pre-computed. The form (6.242) is similar to t he expressions we had obtained in Section 6.1.2 which did not contain the radiation anisotropic stress since Thomson scattering wa not included at this stage. In particular, we have seen how the functions j~OO) an j~lO) for the scalar modes and j?l) for the vector modes arose naturally when th appropriate basis was used for t he expansion. We notice a difference of normalizatio between j?2) and J?2) which that due to the difference between the basis used for th expansion of t he tensor part [see (6. 143)].
6.3.4.3
Freely available codes
Several codes integrating the Boltzmann hierarchy for the temperature and the polar ization numerically are freely available: CMBFAST [34], CAMB [35], CMBEASY [36 and COSMICS [37] . These codes use the advantages from the integration techniqu along the line of sight. They either use a synchronous gauge or the Newtonian gaug and are written in FORTRAN or C++ . The freely available versions integrate th perturbation equations in the simplest case of a ACDM model; all extensions (dar energy, ... ) can be included at will to adapt to one's favourite model.
6.4 6.4.0.4
AnisotrQpies of the cosmic microwave background Observations of the cosmic microwave background anisotropies
We observe the cosmic microwave background on a sphere around us. A map of th cosmic microwave background can t hus always be expanded into spherical harmonic to extract the coefficients ae~,
Anisotropies of the cosmic microwave background
8 obs (e) =
I.: a£~Ycm(e).
357
(6.244)
Pm
The isotropy of the Universe implies that the angular two-point correlation function of the observed temperature fluctuations , (6.245) can be obtained as an average on the sky. This function only depends on cos {) and therefore can be expanded in Legendre polynomials as (6.246) so that
cobs P
= _ 1_ " 2f!.+ 1
~
laObSl2 . Pm
(6.247)
m
The theoretical predictions can only be statistical. Therefore, the aim cannot be to compare the observed cosmic microwave background map to a given simulated map but to compare the statistical properties of the two temperature distributions (observed and theoretically predicted in a given model). The comparison between theory and observations therefore introduces the spatial analogy of an ergodic hypothesis since Gi bs is obtained from the sky average of a unique realization (our observable Universe), whereas Cp is obtained from an ensemble average on some stochastic initial conditions in the frame of a model. This leads to the notion of cosmic variance.
6.4.0.5
Estimator of C p and cosmic variance
From a theoretical point of view, we have access to the angular power spectrum of the temperature field that is a homogeneous and isotropic random field. The coefficients aPm are also independent stochastic fields of vanishing mean value. This implies that
The angular power spectrum Cp is not directly observable but an estimator of Cp can be obtained by summing over m as (6.248) with the variable
Vc
defined as
Gi bs , obtained b~summing the of the estimator Cp .
a£~ over m , as in (6.248), thus represents a measure
358
The cosmic microwave background
In the case of inflationary models, the stochastic fields aim have almost-Gaussi statistics, meaning that their one-point probability distribution function is ) P( aim
=
1
aim) .
(
..j2;c£ exp - 2C£
(6.24
Ve being the sum of the square of 2R. + 1 independent random Gaussian variables w the same variance, its probability distribution function follows a X2 distribution w 2R. + 1 degrees of freedom, i.e.
(6.25
We infer the probability distribution function of
O£,
(6.25
When R. -+ 00 , the central limit t heorem ensures that this distribution becomes Gau sian and that (6 .25 lim O£ = C£. £""' 00
The estimator is then said to be coherent. Since (0£) = C£, from (6.22), this estima is not biased. Its variance can also be computed and we obtain ~2
~
2
2
2
(C£) - (C£) = 2R. + 1 Ce ·
Oe is the best estimator of C£ that we can construct since its variance is the small one can hope to obtain by estimating Ce. The efficiency of the estimator is limited the fact that, for a given multipole, there only exists 2R. + 1 independent modes an 2
2R.
+ 1·
(6.25
This cosmic variance is an unavoidable statistical limitation and is mainly importa at small multipoles. To conclude, no statistical properties of the aem are directly accessible from o servations. To compare predictions to observations, one must construct estimators summing over the aem. In the case where the primordial perturbations do not foll Gaussian statistics , the construction of a good estimator (unbiased and optimal) much more difficult . In particular, the two-point correlation function does not co pletely characterize the distribution. 6.4.0.6
Status of the observations
The measurement of the cosmic microwave background anisotropies has undergon revolution over the past decades and we refer to Refs. [38- 40J for a detailed presentati of these observations.
Anisotropies of the cosmic microwave background
359
The first revolution has been performed by the DMR experiment on board of the COBE sat ellite (COsmic Background Explorer) that has measured for the first time the temperature anisotropies of the cosmic microwave background up to a multipole of the order of £ '" 20, corresponding to an angular scale {) ~ 7 deg . As illustrated in Fig. 6.11, the state-of-the-art has quickly evolved over the past ten years. In 1997, the observations of COBE at small £ and the results of various ground-based experiments were not able to discriminate between two test models. From 1998, t he results of the balloon experiments BOOMERanG and MAXIMA have made it possible to detect the first and then the second peak , which was then confirmed by t he Archeops balloon. Since 2003 , the observations of the WMAP satellite give a good measure of the temperature angular power spectrum up to £ '" 1000. Figure 6. 11 summarizes the complementarity of these experiments. The cosmic microwave background observation experiences can be classified into three categories . 1. Ground-based experiments measure the small-scale temperature anisotropies, mainly
by interferometry (DASI [41 ], CAT [42], VLA [43], CEI for instance). These observations must be performed at altitude or at very dry sites. T heir main problem lies in the atmospherical fluctuations. 2. Balloon experiments are carried onboard balloons flying at an altitude of around 40 km, which reduces the problems related to the terrestrial atmosphere. These experiments are, however, limited in weight and cannot be easily manipulated during the flight that lasts from around t en hours to ten days. The improvement compared to ground-based experiments is, however , important and balloons have led the observational advances in the pre-WMAP era (BOOMERang [44], MAXIMA [45] and Archeops [46], for instance) . 3. Satellite experiments allow a full sky survey to be performed but this solution is expensive. Only two such experiments have been conducted (COBE [47] and WMAP [48]) and a third one, P lanck [49], is scheduled to be launched in 2009. The revolution in this area has been led by t he WMAP satellite, which has measured the cosmic microwave background anisotropies in five frequency bands (22 , 30 , 40, 60 and 90 GHz). Among t he numerous results [50] of the three years of observation, we can note: - the realization of a map of the temperature anisotropies with a resolut ion of 0.3 deg (Fig. 6.12) that gives access to the angular power spectrum up to £ '" 1000. - a 'confirmation of the existence of the first acoustic peak, the position of which is now measured with a precision [5 1] £(1)
= 220.8 ±0.7,
(6 .254)
with an amplitude of 74.7±O.5 ~K. T he second peak is located at £(2) = 530.9±3.8 and has an amplitude of 48.8 ±0.9 ~K. The first and second troughs are located, respectively, at £(-1) = 412.4 ± 1.9 and £(-2) = 675.2 ± 11.1. - a measure of t he cross-correlation spectrum between the temperature and the E-polarization (Fig. 6.11). The spectrum has an antipeak at £ = 137 ± 9 and a peak at £ = 329 ± 19. The existence of this anticorrelation provides a new, even
The cosmic microwave background
360
Angular scale
6000 ,9 ~0__ o ____- 2._o___0~._ S_ o __0~.2_o--
t
10
5
*
OVRO
CAT A MSAM
Temperature ~
¢MAX X SP94 o ARGO • Sas":atoon
- A-CDM (All data)
5000
2,
t WMAP CB I ! ACBAR
;:; 4000 "-
'*
Python • Te nerife o co BE
2003
~ ,...., 3000
2
+
~ <>..,
2000
~
°
b,....,
1 1000
"t<
C'\j
"-
0
cJ ,....,
----+ 10 ~
t OVRO
2000
*Maxim a - 1
3
Reionization
2,
• BOOMERANG
o Saskatoon
5
Temperature ~
E- Polarization
correlation
t< C'\j
X Toco
"-
• QMAP
ill
O COBE
cJ ,...., ---+
2
2
1
~ O
h
1
0
-1 10
100
1000
0
10
40100200 400 800 140
t
Fig. 6.11 Evolution of the observational status between 1997 and 2003 (left). The solid l curve represents the best-fit ACDM model, whereas an open model (no = 0. 3, dotted li is now excluded by observations. (right): The power sp ectra for the t emperature and temperature-E polarization cross-correlation obtained by the WMAP satellite.
more robust , proof of the existence of super-Hubble fluctuations at the time decoupling, since the Sachs- Wolfe plateau of the t emperature power spectru alone could have been generated along the line of sight after decoupling (as, e. in topological defect models) . The sign of the cross-correlation and t he phase the acoustic oscillations compared to that of the temperature power spectrum a strong indications in favour of adiabatic primordial perturbations.
- the 8E cross-correlation at large scales (€ < 20) is explained by an early reioni tion of the Universe, at a redshift of Z re = 1l.0 !~t which gives an optical dep of T re = 0.089 ± 0.03.
Anisotropies of the cosmic microwave background
361
Fig. 6.12 Map of the cosmic microwave background anisotropies provided by WMAP compared to the resolution of the COBE satellite. (WMAP collaboration, 2003).
- The EE correlation was detected at a level of
£(£ + 1) GPE 21f
=
(0.086 ±0.029)(I-lK)2
for £ = 2 - 6. This is interpret ed as the rescattering of cosmic microwave background photons at the reionization. - No evidence for B-modes was seen , hence limiting
for £ = 2 - 6. - From these observations, it can be deduced that the t emperature field is almost Gaussian, as predicted by inflation, but this question requires special attention.
362
The cosmic microwave background
........................
MAX 1-----,-,.---
--'-----1'-.-+--.(,' I PYTHON
COBEDMR
I
S~p~S=A~S"'KA: ----' T-=O.....:0'-l~1
10 104
Fig. 6.13 Summary of t he angular and frequency sensit ivit ies of various cosmic microwav background observation experiments. The hatched zones correspond to the regions where t h foreground emissions exceed 20!-iK in t he cleanest 20% region of t he sky. From Ref. [53J.
After WMAP, cosmology has a new st andard model, the best fit model (or concordanc m odel), which is a fiat ACDM model with nAD rv 0.7. Even though observations ar compatible with the predictions of infiation, some obscure points persist:
- WMAP has confirmed the fact that the quadrupole had less power than expect ed as already pointed out by COBE. - The octopole seems t o be correlated with the quadrupole, - Various statistical analyses seem to indicat e differences between the northern an sout hern hemispheres. The exist ence and origin of this anisotropy is still no completely established.
The first detection of the polarization of the cosmic microwave background wa announced in September 2002 [5 2]. This detection was obtained thanks to an interfer ometer , DASI, working a t 30 GHz a t the south pole. The amplitude of the E and B modes has been constrained by assuming that their spectrum was that of the best-f ACDM model, which made it possible to establish a 5a detection of the E modes .
Anisotropies of the cosmic microwave background
363
The Planck satellite will observe the cosmic microwave background in nine frequency bands spread between 20 and 800 GHz with a resolution of the order of 4 arcmin , which corresponds to £ rv 2000. It uses radiometers , supposed to be 1000 times more sensitive than that of COBE, for the frequencies lower than 100 GHz and bolometers at high frequencies.
6.4.0.7 Statistical isotropy and observation on an incomplete sky When observing the cosmic microwave background , various foreground emissions must be subtracted . This is particularly the case for the emission of our Galaxy, which in the simplest case, amounts to cutting out part of the observed sky. The observations then give access to
8(e)
8(e)W(e),
=
(6.255)
where W is a mask indicating which part of the sky has been cut out. Expanding the mask in spherical harmonics as (6.256) with w£m = (_ 1)mwe _m, then the coefficients aem are related to the primordial coefficients, aem, by
aem =
L
ae,m,
l, m,
L
We2 m2
e2m2
J
d2eYe,m, (e)Ye2m2 (e)Yem(e).
(6.257)
Using (B.23), this expression reduces to
-aem
=
~ ~
e,m, , al,m, K em
(6.258)
elm, where the kernel K i:nm, is explicitly given by
Ki:nm,
=
(_ 1)m
L
we2m2
(2£1 +
1)(2£~7r+ 1)(2£ +
1)
(~ £~ ~)
(:;1
:;2 _~) ,
e2 m2 (6.259) For a constant mask , W = Woo Yoo , we find that K i:nm, = woo6u, 6mm ) v'47f. We can then check that aem does not satisfy the property (aema£'m') = 6w6mm,Ce but (-aem -* ae'm' ) --
~ ~
Ce, Ke,m'Ke,m,* em i'm"
elm, since the statistical isotropy is broken by the mask. This illustrates the difficulty in constructing an estimator of Ce for a partial sky. Moreover, this stresses the difficulty in testing in practice the fact that the aem are independent and isotropic stochastic fields.
364
The cosmic microwave background
6.5
Effects of the parameters on the angular power spectrum
The scalar, vect or and tensor p erturbations leave different signatures on the cosmic m crowave background temperature anisotropies and polarizations. The E-polarizatio is generat ed by both the scalar and t ensor modes, whereas the B-polarization is onl induced by the tensor modes so that , in principle, it allows us to have a direct acces to the properties of the gravitational waves . We can measure the power spectra an cross-correlations of t hese various quantities. Due to symmetry properties , only th correlation between 8 and E are non-vanishing. We thus have access to four observ abIes (8-8 , E- E , 8- E and B-B) ; the first three are generated by the scalar and tenso modes while the fourth is uniquely generated by the t ensor modes. The theoretical curves of these angular power spectra depend on the value of th cosmological paramet ers and on the initial conditions for the perturbations. We illus trat e here the influence of various paramet ers on the t emperature power spectrum. We shall distinguish the cosmological param eters that describe the matter conten of the Universe and that affect the perturbations transfer function and the primordia param eters that describe the perturbations' initial conditions. 6.5.1
6.5. 1.1
Cosmological parameters
B aryon density : DbOh2
As seen earlier (see Section 6.2 .2) , the main parameter that influences the peak struc ture is R. Its value at the time of decoupling is given by
The baryon density
D bO
has four main effects on the angular power spectrum .
- It fixes the oscilla tion frequency of the photon- baryon plasma via the sound speed When DbO increases, Cs decreases and t he frequency of the oscillations is smalle So, DbO influences the position of the peaks and their spacing. - It fixes the relative amplitude of the even and odd peaks. This amplitude is 1 + 6 in the adiabatic case. The greater DbO , the smaller the second peak. It fixes the contrast between the peaks , by influencing the amplitude of th Doppler t erm. If DbO = 0, the peaks disappear altogether and the contrast in creases when D b O increases. - It influences the Silk damping at small scales by changing the photons mean fre path. The greater DbO is, the shorter the mean free path and the more importan is the damping. Figure 6.14 illustrates these effects.
6.5.1 .2
Total m atter density: DmOh2
In t he standard model, the baryon density is small compared to that of dark matter so that the total matter density can be approximated by that of dark matter. Unlik baryonic matter , dark matter only acts gravitationally on the photons. Dmo is t hu only felt via its influence on the time evolution of the Bardeen potentials. In particular
Effects of the parameters on the angular power spectrum
/
/ / /
/
/
/ / /
"
10
" ""
""
/
""
\
365
\ \
\ \ \ \ \ \ \ \
" ""
100
1000
Fig. 6.14 Influence of the baryon density on the temperature angular power spectrum. For the spectra in solid lines , DbOh2 = 0.0125, 0.0250 , 0.0375 , 0.0500 and 0.0875. The two extremal values are 0.0025 (dotted) and 0.1250 (dashed ). From Ref. [3].
an increase in Druo is equivalent to an increase in Zeq and t hus a delay in the matterradiation equality, which affects the contribution of the early integrated Sachs- Wolfe term.
6.5.1 .3 Dark-matter- baryon ratio The ba~yon to dark matter ratio a = Db/Dc has an influence on the relative contribution of both components in the Poisson equation. Since the density contrast of baryonic matter grows only after decoupling, we expect t hat the angular power spectrum will have a lower amplitude when a increases. Moreover, matter perturbations oscillate when they are coupled to photons and perturbations keep a trace of these oscillations after decoupling. These oscillations are then imprinted in the matter power spectrum and are known as baryonic acoustic oscillations.
6.5.1 .4 Hubble constant: h The Hubble constant is not actually a free parameter since, from the Friedmann equations, it cannot be varied independently from the matter content of the Universe. Its influence on t he angular power spectrum depends on parameters that are kept constant when varying its value (the Di or the Wi = Di h 2). As an example, if the curvature
366
The cosmic microwave background
and the cosmological constant are fixed , 1 + Zeq = Dmo / Dro ~ 2.4 x 104Dmoh2 . If h creases, the matter- radiation transition occurs later, which has the effect of increas the amplitude of the acoustic peaks (Fig. 6.15).
104 ~
~
"--' ~ (\J
.......
cJ
",...,+
~
-:::::.
3 ~ 10 .. ... .. . .. .. .
10
100
1000
Fig. 6.15 Influence of t he Hubble constant. The baryon density is fixed to IlbO = 0.0 and that of dark matter varies with h that takes the following values 0.2 (dot ted line), 0 0. 50, 0.65,0.80 (solid lines) and 0.95 (dashed lines). From Ref. [3J.
6.5.1 .5
Cosmological constant: DAO
Varying DAO at fixed Do , the cosmological constant delays the matter- radiation equ ity and modifies the relation between the angular distance and the redshift. The sition of t he acoustic peaks will thus be affected (see Fig. 6.16) . Moreover, the cosmological const ant dominat es today so that the Bardeen tentials decay at late time, which induces a late integrat ed Sachs- Wolfe effect a increases the power at large angular scales.
6. 5. 1.6
Curvat ure: DKOh 2
The effect of the curvature is similar to that of the cosmological constant. In an op Universe , the relation between the angular distance and t he red shift is modified a the Bardeen potentials decay after the matter- curvature equality. The effect of angular distance implies also that the acoustic peaks will be shifted towards hig multipoles for open spaces and smaller ones for closed spaces.
Effects of t he parameters on the angu lar power spectrum
367
/'
I I I
\ \ ,
I I
c:;"
~
I
~
:! ~
1i'
("\J
(\j
"-
cJ
"cJ
".....
''"-< """
+
,,
~
::::, O
h
103
I
I I I I I
I I
+
I I I
""
I /
~
10
I'-...::::::::::::;::~
10
100
/ /
3 f---=::::::::~'?'
1 000
e
,
I I I
~
,,
I,
I
10
100
1000
e
Fig. 6.16 Influ ence of the cosmological const ant on the temp erature angular power sp ectrum. (left) : !lAO varies while keeping p critO and PO fixed , which implies t hat h also varies . !lAO takes the values 0 (dotted line), 0.4, 0. 6, 0.7, 0.8 (solid lines) and 0.9 (d ashed line). (right ): P bO and h are kept constant so that !lmo varies as 1 - !lAO . !lAO takes the values 0 (d ashed line) , 0.4, 0.6, 0.7, 0. 8 (solid lines) and 0.9 (dott ed line) . From R ef. [3] .
Notice t hat we can simult aneously adjust t he curvature and the cosmological constant while keeping the spect rum almost unchanged. These effects are illustrated in Fig. 6.17.
6. 5. 1.7 P arameter degen eracy As illustrated in the previous figures, varying t he cosmological parameters affects, on the one hand , the position of the peaks and their relative amplitude and , on the other , the shape of the Sachs- Wolfe plateau. At large angular scales, the cosmic variance becomes more important and different sets of parameters can lead to ident ical angular power spectra up to the cosmic variance precision. Actually, these degener acies can be understood if we remember t hat t he posit ion of the peaks is mainly dictated by the acoustic scale £A = !K(zLSS )/ rS(zLSS ) (6.74). For inst ance, the sensit ivity of this scale to the st andard cosmological parameters is ~£A
-- =
£A
~Do
~Dmoh2
- 1.1 - - - 0.24 2 Do Dmoh
-
~DAD
0.17- - DAD
O.ll~wQ
~D bOh2
- 0.07 -
- :DbOh2 '
around t he model (Do , Dmoh2, DAO ,WQ, DbOh2) = (1, 0.1 5, 0.65 , - 1, 0.02) , wQ being the equation of stat e of dark m atter. These degeneracies are illustra ted in Fig. 6. 18. 6.5.2
Parameters describing the primordial physics
Primordial physics provides the initial power spectra for scalar and tensor perturbations that are characterized by their amplit udes, spectral indices and variation of
368
The cosmic microwave background
\
€'
2,
~
~
"-
"-
C\j
cJ
~
...... +
~
C\j
,,
,,
"+
......
,
\ \ \ \ \
\ \ \
cJ ,,
\
\ \
,,
,,
,
~
~
~
~
"" ~
~
10 3
,
103
10
100
e
1000
"
"
10
100
e
100
Fig. 6.17 Influen ce of the curvat ure on t he angular power sp ectrum. (left): Spatially hyp bolic spaces. The matter density is constant and no is, respectively, 1 (dotted line), O.S, 0 0.4 (solid lines) and 0.2 (dashed line) . The peaks are shifted towards smaller angular sca and the decay of the gravitational potentials induces a late integrated Sachs- Wolfe effect w opposite sign to that of the ordinary Sachs- Wolfe t erm. (right): Spatially spherical spac The matter density is constant and no is , respectively, 1 (dotted line), 1.2, l. 4, 2.0 (solid lin and 4.0 (dashed line) . The Bardeen potentials increase in time since nm increases b efore t expansion stops . The integrated Sachs-Wolfe term has thus the same sign as the Sachs-Wo term, which translates into an increase in the large-scale power. From Ref. [3].
their spectral indices (Chapter 8) . It also provides an indication on the adiabatic isocurvature nature of the initial conditions for the perturbations.
6.5.2.1
Spectral index
As can be seen from (6 .50) and (6.96), the quantity €(€ + I)C" is proportional to €ns and €n T for the scalar and t ensor modes , respectively. For a red-tilted spectrum (ns < 1, nT < 0) the angular power spectrum has power excess at large angular scales, while a blue-tilted spectrum has a power exc at short angular scales. A deviation primordial spectrum from a scale invariance thus induces a tilt in t Sachs- Wolfe plateau and a modification of the relative amplitude of the acoustic pea compared with the Sachs- Wolfe plateau. This effect is illustrated in Fig. 6.19.
6.5.2.2
Primordial gravitational waves
The contribution of the gravitational waves is dominant on large angular scales, that they tend to decrease the relative height of the acoustic peaks (of purely sca origin) compared to the Sachs- Wolfe plateau.
Effects of the parameters on the angu lar power spectrum
6000
369
n mOh 2 = 0.14 , nboh2 = 0.024 nf(O = 0.0 , n AO = 0 .73 (h = 0.72) nf{O = - 0 .288, n AO = 0.0 (h = 0 .33)
~
~ ~
4000
C\J
"-
cJ ,...., +
~
~
2000
~
[!.; --------------
0 10
100
1000
£ Fig. 6.18 Illustration of the degeneracy of the cosmological parameters . Both models are indistinguishable given the limitation due to the cosmic variance. They have in common (rl mo h 2 , D bo h 2 ) = (0 .14, 0.024). The first model (dotted line) is the best-fit ACDM model with (DKo , D AO ) = (0 , 0.73) and h = 0.72, whereas the second (solid line) is an open model with a low Hubble constant, (DKo, D AO ) = (0.288,0) and h = 0.33 .
The structure of the acoustic peaks is still that of the scalar modes and the normalization of the scalar modes spectrum is lower when the gravitational waves contribute at large angular scales.
6.5.2.3
Adiabatic and isocurvature initial conditions
The choice in the type of initial conditions influences the peaks position and their relative amplitude. This is due to the relation (6.74) that explains how the peaks structure is modified . In particular , the position of the first peak is shifted by a factor 3/2 for an isocurvature model. A second difference arises in the normalization of the spectra since 8s w = /3 for a pure adiabatic model and 2 for a pure isocurvature model.
The cosmic microwave background
370
\
10
4
~
\. 10
'-'
1? C"\J
~
"-
C"\J
c.J 103 ',.... """
~ 102 CJ
',.... """
+
~
~
3
~
+
10
~ 10 1
2
~
10
100
1000
1000
100
10
Fig. 6.19 (left): Influence of the spectral index of the scalar modes on the primordial spe trum. 115 takes the values 0.5 , 0.75 , 1.0, 1.25 and 1.5 (solid lines) from bottom to top in t Doppler region. The dotted spectrum corresponds to the case 115 = 1.5 multiplied by £- 0 in agreement with the spectrum obtained for 115 = 1. (right): Influence of the T / S rat The dotted and dashed spectra represent the contributions from the scalar and tensor mod to 00. Fro alone, and we interpolate b etween these two curves when T / S varies from Ref. [3].
°
;::;'-
~
'-' ~
...................
'""-
cJ ~
.....
+
102
~ <>.0
~ 101 10
100
1000
Fig. 6.20 Comparison of t he temperature angular power spectrum for adiabatic and isocu vature initial conditions. From Ref. [3] .
References [1] R. DURRER, 'The theory of CMB anisotropies', J. Phys. Stud . 5 , 177, 200l. [2] W. Hu, Wandering the background: a cosmic background explorer, Thesis, University of Berkeley, 1995, [astro-ph/9508126]. [3] A. RIAZUELO, Signature de divers modele d'univers primordial dans les anisotropies du rayonnement fossile , PhD thesis , University of Paris XI , 2000 [4] A. KOSOWSKY, The cosmic microwave background, in Modern cosmology, S. Bonometto et al. (eds .), Institute of Physics, 2002. [5] R.K. SACHS and A.M . WOLFE , 'Perturbations of a cosmological model and angular variations of the microwave background ', Astrophys. J. 147, 73 , 1967. [6] M. PANEK, 'Large-scale microwave background fluctuations: gauge invariant formalism ', Phys. R ev. D 34 , 416 , 1986. [7] W. Hu and N. SUGYIAMA, 'Toward understanding CMB anisotropies and their implications' , Phys. Rev. D 51 , 2599,1995 ; W. H u and N. SUGYIAMA, 'Anisotropies in the CMB: an analytic approach ' , Astrophys. J. 444, 489, 1995. [8] A. SAKHAROV, Zh. Eksp. Teor. Fiz. 49 , 345, 1965, [Sov. Phys. JETP 22 , 241 , 1966]. [9] J. SILK , 'Cosmic black-body radiation and galaxy formation', Astrophys. J. 151 , 459, 1968. [10] M. BIRKINSHAW, Th e Sunyaev-Zeldovich effect, Phys. Rep . 310 (1999) 97. [11] J. BERNSTEIN, Kinetic theory in the expanding Universe, Cambridge University Press, 1988. [12] J. STEWART, Non-eq uilibrium relativistic kinetic theory, Lect . Notes in Phys. 10, Springer, Berlin, 1971. [13] J .-P. UZAN, 'Dynamics of relativistic interacting gases: from a kinetic to a fluid description', Class. Quant . Grav. 15, 1063, 1998. [14] M. HILLERY, 'Distribution functions in physics: fundamentals', Phys. Rep. 106, 121 , 1984. [15] P.J.E. PEEBLES and T.J. Yu , ' Primeval adiabatic perturbation in an expanding Universe', Astrophys. J. 162, 815, 1970. [16] J.R. BOND and G. EFSTATHIOU, 'Cosmic background radiation anisotropies in Universes dominated by nonbaryonic dark matter ', Astrophys. J. Lett. 285 , L45, 1984. [17] C.P. MA and E. BERTSCHINGER, 'Cosmological perturbation theory in the synchronous and conformal gauges', Astrophys. J. 455 , 7, 1995. [18] H. KODAMA and M. SASAKI, 'Cosmological perturbation theory ', Prog. Theor. Phys. Suppl. 78 , 1, 1984. [19] C. PITROU, 'Gauge invariant Boltzmann equation and the fluid limit ', Class. Quant. Grav. 24, 6127, 2007.
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References
[20] K. THORNE, 'Multipole expansions of gravitational radiation', Rev. Mod. Ph 52 , 299, 1980. [21] R. D URRER, 'Gauge invariant cosmological perturbation t heory: a general stu and its application to t he texture scenario of structure formation' , Fund. Cos Phys. 14, 209, 1994. [22] M.J. REES , 'Polarization and sp ectrum of the primeval radiation in an anisotro Universe', Astrophys. J. 153, L1 , 1968. [23] W. Hu and M. WHITE, 'A CMB polarization primer ', New Astron. 2 , 323, 19 [24] M. ZALDARRIAGA and D . H ARARI, 'Analytic approach t o the polarization of t cosmic microwave backbround in flat and open Universes', Phys. Rev. D 52,327 1995. [25] A. KOSOWSKY, 'Cosmic microwave background polarization', Annals. Phys. 24 49, 1995 . [26] M. ZALDARRIAGA, Th e polarization of the cosmic microwave background, in Me suring and Modeling the Universe, Ed. W. L. Freedman, Carnegie Observator Astrophysics Series, 309, CUP 2004[astro- ph/0305272].. [27] W . Hu and M. WHITE, 'CMB anisotropies: total angular momentum metho Phys. Rev. D 56 , 596, 1997. [28] J. JACKSON , Classical electrodynamics, Wiley, 1998. [29] E. NEWMAN and R. PENROSE, J. Math. Phys. 7 , 863 ,1966; N.J . VILENKIN, Sp cial functions and the theory of group representations, American Mathemati Society Press, Providence, 1968. [30] S. CHANDRASEKHAR, Radiative transfer, Dover, 1960. [31] M. KAMIONKOWSKI et al. , 'A probe of primordial gravity waves and vorticit Phys. Rev. Lett. 78, 2058, 1997; M. KAMIONKOWSKI et al. , 'Statistics of cosm microwave background polarization ', Phys. Rev. D 55 , 7368, 1997; W. Hu et , 'A complete t ratment of CMB anisotropies in a FRW Universe' , Phys. Rev. 57, 3290, 1998. [32] U. SELJAK , ' A two-fluid approximation for calculating the cosmic microwave bac ground anisotropies', Astrophys. J. 435 , L87, 1994. [33] U. SELJAK and M. ZALDARRIAGA, 'A line of sight integration approach to cosm microwave backgound anisotropies ', Astrophys. J. 469 , 437, 1996. [34] CMBFAST: http://cmbfast . org, developed by U. Selj ak and M. Zaldarriag [35] CAMB: http://camb.info , developed by A. Lewis and A. Challinor. [36] CMBEASY: http://www . thphys. uni - heide l berg. derdoran/ cmbeasy, dev oped by C. Doran. [3 7] COSMICS : http://acturus.mit. edu/ cosmics , developed by E. Bertsching and C.P. Ma. [38] G.F. SMOOT, 'CMB experiments', Phys. R ep. 333 , 269 , 2000. [39] Report 'Le rayonnement fossile a 3 K' , G.R. Acad. Sci. (Paris) 4 (2003); S. E DELMAN et a1. , Particle Data Group , Chap. 23 'Cosmic microwave backgroun Phys. Lett. B 592 , 1, 2004. [40] A.W. J ONES and A.N. LASENBY, 'The cosmic microwave background ', Liv Rev. R elativity (1998) [41] DASL http://astro . uchicago. edu/dasi.
References
[42] [43] [44] [45] [46] [47]
373
CAT: http://www.mnrao . cam.ac . uk/telescopes/ cat/index. html. VLA: http://www . nrao . edu/vla/html/VLAhome . shtml. BOOMERanG: http : //www . physics . ucsb . edurboomerang/ . MAXIMA: http://cfpa . berkeley . edu/ group/ cmb/. ARCHEOPS: http://www . archeops .org. COBE: http : //space.gsfc.nasa.gov/astro/cobe/cobe~ome.html .
[48] WMAP: http://map . gsf c . nasa . gov [49] P lanck Surveyor : http : //astro . estec.esa.nl/Planck. [50] L. PAGE et ai., 'Three year WMAP observations: Polarization analysis', A strophys. J. Suppi. 170, 335 , 2007. [51] D.N. SPERGEL et ai., 'WMAP three year results: implication for cosmology ', Astrophys. J. S uppi. 170, 377, 2007. [52] J. M. K OVAC et ai. , ' Detection of polarization in t he cosmic microwave background using DASI' , Nature 420 , 772,2002. [53] M. T EGMARK CMB analysis centre: http : //space . mit . edu/home/tegmark/cmb/experiments.html.
7 Gravitational lensing and dark matter 7.1
Gravitational lensing and its applications
Light is deflected by any inhomogeneous gravitational field (see Chapter 1). Th property was at the origin of the first tests of general relativity in the Solar System Today, it has become a powerful tool for mapping t he distribution of matter in th Universe, and in particular that of dark matter. The applications of gravitation lensing are numerous. Table 7.1 gives a classification of t he various gravitational lensin systems and their physical applications. Various complementary studies can be fo und in the two monographs [1,2] and the recent reviews [3,4] that detail the historical, theoretical and observational aspect
Table 7.1 Classification of the different (most studied, with well-established observation gravitational lensing systems in terms of t he source and the lens . The effects are classified in strong lensing (SL), microlensing (ML), weak lensing (WL) and statistical (st). Source
Lens
Effects
Star (galactic)
Star (galactic)
ML: light curves
Star (extragalactic) Galaxy
Compact object (galactic) Galaxy
ML: light curves
Galaxy
C luster
stWL : ellipt icity bias, shear- galaxy correlation SL: giant arcs , multiple image WL: arclets, magnification bias
Galaxy
Large-scale structure
StWL: cosmic shear, shear- shear correlation
Quasar
Galaxy
Quasar
Large-scale structure Large scale structure
SL: multiple images, Einstein rings , StWL : cosmic magnification, quasar- galaxy correlation StWL : smoothing of Ge, B modes
Last scattering surface
Applications
Extrasolar p lanets Milky Way inner structure Dark matter in M ilky Way halo Galaxy parameters, ha los properties Total mass of the cluster, DA ,redshifts C luster mass profile, cluster morphology Cosmological parameters Cosmological power spectrum , Mass of galaxies, substructures, Ho , DA cosmological parameters , power spectrum, bias Cosmological param eters, power spectrum
Gravitational lensing and its applications
7.1.1
7.1.1.1
375
Gravitational lensing in the thin-lens regime
Ligh t deflection by a point-like m ass
Lens equation The lens equa tion rela t es the real position of a source to its observed position (see Fig. 7.1 ) . First , we define the optical axis as the line joining the observer to the cent re of the lens (0 L ). The source plane and the lens plane are the two pla nes perpendicular to the optical axis located a t distances D os and DOL, resp ectively, from the observer. I __
I"'~---_-
I: o
I
I I I \
I I I I
'!.. - - -
Fig. 7.1 (left) : Gravitational lens configuration. L is the gravitational lens. e, es and 0: are, respectively, the angular position of the image, the angular position of the source and the deflection angle. DO L and D os are the angular dist ances between the observer and the lens and betw,een the observer and the source, respectively, and DL s is the angular dist ance between t he lens and the source. (right): When the source, t he lens and t he observer are aligned, the image of the source t akes the shape of a circle, called an Einstein ring. The positions of the source and of its image on the source plane are related by + S I , where AS can b e expressed in terms of the angula r position of the source 8s by AS = 8s Dos a nd similarly, AI is given by AI = 8Do s, with 8 the angula r position of the image, a nd S I = o:DLs, 0: being the deflection angle. We thus find tha t in the small-angle limit , the angular position of the image 8 = 8L is
AI = AS
8 = 8s
DLs + -D 0:(8) .
os
(7.1 )
Notice that the distances involved in this equation a re the angular distances since they relate tra nsverse physical distances to their angular diameter. The value of the deflection a ngle by a point-like source was obtained in Chapter 1 [see (l.143) ]; a = 4G N M / bc2 = 2Rs ch/ b, where b = D OL8 is the impact p ar amet er a nd R Sch is the Schwarzschild radius. The lens equation therefore b ecomes
(7.2) where we have introduced the geometrical factor V
== Dos DoL / DLs .
376
Gravitational lensing and dark matter
Einstein radius
The rotational invariance around the optical axis implies that a source loc exactly on this axis (8 s = 0) will be observed as a ring of angular diameter 8 = E
/ 4G N M DLs . c 2 Dos DOL
V
This angular radius, called the Einstein radius, depends both on the lens mass an the factor DLS/ D os DoL that characterizes the geometry of the system . Its phy radius is R E = 8E DOL (see Fig. 7.1). The Einstein radius is a natural angular scale for the problem. If lensing gives to multiple images, 28E will be the typica l angular separation b etween these ima Sources closer than 8 E from the optical axis are subj ect to strong distortion eff giving rise, for instance, to arcs and will be strongly amplified and distorted. Sou with 8s » 8E are almost unaffected by the lens. The typical orders of magnitud the Einstein radius for gravitational lensing by a star in the Galaxy (lVI rv M ev V rv 10 kpc) and by a galaxy (M rv 1012Mev ) in the cosmological context (V rv D are 8E ,star = 0.9
(~) 1/2 Co~pc) - 1/ 2 marcsec, M
8E ,gal
=2(
)
V
1/ 2 (
10 12 M ev
)
(
- 1/ 2
3000 Mpc
(
arcsec.
For inst ance, for a lens at z = 0.5 and a source at z = 2, we obtain 8E for an Einstein- de Sitter Universe with h = 0.7.
rv
2.13 ar
Multiple images and magnification
Equation (7.2) can be rewritten in terms of the Einstein radius in the simpl form
8s
=
82 8 - :-
(
Solving this equation for 8 gives two solutions
8±
=
~ (8s ± J8§ + 48~) ,
(
so t hat each source has two gravitational images . These two images are locat ed either side of the source, one inside the Einstein radius (8_ < 8E ) and the o outside (8+ > 8E ) , and are separated by 6..8 ::::: 28 E . Since the light deflection is not associated with any absorption or emission, gravitational lensing conserves the surface brightness. The surface brightness of image , I (8) , is rela ted to that of t he source, Is (8s), by
1(8) = Is [8s (8)] .
(
The lens gravitational field distorts the light beam and modifies t he solid angle u which the source is observed. The total flux is thus proportional to the ratio betw
Gravitational lensing and its applications
377
the solid angles under which the source and the image are observed. For a symmetric lens, such as a point-like lens, the magnification is given directly by
(7.9) since the surface element is proportional to sin BdB '" BdB, for small angles. In the case of a point-like lens, we obtain, after using (7.6),
(7.10) where u == Bs/BE is the impact parameter in units of the Einstein radius . Since B_ < BE, u_ < 1 and IL < O. This means that t he image inside the Einstein radius has an inverted parity; it is the image of the source in a mirror. Figure 7.2 represents the position and the relative orientation of the two images. 4 r-----------~.-~--------~
8
3
,,
2
,
8- 8s
1
o r---__
,/
("I ' t ',M ' II
-1
-
I
_........"
-3 ~ ~~~~~~~~~~~~~~
-3
-2
-1
0 8
1
2
3
_
--
\.,...,. ........
-2
-4
,,
8_
,,
4
Fig. 7.2 (left) : The solution of the lens equation can b e obtained graphically through t he intersection of the curve a(e) and t he line of equation e - es. (right ): The relat ive p osition of the two images of the source and t heir relative magnification .
The total magnification is defined as t he sum of the magnification of each image .
(7. 11) Moreover , one can check that f..L+ + f..L - = 1. Notice that f..L is always greater than 1, for instance, for a source on the Einstein radius, u = 1 and f..L ~ 1.34. As the source gets closer to the optical axis, Bs tends toward zero and the magnification diverges, hence the magnification of an Einstein ring would be infinite. This divergence disappears for obj ects that are not point-like (and is thus an artefact of this approximation).
Gravitational lensing and dark matter
378
7.1.1.2
General case
Thin-lens approximation
The previous example illustrates the different effects produced by a gravitatio lens, in the approximation in which it is point-like, an approximation beyond wh we need to go. Since a real lens is a priori not symmetric under a rotation around optical axis, we must replace t he angles by two-dimensional vectors representing angular position in the sky, assuming it is flat. As seen in Chapter 1, the total deflection to which a photon is subjected is
0=:2 J
\7 .1. el>
dz,
(7.
where \7 .1. represents a two-dimensional gradient perpendicular to the line of sight el> is the gravitational potential. In the case of a point-like lens of mass M , this poten is explicitly given by el> = -G N M/Vb2 + z2 where b is the impact parameter of undeflected light b eam and z is the coordinate along the optical axis. T his implies t \7 .1. el> = G N Mb/(b 2 + z 2)3 /2 and , aft er integration over z, that 0 = 4G N Mb/c 2 b2 As long as the deflection angle is small , the light r ay is only weakly deviated w respect to the undeflected tra jectory. The integral (7.12) can thus be computed integration along the undeflected trajectory. Such a Born approximation is comple justified for galaxies and galaxy clusters for which the deflection angles are typic I" and a 30" , respectively. a Moreover , the main contribution to the integral (7. 12) comes from the inter /'::,. z '"'-' ±b around the lens. This distance over which the deflection is effective is v small compared to the other distances along the optical axis. We can therefore assu that the lens is thin in the direction of the line of sight and that its three-dimensio mass distribution , PL, can be replaced by a two-dimensional mass distribution , obtained by integrating along the line of sight
:s
:s
~(b)
==
J
(7
pdb , z) dz .
This simplifying approximation is well adapted to the study of the lensing by galax or galaxy clusters. It enco des t he entire information on the deflector in a single fu tion , ~(b) , its surface mass density, localized in the lens plane at DOL. In the cas the point-like lens studied in Section 7.1.1.1 we have ~(b) = M f/2) (b) and thus t erms of angular coordinates, ~(O) = (M/ DbdJ(2) (0). Lens equation
In the approximation discussed above, the lens equation (7.1) takes the vector fo DLS
0 = Os + -D 0(0). os
(7.
To express 0 as a function o f ~ , let us note that the t hree-dimensional Laplacian of gravitational potential can be decomposed into /'::,. el> = /'::,. .1. el> + [Pel> / 2 . The Pois equation (5.5) can then be rewritten as
az
Gravitational lensing and its applications
379
so that the deflection angle is the solution of
(7.16) The integral of the term (EP
(7.17)
which is solved by
(7.18) In order to avoid any confusion, we will denote by ,6. the usual three-dimensional Laplacian" ,6.1- the two-dimensional Laplacian in the physical space (i.e. with respect to b) and ,6.2 the two-dimensiona l Laplacian in angle space (i.e. with respect to 8) so that .6. 2 = DbL,6.1-. We shall use the same convention for the gradient. After introducing the critical surface density
(7.19) the lens equation in the thin-lens regime reduces to
(7.20)
8 = 8s+0(8),
A lens with constant surface density equal to L:crit induces a deflection angle 0 = 8 so that 8 s ' = 0 for all 8. Such a lens perfectly focuses light and a focal length can be associated with it. This is not the case for arbitrary lenses that distort the images and are thus astigmatic. It is also useful to compare L:crit to the typical value of the surface density for a galaxy and a cluster, I: ~ M / R2 ,
L:ga1:c: L:ctuster :c:
0.3C01~0 ) C~~~J 0.03 15
2
-2
g. cm- ,
-2 M ) (Rctuster ) - 2 M g · cm . ( 10 M 0 3 pc
(7.21) (7.22)
380
Gravitational lensing and dark matter
Projected potential It is useful to introduce the projected gravitational potential obtained by integ ing along the line of sight
'IjJ(O) = D
DLS
D
OL
os
2 2"
J
if>(DoLO , z) dz.
(7.
C
Using V 2ln 8 = 0/8 2 , the above potential is related to the deflection angle (7. through
V 2'IjJ( 0) = D LS 0::. Dos
(7.
Using (7.17) and the property ~21n8 = 27T6(2)(O) , we finally obtain ~
1
-~2'IjJ(O) = - . 2 ~crit
(7.
The solution of this two-dimensional Poisson equation gives us the expression for proj ected potential as a function of the surface density 'IjJ(O) =
.!. 7T
J
d 20' ~(O') In 10 ~crit
- 0'1.
(7.
Magnification and distortion
Equation (7.8) implies that
1(0) = Is[Os(O)] = Is[O~
+ A(Oo).(O -
( 0 )],
(7.
where we have expanded the relation Os(O) obtained from the lens equation at f order around O~ = 0 0 - 0:: (0 0 ). This allows us to compare the differential effect neighbouring light rays. The image distortion is thus characterized by the distort matrix A(O) == 80 s (7. 80 ' the expression of which is obtained from the lens equation (7.20)
(7.
This symmetric (2 x 2) matrix involves 3 parameters, which can be decomposed terms of the convergence, K" and the shear, ., = (,1 , ,2) , as
A
=
(1 - ,2 ,I ,2+,1 ) . K, -
1-
(7.
K,
The convergence induces an isotropic focalization of the light beam and is related the projected potential and to the surface density by
Gravitational lensing and its applications ~
21'1,(0) = D.21/;(O) = 2 - ,
381
(7.31)
~cri t
while the cosmic shear describes an anisotropic deformation and is related to t he lens astigmatism. Its two components can be rewritten as 'Yl((}) =
1/;, 11 ;
1/;,22
== ')'(0) cos[2¢(O)], ')'2(0) =
1/;,12
== ')' (0) sin[2¢(O)].
(7.32)
As represented in Fig. 7.3, an initially circular source of unit radius will have an elliptical image with minor and major axes (1- 1'1,+,),)-1 and (1- 1'1,_,),) - 1, respectively, so that the image surface, S, is related to that of t he source, So, by S
= {1So,
{1 =
1 1 detA = [(1 - 1'1,)2 -
(7.33)
')'2]'
Since the differential deflection induces a variation in the shape and the surface of the source, it also modifies the light flux that is proportional to its surface. A is thus the inverse of the amplification matrix, M , which generalizes the expression (7.9) and we have 1 {1 = - - = det M . (7.34)
de t A
Y= 0,
0 1
-
/'
K
K=
1I(1~ 'Y)~\)
0, Y \
1/( 1-,r)
,
I ,
I
"
-- _....
/
Fig. 7.3 Distortion of a circular source under t he effect of t he convergence shear ,.
K,
and the cosmic
Notice that A is different for different multiple images. The comparison of the relative fluxes can in principle be used to constrain the lens matter distribution. For instance, in the case of a point-like lens, when es « eE , the second image is highly attenuated and is usually undetectable. The quantity {1 can be positive as well as negative. Its sign is called parity. In conclusion, the distortion field of a gravitational lens in the thin-lens regime is characterized by the matrix
A=
[1-
1'1,(0)]
(1 0) _ 01
')'
(0) (COS[2¢(O) ] sin[2¢(O)] ) sin [2¢( O)] - cos[2¢(O)] ,
where the projected potential is related to the surface density
~/~cr it
(7.35)
= 1'1,
by
(7.36)
382
Gravitational lensing and dark matter
Different regimes
The amplitude of t he lensing depends on L;jL;crit . Two main regimes can be di tinguished.
- If L;jL;crit > 1, the lens is said to be supercritical or strong. The convergence an the cosmic shear are important . Giant arcs as well as multiple images can appea
- If L;j~cri t < 1, we are in a regime of weak distortion; K « 1 and I « 1. The are no multiple images and t he distortion of the background sources is so wea that it cannot be detected with the naked eye.
- Outside the critical zone, the images can also be subject to strong distortio without necessarily producing multiple images. This intermediate regime is calle the arclet regime.
Figure 7.4 describes the t r ansition between the two regimes in the case of a point-lik lens. If the source is far from the E instein radius, only its shape is affected and all th more so as it gets closer to the deflector. When it enters the Einstein radius, multip images appear , and t hen giant arcs before finally entirely covering the Einstein radi when the source coincides with t he optical axis . Figure 7.5 illustrates these differe regimes in the case of the galaxy cluster Abell 2218.
~
OO Q
o
Fig. 7 .4 Distortion of an extended source by a point-like lens. T he circle represents t E instein radius. T he multiple images a nd the strong distortion effects only appear when th source is inside the E instein radius. From R ef. [3J.
The closed lines on the image plane for which det A = 0 are called critical line They characterize the positions of the strong magnification zones. The locus of th points of the source plane that have their image on the critical line is called a causti It indicates the position of the sources that will be subject to strong lensing and for arcs. One can show that when the source crosses a caustic, the number of images jum by ±2, so that two images appear or blend. A close source inside a caustic produc two very bright images close to the critical line. A source close t o a cusp produc three images (see the illustration in Fig. 7.7). For a lens with spherical symmetry, the critical line is circular. For a point mas it coincides with the Einstein radius and the associat ed caustic reduces to a point
Gravitational lensing and its applications
Galaxy Cluster Abell 2218 NASA, A, Fruchter and the ERO Team (STScl, ST-ECF) . STSc]-PRCOO-08
383
HST· WFPC2
Fig. 7.5 The galaxy cluster Abell 221 8 provides especially sp ectacular lensing effects. The different regimes described in t he text can be observed: giant arcs and multiple images close t o t he heart of t he cluster, strong distortions without multiple images (arclets) and weak distortion effects for sources far away from the deflector. (Picture from the spatial Hubble t elescope, © A . Fruchter , HST/ WFPC2) .
the intersection of the optical axis and the source plane (see Fig. 7.1). One can also show [5] that if the function I; is smooth, bounded (so that ex does not diverge), and non-singular (so that ex is continuous) , then the number of images of (almost) all the sources is odd. Moreover , if I; never becomes negative, then at least one of the images is amplified . In general, these relations and theorems are violated by observations, mainly due to the exist ence of many substructures in the lenses. Furthermore, the very damped images (I); « 1) cannot be det ect ed in general. Time delay
Using the relation (7.24), the lens equation (7.14) can be rewritten as
where the term that appears in bracket s is proportional to the time delay (7.37) between the real trajectory and the unperturbed trajectory. The first term, quadratic in (() - ()s ), has a purely geometrical origin and reflects the length difference between
384
Gravitational lensing and dark matter
the real light path and the unperturbed one. The second term has a relativistic or and is identical to the Shapiro effect (see Chapter 1) , eM = J 2(1 + z) dz/e 2 w the factor (1 + z) takes into account the cosmic expansion ; it is then approximati (1 + ZL) in the thin-lens approximation. All the equations we have obtained ca derived by looking for the extremum of the function T(O, Os). Figure 7.6 comp this approach to that based on the lens equation . This function has various interes properties.
- The difference T(Ol , OS) - T(02 , OS) of the time delays between two station points gives the delay in the reception of the two images . Any variability in source is first observed in the image corresponding to the smallest value of T(O ,
There are three types of stationary points, maxima, minima and saddle po the nature of which depends on the sign of the eigenvalues of the matrix T 2 T/ b , which is simply the distortion matrix. It therefore describes the l curvature of isotime surfaces. We can thus define three types of images.
a aeaae
Type I: images corresponding to minima of T. Both eigenvalues are pos and thus det A > 0 and TrA > O. Type II: images corresponding to saddle points of T. The two eigenva have opposite sign , and thus det A < o. - Type III: images corresponding to maxima of T. Both eigenvalues are n tive, and thus det A > 0 and TrA < O.
Images of type I and II have a positive parity and only t ype III images ha negative parity.
3
8- 8s,'
2
,,
a
3
,,
,,
-2 -3
8/8c 4
,
,, "
,...~2
-3
"
-3
2 r 1 .............
- 4 3 - 2 -1
-5
-1
1
2 3
,, ,
Fig. 7.6 Number of images for a smooth isothermal lens det ermined by t he lens equa (top) or by the time delay (bottom) for a source localized at fJs!fJ c = 0, 0. 3, 1 from to right. The dotted-line parabolas represent the geometrical contribution, while the o dotted curve is the gravitational contribution.
Gravitational lensing and its applications
385
7.1.1.3 Lens models The separation between the multiple images and the position of the critical lines enables us to reconstruct the surface density distribution of the lens. T he aim is to adjust this distribution and the position of the source so that Os = Oi - a.(Oi) for all images Oi. We present here some lens models t hat illustrate the effects discussed in the previous sections. 7.1.1.4
Singular isothermal sphere
A simple model for the mass distribution of a galaxy relies on the hypothesis that stars behave as a perfect gas confined in a gravitational potential. Its equation of state will therefore be P = pkT /m , where m is t he mass of each star, assumed to be all equal,
p(r)
=
(T2
1
(7.38)
_ v_ _ .
27TG N r2
This distribution is singular at the centre and the total mass contained in a sphere of radius r, M(r) , grows as r , so that the Keplerian velocity of a test particle in a circular orbit at distance r from the centre is
(7.39) which is constant. We can also conclude that t he surface density is (7.40) Using (7.26) and (7.24), we find that t he deflection angle is independent of the impact parameter and is given by
a
=
(T2
v 47Tc2
=
1.4
(
(T
k v 1 220 m· s-
)2arcsec.
(7.41)
The Einstein radius is obtained by solving (7.14) for es = 0 and is eE = (DLs/ Dos)a. The lens equation (7.14) takes the simple form e± = es ± eE and there are multiple images only if the source is inside the E instein ring (es < eE) and their angular separation is always 6.e = 2e E . Actually, there is a t hird image with vanishing flux at e= 0 that becomes visible if the profile is smoothed out at r = O. The magnification (7.33) of t he two images is i1± = (1 =f eE/e±) - l. If es > eE then t here is only one image localized at e = es + eE. F igure 7.6 compares t he graphic resolution of this system from the lens equations and that obtained by looking for the extremum of the time delay.
386
Gravitational lensing and dark matter
7. 1.1. 5
Other m odels of halos
T he singular isothermal model is not realistic for various reasons. In particular , lens must have a core where the density does not diverge. Table 7.2 summarizes opt ical properties of some frequently used models with spherical symmetry. Table 7.2 Properties of some models of spherically symmetric halos . 'IjJ( e)
Lens model
a (e) 2
In B (D LS / D os ) 47f((J~ /c 2)B
Point-like mass
(4G N M / D c
Singular isot hermal sphere
)
(DLS/Dos) 47f((J~/C2)(Bc + ( 2)1/2
Smoothed isothermal sphere
(4G N M
47f(J~ / c2
47f((J~/C2) B /(B~
KB2/2
Mass sheet
/c 2 D OL) B- 1
+ ( 2)1
(D os/ D LS)KB
These models are still too idealized to describe real galaxies. In particular, o should go beyond spherically symmetric models. For instance, one can use ellipt i models [6] for which t he surface density takes t he form
~(O )
=
~o [e~ + (1 - E)ei + (1 + €) e~l- 1 /2 .
(7.
In general, t he proj ected potent ial is difficult to obtain. We can t herefore choose directly model this potential by assuming that it is ellipt ical
'1j! (O) = DDLS 47f a~
os
c
[e~ + (1 -
€)ei
+ (1 + €)e~l l/2 .
(7.4
F igure 7.7 represents t he crit ical and caustic lines for a model of an ellipt ic lens.
•
,"
o • /I
-
•
CB
Fig . 7.7 Two exam ples of gravitational lenses. For each one of t hem , the figure on t he represents the image plane wit h t he crit ical lines and t he figure on t he right the source pla wit h t he caustic lines and t he d ifferent source p osit ions. From R ef. [4J.
7. 1.1 .6 Effect of th e en vironmen t
A galaxy is not isolated, in general it belongs t o a cluster or group that induces ext ernal gravitational distortion of the order of a few per cent. Another problem ari
Gravitational lensing and its applications
387
due to the mass-sheet degeneracy. Let us add a mass sheet of constant density while performing the transformation K,
-->
(1 - s)
+ SK" 2
es
-->
ses·
(7.44)
r (0, Os)
1
-2 -1
1
2
3
4
...... ...................
-4
-s Fig. 7.8 Change of the function r(O, Os ) for a smoothed isothermal profile when adding a mass sheet characterized by s = 1 (dashes) s = 0.5 (dots) and s = 1.5 (solid line). The angular positions of the maxima and minima remain unchanged.
The deflection angle then transforms as a --> sa + (Dos/DLs)(1 - s) e, so that the lens equation (7.20) remains unchanged since it is only multiplied by the global factor s. In an analogous way, neglecting the term e§ from the time delay (7.37) (in any case it is not observable since only differences c5r12 = T(e 1 , es) - T(e 2, es ) between various images can be measured) and noticing that the proj ected potential transforms as 'Ij; --> s'lj; + (1 - s )e 2 /2, then c5r12 --> Sc5T12. All the time delays are thus dilated in the same way, conserving the structure of the image. Since the source plane is dilated by s, all magnifications are damped by 1/ s , thus leaving the relative magnifications unchanged (see Fig. 7.8). 7.1.2
Lensing by galaxies and galaxy clusters
This section describes some applications of gravitational lensing in the thin-lens regime. For more details, we refer the reader to Refs. [7,8] for microlensing , to Ref. [9] for giant arcs and the a rclet regime in clusters of galaxies, to Refs . [3,10] for lensing on quasars, and to Refs. [4,11,12] for comprehensive reviews of all observations.
7.1.2.1
Microlenses in our galaxy
If a massive body passes in front of a background source, it can induce a temporary variation of its apparent luminosity even if the mass of the lens is too small to induce
388
Gravitational lensing and dark matter
any multiple images or any important deflections. This phenomena is then calle microlensing. The estimate (7.4) shows that a lens of a few solar masses has an Einstein radiu of a few milliarcseconds, which is well below the spatial resolution of our instrumen (even if multiple images were produced, we would not be able to distinguish each on of them). However, if the source passes near the Einstein radius (Bs c:o: BE), the tota magnification (7.11) would be fJ, c:o: 1.34. So, if the impact parameter of the source smaller than BE, the peak of the light curve is fJ,max > l.34. This corresponds to magnitude variation greater than 0.32 magnitudes, which is in principle detectable. Paczcirisky [7] had the idea to use these micro lensing phenomena to constrain th density of MACHOs (massive astrophysical compact halo object) in our Galaxy. Th characteristic time for the evolution of the luminosity of a source depends on th Einstein radius and on the lens velocity, v, T*
DOLBE
= ---.
(7.45
V
Using the estimate (7.4) for the Einstein radius, we get M
T
*
= 0 214 ( - - ) .
M0
1/2
(D ) (D- - ) --1/2
OL
10kpc
LS
1/2
Dos
V
(200km.s- 1 )
-1
y
ears
.
(7.46
If the lens is located in our Galaxy and the source in the large Magellanic cloud, the DLs/ Dos is of order unity. Then, a sampling of light curves with time intervals rangin between an hour and a day makes it possible to detect masses between 10- 6 M0 an 10 2 M 0 . We should, however , stress that the measurement of T* does not directly giv access to the lens mass.
7.1.2.2
Light curves
T he light curve of the source as a function of time can be obtained using (7.11) fo the amplification by a point-like source,
A(t) =
2+X2
xv'4+x 2 '
(7.4 7
where x(t) == JU;;'in + a 2t 2, Umin being the impact parameter in units of the Einstei radius and a is related to the characteristic time (7.46). Such light curves are depicte in Fig. 7.9 for various impact parameters. Both parameters, a and Umin , can thus be determined by fitting an observed ligh curve. The width of the curve gives access to T* and there is a relation between th amplitude of the peak and Umin. This is , however, not sufficient to determine all of th characteristics of the system since there are three unknowns (M, DOL and v).
7.1.2.3
Cross-section
To evaluate the probability of a microlensing , it is convenient to define a cross-sectio as the surface around the lens through which the light emitted by the source mus
Gravitational lensing and its applications
to
. : . . .: : ~~,:~':~~:~:~~~~~:~:::~~~L:::::~~~~::~: ~:~
........... ____ / ............... "
Umin =
"
Q)
"0
.::
~
0.4
a'" - 1.0 2-
·--·--·--··-· ···-·· "t~ ·U:~~i·I~..;;·O.2
....... _-_._- ; _.. _------_ ....... . __ ._---------_ ........... .. . I
,............ . I ,, I
, I I \
~
"1 _ 0 .5
I \ \
\
,
... ' .........
... _---
, ,,
Um in = 0.2
,-.. - l.S
Um in = l.0
389
I I
~'
-2
-1
o
1
2
Time (in units of RE Iv.1. )
Fig. 7.9 Li'g ht curves d uring a microlensing event for different values of the impact parameter in units of the Einstein radius.
pass in order to be subject to a magnification larger than A. For a point-like source, this surface is 7r DbLe§(1l = A) and thus
0'(> A)
M
= 2~crit
1 , A2 - 1 + AvA2 - 1
(7.48)
using (7.11). The number of lenses with a magnification greater than A is thus given by
N(> A)
=
;n J
r dV n(Dod 0'( > A).
(7.49)
This quantity can be expressed in t erms of the optical depth T , defined as t he integral of the area contained inside the E instein ring of all the lenses in a solid angle ~n , (7.50)
Since dV = dn 2 DbLdDoL and expressing eE , this expression reduces to
47rG N i DOL
DoLDLS Pmacho(DOL) D dDoL , coos
T =
- - 2-
(7.51)
where Pmacho is t he density of MACHOs. By introducing x = Do L! Dos, we finally obtain T
= 47rG N2Dbs c
11
Pmacho(x)x(l - x)dx.
(7.52)
0
We have assumed t hat since space is Euclidean a t these scales, we can make the approximation DLS = Dos - DOL. For a constant density, we will thus obtain T =
390
Gravitational lensing and dark matter
(27f /3)G N PmachoDbs/ c2 . The theoretical estimates predict that 7 ~ 10-6. The numb of events that produce a magnification greater than A is thus given by
N (> A)
27
= --------===
A2 - 1 + AVA2 - 1 '
(7.5
for A = 1.34, N ~ 7 and several millions of stars should be monitored before one c hope to observe any event.
7.1 .2.4
Searching for MACHOs
Several experiments have carried out this idea in order to constrain the density MACHOs in the halo of our Galaxy. The main problem resides in the amount of data needed (several millions of sta are observed) and in the detections that are not associated with any lensing effect. particular, an automated monitoring will detect many variable stars and supernov Two criteria can be used to distinguish between the magnification induced by lensi from such intrinsic variations: the light curve must be symmetric and the magnificati must be achromatic. Figure 7.10 summarizes the results from the two experiments MACHO [13] a EROS [14] that have been looking for microlensing effects on the stars of the lar Magellanic cloud. 7.1.2.5
Gravitational lensing by galaxies
Quasars
The first multiple quasar, QS00957+561 (z = 1.41), was discovered in 1979 [1 and today we know of more than 10 4 of them. Microlensing was initially observed the four images of the quasar QS02237+0305 [17] in 1989. The deflector is a spi galaxy at redshift z ~ 0.04. Uncorrelated variations of the luminosity from the fo images allow us to establish that the microlensing is induced by stars from the galax In this situation, a complex caustic network appears and can be used to put co straints on the structure and the intrinsic size of the source. Moreover, one can dete mine the mass spectrum of these microlenses and thus show that its star distributi is analogous to that of our Galaxy. Let us note some of the results obtained by the observations.
- The central image is absent in almost all the observations. This implies that is very damped and that almost all galaxies must have a very small core radiu rc < 200 pc. - One can show [18] that most galaxies must have a dark-matter halo with a veloc dispersion of the order of 220 ±20 km·s- 1 . If this were not the case, no multip images with angular separation greater than 2" would be produced. The large estimated size for a halo is that of the galactic lens QS00957+561 that is great than 15h- 1 kpc. - The optical depth depends on the redshift distribution of the lenses. If galaxi were formed recently, its main contribution would be from galaxies at low redshi
Gravitational lensing and its applications
391
80
'"'"
60
a'" 0
OJ
..<:: ;?
40
20
OL-~
__- L__- L_ _
~_ _L-~_ _~_ _- L_ _~~
10- 8 10- 7 10- 6 10- 5 10-4 10- 3 10- 2 10- 1
1
10
100
Deflector mass (in M 0 ) Fig. 7.10 Constraints stemming from the results of the MACHO (grey tint) and EROS (hatched) experiments on the contribution of MACHOs to the halo of our Galaxy (modelled by a mass of 4 x 10 11 M 0 and a radius of 50 kp c). The allowed region is the one b elow the U curve. From Ref. [15].
Using the information on the lenses and sources redshift, one can conclude that most galaxies must have b een formed before z ~ 0.8 assuming an Einstein- de Sitter Universe model [19]. - The standard cold dark-matter model predicts the formation of dark halos, which tend to induce more pairs of quasars wit h large separations than that observed. This could be a potential problem for this scenario but the conclusion depends on the properties of these halos (core radius ... ) and on the power spectrum of dark matter [20]. This illustrates the potent ial for this tool to t est the distribution of dark matter. - A cosmological const ant has the effect of increasing d V / dz at large redshifts . This implies [21] that the relative number of lensed sources must rapidly increase with A, which can thus be estimated using the lens probability. The current upper bound obtained from this method is OAO < 0.65 (2a) for a flat Universe [22]. Another approach [23], using the comparison between the observed separations between multiple quasars and that expected given the sources redshifts, has led to 0.48 < OA < 0.93.
392
Gravitational lensing and dark matter
- A last application, not yet exploited , concerns t he differential magnification stars from the bulge, of small Einstein radius, of the quasar accretion disk. T effect might allow us to probe the disk and to study its structure.
Hubble constant
The measurement of t he time delay between different images of the same sou gives access to the Hubble const ant , as noted by Refsdal [24], since the t ime de T( fJ , fJs) is proport ional to Ho· To understand t his, let us consider the model of an isothermal lens. The l equation gives access to 8 - 8s (which is proportional to the deflection angle) for e image. We obtain t hat T( 8, 8s) ex &2 (8) / 2 - 'ljJ( 8). For an isothermal lens, the deflect angle is constant so that the first term does not contribute and OT ex 'ljJ(8d - 'ljJ(8 2 8 1 - 82 = 28E. We finally deduce that COT =
81f (
:~)
2
(1
+ ZL )DOL8E.
(7.
In t his equation, only DOL is proportional to Hal . Of course, the determination Ho depends on the lens model and is very sensitive to the existence of a mass-sh (see Ref. [25] for a modellisation of these effects). For instance, for the quasar Q800957+561 , one can show [26] that h = (0.82 ± 0.06)(1 -
/'i,)
.6. T ( 1.14 year
)-1 ,
(7.
where /'i, is the unknown convergence of the halo containing the galaxy. To meas such time delays, one should therefore monitor these galaxies for several years. Q800957+561 , t he t ime delay is OT
= (417 ±3) days.
(7.
8ince /'i, is positive, we deduce an upper bound for the Hubble constant (h < 0.8 /'i, can be estimat ed from t he weak lensing, giving /'i, = 0.24 ± 0.12 , which allows us conclude [2 7] that h = 0.49 - 0.74. The measurements of the galaxy velo city dispers can also b e used to constrain /'i, and gives [28] h = 0.66 ± 0.07. This method requires that the geometry of the lens and that of its environm are well understood . It has, however, some advantages. For instance, it avoids construction of a distance scale calibrated step by step and to work with sources large redshifts (z ~ 0. 5) . If the deflector potential is perfectly known , the result sho not depend on the physics of the particular object (cepheids, supernov&.. .), and only underlying hypothesis is t he validity of general relativity in the weak-field lim Constraints on the density of compact objects
It has been suggested that a significant part of dark matter could be in the form compact obj ect s. The density of such a component of dark matter can be constrai fro m microlensing effects.
Gravitational lensing and its applications
393
For this, one should estimate the optical depth (7.50) in the cosmological context. It can no longer be assumed that DLs = Dos - DOL and one should use the expressions obtained in Chapter 3. Assuming a constant density PM , we obtain T( ZS )
= 3fl MO 2
tL
io
DLs( z , ZS) Dod z) ~ Dos(zs) DHo E(z)'
(7.57)
+ z)2
and dX = dzl H( z).
where we have used the fact that dV( z ) = DbLdxdfl2 /(1 For an Einstein- de Sitter Universe , we obtain T( ZS ) =
3
flMO
Zs
[(ZS + 2 + 2\1'1 + ZS) In (l + zs ) - 4zs] .
(7.58)
This rate is proportional to fl MO so that the number of events in a given catalogue is a direct estimate of the value of this parameter. Different analyses can be used to exclude IlMO == 1 and give the constraint fl MO < 0.05 for masses M = 10 6 - 10 12 MG. 7.1.2.6 Lensing by galaxy clusters
The first giant arcs were discovered in 1986 in the clusters Abell 370 [29], Abell 2218 and Cl 2214 [30]. The numerous properties of the cluster distortion field allow us to reconstruct its gravitational potential and thus to study its structure. Reconstruction of the dark-matter distribution Together with the analyses of the velocity dispersion and the X-ray emission, gravitationallenses provide an independent method to determine the total mass of clusters. Strong lensing mainly appear in the central zones of the cluster (;S 50h- 1 kpc) and can be used to measure the internal mass. A first estimate of this mass is given by M( < BE) since, for a spherical lens, the position of the critical lines gives a measure of the Einstein radius . This estimate has a precision of the order of 50% and strongly depends on the degree of asymmetry of the cluster . When several arcs and multiple images are present (see Fig. 7.5) , the modelling of t he cluster is refined. One can then proceed by iteration: once a first model is obtained, the position of other multiple images can be predicted and their presence can be checked. In this case, one can deduce the expected morphology deduced from the model and iterate it. These models become precise enough to determine the redshifts of the arcs and arclets [31 ]. The hot gas of the cluster emits Brehmsstrahlung radiation with intensity depending on the ions (mainly protons) and electron number density, and on the gas temperature as well as on its chemical composition. From the measurements of the Xray emission and the temperature profile, one can reconstruct the cluster mass profile, M(r). The discrepancy between X-ray and lensing measurements is mainly attributed to the influence of the substructures on the X-ray emission and to the triaxiality of the three-dimensional potential. These methods have allowed us to determine generic properties of the dark-matter distribution in clusters, independently of any given model. Clusters seem to be dominated by dark matter and their typical mass- luminosity ratio would be MIL > lOOMol Lo · The distribution of dark matter seems to follow that of the luminous
394
Gravitational lensing and dark matter
one, in particular in the central regions . The curvature radius of many giant arcs comparable to their distance to the centre of the cluster, which implies that t he clu t er 's core radius should be of the same order of magnitude. The internal zones of t clusters have many substructures. Magnification bias
The magnification bias expresses two simultaneous and opposite effects to t gravitational m agnification: it increases the flux received from the galaxies that a subj ect to lensing and so allows for the det ection of a larger number of objects, b it also dilates the considered solid angles of the same quantity and thus decreases t density of galaxies. If no(j, z ) is the number of galaxies in t he absence of lenses per unit solid ang wi th flux greater than f a t redshifts between z and z + dz, then t he density of galax in a direction () becomes, because of magnification , n(j, z ) =
fl(;,z) no [fld,z)' z ] .
(7.5
The magnification can thus either locally increase or decrease t he galaxy count. If only have access to the number count integrated over z, we can make the simplifyi z) ~ Assuming that the galaxy number count as a functi hypothesis tha t of t he magnitude has a slope a = dlogno / dm then
fl( (), fl( ()).
n(m , () ) = no(m) [fl(()) f5a -l.
(7.6
An effect thus only appears if a =I- 0.4. The variation of the obj ect density as a functi of the distance to the centre of the cluster therefore gives a depletion curve that allo us to measure the magnification and , for instance, to localize a critical line even if giant arc is associa ted with it [32]. Cosmological parameters
Once a robust model of the cluster is constructed , one can use the informati redundancy to measure the cosmological parameters using geometrical methods. T lens equations can always be decomposed as
The function E = DLs/ Dos depends on the source redshift and f is a function th depends on the modelling of t he cluster. If one has access to different set s of multip images centred on the same gravitational lens, then one can estimate
E(ZSl) E(ZS2 )
f(() 2,2, .. ·) 1 1()2,1 - ()2,21I f(()1 ,1, ... ) - f(() 1,2, . . ·)1· I()l ,l - ()1 ,21I f(()1 ,2 , ... ) -
(7.6
We t hus obtain a measurement of DLs / Dos at different redshifts. This method, whi is simple in principle, is, however , difficult to apply in practice as the function f is n well known.
Gravitational lensing and its applications
395
Detection of distant galaxies A source close to a caustic can be enormously amplified. Giant arcs can thus allow for the observation of galaxies with low surface brightness at high redshift. The lens here plays the role of a gravitational telescope making it possible to observe objects that would otherwise be undetectable. The information on the redshift at which the first structures of the Universe form, which can be obtained in this way, allows constraints to be placed on models of largescale structure formation and on the cosmological parameters. Currently several (5-10) galaxies with redshifts larger than 7 have been detected, amongst which one candidate has a redshift of 10. This galaxy [33], which is close to a critical line, has an estimated magnification of 25- 100. In a flat Universe, with nAO = 0.7 it would thus have been formed around 460 million years after the Big Bang. 7.1.3
Gravitational distortion by the large-scale structure
Up to now, we have only considered the effects of gravitational lensing in the thin-lens regime. Light beams emitted by galaxies are, nevertheless , continuously deformed and deflected by the gravitational field of the large-scale structures of the Universe. These effects are small and do not give rise to multiple images. However , the distortion of the shape of distant galaxies and the statistics of these distortions reflect some properties of large-scale structure.
7.1.3.1
Sachs equation
We start by studying the deformation of a geodesic bundle during its propagation through an inhomogeneous space-time [34]. As seen in Chapter 1, light follows null geodesics. Two geodesics of the same bundle will be subject to a relative deviation described by the geodesic deviation equation (1.118).
o (,.1,=0) Fig. 7.11 D eviation of a geodesic bundle converging at the observer.
The different geodesics of a geodesic bundle can be parameterized as follows
(7.62)
396
Gravitational lensing and dark matter
where xl-' is the worldline of an arbitrary geodesic of the bundle chosen as referenc and ~I-' is a vector characterizing the displacement between two neighbouring geodesi (see Fig. 7.11); A is an affine parameter along the reference geodesic. We assume tha it is zero at (position of the observer) and increases towards the past. In 0 , we can choose an orthonormal basis , {kl-', ul-', ni, n~} composed of the tangen vector to the reference null geodesic ,
°
kl-' = dxl-' -
dA'
of the tangent vector to the observer worldline , ul-', and of two space-like vectors the plane orthogonal to the line of sight. They thus satisfy
(7.63
with a , b = 1, 2. From this basis defined in 0, one can construct a basis at every poin on the reference geodesic by parallel transport (that is according to kl-'VI-'n~ = 0 an kl-'VI-'u v = 0). Any vector ~I-' can then be decomposed as 1
(7.64 a
and one can always set ~o = 0 since two different values of ~o still parameterize th same geodesics. The propagation equation for ~I-' follows from the geodesic deviatio equation (1.118)
(7.65 where Rl-'va(3 is the Riemann tensor and D IdA this equation becomes
== kl-'V w Using the decomposition (7.64
b RI-' a b(3 Ra = va(3 kVk nal-'n,
e
(7.66
e
where we have used the notation = ~a and R· = R~~b. Here a and b are just labe not tensor indices . The optical matrix, R , can be decomposed in terms of the Ric and the Weyl tensors as
(7.67 The linearity of the geodesic equation implies that its derivative by a linear transformation
e(A) = V
de
I.
dA 0
e is related to the initial value
(7.68
1 A priori we should have added a term AuI' in this decomposition. But , it can be shown th kl'EI' = - AE, w ith E = - (kl'ul')' is constant along the geodesic. It follows that this term can be s to zero with no loss of generality.
Gravitational lensing and its applications
The bundle converging in 0, e(O) matrix D
397
= 0 and (7.66) gives an evolution equation for the d2 D = R·D . dA2
(7.69)
The initial conditions in 0 are then given by
D(O)
=
d dA D(O) = I,
0,
(7.70)
I being the 2 x 2 unit matrix. The direction of observation, 0 , and of the un lensed source, Os , are related to the displacement field by
0 = de (0) dA '
(7.71)
where As is the value of the affine parameter at the source and where the distance D A is by construction the angular distance . We then find from (7.68) that Os
= A· 0
with
(7.72)
We recognize the deformation matrix (7.28) that can be decomposed, as previously, into a convergence K, and a cosmic shear, ., [see (7.30)].
7.1.3.2 Application to cosmology The equations we have just derived did not depend on any hypothesis concerning space-time symmetry. We now apply them to the relevant cases for cosmology. Homogeneous and isotropic spaces
For a Friedmann- Lemaitre space-time, the metric takes the form (3.3) and is conformal to the static metric 9J.LvdxJ.LdxV = - d7)2 + l'ij dx i dx j with the notations of Chapter 3. To solve (7.69) we use the fact that two conformal spaces have the same null geodesics (see Section 6.1) and will thus have the same optical matrix. Equation (7.69) then takes the simple form d2 D =- KD (7.73) dA2 ' where K is the curvature of the three-dimensional space with metric l'ij [see (3.4)]. The solution of this equation is
(7.74)
!K being defined by
(3.5) so that
A(O) = I,
where DA has been expressed using (3.74).
Contributions from perturbations
To take into account the effects of inhomogeneities, we work in the framework of the cosmological perturbation theory presented in Chapter 5. The space-time metric
398
Gravitational lensing and dark matter
is assumed to b e of the form ds 2 = a2 (1]) (gl-''' + hI-''' )dxl-'dx" and we expand the matr V in perturbations as V = V (O) + V (I) + ... , the first term being given by (7.74). To first order , (7.69) reduces to
(7.7 which has the integral solution V (1)( A) =
10
oX
jK(A')jK(A - A')R.(1) (A')dA' ,
(7.7
implying that the amplification matrix is given by
(7.7
Finally, one should compute R. (1) and go back to the initial space-time, the expandi one. The definition (7.66) implies that 2 cx 2'0 ,"\.- (1) -- h 1-'",cx/3 kl-'k" n n/3 .
(7.7
Using now the decomposition (5.52) in Newtonian gauge for the scalar modes only, obtain (7.7
where the second equality is only valid if the anisotropic pressure is negligible, whi is a priori the case in the matter era [see (5.ll8)]. The amplification matrix in cosmological space-time can thus be expressed in terms of the gravitational potent as 3
0/'(0 'f/
,x ) -=
!2 c
1 X
0
jK(X')jK(X - X') <1> [j (') 0 ']d I j ( ) K X ,X X K X
(7.81
The distortion is thus obtained by integrating along t he unperturbed geodesic a takes a form analogous to that obtained in the case of the thin-lens approximation, b introducing a projected potential that takes into account the entire matter distributio Note that this is thus a first-order expression so that all the contributions coming fro the lens- lens coupling are neglected. For most practical applications this is , however ,
2Here, we will make a simplifying hypothesis. Since we will b e interested in modes smaller than Hubble scale , we assume that the spat ia l curvature does not influence the perturbations and negl it in the computation of n ( l ) , while we keep it in t he geometrical factors . 3Note that this equation can be rewritten into the equivalent form Aab = Oab -
OOaOb..j;(B, X) ,
_ 2 ( X hdx - X') '" 1f;(B,X) = c 2 Jo fK(x)fK(x') [fK(x )B,X JdX ,
where the derivatives are now t aken with respect to
ea.
(7.
Gravitational lensing and its applications
399
excellent approximation. The thin-lens regime can be recovered by assuming that the gravitational potential is localized on a lens plane at XL , = [fK(X)O , xlo(x - XL). The convergence and the cosmic shear are obtained as
7.1.4
Cosmic convergence
7.1.4.1
Effective convergence
The convergence, obtained from the trace of the amplification matrix, characterizes the density of effective matter integrated along the line of sight (7.25). We deduce from (7.81) and the Poisson equation (5 .117) that it t akes the form 4 (7.82) If the sources have a redshift distribution Pz(z)dz = Px(X)dX, then the effective convergence is obtained by weighting the convergence (7.82) with the source distribution as
1'0,(0) =
J
PX(X)K(O , X)dX-
It then takes the final form 5
D 1'0, (0) = ~2 H6 2 rnO c
l
0
XH
g
(
X )fK (X) O[jK(X)O, () xl d X, a
X
(7.83)
with (7.84) where XH is the value of X associated with the size of the observable Universe. This expression shows that the convergence depends on the properties of the density contrast, on the cosmological evolution, but also on the total quantity of matter, Druo.
7.1.4.2
Limber eq uation for the convergence power spectrum
We are mainly interested in the statistical properties of the convergence that can be related, thanks to (7.83), to that of the density contrast. As discussed in Chapter 5, the 4Here, we have split the 3-dimensional Laplacian to get Ll .L 1> = ~nmo H58la - 0331>. The term involving 0331> gives a vanishing contr ibution when the integra l over X' is computed. 5 After we have used that fOXH dx fox dx ' = fOXH dx ' f ;,H dX, and then exchanged X and X' in our notations .
400
Gravitational lensing and dark matter
density field is a Gaussian stochastic variable whose statistical properties are describ by its power spectrum [see (5.36)]. Just as the density contrast was expanded in Fourier modes as in (5.33) , the convergence can be expanded into bidimensional Four modes as 6
J .e 2
=
d k(.e)ei£e. 271" The convergence power spectrum can then be defined by the relation ",(0)
(k(.e)k(.el ) )
= 0(2) (.e
+ .e1)P",(i!) .
(7.8
(7.8
The relation (7.83) expresses the convergence as a projection of the density contra with weight q(X) as ",(0) = q(X)o [JK(X)O, X] dX· We can then easily convince ou selves that this implies that '" is also a homogeneous and isotropic Gaussian field . angular correlation function ~",(e) == ("'(
J
(7.8
The correlation function of the density field introduced in this integral can be comput by expanding 0 as in (5 .33) and takes the form
J
d3kd3k' 'k (271")3 e -'.L
'P
j' ()'k K
X
e-'
3Xe-
'·k'.L
(
'P +
e)f (') K
kl »
If we only consider small angles (e « 1) then by k -1. We can thus make the approximation
X
e-'·k"3X (o(k, X)o(k, Xl)). (7.8
k~ and the power is mainly carri
(o(k, X)O(kl, Xl)) ~ (o(k-L' X)o(k~, Xl)) 0(k3
+ k~).
The integral over k3 gives 271"0(X - Xl) so that after integration over Xl we obtain
(7.8 Within this approximation, the convergence power spectrum is therefore given by7
(7.9
Such an expression allows us to predict the amplitude of the convergence as a functi of the angular scale and of the cosmological model at hand.
6We are still in a flat-sky approximation. In full generality, K( O) is a function on the celest sphere and should be expanded in spherical harmonics, as , e.g. , the eMB temperature anisotropi For more details on t his approach and on its relat ion to the fl at-sky limit, see, e .g., Ref. [35J . 7Note that if we had not used the Poisson equation, the same derivation would have given
(7.9 Such an expression is more general since it does assume the validity of the Poisson equation .
Gravitational lensing and its applications
401
In practice, the field is smoothed with a filter , that is a window function U(B , Be), with angular radius Be as (7.92) then the variance of the averaged field is given by (7.93) where J o is a Bessel function (B.30).
7.1.4.3 Full-sky versus fiat-sky derivations The previous computation can be performed without making use of the flat-sky approximation. Starting from (7.80) rewritten as
where we recall that the deflecting potential integrated on the line of sight is
~(n) = 2 fo Xg(x)1>[x(n), xjdx
(7.94)
where b.2 is the Laplacian on the 2-sphere. As for any function on the sphere, it can be decomposed in spherical harmonics, in the same way as for the temperature anisotropies of the cosmic microwave background as
~(n) =
L 'l/JRrnYRrn(n). Rrn
Decomposing the gravitational potential in Fourier modes in (7.94) and then the exponential with (B.21) , we obtain
where g(X ) = g(X)/ fK(X), from which we deduce
'l/JRrn
=
87ri
R
fo Xg(X)jR(kX)1>(k , X)Yern(k) (2!3)~/2dX.
(7.95)
The power angular spectrum of ~, defined as ('l/JRrn'I/J;'rn' ) = OU, Ornrn,ct"', is thus given by
ct'" = 167r fo Xg(X)g(X')je(kX)jR(k'X')P1>(k, x, X')dXdX' dkk ,
(7.96)
402
Gravitational lensing and dark matter
where we recall that
The convergence being given by K,(n)
=
~D.2~(n), we deduce that
so that
(7.9
This expression has to be compared to the one we obtained previously in t~e fla sky approximation using the Limber approximation that, rewritten in terms of~, tak the form
and
P,,(£) =
1 4 '4£ P1/J(£),
which is the same expression as in (7.90). On small angular scales (i.e. large £) the tw quantities P,,(£) and Cere coincide. 7.1.5
Cosmic shear
As will be seen, the statistical properties of the cosmic shear can be extracted fro observations. We thus focus on the definition of these quantities and on their link wi the convergence. It will be convenient to use a complex notation for the convergenc 1= 1'1
7.1.5.1
+ i!'2· Kaiser and Squires algorithm
The shear and the convergence are defined from the second derivatives of the potenti ~ and are thus not independent. An efficient method to reconstruct the projected ma distribution (the convergence) from the shear was proposed by Kaiser and Squires [36 The expression (7.81) implies that
(7.9 where the function IC, defined as -1
(7.9
charact erizes the shear induced by a point-like deflector. The convolution of the e fective mass distribution ~ ex K, with this response will thus give the shear associate with this effective distribution. The aim is to invert this relation to reconstruct K, fro f.
Gravitational lensing and its applications
403
For this , it is convenient to switch to Fourier space, where this convolution is converted into a simple multiplication 1 1'(£) = -K(£)k(£), A
(7.100)
7r
for the
Fouri~r -modes,
k(£)
=
for all non-vanishing £
= (£1'£2). This can be rewritten as
~K:*(£)i'(£),
(7.101)
7r
The convergence, and thus the mass distribution, can thus be reconstructed from the cosmic shear as (7.102) The constant "'0 is related to any uniform mass distribution that contributes to the convergence but not to the distortion. Notice that", is real so that the imaginary part of the integral must vanish. What is obvious mathematically is no longer true with data since noise can generate such an imaginary part . Moreover, the integration can then no longer be performed on the entire plane but only on a support of the size of the receptor. The application of this method is thus not straightforward with real data where the noise, the need for smoothing, and the contribution from "'0, related to the mass-sheet degeneracy discussed in (7.44), must be taken into account. An important consequence ofthese results is that [since (7.101) implies that Ikl 2 = 1 . 1'* = I"h 12 + I"b 12] the convergence and the cosmic shear have the same power spectrum (7.103)
7.1.5.2
Two-point statistics for the cosmic shear
The cosmic shear has two components that can be decomposed in various ways. Let us consider a spherically symmetric filter , centred in 0 and of radius Be. The centre of the filter can be used to define the radial and orthoradial (also called tangent) components of the cosmic shear (see Fig. 7.12) . Let If! be the unit vector that joins the centre of the filter to the point where the cosmic shear is defined (that is 0 = Bexpi
404
Gravitational lensing and dark matter
u Compensated /
I
,
Top-hat
\
,------/~+-~\------~ I I
\ \ \
I I
\
,, ()
Fig. 7.12 (left): To compute the variance of the shear, the galaxy ellipticity is average
with a filter of radius ec. Given the centre of the filter, the radial and orthoradial (tangen components of t he cosmic shear can be defined. (right) : Two types of filters are usually use top-hat and compensated filt ers.
(7.10
Using t hese definitions before expanding K(O) into Fourier modes as (7.85) and usin the relations (7 .101) b etween the cosmic shear and the convergence, we obtain
(7.10
Using (7.103), the average of the cosmic shear with a top-hat window function has th variance
(7.10
7.1.5.3
Aperture m ass
For a n arbitrary filter , the effect of an unknown mass sheet (7. 102) biases the reco struction . However, if a comp ensated filter is used , i.e. one satisfying
J
eU(e , ec )d8 = 0,
then any constant contribution to the integration (7.92) would play no role. Definin the aperture mass by
(7.109
one can then show that this quantity can be expressed in terms of t he tangent she alone as
A very popular and useful family of windows [11] is
Gravitational lensing and its applications
405
(7. 111 ) This filter is a bandpass filter of vanishing average, which gives the variance
(M2)
2
=
288 j' dUP (C) [J4(Ce C ) ]
ap
7r
(7.112)
c2e~
K
7.1.5.4 Relations between the power spectra All of these two-point correlation functions are related to the convergence power spectrum. So they are not independent. For instance, one can show, using the properties of the Bessel functions , that
1 deJicJo(eec)~+ (ec) 1 dececJ4(eec)~_ (ec). 00
p",Je) = 27r Similarly
10-7
1O- 11
10- 12 10- 13
(7.113)
(M;p) and (....·ry*) can also be expressed as a function of ~± [37].
........~........,.....,
,....,...-.......,..~
10-8 10- 9 ,.-., 10- 10
'"~
00
= 27r
,
,,
~ 10-7
,,
~
\
- A CDM (lin) --A CDM(nl)
\
1O- 14L.....1..-_.....L..~........L.~........L--'
10
10- 6
... ----
~
':i.""' 10-8 .....-.10- 9
100 1000 10000
'
~
- A CDM (lin) --A CDM(nl) "--_~........J'--~~.....LJ
1
l
10 e (arcmin)
100
-A CDM (lin) --A CDM(nl) 10-6 L.I..-~_.......J._ _~......u 10 1 100 e(arcmin)
Fig. 7.13 Prediction of the different observables for a ACDM model with the contribution from the linear (solid lines) and non- linear (dashed lines) regimes of the power spectrum (see Chapter 5, notice that even in this regime lensing can be treated in a linear way). From left to right , the convergence power spectrum, P",(f) , the aperture mass M ; p(()) , and the cosmic shear top-hat variance, (,2(()) ) . From R ef. [34J.
7.1.6
Measurement of the cosmic shear
The key in the measurement of the cosmic shear relies on the measurement of the shape of background galaxies and on their ellipticity (see Refs. [11,12, 38] for details and problems). If t he image of such a galaxy is represented by an ellipticity c = £1 + iC2 = (1 - r) / (l + r) exp(2i
In the regime of weak distortion ('" Fig. 7.14).
«
1), the ellipticities give access to the shear (see
406
Gravitational lensing and dark matter
The measurement is thus based on the crucial assumption that the source galax have a random and uncorrelated intrinsic ellipticity so that any deviation from random distribution and any correlation in the ellipticities are attributed to lens effects. Assuming that the ellipticities are distributed with a dispersion a e rv 0.3, as s gested from surveys of close galaxies in the Hubble Deep Field, then one can meas an ellipticity, E , induced by a gravitational shear of the order of "( rv 0.1 by averag over the number of galaxies N
>
(:e)
2
~ 10 .
• II _
, ,
BMode
-II'
• 6
Fig. 7.14 (left): Value of he , I'd as a function of the shape of the galaxy with respect to local reference frame. The ellipticity is invariant under a rotation of angle 7r. (right): E B modes . Only E modes are produced by a gravitational field.
It therefore seems easy to measure a gravitational shear of 10%. Neverthel measuring a shear of the order of 1% requires 900 galaxies, which is hardly achieva on small angular scales, unless we average over many fields of 1 arcmin. Nevertheless , many observational problems exist (see Refs. [12 , 38] for detai discussions on the observational aspects). In particular, one should resolve proble (1) from the degeneracy with a mass sheet (for instance, the statistic (M;'p) avo it) , (2) from the angular resolution that is limited by the surface density of galax (3) from the redshift distribution of the sources that is important for calibration, from intrinsic alignments that can bias the measurements, (5) from the atmosphe turbulence that makes every object more circular and thus dilutes the signal and from the optical abberations that introduce artificial ellipticities. Relations such (7. 113) can allow us to control the level of these sources of error for which the effe on the shear do not a priori satisfy these constraints. By analogy to the terminology used for the polarization of the cosmic microw background, one can define E and B modes as
D.E D.B
= (8f - 8~h1 + 28182"(2, = - 28182 "(1 + (8f - 8~h2.
(7.1
Gravitational lensing and its applications
407
As can be ' seen from the definitions of the shear, gravitational lensing induces only E modes (see Fig. 7.14) and the E polarization is in fact simply the convergence. The two components of the shear are actually not independent, which implies, for instance, that the imaginary part of (7.102) vanishes . Since the definitions of E and Bare nonlocal, their reconstruction in real space depends, at any point , on the value of the shear in the whole space. A way to solve this problem is to filter the shear by a filter that eliminates long wavelengths. This can be achieved by considering a compensated filter. In particular, one can show that (M;p) is only sensitive to the E modes while
(7.115) is only sensitive to the B modes. Ml. must strictly vanish if the distortion has a gravitational origin so that any deviation of Ml. from zero can be used to control the data analysis accuracy.
7.1.6.1
Observational status
The first detection of the cosmic shear was performed in March 2000 [39] and then by three other groups [40- 42]. These surveys counted around 105 images of galaxies on a surface of around 1 deg 2 . Figure 7.15 summarizes the results of these various analyses for the variance of the shear and the various two-point correlators.
0.0008
- - I\CDr\'I, Os = 1.0 Zm = 1.0,0.9, O.R c rT-IT v\V+
o eTio
0.0006
+
CHIT K+ VLT
o WilT X Keck
~
liST 0.0004
".!:; 0.0002
(J.()(Joo
10
8 (arcmin )
Fig. 7.15 (left): Compilation of the first observations of the cosmic shear variance (using a top-hat filter). The three curves correspond to the three redshifts 1, 0.9 and 0.8, and data are in good agreement with the behaviour of (7 .116). (right): Constraints for the parameters llmo and Us obt ained from the survey VIRMOS-DECART [43] . The contours are at 1, 2 and 3u and they were obtained after marginalization over Zs and n.
These measurements allow us to constrain the cosmological parameters. The cosmic shear is especially sensitive to the total matter density and to the normalization of the power spectrum. For instance, one can show that for Gaussian perturbations with spectral index n,
408
Gravitationa l lensing and dark matter
(7.11
for sources at redshift Zs' The non-Gaussianity induced by the non-linear dynami (see Chapter 5) has a different dependence
(7.11
This allows us to obtain strong constraints on the parameters nmO and a s (see Fig. 7.1 The latest CFHTLS data [44] span 57 square degrees and allow us to measure the point shear statistics from 1 arcmin to 4 degrees. Its analysis leads to the constrain as
n )0.64 = 0.785 ±0.043, (~ 0.25
assuming K = O. Assuming a mean redshift of 0.5 , the largest physical scale prob by this analysis is 85 Mpc. Using only data with > 85 arcmin, so that only scales the linear regime are taken into account , one deduces that
e
as
"HmO )0.53 = 0.771 ± 0.029. (-0.25
This illustrates the potential of future wide-field observations to probe the large-sca structure. 7.1.7
Lensing on the cosmic microwave background
After decoupling , photons from the cosmic microwave background are decoupled fro matter and propagate along geodesics. As seen several times in this chapter , the geodesics are deviated by the gravitational field of the large-scale structure. The o served CMB t emperature and polarization fields T = (8, Q± iU) in a given directi n are related to their values at the time of decoupling by
T[n]
= T[n + ~(n)],
(7.11
where ~ is the displacement field induced by gravitational lensing. For small displac ments , each quantity can be expanded to second order as •
X(n) = X(n) 7.1.7.1
·
1 ··
+ C(n)Xi + 2C (n)e(n)Xij .
(7.11
Effects on the temperature
The impact of the gravitational lensing on the angular power spectrum of the tempe ature anisotropies can be illustrated in the fiat-sky approximation , in which we c
Gravitational lensing and its applications
409
o 3
0.0001
9
oJ/I LI .1. ..1 .....
1
..
..
5 7 9 Ii (a rcmin )
.
13
1 1..
+
..J. . j
...
3
11
11
5 x lO-s
3
13
5
7 9 Ii (a rcmin )
11
13
0 .0007 0.0004 0.0006
,, ,,
0.0003 0 .0005
~ 0.0004
-;:
0.0003
0.0002
,::
.,.,,
,,
0.0001
0.0000 .......... - 0.0001
0
4
~-----
+"11 :...... ::: :H'i: Ff~ 8
12
16
20
T 24
0.0004 - 0. 0001
0.0006
;;::
0.0004
0
4
8
12
16
20
24
0.0003
L
-;2
,::
0.0001
j!l
-::; 0.0002 0.0000
- 0.0001 0
10
4
Ii (a rcmin)
8
]6 12 Ii (a rc min )
20
24
Fig. 7.16 Meas urements of two-point st atistics in the survey VIRMOS-DE SCART [43]. (top left ): Vari ance of the shear with a top- ha t filter , (top right) ap erture mass of the E and B modes. T he B modes must a priori vanish so that the deviation of with respect to zero can control the data analysis , and in p articular the systematic effect s. (bottom left): Correlation fun ction of t he shear and (bottom right) tangent and radial projections of the shear.
MI
expand the cosmic microwave background temperature anisotropies in Fourier modes as
8 (n ) =
J
2
d £ 8(£) eil n 27f
For a Gaussian field , the two-point correlation function is given by
(7.120)
410
Gravitational lensing and dark matter
c(e) = (8(0)8(n)) =
J(~:~2
Cee
ien
(7.121
,
with (8(£)8(£')) = Ce6(2) (£ + £'). According to (7.118), this angular correlation func tion is modified by lensing effects as
(8(0)8(n)) = (8(0
+ ~(0))8(n + ~(n))) .
.
(7.122 1
·
.
= C(e) + (C(0)e(n))(8 i (0)8 j (n)) + 2(C(0)e(0))(8 ij (0)8(n)) 1
·
.
+2 (C( n)e (n)) (8 ij (n)8(0)).
(7.123
This function has to be expanded to second order in the perturbation decompositio in ~ to obtain the leading contribution to Ceo Indeed , since the average over th displacement vanishes, (C) = 0, it is at this order that the contributions from th lensing coupling in the two-point correlation functions appear. Using the relation between ~ and the gravitational potential and then performin a Fourier transform , we obtain
where P is the power spectrum integrated along the line of sight. Two effects appear, (1) a renormalization that affects the entire spectrum and tha is not directly detectable and (2) a mode coupling. The second term in the previou expression is a convolution that tends to smooth out the angular power spectrum o the t emperature anisotropies. T his term is too weak to affect the first acoustic peak s that its effect will mainly affect the tail of the spectrum. This implies that t he angula power spectrum of the temperature anisotropies , by itself, only gives little informatio on lensing. There are other effects on the anisotropies, for instance, morphologic effects. I particular, the peaks and troughs of the temperature map are modified. T hese effect are related to the departures from the Gaussian properties of the perturbations. Fo instance, the leading term of the four-point function of the temperature field is
(8( nd8(n2)8(n3)8( n4))c
=
(C (nde (n3)) (8i 8( nl)8( n2)) (8j 8(n3)8( n4)) , (7.125
plus its permutations.
7.1.7.2
Effects on polarization
The Stokes parameters are modified according to (7.119) . Only the relation betwee the observation and t he emission points change and gravitational lensing induces n rotation of the polarization vector (Q ± iU) . Moreover, no polarization can be created since if the polarization field vanishes init ially, t hen it will remain so. However, gravitational lensing will change the properties of the polarization field For this, we consider the decomposition of this field into E and B modes, (6.204)
Evidence for the existence of dark matter
411
Furthermore, one should establish the analogue of the relation (7.119) but for the second derivatives (7.126) at first order in B modes,
e.
This relation can then be used to obtain the effects on the E and
D.E = (1 - 2K;)D.E + e· V(D.E) - 25 ij (riD.Pj + V l'i · V Pj ), D.B = (1- 2K;)D.B + e· V (D.B) - 2cij (ri D.Pj + V l'i· V Pj ),
(7. 127) (7.128)
where P = Q + iU. The lensing effects can thus be decomposed as:
e.
- A displacement , given by the term (1 + V)D.(E, B), which is the perturbative expansion of t he Laplacian of (E , B) at the point n + This term is identical to that appearing for the temperature. - A magnification, controlled by the term - 2K; that introduces the convergence. Such a term is not surprising. - A mixture of E and B modes , dictated by the contraction between the cosmic shear and t he polarization vector. This coupling term is composed of two contributions t hat will dominate at different scales, the one involving the gradient of the shear dominating at small scales.
e.
More details on the effects of lensing on temperature and polarization are given in Refs. [45 , 46J. As seen in Chapter 6, the B modes are only generated by primordial gravitational waves. Assuming that initially there is only a scalar polarization, then the gravitational lensing induces B modes, (7.129) This has an important consequence: the detection of B modes will bring much information on the gravitational lensing by large-scale structure but, moreover, it will tend to hide the B modes generated by the primordial gravitational waves . Figure 7.17 illustrates these effects on the power spectrum.
7.2
Evidence for the existence of dark matter
In the modern cosmological model, dark m atter plays a central role in the formation of large-scale structures. We review the different 'proofs' of the existence, or more exactly of the necessity, of this matter . These conclusions rely on the validity of general relativity to describe gravity, which we will assume here. To finish , we will also consider the possibility of a modification of this law of gravity. 7.2.1
Dark matter in galaxies
The most convincing and direct proof of the existence of dark matter comes from the observation of t he galaxy rotation curves in spiral galaxies. Spiral galaxies have a flat disk in which stars follow circular orbits. The rotation curves give the orbital velocity of stars as a function of their distance to the centre of the galaxy.
412
Gravitational lensing and dark matter
c0
~ ~ (\j
'-..
.., CJ ~ ....., +
10 1
cB
.........~ ..................
0.1
~
~
,,'"\',\
,/
.. ./
""-'
0.01
CB, tens
' -' .:.
.................. 10- 3
10
100
1 000
Fig. 7.17 B mode of the cosmic microwave background polarization induced by gravitation lensing (dotted line) compared to that induced by primordial gravitational waves (dashed lin and to E modes (solid line) assuming that T / S = 1. From R ef. [46J.
7.2.1 .1 Rotation curves and dynamical mass
Newtonian dynamics implies that the Keplerian velocity depends on the mass co tained inside the orbit
(7.13
r
where M( < r) = 47f I p(r)r 2 dr . T he surface brightness of spiral galaxies is well fitte by a function t hat decreases exponentially as l(r)
= lo exp ( -
~d) '
(7.13
where the disk diameter, R d , is a characteristic length of the order of a few kpc (4 kp for our Galaxy, 6 kpc for M31, for instance). So, if the stars were the major contributo to the galaxy mass, i.e. if there were no non-luminous matter, then M( < r) wou become constant beyond the optical disk. As a result,
M( < T)
--+
const.,
1
v(r) ex
Jr '
(7.13
Actually, the rotation curves, which are particularly well measured for spiral galaxie do not reproduce this behaviour. The rotation curves tend towards a non-vanishin asymptotic value Voo and the Keplerian regime never seems to be reached. If v --+ V o
Evidence for the existence of dark matter
413
M( < r) ex: r at large distances, which indicat es the exist ence of a non-luminous matter with density profile behaving as p ex: 1/ r 2 at large distances from the centre. At large dist ances, we can express this asymptotic velocity, V oo as (7.133) where ao is a constant with dimensions of an acceleration. This implies in particular that at large dist ances the deflection angle, see (7.1) , is given by (7.134)
7.2.1.2
Description of the rota tion curves
To better understand the implications of the rotation curves , we need to introduce some quantit ies used in the astrophysical literature (see Ref. [47] for a review). The dynamical mass is the mass deduced from the observation of these rotation curves (7.135) The measurement of the dynamical mass relies on the measurement of the stars rotation velocities in the exterior zones of the disk from their Doppler effect that gives access to the line of sight velocity Vr = Vgal
+ v sin i ,
(7.136)
where i is the inclination of the galaxy disk and Vgal is the galaxy velocity. We get the radius from the last observed point , r m ax , at which the velocity is called V max or V oo . However , not all of the dynamical mass is in the form of dark matter . Observations of the CO molecular lines a llow us to reconstruct the internal part of t he rotation curve and the gas cont ribution in this zone where it dominat es. Radio observations from the HI gas, using the 21 cm hydrogen line, allow us to reconstruct the most exterior part, beyond the stellar disk, where dark matter dominates. We can thus decompose the rotation curve into a contribution from the stellar disk and the gas as 2 2 2 (7 .137) V = Vb a ryo n + v D M ' where the baryonic component is itself decomposed into 2 Vb a ryo n
2
= Vgas
+ Vst2 a rs ·
Each of these velocities is rela ted to the mass contribution of the corresponding component , where the link is provided by the Kepler law
Given the mass- luminosity ratio m */ L * of the stellar population of the galaxy, one can reconstruct the m ass profile of the halo (see Fig. 7.18) by distinguishing between
414
Gravitational lensing and dark matter
the contributions from the gas, the stars and, by taking the difference , from the dark matter halo. In fact, from the luminosity profile (7.131) , one can define, for spira galaxies , a surface density profile by
~ = ~o exp ( -
;d) .
(7.138
The optical radius, Rapt, is defined as the radius containing 80% of the luminou energy. It is related to the disk radius by R apt::: 3.2Rd. We obtain equivalent profile for the gas. In conclusion, we see that we have two important notions of mass: the dynamica mass, related to the rotation curves, and the baryonic mass, assumed to be proportiona to the luminous mass. We have a third notion of mass to consider, which is the mas Miens that can be determined by lensing. In the standard dark matter model, thes masses should satisfy (7.139 In what follows , we shall test this relation against observations. 7.2. 1.3
Spiral galaxies
Milky Way
The rotation curve of the Milky Way (MW) is currently known with a great prec sion. In the central part of our Galaxy, the stars velocity dispersion is explained by th presence of a black hole of mass M. ::: (2.87 ±a.15) x 10 6 M (') that was determined from the orbital parameters of stars orbiting close to the galactic centre [48]. The presenc of supermassive black holes is today accepted in numerous galaxies. Our Galaxy can be roughly decomposed as (a) a high-density core, containing black hole, (b) a central region (;S 100 pc) where the density rapidly increases befor (c) it reaches a maximum at around 300 pc and then decreases towards a minimum at around 2 kpc before (d) a gradual pull-up to reach the disk maximum at aroun 6 kpc and (e) a plateau with holes at around 8 kpc and 15 kpc. Taking as a reference the Sun, the dynamical mass (7.135) is then
M«
r)
=
9.6
x 10
10
M(') (
V
220km · s
_1 ) 2
(lO;kpc ).
(7.140
Since the luminosity of the MW is estimated as L Mw = 2.3 X 10 10 L eo), we obtain th mass/luminosity ratio, Q, of the Milky Way
Q MW
Rhala
:::
50 Q (') ( 100 kpc
)
.
(7.141
The last detectable gas is found around 20 kpc, for which there is still no sign of an Keplerian regime, so that Rha la > 20 kpc. The dynamics of globular clusters and o the Magellan clouds allows us to deduce that R hala > 75 kpc , so that
Evidence for the existence of dark matter
415
(7.142) More locally, the comparison between the mass of the close stars and gas to the dynamical mass allows us to obtain that (7.143) in a region of around a hundred parsecs. This limit , obtained from the dynamics of stars normal to the plane of the disk , is called the Oort limit. The Local Group, i. e. mainly the Andromeda galaxy and a few satellite galaxies, can be considered as an isola ted system since the closest galaxy, M81 , is at a distance of around 3 Mpc. Andromeda and the Milky Way are separated by around r = 10 3 kpc and have a relative velocity of v ~ 10 2 km· S - 1 , so that V HO / TC ~ 0(1) , which shows that the system is gravitationally bound. This means that t he dynamical mass is of the order of 10 12 M o . A more precise estimate gives M = (3.2 - 5.5) x 1O l2 M o and the total luminosity of both companions can be estimated to L ~ 4.2 X 10 10 L o so that (7.144) Q g roup = (76 - 130) Qo · Spiral galaxies
Today, the rotation curves of several hundreds of spiral galaxies have been catalogued. They behave differently in the internal zone, depending mainly on the importance of the bulge . Depending of the type of galaxies, these rotation curves have different morphologies but all of them seem to reach a plateau. 1.0 ~.
150
~
.::-'100 I~
'Of)
o
•
o
0.0
..Cl
oo
5
10
15
.
"".
0.5
0
0
0
..
- 1.0
R (kpc) 1.8
2.0 2.2 Log (v oo)
2.4
Fig. 7 . 18 (left) : Rotation curve of the galaxy M33. The observations (points and circles) are compared to a model of dark-matter h alo (solid line) . The contributions from the halo (clashed-dotted line), from the stellar disk (short dashes) and from the gas (long dashes) are also represented . (right) : Tully- F isher relation for various spiral galaxies.
Figure 7.18 represents the rotation curve of the galaxy M33 and many other examples can be obtained , see e.g. Ref. [49]. Notice the fact that the rotation curve remains
416
Gravitational lensing and dark matter
fl at while the dominant matter changes from stars and gas to dark matter; this m look like a conspiracy. It seems that a 'tuning' exists between the baryonic mat and the dark ma tter , without which t he plateau could have had some bumps or ho It seems that the dissipative evolution of the baryons in t he potential wells of d matter is such that both profiles adjust to each other. There exists a relation, without too much dispersion, between the luminosity o spiral galaxy and the maximal velocity, Voo of its rotation curve plateau. This Tul Fisher relation ex = 3.9 ± 0.2, (7. 1 L ex v~,
tells us that for these galaxies, there exists a scaling relation between their mass a size. This relation is no longer valid for galaxies of small luminosity for which the m of the gas must be t aken into account in the luminosity. Variations around the Thlly- Fischer relation
The Tully- F isher law (7.145) relates the total luminosity to the velocity V oo , spiral galaxies. A generalized or baryonic Tully-Fischer law has also been proposed [5 This law relates the total baryonic mass to Voo as
(7 .1
with ex rv 4. This law t herefore includes the contribution of both the gas and t he st and thus becomes general for any kind of spiral galaxy (dwarf, low surface brightne etc). It has been conj ectured [5 1] that dark matter follows the gas distribution and , particular , that their rotation curves were homothetic
(7. 1
With this hypothesis , t he total rotation curve can be decomposed into two contrib tions , that of the stars and that of the gas, and only introduces one arbitrary parame v 2 = V;tars + (1 +l' )vias ' Various studies [52 ,53] have shown that this hypothesis see in to be in good agreement with observations and that the parameter I' must be t he range I' = 3 - 16. Finally, at t he characteristic scales of these galaxies, the ratio between the surf densities of dark and baryonic matter is of the order of [52]
-~DM - - = 8.5 ± 1.5. ~baryon
(7.1 4
These relations have led to the hyp othesis t hat the dark matter from the galaxy ha could be in the form of a gas too cold to be detectable [54]. Universal profiles
The description of the galaxy halos was initially addressed by establishing empiri density profiles [55] that have the property of decreasing as r- 2 at large dist ances. example of these profiles is the isothermal model [see (7.38) for its derivation]. T
Evidence for the existence of dark matter
417
model is singular at the centre and has been improved with an isothermal model with core radius. A commonly used empirical parameterization describes the halo with the profile 1 + 3(r/a)2
(7.149)
POM = Ps 3[1 + (r/a)2J2'
Its only noticeable difference from the isothermal model with core radius is its behaviour in the few central kpc. It seems that all of the halos of spiral galaxies follow a law of this kind [55] . This suggests searching for a general dark-matter halo profile of the form (7.150) PO M( a) = po(a)f (~) ,
f being a universal function. Let us stress that the function Po could depend on other parameters such as the galaxy luminosity. The dark-matter mass is thus given by F(x ) =
foxu
2
f(u)du.
(7.151)
Observations [56] tend to prove that , under the assumption of a profile of the form p ex 1/( r 2 + r~ ) , the dynamical mass Mdyn(a) ex a 2 and a ~ 2.3Rd, which implies in particular that almost all of the baryonic component of the galaxy is in the core of the dark-matter halo since it mainly develops from re' Moreover , in spiral galaxies the baryonic matter is distributed in the form of a disk so that Mbaryon (a) ex R~ ex a 2 , from which we can deduce that MOM(a) ex a 2 . Using the form (7.151), this implies that
po (a) ex
a- I
£0 po(a) == Perit-· a
(7.152)
Furthermore, from these observations one understands that not only do we have a generalized Tully- Fischer law, but in fact it is true for each component since MOM ex a2 , so that v5M(a) ex MO M(a)/a ex a and thus v6M(a) ex a2 ex MOM(a). As will be seen from numerical simulations, every profile mainly differs by its behaviour at the centre and at very large distances. It is difficult to determine from the kinetics of the central region whether the distribution of dark matter is peaked or if the galaxy has a core. T he dynamics of this region is indeed in general domina ted by visible matter. When this is no longer the case , as for galaxies with low surface brightness , the central region is so small t hat the observational resolution becomes a limiting factor .
7.2.1.4
Other observations
Elliptical galaxies Elliptical galaxies are par ticulary interesting for the study of dark matter as they have a simpler geometry and their baryonic mass is almost uniquely in the form of stars (they contain a small amount of gas). Information on these galaxies comes from
418
Gravitational lensing and dark matter
three sources: optical observations, observations concerning dark matter (gravitatio lensing, virial, rotation curves) and numerical simulations. Optical observations have shown that there exists a relation between the luminos L* and the central velocity dispersion, (Jo ,
(7.15
This Fab er- Jackson relation is the analogue of the Tully- Fischer relation. Thousan of galaxies from SDSS [5 7] have been used to est ablish that (JO ~ L~·2 5±O .01 2 .
(7.15
Moreover, these optical observations were able to establish that there was a v precise relation [58] between the effective radius , Reff, i.e. the radius containing half the luminosity, the central velocity dispersion and the luminosity L *. The SDSS [ has established that (7.15 R eff = (Jo1.49±O .05 J-i - 0.75±O.01 ,
with J-i == L* / R;ff . In logarithmic coordinat es, this relation defines the fundamen plane of the galaxy. T he projection of this relation implies that Reff ~ L~ 63±O.0 It also leads to the conclusion that the surface density ~*(r) can be described by function of three parameters . It has indeed been shown observationally that a profile with three paramete called t he Sersic profile, of the form
(7.15
wit h ~eff = ~( Re ff) and X = R / Reff, reproduces well t he observations. In particul for any elliptical galaxy, b(n) = (2n-0.324) with n b etween 0.1 and 100. The dispers of n is today understood as arising partly from the triaxiality of the elliptical galaxi The observational data lead to the conclusion that stars dominate at the cent In particular , the dynamical mass varies very little until 2 Reff which shows that da matter is either sub dominant at the centre or that is has a noticeably constant de sity or fin ally that its density follows that of the light distribution. Observations the stellar velocity distribution tend to prove that the first hypothesis is favour However, we should recall that elliptical galaxies are formed by successive fusion w early galaxies that induces the existence of radial st ellar orbits, causing the dynami analysis of the stars to be more delicate. At large radius , the observation of multi images of quasars shows that dark matter dominates t he matter distribution. Desp the simple geometry of these galaxies, the distribution of dark matter remains poo known , by lack of precise measurements of the stellar dynamics . Low surface brightness galaxies
Low surface brightness galaxies seem to be complet ely dominated by dark m ter , with t he st ellar population contributing only marginally to the total mass. particular , the dark-matter density profile does not appear t o be peaked at the cent
Evidence for the existence of dark matter
419
The relative contributions and distributions of dark matter in various kinds of galaxies contain important information for understanding the relation between dark matter and ordinary matter. Elliptical and spiral galaxies with low surface brightness give access to limiting cases, the consequences of which have not yet been drawn, for lack of precise enough observations. Relation with the mass of the central black hole Just like the Milky Way, it seems that elliptical galaxies and the bulges of most of the spiral galaxies contain a central black hole. A recent empirical law has been established between the mass of the central black hole, M., and the velocity dispersion of the bulge, a v , In
(M. ) M0
= (4.02 ±0.32) In (
v
a 200km· s
-1 ) + (8.13 ±0.06).
(7.157)
This relation has been established from the observation of 30 spiral galaxies for which the mass of the central black hole and the rotation curve could be measured. Since the velocity dispersion is proportional to L;/4 (Faber- Jackson for ellipticals or Tully- Fisher for spirals) we expect t hat L* ~ M • .
The observations seem to indicate effectively that
M. ~ 1O- 3 M*, M* being the stellar mass of the elliptical galaxy or of the bulge of t he spiral galaxy. This relation (M. - av) brings information t hat can be useful for the understanding of galaxy formation and the role of dark matter. It is still too early to assert whether such a relation can be exactly deduced from a model of large-scale structure formation but such a correlation seems 'natural' in a hierarchical model of structure formation.
7.2.2
Dark matter in clusters and groups of galaxies
The first indication of the existence of dark matter was revealed by Zwicky during his study of the Coma cluster in 1930 [60J. Galaxy clusters contain from around a hundred to several thousands of galaxies. 7.2.2.1
Virial theorem
Galaxy clusters are gravitationally bound systems decoupled from the cosmic expansion. One can thus assume that they are in a stationary state. The velocity of the galaxies is small and we can model the cluster as an isolated gas composed of massive particles that interact only through gravitation. The acceleration of a galaxy in the cluster is given by (7.158) the gravitational energy is
420
Gravitational lensing and dark matter
2U
=
'""
- G N L..t iii
mom
IXj J- Xi'I'
(7.15
while the total kinetic energy is given by
(7.16 The second derivative of the total moment of inertia, I == kinetic energy by
L
miX;,
is related to t
(7.16 The equation of motion (7.158) can then be used to deduce that
'"" .. G '"" Xi L..t miXi . Xi = N L..t mimj i
. (Xj -
IXj -
#i
Xi) 3
=
U,
Xil
the second equality being obtained after symmetrization. We thus obtain the relati known as the viria} theorem
Ij
=
2U +
4K. 1
(7.16
For a stationary system, the value of the time average of the moment of inertia constant so that (I) = 0, in which case one has
(U)
+ 2(K ) = O.
(7.16
One can express the mean kinetic energy as (K) = ~M(v2) and the mean potent energy as (U) = aG N M2 jT'h' where M is the total mass, a is a numerica l coefficie t hat depends on the geometry and on the density profile (a rv 0.4 for galaxy cluste and T'h the radius in which half of the mass is contained. The virial theorem can th allow us to estimate the mass of the structure
M. _ Vlr -
2 (V ) T'h
aG
(7.16
• N
7.2.2.2
Coma cluster
The Coma cluster has a mean redshift of z = 0.0232, which places it at arou 100 Mpc of the Galaxy. The velocity dispersion along the line of sight is 880 km· S that is a mean square velocity of (v 2 ) ~ 2.32 X 10 12 m 2 . S-2 assuming that t he veloc distribution is isotropic. The estimate of T'h is difficult, mainly because the distributi of dark matter is unknown. Assuming a r atio M j L constant with the radius and spherical geometry, one can estimate that T'h rv 1.5 Mpc, which gives an estimate the mass of the Coma cluster of M rv 2 X 10 15 M 0 . Since its luminosity is of the ord of L rv 8 X 10 12 M 0 , this means that Qcoma rv 250Q0 ' A refined analysis actually giv
(7. 16 and a similar study in the Perseus cluster gives
QPerse us
rv
600h Q0 '
Evidence for the existence of dark matter
7.2.2.3
421
Mass of clusters
There are two other methods to determine the mass of a cluster. The first one, presented in Section 7.l.2.6, uses either strong lensing at the centre (r 200 kpc), which gives constraints of the order of
:s
QComa rv
(100 - 200) Qo,
(7.166)
or measurements of the gravitational distortions of the background stars (r which leads to QComa rv 300 Qo .
:s 1 Mpc), (7.167)
Another method relies on the X-ray emission of the hot radiation from the intracluster gas. The presence of such a hot gas captured at the centre of the cluster is proof of the existence of dark matter creating a gravitational well strong enough to trap the gas . Without dark matter , the hot gas would be diluted beyond the cluster in a time smaller that the Hubble time. To estimate the dark-matter mass, let us assume that this gas is in hydrostatic equilibrium and that its pressure compensates gravity
dP
GNM( < r)Pgas(r )
dr
r2
(7.168 )
The pressure of the gas is obtained by the measurement of its temperature
kBT
(7.169)
P = Pgas--, fLm p
where fL is the effective mass of its constituents in units of the mass of the proton. As long as fL is constant, i.e. if the chemical composition is uniform, we have
GNMX« r
r) = _ kT(r) (d ln pg as fLm p d lnr
+ dInT)
.
dlnr
(7.170)
The temperature profile is often difficult to measure and the cluster is often assumed to be isothermal and composed of a gas with a radial distribution following a so-called ,B-profile, Pgas = Po [1 + (r /r c)2J - 3.B12. These hypotheses are usually satisfied for clusters with quasi-spherical geometry. The analysis of the Coma cluster with this method gives a mass of the order of M x rv (3 - 4) x 10 14 M o in a region of 0.7 Mpc around the centre and of rv (1 - 2) x 10 15 M o in a region of 3.6 Mpc. These measurements are compatible with the mass determined from the virial theorem and by gravitational lensing. A similar analysis with the Perseus cluster gives Q = (213 ± 60)h Qo .
7.2.2.4
Summary
The above-mentioned studies [6 1J allow us to show that , for clusters, Mgas = (15 ± 5) %, M DM
and the density profiles follow NFW profiles.
M.
-- = MDM
(3 ± 2) %
(7.171)
422
Gravitational lensing and dark matter
To summarize, for r 2: 1 Mpc, we do not know the matter distribution. For 10 kp r ::s 100 kpc, t he dynamical mass , the mass deduced from gravitational lensing a that inferred from X-ray emission mass differ approximately by a factor of 2, proba related to projection effects. For 100 kpc ::s r ::s 1 Mpc, the three kinds of observati give the same value for the mass and the matter distribution is compatible both w a NFW profile and an isothermal sphere with core radius. For r ::s 10 kpc, t here insufficient observational diagnostics . 7.2.3
Cosmological evidence
The evolution of homogeneous space-time and the analysis of the evolution of pert urbations in the linear regime provide two additional arguments for the exist e of dark matter (see Fig. 7.19).
- Primordial nucleosynthesis gives access to the amount of baryonic matter in Universe (see Chapter 4). The shape of t he power spectrum of the matter fluctuations depends on the tio between the baryonic and dark densities, Dba / Dca. In particular when relative quant ity of baryons increases, oscillations appear in the matter pow spectrum. These oscillations are relat ed t o the coupling between baryonic mat and radiation (see Chapter 5) . - The joint analysis of the results of different cosmological observables, for inst an t he cosmic microwave background, type Ia supernov&) and gravitational lensi indicates that around 30% of the matt er must be dark. T hese ana lyses allow us t o est ablish the two constraints D bO h2
= 0.0224 ± 0.0009 ,
(7.1
These measurements imply that 1.0
Supemovre 0.8
Cosmic microwav background
0 .4 0.2
I \
0.0 .........................
10- 2 k (h
10- 1 Mpc- J )
Stars 0.002 - 0.006
0.2 0.4 Baryons 0.04 - 0.05
0.8
Fig. 7.19 (left) : The matter p ower spectrum of large-scale structures is sensitive t o amount of baryons , Pb l pc increasing from the t op to b ottom curve. (right): The combi analysis of cosmological observations also indicat es t he necessity for 30% of d ark matter.
Evidence for the existence of dark matter
Pc = 4.83 ± O.87
423
(7. 173)
Pb
on cosmological scales . 7.2.4
Summary
Table 7.3 summarizes the estimates for t he mass/luminosity ratio of various systems and the quantity of dark matter that can be deduced from t hese systems. These analyses show that it is difficult to understand the structures of t he Universe , regardless of their scale, wit hout introducing dark matter. Much still remains to be understood concerning its physical nature and distribution, in part icular its correlation with ordinary matter. As shown in Table 7.4, however , on most scales t he quantity of baryonic matter is only a fr action of t hat of dark matter that , furthermore, seems to decrease with the scale.
Table 7.3 Summary of the estimated mass/luminosity ratio for various systems and implications for t he dark-matter content at different scales . System Milky Way
E lli ptical galaxies Spiral galaxies Groups C lusters
Oort li m it rotation curves sat ellites Magellan currents tuning of t he core X h alo rotation curves local group others Coma (B ba nd ) P erseus
Cosmology
Scale 100 p c 20 kp c SO kpc 100 kpc 2 kpc 100 kp c 50 kpc SOD kpc 1 Mpc 2 Mpc 2 Mpc 3000 Mpc
n co
Q/ Q0 5(?) 10 40 iS O 12h i 750 i30h 100 260h 400h 600h 0.11 3h -
0.003h - 1 0.006h- 1 0.024h- 1 i O.05h - 1 0.007 0.46h - 1 O.QlS 0.06h - 1 0.16 0.25 0.37 2
Table 7.4 Relative ratio of dark and baryonic matter. T hese constraints indicate that t he ratio varies little as a function of the scale of t he considered structure. System DM vs baryon
7.2.5
Spiral galaxy
I:elI:b
=
S.5 ± 1.5
Cluster
I:elI:b
=
7.2 ±2. 0
Observable Universe
pel Pb =
4.S3 ± 0. S7
Candidates and constraints
Dark matter is said to be hot or cold depending on whether its constituents are relativistic or not at the t ime of decoupling from the cosmic plasma (see Chapter 4). The only example of dark matter whose existence has been proven is the neutrino , which is a hot dark-matter component. We know, however , that neutrinos cannot account for the whole dark-matter component because they are too light.
424
Gravitational lensing and dark matter
We now address the two following aspects: (1) from a structure-formation point view, what must be the characteristics of this dark matter and (2) from a partic physics point-of-view, which are the candidates.
7.2.5.1
Cosmological phenomenology
Standard cold dark-matter model
The standard cold dark-matter (SCDM) model assumes that dark matter is co posed of massive particles that interact very weakly with each other and with the oth components of the Universe. These particles are generically called WIMPs (weakly teracting massive particles) . If these particles are thermal relics , their abundance depends on two paramete their mass and their cross-section. In general , these two parameters determine t nature of dark matter (cold or hot). For cold dark matter , we have established [ (4.76)] that the relic density is given by fl c h
2 rv
0.3
(
10
~26
\O-V;
3
cm· s
~l
) ~l
(7.17
From a cosmological point of view, this model reduces to the addition of a pressu less (i.e. collisionless) fluid , thus with the equation of state w = 0, coupled to ordina matter only through gravity. One of the motivations to consider such a model l in the growth of the cosmological perturbations (see Chapter 5). As we have se in the matter-dominated era, sub-Hubble fluctuations grow as a. It is then diffic to understand how the density contrast can become of order unity today if it were order 1O ~ 5 at the time of recombination at z rv 10 3 . Dark matter resolves this probl since, as it is not coupled to radiation, it can start to develop potential wells bef the m atter~radiation equality. This dark-matter model has been extended to the so-called ACDM model, wh dark matter only represents 30% of the total amount of matter. Today it offers a go understanding of the linear regime and is in agreement with most of the cosmologi observations (cosmic microwave background, large-scale structures, etc.). It is at t basis of the concordance model of cosmology. This dark-matter model is considered successful for three main reasons. (1) predictions are in agreement with most of the observations of large-scale structur (2) There are many candidates motivated by high-energy physics. (3) If dark-mat particles interact via the weak interaction, then their relic density has the correct ord of magnitude [ef. (7.174)]. The SCDM model is a hierarchical model of structure formation (see Section 5.5 in which the small structures are formed before the large ones. So the distribution small halos should not be strongly correlated with that of the larger structures th form later. The properties of the dark-matter halos in this model have been intensiv studied by means of N-body numerical simulations. They have made it possible establish that the dark-matter halos density profiles take the form
(7.17
Evidence for the existence of dark matter
425
These profiles behave as r - f3 = r - 3 at large scales, unlike the b ehaviour as r - 2 of the universal profiles. The empirical behaviour as r - 2 , obtained from the rotation curves , is only an approximation in the transitionary region between the behaviour as r- ' at the centre and r - f3 at the exterior. Independently from other cosmological details, this model predicts the formation of triaxial halos with very dense cores and
- - Moore et al. --- NFW ....... BS
10 '" 1
0.1 ~
'"
:::~~: ,;::::~;,~~S
8..10-2
'0 6, 10-3
'".:,."
'\~'\;
Q..
,,~
M vir = 2XI0 12 M o r vir = 425 kpe
r
",
(kpe)
Fig. 7.20 Comparison between the halo density profiles obtained by numerical simulations and the universal d ensity profiles obtained from a fit to observations.
many substructures. These substructures are natural in any hierarchical model of structure formation. The parameters of these profiles (a , (3, ,,/, rs) are given in Table 7.5. The characteristic density Ps is obtained from the total mass and the size of the galaxy. Figure 7.20 compares the different profiles for a typical galaxy. Table 7.5 Parameters of the dark-matter density profiles (7.175) obtained by differ ent studies of N-body numerical simulations . Model Navarro-Frenk- White (NFW) Moore Kravstov Isothermal with core radius
(kpc)
Q
f3
'Y
1.5 2
3 3 3 2
1 1.5 0.4
20 28
0
3 .5
2
Ts
10
The most recent simulations [59] give other profiles with a form close to the deprojected Sersic profiles
426
Gravitational lensing and dark matter
p(T)
=
po (a) exp
[-2P, ( ~ ) 1/1-'] exp(2p,) ,
(7.1
with P, = 6 ± 0. 2 and a being the radius where t he slope of the profile is equal to Moreover, in these models the profile po (a) behaves as po (a) ex 1/ The value, 'Y, of the slope of the profile in the inner part of the halo is t he sou of much debate. These simulations seem to show t hat the profile does not behave a power law at t he centre. Notice also that these profiles represent a fit to the res of several numerical simulations and do not quantify the dispersion around this fi The CDM model must , however , face unresolved problems at small scales.
va.
- It predicts too many substructures, small halos and galaxies in orbit around lar structures, than what is observed. - The number of these halos varies as t he inverse of the halo mass . Numer satellites, such as the Magellan cloud for our Galaxy, should thus be observ The observation of the Local Group reveals less than a hundred galaxies dwarf galaxies are known in a radius of 0.5 Mpc) while numerical simulati and analytical estimates predict around a thousand satellites. - These satellites should induce a more significant thickening of the gala disk than is observed. - The predicted density profiles of the dark-matter halos are more peaked t what is observed in many self-gravitating systems. - Clusters, observed via gravitational lensing, have less-peaked cores than t predicted for massive halos. Notice that t he observations of the core however , difficult since the baryons dominate . Let us also stress that interpret ation of the rotation curves assumes an axial symmetry, while these regions the stars are not always on Keplerian orbits. - The central density of t he dwarfs and low surface brightness galaxie too weak and the density profile too smooth compared to that obtai from numerical simulations. Surprisingly, the sm allest galaxies (e.g. DD02 DD0154, etc.) are consistent with a quasi-constant dark-matter profile. - The p ersistence of the bars in galaxies of high surface bright ness also imp that these galaxies have cores of small density. - If the core becomes dense too early, then the cooling of the baryons could too efficient, which would lead to galactic disks with angular momenta order of magnitude too small. - Dwarf spheroidal galaxies are also dominated by dark matter , but they sh no sign of a peaked profile. - Dwarf galaxies of weak luminosity seem to be more strongly correlated with minous galaxies than they should . In particular , they do not fill the big vo between the distribution of luminous galaxies and localize on the borders of th voids. - The observation of galaxy forma tion at z ~ 3 - 4, lat er than t hat of the m massive bright galaxies, seems in contradiction with a hierarchical model of str t ure formation. Similarly, there are supermassive black holes (10 9 - 10 10 M 0 high redshifts, which does not seem very underst andable in this scenario.
Evidence for the ex istence of dark matter
427
Nevertheless, one should not draw too hasty conclusions. One of the keys to the debate lies in the behaviour at large distances where simulations predict t hat t he profiles behave as 1/r 3 . These external zones are, to date , difficult to study observationally. Two sources of information can be considered: First, galaxy- galaxy lensing is sensitive to the most massive objects and it seems [62] that observations of the cosmic shear as a function of the distance for galaxies of mass larger than 10 9 M 0 is compatible with the NFW profiles. Second, data from the SDSS survey have allowed for the study of satellite galaxies of various clusters [63], the behaviour of which seems compatible with a NFW profile. Both these observations do not establish the behaviour of t he density profile at large distances in a definitive way but indicate a t endency to be confirmed. Other models The problems of the SCDM model at small scales should not overshadow its successes at large scales and in the linear regime. These difficulties have motivated the development of many phenomenological models of dark matter with the main objective of modifying the properties of the structures at small scales. Each one of these models brings new responses to one or several problems of the SCDM model and has many proper signatures that fut ure observations and numerical simulations should be able to discriminate. T he following list is not at all exhaustive but illustrates the variety of attempts. 1. Hot dark m at t er (HDM): t his model was motivated by the study of neut rinos. This
form of dark matter is relativistic at t he time of decoupling and at the time of the large-scale structure formation. Its main problem is that dark matter particles are too fast to be trapped and to collapse gravitationally. Also, the formation of structures starts too late since equality is delayed. An important scale is the free streaming scale Afs of a relativistic particle before it becomes non-relativistic. In HDM models, fluctuations on scales smaller than Afs are erased so that t he spectrum of matter perturbations has a well-defined cutoff at Afs (see Fig. 5.16). For instance, for massive neutrinos, Afs ~ 40(30eV/ml/) Mpc. This implies that protogalaxies with mass of t he order of 10 15 M0 are t he first objects to form . Smaller objects t hen form by fragm entation. The model of structure formation is then very different from the hierarchical one of the SCDM model. 2. Warm dark matter (WDM): this model [64] is a compromise between the HDM and CDM models. Dark matter would be composed of relativistic thermal relics at the time of decoupling but that would become non-relativistic before equality. The characteristic distance below which density fluctuations are erased is given by 2 1/ 3 ( m ) -4/ 3 (7. 177) R ~ 0.2 (Dco h ) 1 keY Mpc.
A warm dark-matter candidate must therefore have a mass of the order of 0.1 1 keY. T he main predictions of this model are (1) t he smoothing of the core of the halo , which translates into a decrease in the core density and an increase in the core radius, (2) a decrease in the characteristic density of small mass halos, (3) a
428
Gravitational lensing and dark matter
global reduction of the number of small mass halos , (4) a decrease in the num of small mass halos inside larger mass halos , (5) the formation of voids contain no small mass halos , (6) the formation of small halos at redshift z < 4 vi fragmentation process and (7) the suppression of the halo formation at la redshifts (z > 5). The properties (1) and (5) provide solutions to some of SCDM problems while (6) and (7) offer the possibility of specific tests. If delay in the structure formation was too large, the model would then beco incompatible with observations.
3. Strongly self-interacting dark matter (SIDM): this model [65] supposes that d matter has a self-interaction cross-section that cannot be neglected while annihilation rate remains very weak. This interaction only alters the dynamics dark matter when its density becomes important, so that the cosmic microw background and the large-scale structures remain insensitive to this modificati Concretely, the mean free path of SIDM should range between 1 kpc and 1 M when the density is of the order of 0.4 GeV· cm- 3 (typical density at the posit of the Solar System). Indeed, if the mean free path was larger, no interact would occur on scales of the size of the halo and if it was smaller than 1 k dark matter would behave as a collisional gas, which would greatly modify evolution and the structure of the halos. The mass of these particles must ran between 1 MeV and 10 GeV and its cross-section be of the order of a
~ 8 x 10 - 25 (~)
(_.\_) - 1
1GeV
cm 2 .
(7. 1
1Mpc
The halo structure is only modified above a given surface density and indu a matter redistribution in these regions, thus producing a less dense and m spherical core. Notice that if a is too large then the halo can completely eva rate , which can lead to catastrophic effects, but would be observable without a difficulty in the halo demography.
4. Repulsive dark matter (RDM): this model [66,67] supposes that the dark matte a massive boson condensate with a short-distance repulsive potential. The inter part of t he halo then behaves as a superfiuid, which has the tendency to smo out its profile. T he halo cores would then have a minimum size independent their density. In t he linear regime, scales smaller than a few Mpc are suppress
5. Fuzzy dark matter (FDM): this model [68] supposes that dark matter takes form of a very light scalar particle with a Compton wavelength of the order the galaxy core. At small scales, the wave-like behaviour of dark matter preve gravitational collapse so that t he core of t he halo is smoothed . The Jeans leng under which density perturbations are stable (see Chapter 5), is given by m ) .\ '" 55 ( J 10 -22 e V
P
- 1/2 (
2.8 x
1011 M 8
)
Mpc -
3
- 1/4
k c. P
(7.1
A field of mass m '" 10 - 22 eV can then provide a smooth density profile scales of the order of the kpc. This also induces a sudden cutoff around
Evidence for the existence of dark matter
429
wavenumber k ~ 4.5(m/10 - 22 eV)1/2 Mpc - 1 in the linear power spectrum, which can affect the abundance of small-mass objects. This mass scale can, however, seem artificial in the framework of particle physics. Notice that the mass of a particle with a Compton wavelength of the size of the observable Universe is of the order of m ~ 10- 33 eV and that a mass of this order can also be related to the acceleration of the Universe (cf. Chapter 12).
6. Self-annihilating dark matter (SADM): this model [69] supposes that dark matter can annihilate itself in the strong density regions where two-body interactions become important. The annihilation produces a radiation that should not include photons, as their flux would exceed the observational constraints. The annihilation has the effect of decreasing the core density and to induce a global expansion of the halo due to the decrease in the gravitational potential. The annihilation rate depends on the density and on the velocity dispersion, r = n\O"v). In the particular case where r = (plm)O", the density profile takes the form
(7.180) where to is the age of the Universe. A cross-section \O"v) ~ 10- 29 (mil GeV) cm 2 gives a core density of 1 GeV Icm 3 , in agreement with observations . The effects on the number of small halos has not been studied.
7. Decaying dark matter (DDM): if the dense halos can decay into relativistic particles and into particles of smaller mass , then the density of cores formed earlier can be noticeably reduced without affecting the large-scale structure [70j. A potential problem of this model is that the production rate of dense and massive clusters must be far more important than in the SCDM model to obtain a correct distribution after the decay of dark matter. 8. Massive black holes (BH): if the main part of dark matter was composed of massive black holes of around 10 3 solar masses, the dynamical friction between black holes and ordinary matter could lead to the migration of these black holes towards the centre . They can thus give rise to the central black hole of the galaxy, decreasing the quantity of dark matter in the core. But we have to assume that these black holes are present to begin with. 9. Initial spectrum: the breaking of the scale invariance of the primordial fluctuations power spectrum at around k = 5 ~ 10 h - 1 Mpc can marginally modify the structures of halos at small scales [71]. These models introduce two kinds of modifications compared to the SCDM model. (1) The halo is only affected when the interaction rate becomes larger than a critical value (SIDM , BH, SADM). (2) The models have a characteristic time or distance scale below which the halo is modified (WDM, MDM, FDM, DDM). They therefore lead to different observational predictions. For instance, to yield the same structures today, the objects of a given mass must form earlier in DDM, SADM and BH while objects
430
Gravitational lensing and dark matter
of smaller mass form later in FDM and HDM and by fragmentation in WDM. The mography of the halos and satellites is modified together with the core density pro To illustrate these examples, Fig. 7.21 summarizes quantitatively the implication these scenarios. The central structure of t he halos, and mainly that of the dwarfs galaxies of low surface brightness, seem to be the key to characterize dark matter. DDM , SADM
,-
' CCDM, SlDM
C luster
,,:
g
'" E
I I.
J2
"'"' '"
,,
FSB Galaxy
I
"
i
WDM
'0
....
.... WDM
"E'"
.D
BII _ ~;,<.- "'-:"'FDM , RDM
Dwarf
:::E
E o
I
L" Ga laxy
CCDM, SlDM, DDM
C)
I
.~
~
~
I
:l
Z
,,-
FDM , RDM
Dwa rf L· galaxy LSB galaxy C luste r Mass
CCDM , DDM
, C)
..,
,
0. .!<
~
C)
0.
SlDM;,BH
"" '" 0
"'" '" v"
:::E
:::, .<:
-----~.
SADM
, ..........
:'.-.,\
.D
.~. 0" OJ
.
:
WDM , FDM , RDM
WDM
'f
'"
"'0"
,, ,, ,
'" .D
,
\
\ \
, \ ,
\ \.:.,.
Dwarf L* Galaxy LSB Ga laxy Cluster
E
:l
Z
.
CCDM
.....SlDM , SADM ,\ BH, FDM, RDM " Mass
Mass
Fig. 7.21 Predictions and signatures for various models of dark matter. (top left): the mation time of a structure with a given mass for structures of increasing mass (dwarfs, surface brightness, galaxies, clusters). (top right): dependence of the number of objects a function of its mass . (bottom left): density in the inner 1 kpc region as a function of t he m of the system. (bottom right): number of dwarfs in a volume of (1 MpC)3 as a function of mean density in this volume. From Ref. [72].
7.2.5.2
High energy candidates
The standard model of particle physics (see Chapter 2) does not provide any da matter candidate. The only (stable) neutral fermion , the neutrino , has a vanish
Evidence for the existence of dark matter
431
mass , although it is known experimentally to be massive. Nevertheless, almost all the extensions of this standard model provide several candidat es, and in particular massive neutrinos. For detailed reviews concerning these candidates, detection methods and experimental constraints, see R efs. [73- 76]. We only consider here non-baryonic candidates. As explained in Section 7.l.2.1 , micro lensing allows us to put a strong constraint on the abundance of brown dwarfs. Moreover , nucleosynthesis and the large-scale structure formation provide us with strong arguments against the fact that all dark matter could be baryonic. dark-matter candidates can be classified according to various criteria. - The first classification is based on the candidate mass at the time of galaxy formation, at a t emperature of the order of T '" 1 keY. Particles of mass m « 1 keY are called hot (such as neutrinos) , t hose of mass m » 1 keY cold (neutralinos, axions, ... ) and t hose of mass m '" 1 keY warm (sterile neut rinos, gravitinos) . - A second classification can be p erformed on the basis of the production mechanism. One can distinguish between thermal relics, which were in thermal equilibrium during the primordial Universe (neutrinos , neutralinos, ... ) and non-thermal relics, produced by some non-thermal mechanisms (axions produced by cosmic strings , WIMPZILLAs, ... ). - The last classification relies on the framework of particle physics in which they appear. We can distinguish between existing candidates (neutrinos), candidates that are motivated but not yet observed (e.g. light supersymmetric particles , axion), i. e. that naturally appear in t he framework of particle physics independently from the dark-matter problem and that have masses and cross-sections leading to relic densities in agreement with the constraints, and exotic candidates that can, nonetheless, be interesting ideas (e.g., WIMPZILLAs). This last classification is indeed more arbitrary and variable as a function of the progress in particle physics .
7.2.5.3
Ne utrinos
Many, if not all, extensions to the standard model of particle physics allow for the inclusion of a massive neutrino with mass mv < 100 eV. Neutrinos oscilla tions experiments (see Ref. [74]) allow us to est ablish that
D.mi2 '" 7 x 1O - 5 eV 2. (7.181) This leads to the conclusion that the mass of the heaviest neutrino must be larger than 0.05 eV and laboratory experiments show that the mass of each of the three neutrinos must be lower than 2.8 eV. Neutrinos t hus contribute to hot dark matter and their abundance (see Chapter 4) can be evaluated as (7.182) Ve
f->
(V""VT
):
with gi = 1 for Majorana neutrinos and gi = 2 for Dirac neutrinos. So with only one Majorana neutrino, the constraint on the neutrino mass implies that
432
Gravitational lensing and dark matter
(7.1
The analysis of data from t he cosmic microwave background , from primordial nuc osynthesis and large structures implies that
(7. 1
which requires that L 9imi < 0.7 eV. To conclude, the combination of laboratory a cosmological constraints gives 0.0006 < DvOh2 < 0.0076 ,
(7.1
These constraints imply t hat neut rinos cannot represent the main part of dark mat (for a review on cosmological neutrinos , see [77,78]) . We are t herefore led to consi particles whose existence is only suggested from the extensions to the standard mo of particle physics. 7.2.5.4
Other kinds of neutrinos
Neutrinos are fermions. The Pauli exclusion principle implies that light neutrinos c not represent the main part of dark matter. Indeed , in a dwarf galaxy, which must dominated by dark matter , these neut rinos must be confined in phase space, with volume determined by the P auli principle. One can show t hat in order to explain distribution of dark matter in dwarf galaxies, we need that t he mass m v of at le one species of neutrino satisfies [79] mv
~
120
~v
COOkm.s- 1 )
- 1/ 4 (
Tc
- 1kpc
) - 1/ 4
eV ,
(7. 1
which is in contradiction with the constraints (7.185) . Sterile neutrinos [80], that are right-handed neutrinos coupled only to left-hand neutrinos (active neutrinos), have also been considered. T hese neutrinos are mai produced by neutrino oscillations. Depending on their masses, they can behave as h warm or cold dark m atter. As an example, t he mass term for a model wit h only o generation of right-handed neutrinos is given by the Lagrangian (7.1
where ¢ is the Higgs field and v its vacuum exp ectation value (VEV) . The case of h dark matter corresponds to f.-L '" 90h 2 eV and M « f.-L or f.-L2/M '" 90h 2 eV and M» When sterile neutrinos are candidates for dark matter , M is the relevant mass. 7.2.5.5
Axions
The combined action of charge conjugation and parity (see Chapter 2) is not an ex symmetry even in the standard model of particle physics described in Chapter 2. T
Evidence for the existence of dark matter
433
CP violation in t he strong interaction sector is expected in the framework of quantum chromodynamics. The effective Lagrangian then contains a term of the form [81]
.ce ex eg32cap,vE. /.wpaC apa,
(7.188)
G~v being t he tensor of the gluons. This indeed violates parity (p) a nd time reversal
(7) and therefore cP as CPT must be conserved. Such a term induces a permanent electric dipole for t he nucleons, which is, however, not observed down to the current experimental sensitivity, corresponding to the constraint < 10 - 10 . Axions were introduced by Peccei and Quinn [82] to solve this problem. For this , the symmetry group of the standard model is extended by a global symmetry, U(1)PQ' which is spontaneously broken. The symmetry being global, its Goldstone boson cannot be absorbed by the Higgs degree of freedom, and so there must exist an additional scalar field , called the axion, a, which is associated with the phase of the U(1)PQ transformation . Notice that such additional global symmetries U(l) also appear in some compactification schemes of type I/Il/ heterotic string theories . The phenomenology of the axions is determined by a unique paramet er, f a which characterizes the scale of the Peccei- Quinn symmetry breaking, and in terms of which the axion mass is given by
e
ma
=
0.62 (
10
7
Gev)
fa
eV.
(7.189)
Moreover, one can show that t he axion is coupled to two photons since its effective action contains a term of the form
.c a, --
CY
fa
F
p,v
F ~p,v
a
(7.190)
where Fp,v is the Faraday tensor and Fp,v = ~E.p,vpaFpa, its dual. This coupling implies that the axion can decay into two photons . This process dominates unless ma > 2me as the axion can then rapidly d ecay into electron- positron pairs. Moreover, in t he presence of a m agnetic field, an axion can oscillate into a photon of energy ma. Laboratory and astrophysical constraints imply that ma < 0.01 eV. Its coupling to other fields is thus very weak and the axion therefore b ehaves as cold dark matter. The original Peccei-Quinn axion is thus very constrained, mainly since it interacts with ordinary matter and because one needs fa rv 250 GeV. The properties of the axion have b een ext ended in order to introduce it into other sectors, making it an invisible axion.
7.2.5.6
Candidates motivated by supersymmetry
From a theoretical point of view, the best motivated dark matter candidates are probably those that appear in the framework of supersymmetric theories (see Chapter 10). In order to be a WIMP candidate, the lightest supersymmetric p article (LSP) must be electrically neutral and a colour singlet .
- sneu trino: the supersymmetric partner of the neutrino, V, has b een considered as a good candidate for a long time and its relic density has the correct order
434
Gravitationa l lensing and dark matter
of magnitude if its mass is of the order of m ;; = 550 - 2300 GeV. Neverthele the cross-section of its interaction with nucleons exceeds constraints obtained dark-matter detection experiments tha t impose that m;; 2: 20 TeV. - gravitino: the supersymmetric partner of the graviton can be the LSP in loca supersymmetric theories (supergravity) . Nevertheless, since it only interact s gra itationally, it is almost impossible to detect. The gravitino has a spin equal to 3 and gets a mass , m 3 /2 , when supersymmetry is broken. Since its interaction purely gravitational, its interaction rate is of the order of
(7.19
Decoupling occurs when r 3 / 2 rv H rv T 2 / M p , i. e. at around T rv M p. Using t rela tion (4.80) , we expect a relic density of the order oH2 3 / 2 h 2 rv 2.25(m3 /2/ 10 e [using g3 /2 = 4 and evaluating q*(xt) rv 200j. If h rv 0.7, this requires that m 3 / 2 ,:S 4.42 eV.
(7.19
This value is too small since m 3 /2 controls the supersymmetry-breaking scale a such a value does not allow supersymmetric particles to have a mass larger than TeV. This is what is called the gravitino problem. It can be avoided by assumi tha t the gravitino is heavier and that it decays during the cosmic evolution. T decay rate is then of the order of r 3/2 rv m~/ 2 /M; instead of (7.191). It becom comparable to the expansion rate, H rv T2 / M p , at a temperature of the order Td rv m~~; M; /2 which corresponds to the temperature at which gravitinos deca The Universe is then reheated and constraints are then set by nucleosynthesis this decay should not modify the light-element abundances . Finally, gravitin could be overproduced during the primordial Universe [83]. - n eutralino: there are four neutralinos, X~ with n = 1 .. . 4 in order of increasi mass , in the minimally supersymmetric model (MSSM) that are linear combin tions of the Wino , W 3, the Bino, B, and of the two Higgsinos, lid and liu , the particles being the supersymmetric partners of, respectively, the ga uge bosons W and B, and of neutral Higgs bosons, Hd and Hu. The four neutralinos sta tes a thus given by
/
(7.19
where the coefficients N in are the eigenvectors of the neutralino mass matr This mass matrix depends on Bw and mz (see Chapter 2) and four of the undetermined parameters from the MSSM [Ml and M 2 , the masses of the gaugin of the groups U(l)y and SU(2)L' tan /J , the r atio of the VEVs of the two Hig fields, and IL , the mass parameter for the Higgsinosj . The problem one has to solve is understanding whether , for a given set of valu of these parameters, the cross-sections for the annihilation and interaction w the nucleons (see Ref. [84]) , as well as the mass of X~ , are compatible with t relic abundances. Since neutralinos are non-relativistic, their cross-section can
Evidence for the ex istence of dark matter
435
expanded as ((Jv) = a + bv + ... . Using the analytical approximation (4.76), we obtain a lower bound for t he relic density [84]
D
>
xO ~
(
mx 200 TeV )
2
(7.194)
This estimate can be modified by a factor of 2 if coannihilations are taken into account. Cosmological and accelerator constraints lead to t he conclusion that t he neutralino mass must satisfy [73] 108 GeV :::; mx :::; 370 GeV
(p, > 0)
160 GeV :::; m x :::; 430 GeV
(p, < 0).
(7.195) - axino: the supersymmetric partner of t he axion usually behaves as hot or warm dark matter. It can only behave as dark matter if t he reheating t emperature is small enough.
7.2.5.7 WIMPZILLAs The hypothesis that dark matter is a thermal relic is somehow very restrictive. It for ces the mass of the possible candidate to be lower than a few TeV, i.e. of the order of the electroweak scale. T he reason for t he existence of such a limit lies in the fact that for a particle of mass mDM, the annihilation cross-section has a maximal bound of the order of mD~I ' called the unitary bound [85]. This imposes the mass of the particles [84] to be of the order of mDM
;S 340 TeV.
(7.196)
WMAP measurement s have reduced this constraint to mDM
;S 34 TeV.
(7.197)
It can only be avoided if dark matter is not a thermal relic, since then the relations (4.76) and (4.80) cannot be applied. WIMPZILLAs (X) are ultraheavy relics produced through a non-thermal mechanism. In order for t his mechanism t o be viable, the particles should be stable, or at least t heir lifetime larger than the age of the Universe that, for all practical purposes, is equivalent . In particular , the decay of WIMPZILLAs could explain the origin of ultrahigh-energy cosmic rays if t heir lifetime happened to be comparable wit h the age of the Universe. Furthermore, we have to require that WIMPZILLAs were not at equilibrium before their abundance froze , since otherwise Dx » l. A sufficient condition is that (see Chapter 4),
rx = nX((Jv)
< H.
(7.198)
Reciprocally, if the particles X were produced during the radiation era, at a temperature T. , then t heir relic abundance would be pxo = mxnx(T.) [so/s(T.)], and therefore that nx(T.)/H . ~ (Dxo/ D,o)ToMpT./ mx . Thus, taking into account the
436
Gravitational lensing and dark matter
unitary bound (CJv ) < m;/, we obtain that r x/ H . < (nxo/ n,o) ToT. Mp/m~. T non-equilibrium condition then becomes mx ( 200TeV
)3> ( 200TeV T. )
.
(7.19
:s
m To sum up, a non-relativistic particle of mass mx ;::: 200 TeV, created at T. with an abundance small enough so that nxo < 1, actually has such a small dens that it could not have been in equilibrium. Various scenarios of WIMPZILLAs product ion have been proposed [86,87] amo which one relies on the amplification of perturbations of a non-minimally coupled sca field during the reheating phase. Just as the inflaton fluctuations (see Chapter 8), t scalar field X(x, TJ) can be expanded in modes as
(7.20 The evolution equation of the mode hk is then
(7.20 The initial conditions are set in the inflationary era by assuming that
At the end of inflation, at TJ = TJI , the field contains modes of positive and negat frequencies,
Thus, at the end of inflation, particles have b een produced. T heir density is related the value of the Bogoliubov coefficient, 13k, by
(7.20
Particles with masses comparable to the Hubble constant at the end of inflatio rv HI , can be created with a density PX(TJI) = mxnx(TJI)' Figure 7.22 describ the time evolution of the Bogoliubov coefficient and the relic density obtained a function of mx / HI· Various scenarios can be constructed depending on the w inflation ends.
mx
7.2.5.8
Other candidates
There is a long list of other candidates [73- 76] t hat we will not detail here. We simp mention the following possibilities.
Evidence for the existence of dark m atter
437
10-2
10- 5 10-6 L......~L........~~-'---'-~~~~~---'-~---.J
-10
10
Fig. 7.22 (left) : Conformal time evolution of the Bogoliubov coefficient for various wave numbers. flI corresponds to the end of inflation. (right) : Contribution to rlx oh 2 of gravitationally produced WIMPZILLAs as a function of the ratio between its mass, mx , and the Hubble constant at the end of inflation , HI. Trh is the reheating temperature and the shaded region corresponds to parameters for which thermalization can occur. Inflation is assumed to be smoothly followed either by a m atter-dominated era (solid line) or a radiation era (dashed line). The dotted line shows a thermal density with the temperature T = HI I27r . From Ref. [86J.
- Light scalar particle: the Lee-Weinberg bound [85] can be bypassed for nonfermionic dark matter. A model with a scalar particle having a mass in the range 1-100 MeV has recently been proposed [88]. Although this candidate is relatively ad hoc from a particle-physics point of view, it is motivated by the fact that it may offer an explanation of the "(- ray emission at 511 keY observed in the bulge of our Galaxy. This emission would be due to the annihilation of these light scalar particles into positrons, which would then produce the recently observed ,,(-rays when annihilating. - Model of the 'little Higgs ': the 'little Higgs ' model [89] is an alternative model to super symmetry in which the Higgs from the standard model is a pseudo-Goldstone boson, the mass of which is approximatively protected by a global symmetry (see Chapters 2 and 9). This makes it possible to stabilize the electroweak scale at around 10 Te V. Various scenarios of this model contain scalar particles that could be good dark-matter candidates. In one of t he versions [90], a new symmetry is introduced at the Te V scale and implies the existence of WIMPs with a mass of a few TeV. - Kaluza- Klein modes: in theories with higher dimensions compactified a la KaluzaKlein (see Chapter 13) where all the fields can propagate along the extra dimensions (these models are called universal extra dimensions), the Kaluza- Klein modes of the standard model fields can act as dark matter. The lightest KaluzaKlein particle [9 1] can be stable if the extra dimension is compactified on an orbifold, such as §l/Z.
438
Gravitational lensing and dark matter
- Vortons: vortons are superconducting cosmic string loops stabilised by t heir c rent [92]. They are formed during a phase transition associated with spontane symmetry breaking. As for monopoles, they have the tendency to dominate v rapidly the matter content of the Universe, which is known as the vorton exc problem. Light vortons , however , were also suggested as candidates of ultrahi energy cosmic rays. 7.2.5.9
Direct searches
From a phenomenological point of view, WIMPs are characterized by their mass the cross-section of their annihilation and interaction with the nucleons (see Fig. 7.2 These parameters make it possible to compute the relic density of the candidate that they are constrained by cosmological observations (see Fig. 7.24 left).
-
Neutrino v
-5
Neutrinox ~
-10
-WIMP
- 15
bl~
-20
~
.....:l
~'I( i on
~'I(ino
a
a
-25
-30 Gravitino G
-35 -40 -15 -12 - 9
fa -6
-3
0 Log
0
6
9
12
15
18
m .<
1 GeV
Fig. 7.23 Different d ark-matter candidat es in para meter space (cross-section , mass ). F Ref. [94]
Many experiments [93] have attempted to directly det ect the elastic scatter of WIMPs on nucleons. The principle of the experiment is based on the fact t the WIMP scattering induces a recoil of the nucleon that can be det ected by th methods. (1) If the detector is a semiconductor (for instance, germanium) , the re can release free charges through ionization (electron- hole pairs). These charges then be collected by an electric field applied at t he boundary of the cryst al. (2) So crystals and liquids release flashes of light when the nucleus slows down. This is
Evidence for the existence of dark matter
439
scintillation phenomena. The quantity of light produced depends on the recoil of t he nucleus. The materials used for these detectors are mostly sodium iodide, NaI , and liquid xenon. (3) In crystals, the recoil energy is transformed into vibrations of the crystal lattice (phonons) which can be detected, at very low t emperatures , through the increase of temperature that they induce. Bolometers made of sapphire, Te02 and germanium, LiF have, for instance, been used. One of the main difficulties of these methods lies in the existence of a natural radiation that is more intense than the source of WIMPs we are looking for (cosmic r ays or natural radioactivity), which generates an intensive flux of parasitic events. The incident particles are then mainly photons, electrons and neutrons. To prevent these noises, the experiments are installed in deep underground sites. Two parameters determine the efficiency of such a method , the incident WIMPs flux and the WIMP-nucleon cross-section. In a given model , this cross-section can be computed and the incident flux is constrained by astrophysical data. In the most optimistic models, the expected rate of events is of the order of 10- 7 p er day per kilogram . We can thus put constraints on these two parameters (see Fig. 7.24). Moreover, when the model is well specified (for instance, in the minimally supersymmetric model) one can put constraints directly on the parameters of this model (in this case on j.i , (3, Nh and Nh). These constraints are then compared with accelerator experiments limits. 105 .-~~~-.~mmr-~~~-.~
10 4 103 10 2 10 N
~1
c:
10- 1 10- 2 10-3
10-4 10-45 L-_ _ __ -' 10 J
10-5 L-~~~~~~L-~~~~~
10
Neutrino mass (GeV) Fig. 7.24 (left): Experimental limits obtained by the CDMS (black) and Edelweiss (grey) experiments in the (cross-section, mass)- plane. The shaded region corresponds to the predictions from the constrained MSSM and the light region to that from the so-called 'non-universal Higgs mass' model. From [95]. (right): Predictions of the LSP relic abundance as a function of the mass and the cosmological constraints. For each mass many relic densities are possible as a function of the value of the other parameters of the model. From R ef. [75].
440
Gravitationa l lensing and dark matter
7. 2. 5.1 0
Gravitational alternatives
All of the conclusions we have drawn on the existence of dark matter rely on validity of general relativity to describe gravity, and in fact mostly on the validit Newton's law. The behaviour of the galaxy rotation curves can, however , be explai without requiring any dark matter if the gravitational potential behaves as
(7.2
at large distances, assuming 1"0 to be a characteristic scale of the order of a few k In this regime, the velocity of a t est body in Keplerian orbit tends t o a const v 2 = GNM / 1"o. So, if the modification of gravity occurs at a fixed distance sc we find that L ex M ex v 2 , which is in contradiction with the Tully- Fischer relat T his conclusion is generic to any modification associated with a characteristic dista scale. With the definition (7.133) , we can rewrite the gravitational potential as
(7.2
where ao is a constant with dimensions of acceleration. We see that the transi between the Newtonian and modified Newtonian regimes has to occur at a scal order r rv VG NNI / ao , which depends on the mass of the galaxy.
7.2.5.11
MOND: formulation
As illustrat ed in Fig. 7.25, there is no correlation between the ratio M / L and the of the galaxy. The Newt onian ratio M / L is obtained by computing the dynamical m from the rotation velocity, v 2 1"/G N, obtained from the farthest point to t he centr the galaxy. However , t here seems to be a correlation with the centripetal accelera a = v 2 /r, M / L ex l / a for a < 1O -8 cm · S-2 . This is only a rephrasing of the Tu Fischer relat ion. Milgrom [96] noticed that a way to recover this relation was to mo t he Newtonian theory of gravity in the small acceleration regime, thus its name MO (modified Newtonian dynamics) . MOND can be formulated either as a modification of Newton 's second law motion , or as a modification of gr avity. In the first version , the relation between fo and acceler ation, F = ma, is modified to
ma~ (:0)
(7.2
= F,
where the constant ao has to be of the order ao rv 1Ofunction ~ should satisfy t he asymptotic conditions
x«
1,
x» l
lO m /s 2
and where the fit
(7.2
In the strong acceleration regimes, Newton's second law of motion is indeed recove In the second formulation , we assume tha t the gravitational acceleration , g, is rela to the Newtonian gravitational acceleration, g N, by
Evidence for t he ex istence of dark matter I
•
1.0
I
• • • • I • • • • • • • • •• • • •
,-...
t:3
$
'0.{)
0
•
..
••••
0.5
....:l
0.0
1.0 - .
•
••
•.~ 0.5 -
1 1.5 Log [r(kpc) ]
0.0 -
-1
-
e.
-
• • • • • • •• •
I
0.5
I
..... . ••
• •
0.5
441
-
I
o
-0.5
Fig. 7.25 Newtonian mass to luminosity ratio for spiral galaxies of Ursa Major as a function of the size of the galaxy (left) and as a function of the Newtonian acceleration (right). From Ref. [97].
(7.207) This corresponds to modifying the Poisson equation according to (7.208) Even though there are differences in principle and in practice between these two formulations, they lead to the same predictions in the small-acceleration regime. In particular , the effective gravitational field becomes 9 = vgNaO and the Keplerian velocity of a test body orbiting around a body of mass M is (7.209) The constant ao must be the same for all galaxies. It is associated with a characteristic scale eo = c2 / ao rv 9 X 10 26 m that , surprisingly, is of the same order of magnitude as c/Ho. The transition scale between both kinds of regimes depends on the mass of the galaxy and is typically given by
e. = Jeo~~M 7.2.5.12
= 10
20
C01~M(J 1/2 CO_1~~.s_2 r1/2 m.
(7.210)
Some predictions
In the MOND context , the existence of flat rotation curves has an absolute character. In particular, this hypot hesis can be disproved by the observation of isolated galaxy rotation curves t hat would decrease in a Keplerian way at large distances , thus merely indicating the end of the dark-matter halo. This model also has a number of predictions that we briefly summarize (see Ref. [97] for the current observational status) .
442
Gravitational lensing and dark matter
- The Tully- Fisher relation L ex v~ is absolute, having an exponent 0' = 4.0, a gives a relation between the baryonic mass and the asymptotic velocity of t galaxy, namely
lnL = 4lnvoo - In (GNao \
~b))
,
(7.21
in agreement with the observed slope, 0' = 3.9 ± 0.2. - One can define a critical surface density I;m = ao/G N delimiting between tw regimes. For I; > I;m (such as for galaxies with high surface density) , there little difference between the dynamical and visible mass , i.e. the galaxy does n seem to include any dark matter. For I; < I;m (such as for galaxies with a lo surface density), the acceleration is weak and we are in the MOND regime. F these galaxies, there must be a large difference between the dynamical and visib mass; it seems to be dominated by dark matter. This is the case observational - I;m/27f ~ 140M8 /pc 2 must be the limiting value of the galaxy surface densi supported by the rotation. This observed limit appears naturally in the MON context. - The rotation curve of a galaxy can be obtained uniquely from the knowledge the baryonic matter distribution.
These four predictions have been verified observationally and in particular , all of t galaxy rotation curves seem to be reproducible with a unique parameter, the r a M b / L between the baryonic and luminous mass (which is in fact almost a constant Concerning lensing, it is clear from the form (7.204) of the potential that , at d tances larger t han r ~ VGNM/aO, the deflection angle is
that is exactly the same as in the case of a dark-matter halo when gravity is assum to be described by general relativity.
7.2.5.13
Some problem s
Because of the simplicity of its formulation and of its predictions, the MOND mod remains a seductive alternative to the existence of dark matter. Nevertheless, seve pro blems are present .
- From a theoretical point of view, there are no satisfactory field theories leading MOND in t he weak-field limit, despite a recent proposal of TeVeS theories [98]. particular such a theory seems to require an acceleration ao and requires us to a reference frame. The existence of such a preferred frame is very constrained tests of gravity in the Solar System. See also Ref. [99] for an in depth discussi of the difficulties to construct field t heory versions of MOND.
- Galaxy clusters have a surface density that is important enough to be in t Newtonian regime for which MOND does not differ from Newtonian theory. B strong gravitational lensing indicates the need for dark matter in this regime.
Evidence for the existence of dark matter
443
- The required value of ao needed to explain observations of rich clusters is twice as big as that required to explain the galaxy rotation curves. It seems that the dynamics of elliptical galaxies requires ao ~ 10- 9 m . S- 1. Table 7.6 summarizes the comparative predictions from the MOND and SCDM models on a rating scale from 0 to 3. This summary suggests that the construction of a field theory capable of including MOND represents an important challenge. Moreover, construction of tests of the theory of gravity at cosmological scales is also a priority in order to be sure of the conclusions on the properties of dark matter. Table 7 .6 Summary of comparative predictions from MOND and SCDM models. Crosses represent difficulties and question marks are questions that cannot be rigorously addressed in the current framework. Test I predictions Rotation curves of spira l galaxies Rotat ion curves fr om v isible matter Properties of elliptical galaxies
I?
Thlly- Fischer relatio n
7.2.5.14
MOND
3 3 3
SCDM 1 x x x
Temperature profile of clust ers
x
1
Cosmic microwave backgro und
?
Large-scale structures a nd nucleosynthesis Correct light d efl ection
? x?
3 3
Satisfy ing theoretical framework
x
2 2
Dark matter vs. modified gen eral relativity
The former discussion concerning MOND relies on a Newtonian version of the law of gravity. Thus , a complete theory leading to MOND should arise from a modification of general relativityB since the la tter weak field limit is indeed Newton gravity. Various proposals have been investigated in the literature and we refer the reader to Ref. [99] for an extensive review. In general relativity, the metric propagates as a pure, massless spin-2 field and matter is universally coupled to this metric. To extend general relativity, one must introduce new degrees of freedom into the theory, besides the graviton and the various matter fields entering the standard model of particle physics. This already renders the distinction with dark-matter models actually rather subtle. The principal difference is that in the dark-matter model the amount of dark matter is imposed by initial conditions and its clustering generates gravitational wells in which baryonic matter fall to form large-scale structures. In a modified general relativity model, the baryonic matter generates by itself an effective dark-matter halo , scaling as MDM C( jMbaryon ' Such a halo may thus merely be an artefact of the way 8We do not use the term ' m odified gravity' since we define gravity as the only long-range for ce that cannot be screen ed. What could be modified is only the theory that describ es gravity, which we assumed to be general relativity.
444
Gravitational lensing and dark matter
we interpret the gravitational field of baryonic matter alone at large distances. it may also be a real dark-matter halo , made of the new (gravitational) fields t generates itself a Newtonian potential. Whatever the class of models, the theory has to involve new fields, let us call th generically 'IjJ . T here is a stress-energy tensor associated to these fields and it will en the 'Einstein ' equations. We have various possibilities:
-
I
I
I TS~) » ITs~at) and
'IjJ is minimally coupled to the metric , in which case
have a dark-matter model. In the Newtonian regime, it is equivalent to add a new component of matter and to assume the validity of the Poisson equat sourced by this new component .
-
I
I,
ITS~) « I Ts~at) in which case the contribution of the new field to the ene
density is negligible. This new field must then be responsible for a new interact We can say that we have a modified general relativity model. In t he Newton regime, it modifies the Poisson equation and assumes that the energy-densit that of the baryons.
-
I
I
ITS~) » I TS~at) and the new field is responsible for a new interaction. It is t difficult to decide whether we have a dark-matter model or a modified gen relativity model.
Whatever the class of models, it is important to realize that we have to add n degrees of freedom in the currently accepted theories of physics. This ought to b e most fundamental conclusion that can be drawn from observational lensing.
References [1] A.O . PETERS, H. LEVINE and J. WAMBSG ANSS, Singularity theory and gravitationallensing, Birkhauser , Boston , 1991. [2] P. SCHNEID ER, J. EHLERS and E.E. FALCO, Gravitational lenses, Springer , Berlin, 1992. [3] J. WAMBSG ANSS , 'Gravitational lensing in Astronomy', Living R eviews R ei. 1 , 1, 1998. [4] R. NARAYAN and M. B ARTELMANN, 'Gravitational lensing', in Form ation ofstructure in the Universe, A. Dekel and J . Ostriker (eds. ), Cambridge University Press, 1999, pp. 360. [5] W.L. B URK E, 'Multiple gravitational imaging by distributed masses', A strophys. J. 244, Ll , 1981. [6] R.D . BLANDFORD et ai. , ' Gravitational lens opt ics', Na ture 245 , 824, 1989. [7] B. PACZYNSKI, 'Gravitational microlensing in the local group ', Ann. R ev. Astron. Astrophys. 34, 419, 1996. [8] E. RO ULET and S. MOLLERACH, 'Microlensing', Phys. R ep. 279 , 67, 1997. [9] B. FORT and Y. M ELLIER, 'Arc(let) s in clusters of galaxies' , A stron. A strophys. Review 5 , 239 , 1994. [10] F. COURBIN, 'Quasar lensing', Lect. No tes Phys. 608 , 1, 2002. [11] M. BARTELMANN and P . SCHNEIDER, 'Weak gravita tional lensing', Phys. R ep. 340 , 291 , 200l. [12J Y. MELLIER, 'Probing the Universe wit h weak lensing' , Annu. R ev. A stron . A strophys. 37, 127, 1999. [13J C. ALCOCK et al., 'Possible gravitational micro lensing of a star in t he large Magellanic cloud ', Nature 365 , 621 , 1993. [14J E. AUBO URG et ai. , 'E vidence for gravitational microlensing by dark obj ect in the galactic halo', Nature 365 , 623, 1993. [15] A. MILSZTAJN, 'The galactic ha lo from microlensing', Space Sci. R ev. 100, 103, 2002. [16J D. WALSH, RF. CARSWELL and RJ. W EYMAN N, ' 0957+ 561 A , B: twin quasistellar objects or gravitational lens?', Nature 279, 384, 1979. [17] M.J. IRWIN et ai. , 'Photometric variations in the Q2237+ 035 syst em : first detection of a microlensing event ', A strophys. J. 98 , 1989, 1989. [18J C.S. KOCHANEK, 'The analysis of gravitational lens survey. II. Maximum likelihood models and singular p ot entials', A strophys. J. 419, 12, 1993. [19J S. MAO and C. K. KOCHANEK, 'Limits on galaxy evolution' , Month Not. R. A stron. So c. 268 , 569 , 1994. [20J R NARAYAN and S. WHITE, 'Gravitational lensing in cold dark-matter Universe', Month Not. R. A stron. Soc. 231 , 97, 1988 .
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8 Inflation Inflation is a postulated period of accelerated expansion occurring before the radiation era of the standard Big-Bang model. It was initially proposed as a tentative solution of the problems of the hot Big-Bang model (Chapter 4). In particular, it provides a simple explanation for the homogeneity and flatness of our Universe. Before the first models of inflation were proposed, precursor works appeared as early as 1965 by Gliner [1], who postulated a phase of exponential expansion. In 1978, Englert, Brout and Gunzig [2], in an attempt to resolve the primordial singularity problem and to introduce the particles and the entropy contained in the Universe, proposed a ‘fireball’ hypothesis, whereby the Universe itself would appear through a quantum effect in a state of negative pressure subject to a phase of exponential expansion. Starobinsky [3], in 1979, used quantum-gravity ideas to formulate the first semirealistic rigorous model of an inflation era, although he did not aim to solve the cosmological problems. A simpler model, with transparent physical motivations, was then proposed by Guth [4] in 1981; this model, now called ‘old inflation’, was the first to use inflation as a means of solving cosmological problems. It was soon followed by Linde’s ‘new inflation’ proposal [5]. The formulation by Mukhanov and Chibisov of the theory of cosmological perturbations during new inflation, which links the origin of the large-scale structures of the Universe to quantum fluctuations, quickly followed [6, 7]. All these important works were achieved in the early 1980’s, which can thus be considered as the ‘date of birth’ of inflation The first models of inflation were actually only incomplete modifications of the BigBang theory as they assumed that the Universe was in a state of thermal equilibrium, homogeneous on large enough scales before the period of inflation. This problem was resolved by Linde with the proposition of chaotic inflation [8]. In this model, inflation can start from a Planckian density even if the Universe is not in equilibrium. A new picture of the Universe then appears. The homogeneity and isotropy of our observab