Cosmology (Springer Tracts in Modern Physics)

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Author: Dierck-Ekkehard Liebscher

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Springer Tracts in Modern Physics Volume 210 Managing Editor: G. Höhler, Karlsruhe Editors: J. Kühn, Karlsruhe Th. Müller, Karlsruhe A. Ruckenstein, New Jersey F. Steiner, Ulm J. Trümper, Garching P. Wölfle, Karlsruhe

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Dierck-Ekkehard Liebscher

Cosmology With 100 Figures

ABC

Dierck-Ekkehard Liebscher Astrophysikalisches Institut Potsdam An der Sternwarte 16 14482 Potsdam, Germany E-mail: [email protected]

Library of Congress Control Number: 2005921920 Physics and Astronomy Classification Scheme (PACS): 98.80.-k, 04.20.Cv, 01.55.+b ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN -10 3-540-23261-3 Springer Berlin Heidelberg New York ISBN -13 978-3-540-23261-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in Germany

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Preface

This book is intended to explain the theoretical issues of cosmology. To some extent, it is a complement to presentations that emphasise the progress and success of our observational power and the importance of determining phenomenological parameters. Since Einstein solved Newton’s paradox through use of the theory of general relativity, the universe has been described in that framework. The present standard model is the result of many observations, which culminated in the celebrated determination of the structure in the microwave background. This book will acknowledge this development, of course, but it will emphasise the theoretical aspects of cosmology that can be easily forgotten in view of such a success. Just as stars and black holes have been found through mere solutions to theory (i.e. without observation), cosmology too is a scientiﬁc topic, and was so even before anyone knew what to observe. The central issue of cosmology is the compatibility of diﬀerent areas of physics. This compatibility ought not be bounded or restricted by horizons of any kind. In contrast, the realm of experience is always limited by the observational or experimental means currently available or possible in principle. The compatibility, or universality, of physics is nonetheless indispensable for our trust in physics itself. Cosmology is an exciting science, even though and indeed because neither its subject nor its epistemological status can be described in a way that is generally accepted. On the one hand, cosmology is the science of global extrapolation; on the other, it is understood as the science of the gigantic history of the evolving universe. This evolution is due to the overall expansion, which pushes many systems away from equilibrium even though the expansion itself is adiabatic. Modern cosmology is closely connected with the physics of elementary particles and their interactions. The energy scale of elementary particles, atoms and molecules is reﬂected in a timescale in the history of the universe. The expansion of the universe must have begun from a time when it was so hot, that any value of energy that we might consider at present (from the nuclear binding energy to the grand uniﬁcation scale) has existed as a thermal energy in the distant past. In particular, the processes that are believed to occur at the energies of grand uniﬁcation will only have taken place in a past

VI

Preface

so distant that their consequences will, at best, be observed as relics of that stage. Modern cosmology is closely connected with the theory of general relativity, and the fundamental question of its relation to quantum theory is reﬂected in the question of the origin of our universe. Cosmological models reveal the abstract features of the singularities of general relativity and all their consequences. General relativity has to understand other exciting objects, too, but the universe is the simplest object that can be consistently described only by curved space-time, which is the essential invention of general relativity. Nevertheless, the notions of cosmology can be explained without going too deeply into general relativity or quantum theory, because the simplicity of the overall motion allows us to reduce most of the arguments to phenomenological thermodynamics, and because the general-relativistic equations for the expansion of the universe can be reduced in the case of our cosmological models to equations that are formally equivalent to Newtonian ones. This book will introduce the reader to cosmology not only with general arguments, but also with the key formulae. These formulae will be given with their derivations. The reader not only should get an overall impression, but also should be enabled to read more advanced literature without too much trouble to put everything into place. The network of basic arguments and basic facts will be presented in a formal context, too. In this way, the book will be not only an introduction, but also a kind of revision course. Modern cosmology rests upon the continuing expansion of the universe, which was identiﬁed ﬁrst by Hubble through an appropriate interpretation of the redshift that increased with the apparent distance. Friedmann had already found the correct geometrical form of the cosmic expansion as a solution of the ﬁeld equations of Einstein’s theory of general relativity. The expansion triggers the decay of various kinds of equilibria. The slow formation of structure by gravitational condensation and the fast nuclear binding reactions can be understood as the result of such decays. Two basic observations support the picture of the Friedmann universe. First, we observe a comparatively large concentration of deuterium, which can only be explained by some suddenly halted nucleosynthesis in a universal neutron-rich environment. The result of primordial nucleosynthesis is explained by reaction kinetics that describe processes that start after cooling and are halted after rarefaction, both of which result from the expansion. Second, the heat bath of the universe, now decoupled and cooled by the expansion, forms the microwave background. This background is astonishingly isotropic. The density perturbations that are reﬂected in its ﬂuctuations are very weak and have a remarkably featureless power spectrum. After 1980, a loop of argument was found that makes the arguments about the evolution of the universe very strong (Fig. 1). When applied to baryon– antibaryon annihilation in a charge-symmetric universe, the nucleosynthesis paradigm leads to a present baryon concentration that is 10 orders of

Preface

Inflation

Monopole problem

Symmetry breakdown

Seed suppression

Baryon asymmetry

Structure evolution

Gamow’s conjecture

Microwave background

Friedmann expansion

VII

Helium and deuterium

Hubble expansion

Fig. 1. The cosmic loop

magnitude lower than the observed value. An initial baryon asymmetry must be assumed, and grand uniﬁed theories provide an explanation of how such an asymmetry can result from asymmetric decays of GUT particles. In grand uniﬁed theories, symmetries may be spontaneously broken. This is ﬁne for the baryon asymmetry, but is bad for the present density of the universe, which would exceed the observed value by far since the breakdown of symmetry leaves too many topological quasi-particles (monopoles) behind. How do we get rid of these monopoles? We need a dilution in a phase of exponential increase of the size of the universe. By coincidence or not, the grand uniﬁed theories contain the concept of a high-temperature vacuum that drives such an inﬂation when the temperature falls below the temperature of symmetry breakdown. The universe cools down to nearly absolute zero before the hightemperature vacuum decays into a low-temperature vacuum, releasing its energy density into the degrees of freedom of ordinary GUT particles. Now, the zero-point ﬂuctuations that remain lose their coherence and turn into ordinary perturbations, which show correctly the power spectrum inferred from observations. Grand uniﬁed theories also keep ready exotic particles

VIII

Preface

that may be candidates for the dark matter that is suspected to exist in galaxies, clusters and large-scale structures. The loop is closed by the phenomena in the microwave background and the evolution of structure in the transparent universe. This book will follow this cosmic loop. After a short introduction to the geometry of space-time in general and the geometry of an expanding universe in particular, we explain the nuclear-synthesis paradigm, follow the argument about inﬂation, describe the inﬂationary scenario, consider the origin of perturbations and follow their evolution to the recombination time and after. This closes the loop. Finally, we return to the basic questions of consistency and the cosmological singularity, i.e. we consider the basic notions of quantum gravity and other extensions of the standard picture.

Potsdam February 2005

Dierck-Ekkehard Liebscher

Contents

1

Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Cosmos, the Universe and the Metagalaxy . . . . . . . . . . . . . 1.2 Explanation and Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Cosmological Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Observation and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Newton’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Olbers’ Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Lambert’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Einstein’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Friedmann’s Solution and the Hubble Expansion . . . . . . . . . . . 1.10 Gamow’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Jeans’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Lemaˆıtre’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 6 8 8 10 11 12 16 18 19 22

2

Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox . . . . . . . . 2.2 The Equivalence Principle, Deﬂection of Light and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 General Relativity: Solving Galileo’s Paradox . . . . . . . . . . . . . . 2.4 Positive Curvature: Solving Newton’s Paradox . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25

3

Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Friedmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Worlds of Constant Curvature . . . . . . . . . . . . . . . . . . . . . . . 3.3 Worlds with Matter Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Barotropic Components of Matter . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Friedmann’s Universe: Solving Olbers’ Paradox . . . . . . . . . . . . . 3.6 The Cosmological Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 58 65 67 69 74 77

4

Cosmometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.1 The Past Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

34 42 46 50

X

Contents

4.3 The Cosmological Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Distance Deﬁnitions in the Expanding Universe . . . . . . . . . . . . 4.5 Distance Determinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Determinations of the Expansion Rate . . . . . . . . . . . . . . . . . . . . 4.7 Curvature Determinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Gravitational Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Quasar Absorption Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 88 92 98 101 104 111 115

5

Matter and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Average Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Counting Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Microwave Background Radiation . . . . . . . . . . . . . . . . . . . . 5.4 The Photon Bath and the Notion of the Ether . . . . . . . . . . . . . 5.5 The Homogeneous Large-Scale Structure . . . . . . . . . . . . . . . . . . 5.6 Correlations and Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Fractal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 123 125 127 129 131 137 139

6

Standard Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Gamow Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Standard Process and the Decay of Equilibrium . . . . . . . . 6.3 The Primordial Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Weakly Interacting Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Problem of the Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 146 154 160 161 166

7

Inﬂation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Implications of the Monopole Problem . . . . . . . . . . . . . . . . 7.2 The Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Inﬂaton Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Homogeneous Inﬂation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Concomitant Solutions to Fundamental Problems . . . . . . . . . . . 7.6 Inhomogeneities and Inﬂation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 168 170 173 176 182 184 188

8

Structure Formation in the Opaque Universe . . . . . . . . . . . . . 8.1 Perturbations in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Relativistic Approach and the Evolution on Large Scales . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Inhomogeneities and Inﬂation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Evolution of Small Scales and the Microwave Background . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189 194 199 205 212

Contents

9

XI

Structure Formation in the Transparent Universe . . . . . . . . . 9.1 The Newtonian Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Decoupling of Condensations from the Cosmic Expansion . . . . 9.3 Jeans’ Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Peculiar and Bulk Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Models for Non-Linear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 The Nails of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 215 223 224 227 231 233 236 239 241

10 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Why Three Dimensions of Space? . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Why More Than Three Dimensions of Space? . . . . . . . . . . . . . . 10.3 Multidimensional Cosmological Models . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243 243 245 247 251

11 Topological Quasi-Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Topology of a Field Distribution . . . . . . . . . . . . . . . . . . . . . 11.2 Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Monopoles and Textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 253 255 256 260 262

12 Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Why Quantum Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Canonical Formulation of General Relativity Theory . . . . 12.3 The Wheeler–deWitt Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Minisuperspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 266 269 271 272

13 Machian Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Mach’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Machian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Time without Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Measure by Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 The Assumed and the Explained . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273 273 275 278 279 279 283

14 Anthropic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Cosmological Principle and Cosmological Result . . . . . . . . . . . . 14.2 Life as an Observational Datum . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 287 288 291

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Notation

a, b []

() {} a c d D D∈ d D ds2 dτ 2 dσ 2 dω 2 e ei E E Elm f2 F g g gik G h h ¯ h[z] H H0 I

expectation value scalar product list of variables, in particular list of variables indicating functional dependence. The variable of any function is set in these brackets in order to distinguish it from a factor. factors are always set in round brackets matrix of expressions expansion parameter velocity of light distance distance, dimension variance diﬀerential operator covariant diﬀerential operator line element of space-time element of porper time line element of space line element of the unit sphere electric charge electric charge energy expectation value Einstein curvature tensor degrees of freedom (Chap. 6) force multiplicity, number of degrees of freedom acceleration by gravitation general metric tensor Newton’s gravitational constant Planck’s constant, present Hubble constant in 100 km/(s Mpc) Planck’s constant h ¯ = h/(2π) reduced Hubble constant and its dependence on the redshift Hubble constant, dependent on the redshift present Hubble constant intensity (energy per unit time and unit area)

XIV

Iab I k k˜ k l L L LX ln lg m, M n o[x] O[x] On [x] p P P [k] q qi Q r r R RH Ri klm Rlm R s S t T Tik ui U v V w W x x z Z

Notation

tensor of inertia (Sect. 9.4) information (Sect. 5.7) Boltzmann’s constant, curvature index, wave number curvature index wave-number vector in space length, Compton length Lagrange function, luminosity (energy per unit time) Lagrange density Lie derivative (Chap. 2) natural logarithm decadic logarithm mass density of particles (negligibly) small with respect to x term of the order of x term of the order of xn momentum, pressure probability density power function deceleration parameter generalized coordinates (Sect. 9.4) eigenfunctions of the Laplacian radius, Gauss’s radial coordinate position in three dimensions, in particular physical position radius, baryon-to-photon relation (Sect. 8.4) Hubble radius Riemann curvature tensor Ricci curvature tensor radius entropy per unit comoving volume, entropy per baryon, number of photons per baryon action integral time, cosmological time temperature, time (Chap. 3), transfer function energy–momentum tensor four-velocity, special combination of perturbations circumference velocity volume, potential of the inﬂaton, perturbation of velocity relative pressure w = p/ε window function (Chap. 5) position, process variable (Chap. 6) position in three dimensions, in particular comoving position redshift special potential of perturbations

Notation

α Γ δ δ[x] ε iklm ζ η ηik Θ κ λ Λ µ ν ξ Π σ σab Σ τ φ Φ χ ψ Ψ ω Ω

XV

angle, Sommerfeld constant coeﬃcients of the linear connection (Chap. 2) operator which selects the perturbation of a quantity relative density enhancement energy density small number permutation symbol relative reciproqual of the expansion parameter (ζ = 1 + z), special combination of perturbations conformal time Minkowski metric relative perturbation of the temperature reduced curvature (3.20) reduced cosmological constant (3.20), wavelength cosmological constant mass of particles frequency, baryon-per-photon ratio (Chap. 6) correlation function (Chap. 5) perturbation of stress density of mass, radial coordinates cross-section tensor of shear surface brightness (energy per unit time, unit area, and steradian) proper time scalar ﬁeld (inﬂaton) potential comoving spatial distance scalar function, matter ﬁelds potential surface area, reduced radiation density (3.20) surface area, reduced matter density (3.20)

1 Basics

1.1 The Cosmos, the Universe and the Metagalaxy Telescopes of all kinds look deep into the sky and ﬁnd planets, stars, the Milky Way, other galaxies, clusters of galaxies, the large-scale structure in the distribution of galaxies and quasars and, ﬁnally, a distant region: the ﬁreball. Its radiation is that of a black body, with a temperature of 2.73 K now. The ﬁreball is apparently opaque; we cannot see behind it. With consideration and care, we call the space within this ﬁreball and the objects that we can identify with our telescopes the metagalaxy. Is this all? Is this the universe? The universe is deﬁned as the system (or object) that is not part of any metasystem. There may be many things outside or behind the metagalaxy, of course. There can be nothing outside, behind or in addition to the universe: the universe contains it all, by deﬁnition. Nobody observes the universe from the outside. The universe, is it a subject of physics? We describe and construct the natural and artiﬁcial objects of our experience by means of mathematical structures that may be called laws. The notion of a law may be a subject of misunderstanding if it is confused with the notion of civil law. Civil law is a matter of compromise in order to ensure more or less peaceful contact, tolerance and exchange between people, and is dependent on how the people who make the law live and think. A law that describes a mathematical structure by no means depends on us. We can detect it and describe it, but – in general – we have no choice. The structures that we detect, the laws, may be a viable approximation only in the circumstances where we ﬁnd them, may be valid in a larger realm or may be valid universally. These knowledge about these structures that we detect is meant to be used in the construction of technical devices, and in order to avoid disappointments and catastrophes we must be sure of their validity and applicability. We must know where the structures become invalid, where they have to be replaced by more precise ones and where they cannot guide us any more. When observational relief from this diﬃculty cannot be obtained, we have at least one condition: laws that we believe to be universal have to be consistent. That is, they ought not lead to internal contradictions. We see the theoretical task as being to explore the consistency of laws by trying to extrapolate them beyond any limit, beyond limit on size, velocity and energy. Dierck-Ekkehard Liebscher: Cosmology STMP 210, 1–23 (2004) c Springer-Verlag Berlin Heidelberg 2004

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Following our Greek ancestors in science, we call any such complete system of laws consistent beyond any limit a cosmos. It is an open question whether such a cosmos exists at all. If it exists, it is an open question whether it is unique. It is another question whether there exists something that follows these laws. The object of the cosmos of laws is the universe. The universe must contain all imaginable objects. If the cosmos is consistent and closed as required, the universe cannot have an outside (in particular, it cannot be one of many). Again, it is an open question whether the universe has to be subject to some cosmos of laws. In order to know whether a cosmos may be applied to the universe, we have to observe the universe. However, the largest subject of observation is the metagalaxy, the part of the universe inside some horizon. The universe becomes an object of physics if we assume that the metagalaxy is characteristic of all the universe, i.e. that the physical laws and the physical state of the metagalaxy represent the physical laws and the physical state of the other parts of the universe. In this sense, the universe has to be homogeneous in the large, with respect to the laws that govern it and with respect to the average state of matter as well. The metagalaxy can be observed and measured well with our earthly theories and measures. We observe and measure the universe if the metagalaxy characterises this universe. Homogeneity beyond the horizon may not be the case. However, we cannot prove or disprove it; we simply have to assume it. If we do not want to assume homogeneity in the laws and the state of the universe or when we are not able to do so, we have to abandon the goal of explaining the existence of the universe even partly. We would be limited to developing a picture of the metagalaxy as it is at present and to take it as a panorama of interesting and maybe instructive objects that does not admit the question of a synthetic understanding. Of course, we feel pushed to explain it, to ﬁnd a connection between the phenomena which we observe and to suppose the existence of the lawful order that our ancestors called the cosmos. The cosmos was the opposite of chaos – lawlessness.1 The belief in, and hope for, such an order leads us to assume the existence of a universe that can be validly observed in the metagalaxy. We recall that the possibility of deﬁning a science named cosmology as the consideration of the largest possible system is not bound to the existence of a universe or of a metagalaxy that represents it. Although a universe of the kind that we are referring to might not exist, cosmology would be the science of a cosmos in the ancient sense. Its subject would be the pure question of the global consistency of the known laws of physics. There is no way out of the task of testing the credibility of the laws and methods of prediction that we have found in objects of laboratory dimensions. Even in the case of the simple task of designing larger machinery or buildings, in order not to endanger people and the environment, we must extrapolate 1

In modern mathematics, the term ‘chaos’ has a diﬀerent meaning, which admits the formulation and testing of laws again.

1.2 Explanation and Evolution

3

and know how valid the extrapolation is beforehand because we can make measurements on a system of the true size only after construction. We must make sure that extrapolations do not reveal contradictions in our knowledge. It is cosmology that considers the ultimate extrapolation. When limits on a theory exist, it is here where they must be unveiled [11]. Contradictions in extrapolations to larger sizes, temperatures or densities do not necessarily reduce the applicability of a theory in a smaller domain. The contradiction may hint at the points where the theory has to be modiﬁed or made more precise. Once these limits and the circumstances where this is necessary are found, we may be sure about the reliability of the theory within these limits. The viability of Newton’s theory of gravitation was proven by the theory of relativity, because the velocity of light was identiﬁed as the limit that demands the ﬁrst changes. For small velocities, Newtonian mechanics has now been proven to be valid: the errors are of the order of the velocities measured in units of the velocity of light. From this point of view, cosmology is the task of ﬁnding global solutions and global models for the physics that we believe after tests on earth. The possibility that these eﬀorts might fail makes cosmology a positive science even without the hope that these models might describe a real universe or the metagalaxy. We have only to show that the metagalaxy and its phenomena ﬁnd their place in these models. Although the progress in observational facilities has been impressive [17], it not the main task of cosmology to ﬁnd the parameters of the metagalaxy. The possibility of cosmologically important observations is limited in many respects. Because of the known ﬁnite value of the velocity of light and because of the large distances in the universe, we have to accept that we know only about a thin slice of space and time. In space-time, this slice looks like the surface of a cone that opens into the past. On the one hand, it would not be allowed for us to take our observations as observations of the universe if we did not accept the assumption of homogeneity in laws and state. On the other hand, we can still try to ﬁnd out whether our observations ﬁt with our previous knowledge. We intend to combine physics and astronomy from a global point of view, to ask about the limits of the applicability of our knowledge in physics, and to look for hints indicating the points where our knowledge has to be modiﬁed or even newly created.

1.2 Explanation and Evolution In physics, we try to ﬁnd symmetries or structure in the form of laws in order to explain the observed state of an object in question. The laws will have many diﬀerent solutions. One solution to these laws should be represented by the object studied, of course. Can we trust the laws if this is the case? We become more assured if we ﬁnd other systems that show the same structure, i.e. that obey the same laws. In experimental physics, we can test our hypotheses by preparing many similar or even identical objects. In cosmology, the universe

4

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exists only once – by deﬁnition. We can no longer refer to similar but diﬀerent objects that can be suspected to be subject to the same laws. Of course, we have to consider the notion of structure as given in space and time. When we separate the time, the laws that correspond to the structure in space-time become equations of motion in space. The goal now changes into understanding the observed state as the result of the action of laws of motion. In order to ﬁnd laws of motion, the system must be in motion, must evolve. It is a pointless eﬀort to search for laws of motion of a system that is always at rest, a stationary system. Evolution is a prerequisite for explanation. Laws of motion explain the state at a given time as a consequence of the state at an earlier time. For an object in our laboratories which can be prepared many times for an experiment and can be prepared in diﬀerent ways, this is no problem, it is the solution. We can prepare diﬀerent initial states and follow their evolution in time. In the case of the universe, nothing can be prepared for an experiment, and the projection of the state that we observe now onto the states of an earlier time only shifts the problem of explanation to the past. In this case, the question of why we observe what we see now is transformed into the question of why the universe was in the state that led to the state that we observe. Instead, an explanation of the universe requires laws of evolution that provide for a more and more diﬀuse projection into the past, laws that make the recent state more and more independent of the state in the distant past. We hope for laws of evolution that generate unstable solutions when followed into the past. If most of the past states were to evolve into the state now observed, it would be the laws of evolution that explained the universe. However, this seems not to be the case. When we follow the evolution back into the past, some properties of the solution are stable. The best-known example is the existence of a characteristic singularity, the cosmological singularity. This is the expression of the attractive action of gravitation. In addition to the expectation of ﬁnding laws with solutions that are unstable in the past, the solutions should be unstable in the future, too. How, otherwise, could we expect to understand the condensation of matter in the form of galaxies, stars and planets? It is curious that, at ﬁrst sight, the instabilities of the solutions are too weak, also. As we shall see, an age of 15 billion years is rather short for the formation of large-scale structures in the expanding universe. The last problem in this sequence is the question of boundary conditions. In most cases, the solution of laws of motion requires the ﬁxing of a condition in extremely distant parts of space. Maybe we shall never be able to consider the boundary conditions properly because no information has reached us from so far away. We may, however, hope for a consistency test. This may involve boundary conditions as well as initial conditions. Up to now, we have considered only the explanation of observations and their consistency. The immediate applicability of cosmology to a real, ordinary prediction cannot be

1.3 The Cosmological Principle

5

tested as long as we understand the idea of prediction purely as an enlargement of the realm of experience in time. Time is only one of the dimensions of this realm. All of these questions will be postponed in order to make a start. We shall see that the conditions that are necessary for cosmology allow us to take this point of view.

1.3 The Cosmological Principle After all, we need only one principle. It is usually called the cosmological principle or Copernican principle, but it is more precise to call it the Cusanus principle [3, 23]. This principle states the equivalence of all points of space with respect to governing laws, and to the average state and average sky as well. We suppose the following: The observed part of the universe (the metagalaxy) represents the universe in its properties and structures. The universe is homogeneous in the large. Cusanus did not see any possibility of ﬁxing the earth as the centre of the universe, and Copernicus did not see any reason to to do it because the centre could be anywhere: Unde erit machina mundi quasi habens undique centrum et nullibi circumferentiam, quoniam eius circumferentiam et centrum esse deus, quid est undique et nullibi2 (Nicolai de Cusa, De docta ignorantia [3], paragraph 162). We proceed even further and add that there is no preferred direction either. The universe is isotropic in the large. There is no velocity of the universe. Of course, there might be velocities deﬁned with respect to the universe. The motion of the earth around the sun, the motion of the sun in our galaxy and the motion of the galaxy itself deﬁne small anisotropies that have to be subtracted appropriately in equally ﬁne observational data in order to judge the metagalaxy in the large. The Cusanus principle makes cosmology possible in the sense that we are allowed to test theoretical conclusions by observations in and of the metagalaxy (Fig. 1.1). In a certain sense, it allows us to discuss boundary conditions. Evidently, we not only ﬁnd the metagalaxy to be the image of the 2

‘The world has a centre everywhere and nowhere a border, because the centre and border are God himself, who is everywhere and nowhere’. The proverbs of many a language formulate this idea in earthly terms. In Sicily, it reads ‘Tuttu lu munnu `e comu casa nostra’.

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1 Basics

Cosmos

Metagalaxy

some system of globally consistent physical laws, that is assumed to exist

the region inside our horizon, i.e. the maximal region of observational experience

Universe the ideal object for a cosmos, also assumed to exist

Cusanus’ principle The metagalaxy is characteristic of the universe, in particular for the regions beyond the horizon

Fig. 1.1. The cosmological principle. A cosmos is a system of globally consistent physical laws that is assumed to exist. The universe is the ideal object (also assumed to exist) for a cosmos. Testing is possible in the metagalaxy only. This is the region inside our horizon. A test in the metagalaxy is a test of the universe only in the case where the metagalaxy is characteristic of all the universe, that is, when the universe is homogeneous at least on and beyond the scale of the horizon

universe, but also see that any volume with a border that expands with the universe becomes virtually an adiabatically closed system. Because of the homogeneity of the universe, no exchange across the border of the volume under consideration can aﬀect the thermodynamic balances. Every representative part of the universe can be envisaged as a thermodynamically isolated and homogeneous system. It can consist of diﬀerent components that are not necessarily in reciprocal thermodynamic equilibrium, but are mixed homogeneously in the large. The history of the universe turns out to be a sequence of reactions between these components that may be interpreted quasi-chemically. Owing to the enormous size of the metagalaxy, it is necessary anyway to think in terms of thermodynamics. The mass in the metagalaxy that is found by counting galaxies and estimating their mass through some mass–luminosity relation turns out to be that of about 1080 baryons. Any corresponding number of degrees of freedom can be managed only by selecting some of them and by combining the others by use of the methods of thermodynamics.

1.4 Observation and Measurement Homogeneity is only one side of the coin. When we intend to observe, there have to be objects to observe, i.e. objects that can be identiﬁed against their environment. We observe stars, galaxies and clusters of galaxies, to mention

1.4 Observation and Measurement

7

the simplest of these objects. In an ideally homogeneous ﬂuid, there are no observed quantities except for the thermodynamic intensive variables such as temperature, pressure and density. In short, observability requires inhomogeneity. Even homogeneity is observed against an inhomogeneous background and in contrast to it. Inhomogeneity subject to laws is structure. Again, the structure has to be in some sense homogeneous, that is, it needs a kind of explanation through underlying homogeneous statistics and a homogeneous law of its genesis. The Cusanus principle must be invoked both to describe the structure and to explain its evolution. Most of the cosmologically relevant observations share a peculiarity of all observations in astronomy: they have been performed over incredibly large distances. The analysis of the electromagnetic radiation that is collected in telescopes in various ranges of wavelength is the main source of the facts that are at our disposal. Again, it is the assumption of homogeneity that allows us to make cosmologically relevant measurements in the vicinity of our earth. An example of such measurements is the experiments that search for and identify unfamiliar, rare kinds of particles. Again, we have to refer to the cosmological principle when we intend to derive the physical state of a source of radiation from the spectrum of the latter. We have to suppose that the laws for the generation, absorption and dispersion of radiation very far away are just the same as in our laboratories. This argument immediately concerns the physics of the atom, quantum mechanics, which is at the heart of all microscopic analysis. If quantum mechanics with all its ingredients, especially the values of the constants that enter the calculations of spectra, is everywhere the same, these constants should not change with time. We shall consider this question later in more detail, but we note now that only dimensionless numbers can be questioned at all. Any quantity with dimensions can change numerically if the deﬁnition of its units changes. Only the relative change of the quantity in question with respect to the quantity deﬁning the units can have a physical meaning. Let us ask, for example, about the constancy of the velocity of light. Together with this question, we have have to consider the units of velocity. The latter is deﬁned by structural elements of certain atoms, which may themselves depend on the velocity of light. In the simplest case, we can deﬁne time by the frequency of some spectral line, in eﬀect by the Rydberg constant me4 α2 mc2 = α2 c/2lCompton . = Ry∞ = 2h3 2h If the unit of length is given by the Bohr radius rBohr =

h2 = lCompton /α , me2

the unit of velocity becomes v = 2rBohr Ry∞ = αc .

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The question of the invariability of the velocity of light turns into the question of the invariability of Sommerfeld’s constant α, which is a dimensionless number. Today, in the International System (Syst`eme International, SI), the velocity of light is deﬁned as a unit itself; the question now is only of the constancy of all the other velocity units which might be invented with respect to the speed of light. The time dependence of the fundamental constants can be estimated from various observations [21]. All arguments show that no change is found. To be more speciﬁc, any possible rate of change must be much smaller than the expansion rate of the universe. In particular, an analysis of the ﬁne structure of molecular lines in quasar spectra [1, 20] shows that 1 dα < 4 × 10−14 year−1 . α dt This bound is less than 10−4 of the characteristic cosmological scale, which will be explained in Sect. 1.8. The question of the variation of fundamental constants will be resumed later with the discussion of higher dimensions.

1.5 Newton’s Paradox The invariability of seasonal and planetary phenomena led to the impression of a stationary universe. The ﬁrst quest in cosmology was for a static or at least stationary universe. Newton, however, did not obtain any useful result. Newton’s theory of gravitation, which states that the force on an inﬁnitesimally small particle of mass m is (˜ x) ˜ , m¨ x = − grad Φ = −4πG grad d3 x ˜| |x−x R3

is incompatible with the cosmological principle: the potential diverges, and the force does not converge absolutely. No consistent model universe can be found if we do not accept that the universe is an isolated cloud in an otherwise empty space. In addition, an isolated cloud will not be stationary in time; it will expand or contract, or both, one after the other. This consideration does not eliminate a model of a homogeneous spherical cloud of particles that is forced to conserve homogeneity. In this case, it has only one degree of freedom (its radius), and its Newtonian equation of motion resembles the precise equation for a homogeneous unlimited universe to a certain degree (Chap. 3).

1.6 Olbers’ Paradox Even if the theory of gravitation theory were to allow it, an inﬁnite and homogeneous universe could not be stationary: the average surface brightness

1.6 Olbers’ Paradox

9

of the sky would diverge. We use the following argument. A star of radius 2 R and distance r covers about R /r2 of the sky. If we integrate over a presumably homogeneous and static universe, we obtain ∞ 4πr2 dr

Ω=

2 R →∞. 2 r

(1.1)

0

The stars cover the sky inﬁnitely often. Their surface brightness must be assumed to be independent of their distance, and the night sky will be inﬁnitely bright (Fig. 1.2). We might try to argue that most of the stars would shine behind other foreground stars. Nevertheless, the surface brightness of the sky should be equal to the surface brightness of the sun at least. This paradox carries the name of Olbers, although he had precursors both in the

Fig. 1.2. The fundamental paradoxes. We partition space into shells of equal thickness ∆ around our position. The nth shell contributes more than 4π(n − 1)2 ∆3 to the mass, more than (L/M )((n − 1)/n)2 ∆ to the intensity at the centre and more than 4π(n − 1)2 /n∆2 to the potential at the centre. All three quantities increase beyond any limit if n is large enough. A homogeneous (Cusanus) universe does not exist in Newton’s theory of gravitation

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question3 and in the answer: Jean-Philippe Loys de Cheseaux speculated about an interstellar absorption of the light of very distant stars, so that stars are not seen when they are too far away. This is the explanation of the darkness of the night sky adopted by Olbers, but it does not work. The reason is straightforward: no stationarity is possible when a non-equilibrium process, the emission of radiation by stars, is occurring uniformly. Any absorption of the radiation would heat the absorbers to a temperature at which they would radiate as brightly as the absorbed radiation. A ﬁnite lifetime for the stars does not help to solve the paradox, if we assume a homogeneous and constant formation rate. There has to be a history: that seems to be the lesson of Olbers’ paradox. Something must prevent us from seeing stars that are too distant. The integral (1.1) ought not to grow unacceptably.

1.7 Lambert’s Solution However, there is still one explanation diﬀerent from evolution. We can imagine a stationary, homogeneously structured universe where the average density is zero, and the average luminosity per unit volume is too. This is the case in the hierarchical models which were considered ﬁrst by Lambert [15, 22] and later by Charlier (Fig. 1.3). We simply assume a universe that consists of systems of diﬀerent order. A system of order n contains gn systems of order n − 1, where gn is the multiplicity, and occupies a volume of typically Rn3 , and hence the average distance between its parts is of the order of Rn . Now every integral over the universe may converge, as we shall show below. The density in the system of order n is bounded by n = gn gn−1 . . . g1

M , Rn3

the Newtonian potential is bounded by Φ=

∞

gk gk−1 . . . g1

k=1

GM , Rk

and the part of the sky covered is bounded by Ω=

∞ k=1

gk gk−1 . . . g1

R02 . Rk2

All of these sums converge if some exists such that 3

In 1570, Marcellus Palingenius speculated in his Zodiacus vitae about a wall of unseen light, and in 1720 Edmond Halley asked the question of the eﬀect of the existence of an inﬁnite number of stars.

1.8 Einstein’s Solution

11

Fig. 1.3. A simple hierarchical structure. As proposed by Lambert and constructed by Charlier, one could consider a fractal universe of dimension less than 3, which can lead to a ﬁnite gravitational potential and light intensity. However, its average density approaches zero when the volume increases without limit

Rn−1 Rn−1 ≤ < 1 and gn ≤<1. Rn Rn We obtain ∞

∞

k=1

k=1

GM R0 R0 n GM n < 2 → 0 , Φ < , Ω< k = k →0. M Rn R0 R0 1 − Rk The average density is zero in the limit of n → ∞. However, this scheme does not apply, because there is actually a uniform surface brightness of the night sky given by the microwave background. Beyond some hundreds of megaparsecs, we observe only homogeneity with a non-vanishing average density.

1.8 Einstein’s Solution In a space-time curved by gravitation, we might imagine a universe without a boundary, but ﬁnite – just as the surface of a sphere is ﬁnite but without a boundary. In such a universe, gravitational dynamics will be free from Newton’s paradox. Einstein [4] showed that a three-dimensional sphere of constant curvature and density could solve his equations for the gravitational ﬁeld if these equations contained a constant (called the cosmological constant)

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that represents some kind of a ground-state value of the scalar curvature.4 The radius appears to be related to the average mass density by the equation c2 R= . (1.2) 2πG In this model, the universe is closed and ﬁnite because of gravitation and not simply ad hoc by various periodicity conditions which lead to nonstandard global topologies (See Chap. 13). However, Olbers’ paradox still puzzles us: in such a universe, the inﬁnite number of stars is substituted by a ﬁnite number, but with an inﬁnite number of images. The ﬁniteness of the universe makes the thermodynamic argumentation even more important. We obtain the result that even a gravitationally consistent universe needs evolution and history to overcome Olbers’ paradox, and this history can only be one of expansion. Such an expansion was found as a solution of Einstein’s equations by Friedmann, and was observed by Hubble.

1.9 Friedmann’s Solution and the Hubble Expansion Einstein did not literally solve his equations for the universe, but only tested his static curved space to ﬁnd out whether it solved the equations. The model did so when the supplementary cosmological constant was included in the equations. Einstein did not consider any evolution. It was Friedmann who literally solved the equations [6, 7] and demonstrated the following results: – Positive curvature, i.e. ﬁniteness of the volume of space, is not necessary to solve Newton’s paradox. For instance, models with negative or zero curvature of space also solve the paradox. – Evolution, i.e. variation of the curvature of the universe with time is the rule, and a static universe the exception. – An evolving solution does not require any cosmological constant. In 1927 Hubble5 found a redshift in the radiation from distant galaxies (Fig. 1.4), which he compared with the apparent magnitude6 , and this 4

5

6

This vacuum value of the curvature is interpreted today as the vacuum contribution to the energy density and has been termed dark energy, although it is dark only in the sense of ‘obscure’. With respect to light, it is transparent, of course (see Sect. 7.2). The ﬁrst redshift was noticed by Slipher in 1912, and Wirtz found the ﬁrst velocity–distance relation in 1922. The apparent magnitude is deﬁned in astronomy by the logarithm of the intensity at the position of the observer, such that m = m0 + 2.5 log(f0 /f ). Here, the intensity f itself is used to represent the apparent brightness.

1.9 Friedmann’s Solution and the Hubble Expansion

13

Fig. 1.4. Sketch of some spectra. Slit spectra of small objects look like lines of variable density combined with slit images for the lines of the atmospheric background. The structures in the spectra show collective shifts due to the relative velocity of the source. In the light from galaxies, all kinds of spectra are mixed together. Only the Ca lines H and K survive, because of their universality. These lines show the radial velocity of galaxies. Quasars and Active Galactic Nuclei (AGN) in general show emission lines

redshift was interpreted as a recession velocity7 [13]. The recession turned out to be deﬁnitely faster for more distant galaxies, and he found a linear relation (Fig. 1.5): dr = H0 r , (1.3) dt 7

This interpretation supposes the context of Newtonian kinematics and Euclidean space. It must be modiﬁed for relativistic space-time and also for curved, expanding space. It is nonsense to suggest that galaxies distant enough recede with a velocity faster than light. The Doppler eﬀect is produced through an increase in distance, not through a literal velocity.

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Velocity (km/sec)

40000

30000

20000

10000

0 0

100

200

300 400 500 Distance (Mpc)

600

700

Fig. 1.5. The Hubble relation for SN Ia. This Hubble diagram is based upon distances to supernovae of type Ia. The linearity of the more distant entries is spectacular. The slope is the Hubble constant, H0 = 64 km s−1 Mpc−1 . Reproduced from [5] with kind permission from A. Riess (STScI) and Elsevier

where r is the distance from the galaxy in question, and H0 is the Hubble constant.8 First, we are interested in homogeneity: can we deﬁne a centre in an overall expansion by (1.3)? We cannot: when we change the basis galaxy in an universe expanding in accordance with (1.3), the expansion law remains invariant if the expansion is not accompanied by lateral motion. The form of the equation d (r i − r 0 ) = H0 (r i − r 0 ) dt remains unchanged when we choose another basis r 1 by subtraction of the equation corresponding to r 1 from all others (Fig. 1.6). By formally calculating back in time, we ﬁnd an instant where all distances between galaxies vanish.9 This instant is the ﬁrst sign of the cosmological singularity, and deﬁnes the formal age of the universe. Its order of magnitude is given by the Hubble age, which is simply the reciprocal of the Hubble constant: tHubble = 1/H0 . Diﬀerent models of the history of the expansion (cosmological models) give diﬀerent values for the age, which coincide with the value tHubble only in particular cases. Again, we have a possibility of checking consistency: The age of objects in the vicinity of the earth can also 8

9

H0 is independent of r, but not of time. We add the subscript 0 to remind ourselves that the present value of the factor is meant here. This does not imply that the universe was in one point. A point is not deﬁned by its metric relation to its neighbourhood, but by its topological properties, i.e. by its being distinguished from other points. The points of the universe can be distinguished even in the case of vanishing distance [12].

1.9 Friedmann’s Solution and the Hubble Expansion

15

Fig. 1.6. Homogeneous expansion. This ﬁgure shows the velocity vectors for the Hubble expansion in a plane. The reference point is O. The direction of the vector at each point is that from the origin. We now change the reference point from O to A. The lines AC and BD are parallel because DC/OC = AB/OA. We construct the parallelogram ABDE and see that the velocity diﬀerence CD − AB = CE lies in the direction of AC and that CE/AC = DC/OC is equal to the common constant H0

be determined. We can measure the age of meteorites and of rocks, and ﬁnd an age of about 5 billion years for the earth. The recession velocity is deﬁned by the redshift by analogy with the Doppler eﬀect, and to lowest order we obtain z=

v ∆λ = . λ c

The question of determination of the distance is a separate problem (Chap. 4). Hubble at ﬁrst found a factor H0 that was too large because he interpreted the brightest resolved objects in distant galaxies as the brightest stars. Today we know that these objects are clouds of ionised hydrogen (H II regions). When their brightness were compared with that of the brightest stars, the distances determined were too small. The formal age H0−1 that followed from the ﬁrst value estimated by Hubble (H0−1 ≈ 2 × 109 years) turned out to be too small with respect to the age of the earth (5 × 109 years) and the estimated age of some globular clusters (15 × 109 years). The determination of the present expansion rate remains a diﬃcult problem. The accepted value in 2002 was H0 ≈ 65 km/(s Mpc), H0−1 ≈ 15 × 109 years .

16

1 Basics

In a homogeneous expansion, all distances vary at the same rate. Hence all angles (and the appearance of the deep sky in particular) remain constant. The homogeneous Hubble expansion does not alter our view of the deep sky. It changes the colour and the apparent magnitude, but not the apparent position of a galaxy. This can be used to introduce comoving coordinates that remain constant during the expansion, just as the spherical coordinates of a point on a balloon do not vary when it is blown up.10 Comoving coordinates are spanned by the ﬂuid particles themselves. The deﬁning particles do not change position in comoving coordinates. Any motion that is deﬁned with respect to this coordinate system is called a peculiar motion. The expansion can be described by the expansion parameter a[t], which depends on time, but not on position. Its rate of change is the Hubble constant.

1.10 Gamow’s Conjecture Gamow made the most important conjecture of cosmology by seriously extrapolating the expansion into the past [8, 9, 10]. The argument is not complicated. Let us take the average diameter of a galaxy to be 10 kpc, and the average distance between galaxies to be 10 Mpc. At some time, which we call recombination time, the distances in the universe were only one-thousandth of the distances today: the galaxies were virtually all in contact, and we may assume that all the matter components were ionised and in thermal equilibrium, and that the universe was opaque. Hence, the radiation should have had a temperature of about 5000 K. If the universe had been transparent after that time, that is, if the radiation was decoupled in eﬀect from the bulk of the matter, it should have cooled down, by Wien’s law (such that aT is constant) to about 5 K, still homogeneously ﬁlling the universe. This was Gamow’s claim. Twenty-ﬁve years later, the radiation was found by Penzias and Wilson [19] with an actual temperature of 2.7 K. The maximum intensity at this temperature is found at a wavelength λ ≈ 1 mm. The discovery of this background radiation shows two things. First, the reasoning connected with Olbers’ paradox is justiﬁed, and there has been evolution since those times. Second, the ionisation energy of hydrogen marks a speciﬁc time in the past at which the processes occurring were decisive for the universe. This is the ﬁrst example of what we must expect if we suppose that the expansion occurred in the even more distant past: The characteristic energies of atomic and elementary-particle physics mark characteristic instants in the past of the universe that correspond to those energies. This is the cosmological content of Wien’s law: in any expansion law, we may substitute the expansion parameter a by a ﬁxed multiple of h ¯ c/kT to obtain 10

If position is deﬁned with respect to the raisins in a cake, the raisins do not move when the cake rises.

1.10 Gamow’s Conjecture

17

¯hc t = t[a] = t a . kT We obtain a mapping of the temperature, or energy, scale onto the time scale. The ionisation temperature of hydrogen is of basic importance here because it marks the time of the transition from the opaque to the transparent state of the universe. Gamow’s conjecture reaches a still more distant past where the temperature is of the order of 1010 K. At the corresponding time, no atomic nuclei can exist, in particular no deuterium. Around this time, in a universe that subsequently cools down, all atomic nuclei had to be synthesised, in particular the large fractions of He (approximately 25% in mass) and D (approximately 10−3 %) that are found in basically every object in the universe. Both elements are synthesised in stars too, but the result of stellar nuclear burning is far too ineﬃcient to explain the universal concentration of He and D. The He content of stars and galaxies seems to be nearly independent of their age. Taking into account the fact that D is produced very slowly in stellar interiors and used up very rapidly in subsequent processes, the observed concentration of D in hydrogen clouds is very large, not small (Fig. 1.7). In stellar nuclear burning,

intensity erg cm in

s

nm

30 interstellar D

H

20

10

wavelength in nanometer

121,4

121,5

121,6

121,7

Fig. 1.7. Primordial deuterium in interstellar clouds. In the light of the Lyman-α emission line of the star Capella we see the Lyman-α absorption line of an interstellar hydrogen cloud. This cloud is so cold and the wings of the absorption line are so steep that the line of the cosmological deuterium can be seen as a weak companion. The spectrum was recorded by the Goddard High Resolution Spectrograph. Reproduced from [18] with kind permission of J.L. Linsky and the AAS

18

1 Basics

deuterium is an intermediate product that is processed fast. It cannot be convected to the surface or blown away by stellar winds in the observed amount. The observed deuterium must be explained as a relic of a global nuclear synthesis in a neutron-rich environment that was halted (through dilution by cosmic expansion) before it was completed. This primordial synthesis will be the subject of Chap. 6. Gamow’s conjecture of was nicknamed the ‘Big Bang’ by Hoyle. This nickname has become a label and today refers to either the whole scenario or the cosmological singularity. The microwave background is a weak remainder of the heat bath that governed the expansion of the early (opaque) universe.

1.11 Jeans’ Problem At any given point in a precisely homogeneous distribution, only the intensities can be measured (i.e. pressure, radiation density, temperature and concentrations), and no observation of distant points is possible. We are happy that the universe is extremely inhomogeneous, ﬁlled with identiﬁable objects that admit observation, even the observation of their homogeneous distribution. Objects and structures are necessary for observation, and homogeneity must be understood as a homogeneous structure of inhomogeneities or, at least, homogeneous laws that generate the structures and inhomogeneities. In particular, we interpret the distribution of galaxies as a realization of a statistical law, i.e. a stochastic variable that itself has some probability distribution. This is the homogeneity that is supposed by the Cusanus principle. The concentration of matter in galaxies and large-scale structures shows the extreme inhomogeneity of the distribution of matter on sub-horizon scales. The requirements of cosmology let us suppose that this is due to an evolution, a concentration process that was driven by gravitation and began from small ﬂuctuations of a nearly homogeneous distribution of matter. The background radiation should show these ﬂuctuations to some extent. Such ﬂuctuations in temperature have been found with the COBE satellite. Their relative amplitude is approximately 10−5 . Because of the smallness of this value, the evolution of these ﬂuctuations can be followed by linear calculations. After they grow to an order of magnitude of about 10−1 , non-linear growth can be expected and should lead to the recent state that is observed. In fact, for an isolated system, the equations for the evolution of unstable perturbations show an exponential growth with a fast transition to a non-linear stage. However, the universe expands, and the average density decreases in proportion to a−3 . The solutions of evolution equations do not show an exponential growth any more. The amplitudes grow, in the linear approximation, nearly in proportion to a only. The time interval from the time the universe became transparent to the present day is characterised by an expansion factor of a ≈ 103 . This interval appears to be much too short to allow ﬂuctuations

1.12 Lemaˆıtre’s Problem

19

of less than 10−4 to grow to a non-linear stage. Nevertheless, we observe condensations with a non-linear density contrast at high redshifts (z > 6). In addition, the giant elliptical galaxies should have an age that admits a long history of events in which smaller constituents merged. Therefore, the time left for the ampliﬁcation of small ﬂuctuations is too short. For the simplest model of the history of expansion there is no simple model that applies to the evolution of galaxies and of large-scale structure. A successful numerical simulation must suppose the existence of some matter component that does not interact with electromagnetic ﬁelds (otherwise it would show its existence in the microwave background explicitly), does not take part in the strong interaction (otherwise it would alter the structure of atomic nuclei), and interacts mainly by its gravitational action, that is, by an acceleration of the evolution of ﬂuctuations (Chap. 8). This matter was christened dark matter, although it is not dark like dust but transparent.11 Cosmological models with a much larger age of the universe (Chap. 3) do not help, because of other drawbacks. The smallness of the ﬂuctuations of the microwave background poses an inverse problem, too. Most of the simpler estimations of the genesis of ﬂuctuations yield amplitudes that were much too large in the distant past and never develop to the small values observed in the microwave background.

1.12 Lemaˆıtre’s Problem The gravitational interaction is attractive only. It follows that in all simple12 models of the history of the expansion there was a singularity, the cosmological singularity, a zero of the expansion parameter a. In Einstein’s theory of gravitation (the theory of general relativity), there exist exact theorems which conﬁrm this expectation. Lemaˆıtre christened the singularity the Uratom. What interpretation can be given to such a singularity? The ﬁrst, simplest answer is that the universe had a beginning, not necessarily in time but with time. If the intention of cosmology of ﬁnding an explanation for the observed state of the universe in its evolution and not in its initial conditions is successfully achieved, we cannot tell anything observable about this beginning: it is not essential. The second answer is that the cosmological singularity shows the limitations of the physical laws whose extrapolation leads to this singularity. Most proposals of this kind consider the question of whether an appropriately constructed quantum theory of gravitation can avoid singular phenomena. The answer could be that the singularity indicates a transition from a Euclidean 11

12

Nevertheless, we shall use the familiar term dark matter and its abbreviation DM later. Simple means here models with conservative matter components that show positive or zero pressure only.

20

1 Basics

(and therefore timeless) state of the universe to a pseudo-Euclidean state with time and expansion, through some kind of tunnel eﬀect. The beginning of our universe is then interpreted then as a transition. This picture turns out to be a variant of the ﬁrst answer (Fig. 1.8).

Fig. 1.8. A metaphor for the split of time. The part of the universe that evolves in time may be complemented by and connected to a part where no time and no evolution in time is deﬁned. This connection could be constructed in diﬀerent ways

The third answer is that the singularity indicates that our universe is part of some meta-universe, and its beginning is a kind of process that isolates it from this meta-universe. This would be an answer in the realm of classical physics, but it prevents the applicability of the cosmological principle and destroys the connection between cosmology and observation that was described in Sect. 1.1. It is remarkable that the commonly accepted picture of a universe with a very early stage of exponential expansion (inﬂation, Chap. 7) contains aspects of all three interpretations. First, this stage wipes out all traces of the state before (even if it does not change the existence of a singularity). Second, the quantum construction of a tunnelling process from a timeless state to the expanding state of the universe seems to imply subsequent inﬂationary stages. Third, these inﬂationary stages can lead to a universe that is no longer homogeneous on the scale of the metagalaxy, and reaches homogeneity far

1.12 Lemaˆıtre’s Problem

21

Exotic relic particles

Dark matter

Quantum cosmology Symmetry breakdown

Ination Monopole problem

Baryon genesis

Seed suppression

Jeans’s problem

Evolution before recombination

Evolution after recombination

Microwave background

Horizon problem

Gamow’s solution

Friedmann expansion

Homogeneous structure

Hubble expansion

Einstein’s GRT

Inhomogeneity

Olbers’ paradox

Newton’s paradox

Observability

Cusanus’ principle

Helium and deuterium

Computability

Fig. 1.9. Guiding map through cosmology

beyond any horizon at best [2, 16]. We shall not consider the developments related to supersymmetry [14]. Where to start? First, we take the Cusanus principle for granted. We neglect the question of why we may calculate back, although an explanation of the state today requires the instability of such a calculation, and the question of why the universe is simple enough to allow few-parameter models. We postpone the question of how to avoid or at least interpret the cosmological

22

1 Basics

singularity, and the question of why the universe is curved in the way observed. Second, we consider the abundances of helium and deuterium to be of primordial origin. This enables us to determine the expansion rate at the nucleosynthesis time from the abundance of helium, and to determine the nucleon density at this time from the abundance of deuterium. We postpone the question of why we observe nucleons at all, i.e. why the universe shows a nucleon–antinucleon asymmetry, the question of the origin and interpretation of the large numbers connected with the exceedingly large diﬀerences in the values of the various coupling constants, and the question of why the universe contains just the matter components it does. Third, we assume the large-scale structure in the universe to be generated by gravitational instability. We postpone the question of where the inhomogeneities come from (we may handle them like other matter components, of course). The various paths to follow the arguments are summarised in Fig. 1.9.

References 1. Bahcall, J. N., Steinhardt, C. L., Schlegel, D.: Does the ﬁne-structure constant vary with cosmological epoch? Astrophys. J. 600 (2004), 520–543. 8 2. B¨ orner, G., Ehlers, J.: Was there a big bang? Astron. Astrophys. 204 (1988), 1–2. 21 3. Nicolaus de Cusa: De docta ignorantia, liber secundus, P. Wilpert (ed.), Akademie-Verlag, Berlin (1440). 5 4. Einstein, A.: Kosmologische Betrachtungen zur allgemeinen Relativit¨ atstheorie, SBer. Preuss. Akad. Wiss. (1917), 142–152. 11 5. Filipenko, A. V., Riess, A. G.: Results from the High-z Supernova Search Team, Phys. Rep. 307 (1998), 31–44. 14 ¨ 6. Friedmann, A. A.: Uber die Kr¨ ummung des Raums, Z. Phys. 10 (1922), 377– 386. 12 ¨ 7. Friedmann, A. A.: Uber die M¨ oglichkeit einer Welt mit konstanter negativer Kr¨ ummung des Raums, Z. Phys. 12 (1924), 326–332. 12 8. Gamow, G.: Expanding universe and the origin of elements, Phys. Rev. 70 (1946), 572. 16 9. Alpher, R., Bethe, H., Gamow, G.: The origin of chemical elements, Phys. Rev. 74 (1948), 505. 16 10. Gamow, G.: The creation of the universe,, Viking Press, New York (1952). 16 11. Goenner, H.: Was f¨ ur eine Wissenschaft ist die Kosmologie? Proceedings of the 3rd FEST workshop on physics and philosophy, Heidelberg (1990). 3 12. Hawking, S. W., Ellis, G. F. R.: The Large-Scale Structure of Space-Time, Cambridge University Press (1973). 14 13. Hubble, E.: A relation between distance and radial velocity among extragalactic nebulae, Proc. Natl. Acad. Sci. USA 15 (1929), 168–173. 13 14. Kallosh, R., Prokushkin, S.: SuperCosmology, SU-ITP-04/09, hep-th/0403060 (2004). 21

References

23

15. Lambert, J. H.: Cosmologische Briefe, in: F. L¨ owenhaupt (ed.): Johann Heinrich Lambert. Leistung und Leben Braun, M¨ ulhausen 1943 (1761). 10 16. Levy-Leblond, J. M.: Did the big bang begin? Amer. J. Phys. 58 (1990), 156– 159. 21 17. Liddle, A. R.: How many cosmological parameters? Mon. Not. R. Astron. Soc. 351 (2004), L49 (astro-ph/0401198). 3 18. Linsky, J. L., Brown, A., Gayley, K., Diplas, A., Savage, B. D., Ayres, T. R., Landsman, W., Shore, S. N., Heap, S. R.: Goddard high-resolution spectrograph observations of the local interstellar medium and the deuterium hydrogen ratio along the line of sight toward Capella, Astrophys. J. 402 (1993), 694–709. 17 19. Penzias, A. A., Wilson, R. W.: A measurement of excess antenna temperature at 4080 Mc/s, Astrophys. J. 142 (1965), 419–421. 16 20. Potekhin, A. Y., Varshalovich, D. A.: Non-variability of the ﬁne-structure constant over cosmological time-scales, Astron. Astrophys. Suppl. Ser. 104 (1994), 89–98. 8 21. Sisterna, P., Vucetich, H.: Time variation of fundamental constants: bounds from geophysical and astronomical data, Phys. Rev. D 41 (1990), 1034–1046. 8 22. Treder, H.-J.: Elementare Kosmologie, Akademie-Verlag, Berlin (1975). 10 23. Wahsner, R.: Mensch und Kosmos, Akademie-Verlag, Berlin (1984). 5

2 Relativity

2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox Astronomical observations bridge distances so large that the time taken by the light in passing from its source to the observer exceeds by far the historical time span of observation itself. In addition, nearly all observations concern light. This situation requires a consistent theory of motion for speeds of the order of magnitude of the speed of light, and that theory is the theory of (special) relativity (special relativity theory, SRT). Why is Newton’s mechanics insuﬃcient in this respect? It takes for granted what everyday experience seems to tell us about time, that simultaneity is independent of any observational or other condition. This allows us to state a law for gravitation that requires us to specify the distance between the positions of two bodies at a given time. It follows that velocities have to be combined by addition and subtraction. The velocities of everyday life show no deviation form this rule, but light is something else. The speed of light does not vary through combination with other velocities. This fact generates the relativity of simultaneity: a statement of the simultaneity of two events depends on the motion of the observer whenever the events have a distance between them in space. Newton’s laws deﬁne mass and force and their relationship in certain systems of reference, which are called inertial. First, a linear system of reference is virtually deﬁned by the free motion of point particles. Newton’s ﬁrst law states that the world lines of free motion constitute a set of straight lines. Hence we can ﬁnd and use a reference system in which free motions yield linear relations between the coordinates of the world lines. Further, Newton’s third law (in Huygens’ form) states that there exists a kind of an integrating factor for the velocities that yields a conservation law for the sum of the velocities when they are weighted with that factor. This factor is the (inertial) mass of the bodies in question. The velocities weighted by mass are called momenta. Both the sum of the momenta and the sum of the masses are conserved. This introduction of the notion of mass gives a scale to linear reference frames and reduces them to inertial frames. Deviations from linear world lines require explanation, i.e. deﬁne force. Owing to the third Dierck-Ekkehard Liebscher: Cosmology STMP 210, 25–51 (2004) c Springer-Verlag Berlin Heidelberg 2004

26

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law, forces are proportional to changes of momentum, and they are subject to restrictive relations such as the principle of action and reaction. It is non-uniform motion that requires explanation. Uniform motion is relative: it needs external reference objects to be detectable at all. An inertial system of reference cannot be deﬁned as moving without reference to some other object or inertial system. Because of the fact that acceleration indicates force, i.e. physical interaction, no physical interaction that makes a referencefree measurement of a velocity possible exists in Newtonian mechanics. Velocity measurements have to refer to other objects. This is the content of Galileo’s principle of relativity. In Newton’s mechanics, simultaneity is assumed to be obviously absolute, and this is equivalent to the assumption that inertial masses do not depend on velocity, and that velocities obey a simple additive composition law: if the velocity vAC of an object C relative to another object A is made up of the velocity vAB of B relative to A and the velocity vBC of C relative to B, we have (2.1) vAC = vAB + vBC . We shall call this formula the Galilean composition law.1 One of the ﬁrst successes of this theoretical scheme was the detection of the aberration of starlight (Fig. 2.1). Light, in particular stellar light, had been understood as an emanation of particles that move with some huge velocity. When observed from the earth, this velocity had to be combined with the velocity of the earth in its orbit around the sun, with the result that all apparent positions seem to be shifted towards the direction of motion (which changes during the year). The observation of aberration by Bradley 1727 ﬁtted this expectation. It conﬁrmed the ﬁnite velocity of light, the motion of the earth around the sun and the additive composition law of velocities, i.e. the absoluteness of time. 1

The addition theorem follows from the Galilean transformations between inertial systems. The two-dimensional form of these transformations is given by the special Galilean transformations t∗ = t + t0 , x∗ = x − vt + x0 . These transformations form a group, speciﬁcally the group of motions in the space-time plane. This group is the product of the reﬂections in straight lines such as x = vt + x0 . The form of such a reﬂection S is described by tS = t , xS = 2(vt + x0 ) − x . In turn, the Galilean transformations are determined by the invariance of the conservation of momentum when the masses do not vary with velocity.

2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox

27

Fig. 2.1. Aberration of particles. The world line of a photon is projected parallel to the time axis onto space. If the time axis is tilted (the observer moves with respect to the initial system), its direction is rotated away from the original direction

When Young and Fresnel considered the wave theory of light in order to explain interference phenomena, they could easily explain refraction and reﬂection by the Huygens procedure that we know from school. However, they found the paradox that wave crests do not show any aberration at all (Fig. 2.2). Today, we know that the absoluteness of time is the ‘bug’. Fresnel could not know this, and he avoided the paradox by mentioning that any observation of starlight is made by telescopes that cut oﬀ only a small part of the wave ﬁeld. This part behaves like an ordinary particle and is subject to the addition of velocities if the waves are carried by some medium that sweeps through the telescope without any resistance or drag. This hypothetical medium was called the ether, and its curious drag-free sweeping through matter vexed physicists all through the nineteenth century. Today, we can measure wavefronts themselves (very long-baseline interferometry and active optics do this with success), and Fresnel’s solution would not help. In fact, the observed aberration requires us to abandon absolute time. Michelson’s famous experiment showed explicitly that Fresnel’s explanation does not work, i.e. it showed that the ether is dragged along with the motion of the earth just as everybody expected at the beginning of the 19th century.

28

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Fig. 2.2. Fresnel’s paradox: the aberration of wavefronts. The world sheet of a wavefront cuts the plane t = const. The normal of the intersection does not depend on the tilting of the time axis, but it does depend on the deﬁnition of simultaneity. We have to accept the relativity of simultaneity if the aberrations of photon world lines and wavefronts are identical

Fresnel’s paradox was only a forerunner of the diﬃculty of formulating a theory of the electromagnetic ﬁeld in a Newtonian context. Finally, this diﬃculty led to an experimental question about the combination of the velocity of light with other velocities, and with the orbital velocity of the earth in particular. Michelson’s experiment was interpreted now as showing that the velocity of light does not obey the Galileo composition law. Instead, the combination of the velocity of light with any other velocity turns out always to produce the velocity of light.2 We might call this the light-velocity paradox. 2

Michelson’s experiment, in its original interpretation, was not very important for Einstein, who often told that he was inspired by the question of the behaviour of light reﬂected from fast-moving mirrors. Einstein turned the argument into a rigorous statement about the ‘addition’ of velocities. The limitations of Michelson’s experiment (which measures only the harmonic mean of the velocities back and forth) were overcome by Isaak, who set up an experiment comparing unidirectional velocities of light that reached a sensitivity down to 1 cm/s for Fresnel’s ether drift. At present, the SRT is supported most strongly by its successful application to complex problems such as the physics of elementary particles. A typical success was the prediction of the existence of antiparticles that transfer, in reactions with their partner particles, all of their mass into the mass of radiation.

2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox

29

In Newton’s theory it is a paradox, but now it becomes the key principle: when we develop a consistent theory on the basis of the invariance of the speed of light when it is combined with other velocities (of mirrors, observers or reference frames), we have to construct a kinematics and mechanics in such a form that this invariance applies there too. The theory of (special) relativity is the result of pursuing this aim. It turns out that the apparent ﬂow of time is no longer independent of the choice of the inertial system, that inertial masses depend on their velocity with respect to the observer and that this dependence is governed by the mass of the kinetic energy. In general, all forms of energy contribute to the inertial mass via the famous formula E = mc2 .

(2.2)

The photon too has a mass proportional to its energy. This mass allows us, for instance, to derive the radiation pressure formula from the kinetics of photons.3 The theory of relativity declares the conservation of energy to be a consequence of the conservation of mass. We deﬁne a space-time or world as a four-dimensional product of space and time. A point in the space-time is called an event, and a curve a world line. The history of a point particle is represented by such a world line. The special theory of relativity implies that horizontal lines in the spacetime plane are no longer invariant under reﬂection. Instead, the manifold of light-like lines (which describe the world lines of light signals) is invariant under reﬂection (Fig. 2.3). The world lines of light remain world lines of light after reﬂection. The line A1 A is reﬂected in A1 S[A], and AA2 in S[A]A2 . The image of the intersection point is the intersection of the images of the lines. Hence the event S[A] is the reﬂection image of A. The light-ray quadrilateral deﬁnes orthogonality: the line connecting A and its image S[A] is deﬁned to be orthogonal to the world line of the mirror.

Reﬂections now show the relativity of simultaneity (the events A and S[A], which have to be simultaneous with respect to the mirror g, happen at diﬀerent times in our plane, in which the mirror moves). For the momentum analysis of a symmetric decay, we obtain the dependence of mass on velocity with a formula that – with Young’s deﬁnition of energy – immediately leads to 1 1 dE = F , dx = p, dp = m dm c2 , m m where F is the force, dx the path increment and dp the momentum increment (Fig. 2.4). 3

One can invert the argument and state that when energy and mass are conserved separately, and when one type of objects exists for which energy and mass are proportional (classical electrodynamics states that light is such an object), then any energy must be proportional to a mass with the same factor.

30

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Fig. 2.3. Physical reﬂection determines geometrical reﬂection. The basic rule requires that the set of world lines of light remains invariant after reﬂection. An event A and a world line s deﬁne a ‘light-ray quadrilateral’. Its fourth point is the reﬂection image S[A], and its second diagonal is deﬁned to be orthogonal to the mirror line s. This deﬁnition of reﬂection deﬁnes the geometry of space-time

Fig. 2.4. The variation of mass with velocity. We assume the symmetric decay of some body into two equal parts. We choose the motion of the decaying object in such a way that one fragment remains at rest (momentum OA). The momentum vector (OB) of the other fragment must be symmetric with respect to the total momentum (parallel to the world line OC of the decaying object). The mass of the moving fragment is larger than that of the fragment at rest

2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox

31

Fig. 2.5. Pythagoras’ theorem in space-time. Orthogonality is deﬁned by reﬂection maps. In Fig. 2.3, the line connecting A and S[A] is orthogonal to g. Euclidean squares now correspond to diamonds with diagonals that are world lines of light signals (light-like lines). The angle at C is a now right angle. We obtain ABBC AC + CAAB CB = ABBC AC + CAAC PC = QA QB BQ + QC QA QC = BCCA BA , that is, c2 = a2 − b2 The construction can be evaluated to yield (2.5) as follows. m[v] OA + HC − GD v HC + GD = , = . m[0] OA c OA + HC − GD Since O, C and D are collinear, GD HC = . OA − GD OA + HC This yields m2 [v] m2 [0]

1−

v2 c2

= 1 , m2 [v]c2 = m2 [0]c2 + p2 .

The new prescription for reﬂection leads to a pseudo-Euclidean measure in the plane (Fig. 2.5, [6]). Reﬂections generate the (special) Lorentz transformations ct − (v/c)x x − (v/c)ct ct∗ = + ct∗0 , x∗ = + x∗0 . 2 2 1 − v /c 1 − v 2 /c2

(2.3)

32

2 Relativity

Velocities turn out to be the hyperbolic tangents of some angle that is the additive parameter of the group of these transformations. Therefore, the Einstein composition (or addition) theorem of velocities has the form of the addition theorem for the hyperbolic tangent:4 vAC =

vAB + vBC . 1 + vAB vBC /c2

The transformation of time reveals the relativity of simultaneity. It is this relativity that is the crucial point in all of the popular misunderstandings of the geometry of space-time. The basic geometric quantity is the line element of space-time. We write this quantity as a quadratic form and use Einstein’s summation convention:5 ds2 = c2 dt2 − dx2 − dy 2 − dz 2 = ηik dxi dxk . The invariance of the line element is the direct expression of the invariance of the wave equation 2 ∂ ∂2 ∂2 ∂2 ∂2 − − − Φ = source . η ik i k Φ = ∂x ∂x c2 ∂t2 ∂x2 ∂y 2 ∂z 2 The matrix ηik and its inverse η ik determine the metric properties of spacetime. The Lorentz transformations mediate between systems of coordinates; here, the matrix ηik has the numerically ﬁxed form   1 0 0 0      0 −1 0 0 . ηik = 0 0 −1 0      0 0 0 −1 The lines ds2 = 0 are the bicharacteristics of the wave operator. They model the rays of the wave ﬁeld, in particular light rays. The line element characterises space-time as a pseudo-Euclidean space called Minkowski space or the Minkowski world. In a physical interpretation, the line element must be identical to the element of proper time: c2 dτ 2 = ds2 . The length of a world line with positive line elements (called a time-like line) is the integral of the time steps that that would be measured in locally comoving inertial frames, or by a clock that is insensitive to acceleration (pendulum clocks are not advisable). The line element is the pseudo-Euclidean expression for the Pythagoras theorem derived from the invariance of the light cone [6]. 4

5

The three-dimensional space of velocities has a negative curvature and is isometric to the intersections t = const of the Milne model (Chap. 3). We sum, without explicit mention, over indices that appear twice in a product. x0 stands for ct.

2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox

33

A vector ui is called time-like when its norm ηik ui uk is positive. In this case, we can ﬁnd an inertial system in which the space components of the vector vanish. The sign of the time coordinate of a time-like vector can be changed only by reﬂections of time itself. The tangent vector to a world line of a slow motion is time-like. A time-like world line has time-like tangent vectors only. A timelike straight line is the longest connection (measured in proper time) between two events. Any other time-like connection will be shorter in proper time.6 A vector is called space-like when its norm is negative. In this case, we can always ﬁnd an inertial reference frame for which the time component of the vector vanishes. A vector is called light-like when its norm vanishes. Hypersurfaces on which the solutions of the wave equation and/or their derivatives can change discontinuously must have light-like normal vectors.

From a technical point of view, all of the theory of relativity is obtained from the requirement that the invariance group of the wave operator (or the inhomogeneous wave equation) is universal, i.e. has to apply for any phenomenon of physics. In constructing and reconstructing physical laws, we must use only quantities that are invariant with respect to the (inhomogeneous) Lorentz group, and covariant quantities (vectors, tensors and spinors) in space-time that yield invariants through appropriate simple multiplication. The vector of the four-velocity of a massive point is tangential to a world line and therefore normalised: ui =

dxi dxi =c , ηik ui uk = c2 . dτ ds

The vector of the four-momentum is proportional to the four-velocity. The factor of proportionality is an invariant mass, the rest mass m0 . The fourmomentum has an invariant norm, too: pi = m0 ui , ηik pi pk = m20 c2 .

(2.4)

In a Newtonian interpretation, the four-velocity and the four-momentum are given by ui =

1 1 − v 2 /c2

[c, v 1 , v 2 , v 3 ] , pi =

m0 1 − v 2 /c2

[c, v 1 , v 2 , v 3 ] .

The inertial mass of Newtonian mechanics (i.e. the weighting factor of the ordinary velocity in the conservation law of total momentum) turns out to vary with velocity: 2 v m0 1 2 m= = m0 + 2 m 0 v + o 2 . (2.5) 2 2 2c c 1 − v /c 6

This fact is used to formulate the twin paradox, which is paradoxical only for our Galilean experience.

34

2 Relativity

The variation arises through a contribution to the inertial mass that has to be attributed to the kinetic energy. One can show that all the energy contributes to the inertial mass in a way that corresponds to the famous formula (2.2). The photon too has a mass proportional to its energy, which allows the derivation of radiation pressure through photon kinetics. The diﬀerences between Newtonian mechanics and Einstein’s mechanics are summarised in Fig. 2.6. The theory of relativity identiﬁes the conservation of total mass with the conservation of total energy. The total energy itself turns out to be the time component of the four-momentum. The sum of the rest masses is not conserved. Any inelastic process turns kinetic energy into internal energy that contributes to the rest mass. In Newtonian mechanics, the time component of the momentum is the mass, too. The total mass is conserved; however, the sum of kinetic energies is not (as in the SRT). The conservation of energy requires additional consideration here. Inelastic collisions and scattering shift the balance between external kinetic energy and internal (heat) energy. In terms of the SRT, this is a shift of the boundary between kinetic energy and rest energy, and it changes the contribution of the rest mass to the total inertial mass. We know from quantum theory that the internal energy (now proportional to the rest mass) of an object cannot change continuously. The value of the rest mass becomes a property that characterises the object or the state of the object. Any external force that does not change this state may change the four-momentum, but not its norm (i.e. the rest mass; see (2.4)). Such a force is a space-like vector. Any object that has a positive rest mass m0 > 0 in its ground state can have only a positive rest mass in its excited states. Therefore, no force can change a time-like momentum into a light-like or space-like vector if the object retains its identity. No object that can be studied at rest can be accelerated to the velocity of light or beyond.

2.2 The Equivalence Principle, Deﬂection of Light and Geometry Galileo stated clearly the paradox that all bodies fall with the same acceleration if other forces such as air resistance are removed. The inertial mass is strictly proportional to the gravitational charge. Direct experiments [1, 2, 9] and lunar laser ranging [8, 10] conﬁrm this statement down to a precision of 10−14 . This fact – called the equivalence of inertial and gravitational mass – allows us to represent motion in a pure gravitational ﬁeld as uniform motion in a curved four-dimensional union of space and time (Fig. 2.7). Now light is also inﬂuenced by gravitation. Light is deﬂected by gravitating objects from the path expected for ﬂat space-time. For instance, the sun attracts light passing by and produces an eﬀect whereby the apparent pattern of stars

2.2 The Equivalence Principle, Deﬂection of Light and Geometry

NEWTONIAN MECHANICS Absolute simultaneity Additive combination of velocities; the velocity of light depends on the system of reference

Masses constant, i.e. independent of of velocity Wave equation not invariant, Fresnel’s ether concept

EINSTEIN’S MECHANICS Invariance of the wave equation Velocity of light absolute, independent of the system of reference.

Masses depend on velocity m = Ec−2

Reﬂection symmetry of the light cone

Reﬂection symmetry of the mass shell

Non-additive combination of velocities

E = c m20 c2 + p2 Dispersion relation h−2 + c2 k 2 ω 2 = m20 c4 ¯

Relativity of simultaneity

Fig. 2.6. Newtonian versus Einstein’s mechanics

35

36

2 Relativity

Fig. 2.7. Uniform acceleration. The set of world lines of uniformly and equally accelerated particles is a set of straight lines. The parabolas in the ﬁgure form such a set: two parabolas intersect once at most, and two events determine a parabola uniquely. When we refer the distances to another parabola (a ‘freely falling observer’, dashed–dotted line), we obtain ordinary linear relations. If we shift F to F ∗ and E to E ∗ , the apparently curvilinear triangle ABC is shifted into the ordinary triangle A∗ B ∗ C ∗

behind the sun is spread out. This deﬂection of light was observed in 1919 and initiated the popularity of Einstein and his theory of relativity. The reason is that the light, because of its energy, carries inertial mass. Through the equivalence of inertial and gravitational mass, it must be inﬂuenced by gravitational ﬁelds. Galaxies, clusters of galaxies and black holes can act as gravitational lenses (Sect. 4.8). The important point is that a straight line can be physically deﬁned best by a light ray – and, technically, a straight line is always deﬁned this way. Hence we are forced to accept an eﬀective curvature of space-time that is immediately derived from a wave equation with coeﬃcients that vary with position and time. Any wave equation7 g ik [x] 7

∂2 Φ = source ∂xi ∂xk

Diﬀerent metric tensors are anisotropic with respect to each other. Relativity is based on the isotropy of light propagation with respect to the geometry of mechanics, so the metric tensor of the wave equation for light must coincide with that of mechanics. Universal Lorentz invariance requires all metrics of (vacuum) wave equations to coincide.

2.2 The Equivalence Principle, Deﬂection of Light and Geometry

37

produces a metric ds2 = gkl dxk dxl , where gkl is given by g ik gkl = δli . On the one hand, non-linear propagation of light implies non-linear bicharacteristics of the wave equation. The coeﬃcients of the wave operator cannot remain constant. On the other hand, light rays are the straightest lines that can be deﬁned. Hence, the metric tensor corresponding to the bicharacteristics of the wave operator must be substituted for the metric tensor of space-time. We obtain a curved space-time. Can we ﬁnd it also in mechanical motion? Already Newtonian mechanics contains curved space-times. However, this aspect was not so popular, and the curvature was hidden in the time direction. This found its expression in the correspondence of Fermat’s principle for rays of light with other integral principles for mechanical motion, and the refractive index corresponds to a function of the potential energy. Figure 2.8 illustrates refraction and reﬂection in mechanics.

Fig. 2.8. Refraction and reﬂection. Refraction and reﬂection in mechanics are eﬀects of discontinuities in the potential. This ﬁgure shows orbits of particles near to such a discontinuity. Mechanical refraction and reﬂection correspond to analogous phenomena in optics that are caused by discontinuities in the refraction index. When a gravitational ﬁeld acts in the same way on both waves and particles, the refractive index and potential represent a curvature of space-time due to a positiondependent measure of time (more precisely, a measure of derivatives with respect to time). It is relativity that shows that, in this case, space has to be curved too

38

2 Relativity

The geodesics of the curved space-time replace the world lines of any otherwise force-free motion in the gravitational ﬁeld. We can state that the equivalence principle allows us to consider motion in a pure gravitational ﬁeld as uniform motion in a curved space-time. This curved space-time determines, by its wave operator, the form of the light rays. From now on, the metric tensor gik is the central quantity that we must use to describe the gravitational ﬁeld, and the objects of gravitational physics are the invariant quantities of Riemann geometry. The coordinates become, basically, freely exchangeable. This makes the geodesic line into the equivalent of the straight line, and makes gravitation become locally only a tidal action. When we admit coordinate-dependent wave equations, we cannot insist on linear coordinate transformations any more. We turn this necessity into an virtue and formulate the equations so that they are free of coordinates or invariant with respect to any substitution of coordinates. The requirement that scalar products do not vary in such substitutions determines the transformation law of the factors of these products. An inﬁnitesimal translation is described by dP = dxi ei , a = ai ei . The scalar product a, b deﬁnes the metric tensor gik : a, b = ei , ek ai bk , ei , ek = gik . In the inertial coordinates of a Minkowski world, gik = ηik . We may interpret the base vectors ei as operators for the partial derivatives ∂/∂xi and write df = dxi (ei f ) . The partial derivatives and with them the base vectors are transformed through the chain rule ∂x∗k ∂ ∂ = . ∂xi ∂xi ∂x∗k A quantity with this transformation rule is called covariant, and the index i is called a covariant index; the index is conventionally written in the lower position. The diﬀerentials of the coordinates are transformed inversely: dxi =

∂xi ∂f dx∗k , i.e. df = dxi is scalar . ∂x∗k ∂xi

A quantity that obeys the transformation law of a diﬀerential is called contravariant, and the index i is called a contravariant index. This index is written in the upper position in order to distinguish it from a covariant index. The product of a covariant and a contravariant quantity can be made scalar if we sum up over the index. We can generate a whole tensor algebra by multiplication and linear combination of products of the same degree (i.e. with identical pattern of upper and lower indices). If we require that index pairs that are summed do not contribute to transformation matrices, and require the product rule (AB)transformed = Atransformed Btransformed ,

2.2 The Equivalence Principle, Deﬂection of Light and Geometry

39

the general transformation law is found to be ∗a1 2 m 2 ,...,im Tki11 ,i ,k2 ,...,kn = T b1 ,b2 ,...,bn ,a ,...a

∂xi1 ∂xi2 ∂xim ∂x∗b1 ∂x∗b2 ∂x∗bm . . . ∗a ... . ∗a ∗a k k ∂x 1 ∂x 2 ∂x m ∂x 1 ∂x 2 ∂xkm

Consequently, the coeﬃcients of the line element form a covariant tensor of degree two, the metric tensor. The Kronecker symbol δki is a numerically invariant mixed tensor of degree two. A volume integral of a scalar quantity is not a scalar, because a substitution leads to

L[x[x∗ ]] d4 x∗ =

L[x] det

∂x∗i ∂xk

d4 x .

The functional determinant, however, can be compensated by other determinants, for instance the determinant g of the metric tensor gik : ∗

g =

∗ det(gik )

∂xm ∂xn = det gmn ∗i ∂x ∂x∗k ∗

L[x[x ]]

√

−g ∗

4 ∗

d x =

=g

∂xi det ∂x∗k

2 ,

√ L[x] −g d4 x .

√ Hence the product of −g and the permutation index (Levi–Civita symbol) iklm (which can generate any determinant) is a tensor of degree four. A partial derivative of a vector cannot be a tensor. The dependence of the transformation matrix on position generates an inhomogeneous transformation law for the partial derivatives. We have to introduce a new universal ﬁeld described by the coeﬃcients of the aﬃne connection, Γ ikl , that compensates the inhomogeneous terms. We write d(ai ei ) = ei dai + ak dek = ei (dai + ω ik ak ) , ω ik = Γ ikl dxl . The covariant derivative is deﬁned by dai = dai + Γ ijk aj duk . The ordinary partial derivative is indicated by a subscript comma, and the covariant derivative by a semicolon: dA . . . = A . . .,k dxk , dA . . . = A . . .;k dxk . The extension of this deﬁnition to the whole tensor algebra is performed through two rules. First, the supplementary terms must compensate each other when a pair of indices is summed over: d(ai bi ) = d(ai bi ) . Second, we require a general product rule, d(a . . . b . . .) = d(a . . .)b . . . + a . . . d(b . . .) . In particular, for a covariant quantity, we obtain k dbi = dbi − bm ω mi = dbi − bm Γ m ik dx .

40

2 Relativity We obtain the coeﬃcients of the aﬃne connection as a function of the ordinary and covariant derivatives of the metric tensor and of a new tensor, the torsion of space-time: m nikl = gik,l − gmk Γ m il − gim Γ kl ,

S ikl = Γ ikl − Γ ilk , =⇒ 1 (−gkl,i + gil,k + gik,l ) 2 1 − (−nkli + nilk + nikl ) 2 1 − (−gim S mkl + gkm S mil + glm S mik ) . 2

gim Γ m kl =

The tensor S mkl is called the torsion of space-time. It describes a closure error of rhomboids. The tensor nikl is called the non-metricity of space-time. In general relativity theory, the torsion is assumed to vanish, and the metric tensor is required to be covariantly constant.

When the metric is not covariantly constant, the length of a vector can change when it is transported from point to point in such a way that it remains parallel to itself. For the momentum vector of a freely falling particle, this means that its norm (the rest mass) can change, and depend on the history in particular: k m n k m n dx (gmn p p ),k = dx (gmn p p );k = dxk gmn;k pm pn = 0 . Einstein argued against such considerations by referring to the narrowness of spectral lines. If the rest mass of an electron could depend on its history (i.e. its world line through the history of the universe), two hydrogen atoms with diﬀerent histories would not show the same spectral lines. Hence one should use always a metric tensor that is covariantly constant, i.e. nikl = 0 (Ricci’s lemma). The part played by torsion in the theory of gravitation is considered in various generalisations of general relativity theory (GRT). In Einstein’s GRT, it is assumed to be zero. The analogy of gravitational theory to other gauge theories seems to indicate that this assumption is not obvious [4]. The covariant derivatives do not commute. When the torsion vanishes, we obtain a linear combination in the form Ai ;kl − Ai ;lk = −Ri mkl Am − S m kl Ai ;m .

(2.6)

The quantity Ri klm is called the Riemann tensor. It is calculated from the coeﬃcients of the aﬃne connection through Ri mkl = Γ i ml,k − Γ i mk,l + Γ i kn Γ n ml − Γ i ln Γ n mk . The geometric implication of the non-commutativity of the covariant derivatives is a rotation that remains when a vector is propagated along a closed curve parallel to itself:

2.2 The Equivalence Principle, Deﬂection of Light and Geometry

A∗i = Ω ik Ak =

dAi DAi =0 =

∂S

41

Ri

j jkl A

df kl .

(2.7)

S

For inﬁnitesimal curves, we obtain A∗i = Ai + dΩ ik Ak , dΩ ij = Ri

jkl

df kl .

(2.8)

These residual rotations generalise the theorem that the excess of the sum of angles in a geodesic polygon is equal to the integral of the Gaussian curvature over the enclosed surface (Fig. 2.9). The curvature vanishes in the gravitationfree Minkowski world only. The rotation of the vector space is the deﬁning quantity for the curvature, because it does not refer to any embedding of the space in a higher-dimensional space, and can be measured by internal procedures only.

Fig. 2.9. Parallel transport and curvature. We transport a vector v0 at P around a geodesic triangle. Parallel transport is deﬁned here by a constant angle between the vector and the geodesic. After returning to the initial position, we obtain a rotation of v1 with respect to v0 through an angle equal to the excess of the sum of angles of the geodesic triangle

For any isolated event (for a whole geodesic line, to be more speciﬁc), coordinates can be found in which the metric tensor is numerically equal to ηik , and in which the ﬁrst derivatives vanish (normal coordinates). The second derivatives, however, cannot be eliminated completely. There always remains a combination that is equivalent to the Riemann tensor of curvature.

42

2 Relativity The transformation xi = ai k x∗k + O2 [x∗ ] can be used to transform the metric tensor at the event (P : xk = 0) to Minkowski values: ∗ = gmn ami ank = ηik . gik

In the next step, the transformation xi = x∗i + (1/2)ai kl x∗k x∗l + O3 [x∗ ] is used to eliminate the ﬁrst derivatives of the metric at P : ∗ = (ηmn + gmn,r xr )(δim + amiu x∗u )(δkn + ankv x∗v ) , gik ∗ gik,l = gik,r + ηim amkl + ηkn anil = 0 .

With the choice

1 (gkl,i − gik,l − gil,k ) , 2 the ﬁrst derivatives of g vanish in the new coordinates. Finally, we choose a transformation 1 xi = x∗i + ai jkl x∗j x∗k x∗l + O4 (x∗ ) . 6 The second derivatives of g ∗ at the event P yield ηim amkl =

∗ = gik,jl + ηim amkjl + ηkm amijl . gik,jl

When we now choose ai

jkl

1 1 = − η im (gmj,kl + gmk,lj + gml,jk ) + η im (gjk,lm + gkl,jm + glj,km ) , 3 6

we ﬁnally obtain ∗ gij,kl =

1 ∗ ∗ − gil,kj . (Riklj + Rlijk ) , Riklj = gij,kl 3

The tensor parts of the second derivatives of the metric are determined by the curvature tensor. Vice versa, the curvature tensor and its concomitants are the only tensors that can be constructed from the metric and its ﬁrst and second derivatives.

The determination of the curvature tensor and the Einstein equations for any particular metric is performed most easily with computer algebra programs [5].

2.3 General Relativity: Solving Galileo’s Paradox Einstein found the solution (which is still valid) to the problem of constructing a consistent relativistic theory of gravitation. This is the theory of general relativity [3]. It interprets the action of gravitation as the result of the pure curvature of space-time, and realizes the equivalence of inertial mass and gravitational charge by construction. The metric tensor becomes the central quantity for describing the gravitational ﬁeld. Coordinates can be substituted, basically without limitation [7, 11]. Any physical quantity must be independent of the coordinates, i.e. it must be invariant against substitutions. Other

2.3 General Relativity: Solving Galileo’s Paradox

43

than the metric, there is no independent tensor connected with gravitation. That is, the covariant derivative is free of torsion and non-metricity: i def 1 i i = (−gkl,i + gil,k + gik,l ) . (2.9) gik;l = 0 , Γ kl = Γ lk = kl 2 The geodesics play the role of straight lines. The world line of a particle free from non-gravitational interactions is a time-like geodesic. The component g00 plays the role of Newton’s potential in the case of slow motion in a weak ﬁeld: Φ Φ gik = ηik + O 2 , in particular g00 = 1 + 2 2 . (2.10) c c On the geodesic, we can construct normal coordinates for which the ﬁrst derivatives of the metric (now Newton’s force) vanish. If the mass point in question is surrounded by a laboratory that moves together with the mass point freely in the gravitational ﬁeld, we do not observe a force from the external ﬁeld any more. The curvature tensor of the external ﬁeld is present as before. It can be measured, and it produces tidal eﬀects.8 The Poisson equation for the Newtonian potential must be replaced with a second-order equation for the metric, and the mass density by some combination of the energy–momentum density. The terms of the equation should be invariant with respect to coordinate substitutions. The curvature tensor is the only candidate for this construction. Einstein’s equations, Ri k = g lm Ri lmk , R = Rk k , 1 8πG Ri k − δ i k R = 4 T i k , (2.11) 2 c are constructed from the Riemann curvature tensor. In some sense, the solutions of Einstein’s equations in vacuo correspond to minimal surfaces, which are characterised by zero mean curvature in two dimensions. The non-linear character of the full Einstein equations allows us to deﬁne particles as ﬁeld concentrations and to derive the equations of motion of particles in a gravitational ﬁeld from the ﬁeld equations themselves. For simple particles without internal structure, we obtain the law of geodesics as expected. Structureless particles move on geodesics in space-time. The bicharacteristics of the wave equation are light-like geodesics. 8

It is a common error to confuse space-time with space. Gravitation curves spacetime. Instead, the curvature of any space in space-time depends mainly on the choice of the space, which is now a hypersurface of space-time. Even ﬂat spacetimes contain curved spaces, just as curved space-times may contain ﬂat spaces. In addition, classical gravitational mechanics can be interpreted as a theory of curved space-time, with the restriction that spatial sections are always chosen to be ﬂat. Relativity intertwines time with space, and curvature of space cannot be avoided any longer.

44

2 Relativity

As Hilbert remarked, the theory can explicitely include a free constant (later called the cosmological constant). Equation (2.11) has then to be replaced with 1 8πG g jk Ri jkl − δli g jk Rmjkm − Λ δli = 4 T il . 2 c This is the most general equation that contains only the metric up to its second derivatives, is linear in the second derivatives and is fully covariant against coordinate substitutions. It is the Euler–Langrange equation of the action integral √ c3 (2.12) d4 x −g(R + 2Λ) . Sgrav = − 16πG We have √ c3 1 d4 x −g Rik − Rg ik − Λg ik δgik , 16πG 2 √ 1 ik d4 x −gTmatter =− δgik . 2c

δSgrav = − δSmatter

Variational calculus with covariant derivatives contains some peculiarities which reﬂect the properties of a variable metric. The variation of a ﬁeld Ψ [x] is given as usual by a one-dimensional set Ψ [u, x], with Ψ [0, x] = Ψ [x] and δΨ = (∂Ψ /∂u)δu. Let the general covariant derivative be Ψ;k = Ψ,k + Γk [gmn ]Ψ . The action integral

Smat =

√ L[Ψ, Ψ;k , gik ] −g d4 x =

L d4 x

yields the variation

δSmat =

∂L ∂L ∂L ∂L δgik + δΨ,k + Γk δΨ δΨ + ∂expl Ψ ∂expl gik ∂Ψ;k ∂Ψ;k

∂L ∂Γm ∂L ∂Γm + Ψ δgik + Ψ δgik,l ∂Ψ;m ∂gik ∂Ψ;m ∂gik,l

=

∂L ∂expl Ψ

−

∂L ∂Ψ;k

;k

δL δΨ + δgik δgik

d4 x

d4 x .

The Euler–Lagrange equations turn out to be ∂L − ∂Ψ

∂L ∂Ψ;k

=0. ;k

The variational derivative of the integral Smat with respect to the metric yields terms proportional to δΨ;k , which are, however, linear combinations of diﬀerent (δgik );l . In each term, the derivatives of the δgik can be shifted to their factors

2.3 General Relativity: Solving Galileo’s Paradox

45

through partial integration, and we ﬁnally obtain a linear combination of the δgik . The coeﬃcients of this expression form what is called the metric energy– momentum tensor, √ √ δ(L −g) c = 2 −gT ik . δgik The invariance of the form of the Lagrange density with respect to coordinate substitutions √ √ ∂x (L −g)[x∗ ] = (L −g)[x[x∗ ]] ∗ , ∂x together with the ﬁeld equations, implies the covariant conservation theorem ∂T ik + T mk Γ imk + T im Γ kmk = 0 . ∂xk To demonstrate this, we have only to replace δΨ and δgik in the action integral Smat with the form variations9 LX Ψ and LX gik for the one-dimensional inﬁnitesimal transformation T ik;k =

x∗i = xi + X i [x]t . For the metric, we obtain LX gik = (gim X m );k + (gkm X m );i = Xi;k + Xk;i . If we suppose the Euler–Lagrange equations for Ψ to be satisﬁed and the X k to vanish on the boundary of the integration domain, we can shift the derivatives of X to the variational derivatives of the metric and obtain the conservation theorem. In Einstein’s equations, we can take only such sources into account because the left side fulﬁls the condition (Rik − (1/2)Rg ik );k = 0 by construction. The conservation law T ik;k = 0 is a integration constraint on Einstein’s equations.

Let us summarise. – Because of the relativity of simultaneity, space and time cannot be kept separate any more. We ﬁnd a space-time union, or world. A point of this world is an event, and angles represent relative velocities. The kinematics of relativistic mechanics is the geometry of a pseudo-Euclidean world. – Energy carries inertial mass, E = mc2 , and inertial mass indicates energy. – Because of the equivalence of inertial and gravitational mass, photons are subject to gravitation, too. Triangulation with photons (and there is nothing more appropriate) shows us a curved world, where the sum of angles depends on the gravitational ﬁeld and does not coincide with the Euclidean value. – Because of the equivalence of inertial and gravitational mass, curvature represents the gravitational ﬁeld, at least partly. Einstein’s general relativity theory supposes that curvature describes the gravitational ﬁeld completely. 9

The Lie derivative LX is deﬁned as an inﬁnitesimal variation of the form of a function. For a function with the transformation law Ψ ∗ [x∗ ] = T [x∗ [x]]Ψ [x[x∗ ]] and the inﬁnitesimal transformation x∗i = xi + X i [x]t, we obtain Ψ [x∗ ] = T [x∗ [x]]Ψ [x[x∗ ]] + LX Ψ dt.

46

2 Relativity

2.4 Positive Curvature: Solving Newton’s Paradox A universe with a boundary clearly does not ﬁt the Cusanus principle. How do we obtain a ﬁnite universe without a boundary? We have to assume the space to be curved, just as a sphere is a ﬁnite two-dimensional space without a boundary. The theory of relativity now shows that the curvature of the world implies that space can be ﬂat only in particular cases, i.e. that it is quite natural to assume a homogeneous universe to be positively curved and ﬁnite, but unbounded. We shall ﬁrst construct the line element for a space of constant curvature. In the case of the sphere, we know of the polar coordinates that consist of lines of constant latitude θ and the meridians. We know of the metric dω 2 = R2 (dθ2 + sin2 θ dϕ2 ) ,

(2.13)

where R is the radius of the sphere. The length of a circle around a pole is not U = 2πRθ, but U = 2πR sin θ (Fig. 2.10). If we choose r = U/2π as the radial coordinate, we obtain the form dr2 2 2 + r dϕ . (2.14) dω 2 = R2 1 − r2

Fig. 2.10. Circle and positive curvature. Let M be the centre of a circle on a sphere. Its radius is the arc Rχ from M to P √. The Gaussian radial coordinate is r = sin χ, and we see that F = 2πR2 (1 − 1 − r2 ) and U = 2πRr. Therefore, neither U = 2π(Rχ) nor F = π(Rχ)2 . We obtain the inequality U 2 < 4πF

2.4 Positive Curvature: Solving Newton’s Paradox

47

We can also make the choice of locally isotropic coordinates, dω 2 =

R2 (d2 + 2 dϕ2 ) . (1 + (1/4)2 )2

The substitutions are r = r[θ] = sin θ, and r = r[] = 1+(1/4) 2. Now we translate this result to the three-dimensional case. Again, we refer our coordinates to one freely chosen point, the pole, and introduce coordinates by means of spheres around this pole. On these spheres, we use the ordinary spherical coordinates θ and ϕ. The spheres themselves are characterised by a radial coordinate, which we obtain from the surface area of the sphere (Ω = 4πR2 r2 ). In an ordinary ﬂat space, the metric is then given by

dσ 2 = R2 (dr2 + r2 (dθ2 + sin2 θ dϕ2 )) . If the space has positive curvature, any section ϕ = const is a sphere with a metric similar to that of (2.14). Hence we obtain, in general dr2 2 2 2 2 dσ 2 = R2 + r (dθ + sin θ dϕ ) . (2.15) 1 − r2 The curvature of such a space is uniformly R−2 . The coordinate r is normalised in such a way that the size parameter R can be split oﬀ as a factor. We may generalise the metric (2.14) to negative curvature by the introduction of a curvature index k, so that dr2 2 2 2 2 + r (dθ + sin θ dϕ ) . dσ 2 = R2 1 − kr2 This index k is 1 for positively curved space, 0 for ﬂat space, and −1 for negatively curved space. We can again choose two other types of radial coordinate. The reduced distance χ can be introduced by means of   sin χ for k = 1 , for k = 0 , r = r[χ] = χ (2.16)  sinh χ for k = −1 . This yields (corresponding to (2.13)) dσ 2 = R2 dχ2 + r2 [χ](dθ2 + sin2 θ dϕ2 ) .

(2.17)

Again, the isotropic coordinates can be found through the substitution r = r[] =

. 1 + (k/4)2

(2.18)

48

2 Relativity Curvature is seen best in the relation between the volume V and surface area Ω of a sphere, which is similar to the two-dimensional relation between the circumference U and area F of a circle. When, for any circle, U 2 = 4πF , the surface is ﬂat; when U 2 < 4πF , it is positively curved; otherwise (U 2 > 4πF ), it is negatively curved (Fig. 2.11). We can ﬁnd the curvature of a three-dimensional space in a corresponding way. When it is ﬂat, we know that for any embedded sphere, Ω 3 = 36πV 2 . Positive curvature yields Ω 3 < 36πV 2 ; negative curvature yields Ω 3 > 36πV 2 . The invariant radius is given by the product Rχ. The volume can be calculated through the integral

r V [r] = R3

4πr2 dr √ , 1 − kr2

0

which yields

V = R3

√  (4π/2)(χ[r] − r 1 − r2 )    

   

(4π/3)r3

√

(4π/2)(r 1 + r2 − χ[r])

for k = 1 , for k = 0 ,

(2.19)

for k = −1 .

Fig. 2.11. Circle and negative curvature. A time shell in a Minkowski space-time is the locus of a constant time-like distance to a centre O. It is a negatively curved space. This time, the length of the arc is smaller than the value which the Gaussian radial coordinate suggests, and we obtain the inequality U 2 > 4πF

2.4 Positive Curvature: Solving Newton’s Paradox

49

For a positively curved space, we ﬁnd a ﬁnite total volume V = 2V [1] = 2π 2 R3 . In the ﬁrst post-Euclidean approximation, we obtain

3V 4π

2

Ω 4π

−3

=1+k

Ω 3 Ω + O2 10 4πR2 4πR2

.

(2.20)

A comparison of the numbers N ∝ V and apparent magnitudes f ∝ Ω −1 of homogeneously distributed standard candles would show the curvature of space – if there was not the cosmological expansion (Chap. 4). For higher dimensions, we refer to the formula for the volume of a sphere in an n-dimensional ﬂat space: Vn [R] =

ωn n π n/2 R , ωn = 2 . n Γ [n/2]

The surface of a sphere in an (n + 1)-dimensional ﬂat space is an n-dimensional space of constant positive curvature with a total volume Vn =

dVn+1 [R] = ωn+1 Rn . dR

The measure of the points at a ﬁxed distance χ is, in turn, a surface of a sphere with radius r = R sin χ in an n-dimensional space, that is, Ω[χ] = ωn rn−1 . The measure of the corresponding spherical cap is the integral of dV = Ω[χ]R dχ which can be transformed into

1 dV = dΩ r[Ω] n−1

r2 [Ω] 1− R2

−1 .

The Gaussian radial coordinate is determined by the Euclidean relation Ω[χ] = ωn rn−1 . For zero curvature (R → ∞), we obtain what is expected in the Euclidean case, i.e. VEuclidean [Ω] for the function V [Ω]. Any observation of V [Ω] > VEuclidean [Ω] would indicate positive curvature.

Einstein checked whether a space-time with positively curved, closed space would solve his equations for some density of pressure-free homogeneous matter. He replaced the general line element with the static line element dr2 2 2 2 2 2 2 2 2 ds = c dt − R + r (dθ + sin θ dϕ ) (2.21) 1 − r2 and took Tki = 0 except for T00 = c2 as the matter tensor. He found a solution

50

2 Relativity

1 = 4πG 2 , 2 R c but it implied

, c2 too. Hence, Hilbert’s trivial term in the action (2.12) seemed to be a necessary ingredient of the solution of Newton’s paradox. The cosmological density of mass determines the curvature as expected, but this density is determined in turn by the cosmological constant. A density of = 10−27 kg m−3 determines a curvature radius R = 3.45 × 1026 m and requires a cosmological constant Λ = 8.38 × 10−54 m−2 . All quantities in Einstein’s solution are deﬁned and ﬁnite. Hence, the Einstein universe was the ﬁrst solution of Newton’s paradox. The curvature induced by gravitation enables the construction of a ﬁnite model of the universe without any boundary, that is, homogeneous throughout. Einstein deliberately constructed a static model. In space-time, Einstein’s universe is a kind of cylinder with the time direction as the axis. The cosmological constant and the curvature of space are of the same order of magnitude. Olbers’ paradox, however, ﬁnds no solution. In a stationary closed system, thermodynamic equilibrium must be expected. Consequently, the night sky should be as bright as the surface of the sun. In addition, as we shall see in the next chapter, Einstein’s solution is an unstable one. Any virtual decrease in R generates a further contraction, and any virtual increase generates a further expansion. No stable stationary solution for the universe exists. Λ = 4πG

References 1. Berg, E. C., Steﬀen, J. H., Bantel, M. K., Boynton, P. E., Cross, W. D., Inoue, T., Moore, M. W., Newman, R. D.: Laboratory tests of gravitational physics using a cryogenic torsion pendulum, Proceedings of the 10th Marcel Grossman Conference, Rio de Janeiro (2003). 34 2. Braginskij, V. B., Panov, V. I.: Veriﬁcation of the equivalence of inertial and gravitational mass, JETP Soviet Physics 61 (1972), 873–1272. 34 3. Einstein, A.: Die Grundlagen der allgemeinen Relativit¨ atstheorie, Ann. Phys. (Leipzig) 49 (1916), 769–822. 42 4. Hehl, F. W., McCrea, J. D., Mielke, E. W., Ne’eman, Y.: Progress in metricaﬃne gauge theories of gravity with local scale invariance, Found. Phys. 19 (1989), 1075–1100. 40 5. Hehl, F. W., Winkelmann, V., Meyer, H.: Computer-Algebra, Springer, Berlin (1992). 42 6. Liebscher, D.-E.: The Geometry of Time, Wiley VCH, Weinheim (2005). 31, 32 7. Misner, C. W., Thorne, K. S., Wheeler, J. A.: Gravitation, Freeman, New York (1973). 42 8. Nordtvedt, K.: Lunar Laser Ranging – a comprehensive probe of postNewtonian gravity, Proceedings Villa Mondragone International School of Gravitation and Cosmology, gr-qc/0301024 (2002). 34

References

51

9. Roll, P. G., Krotkov, R., Dicke, R. H.: The Equivalence of Active and Passive Gravitational Mass, Ann. Phys. (NY) 26 (1964), 442. 34 10. Turyshev, S. G., Williams, J. G., Nordtvedt, K. Jr., Shao, M., Murphy, T. W. Jr.: 35 Years of Testing Relativistic Gravity: Where do we go from here? W.E.Heraeus Seminar: Astrophysics, Clocks and Fundamental Constants, grqc/0311039 (2003). 34 11. Weinberg, S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York (1972). 42

3 Expansion

3.1 The Friedmann Equations It was Friedmann who ﬁrst solved Einstein’s equations for a cosmological model. To Einstein’s surprise, Friedmann showed that a closed, positively curved space is not necessary to solve Newton’s paradox. In particular, he found negatively curved spaces that solved Einstein’s equations at the cost of a time dependence of the scale. If one accepts such a time dependence, and Einstein accepted it reluctantly, the cosmological constant can be set to zero. Some years elapsed before the expansion of the universe, i.e. its variation with time, was conﬁrmed through observation by Hubble. Nowadays, the homogeneous universe with a varying scale is a kind of standard model. It is called the Friedmann model because it is the direct descendant of Friedmann’s work. In the Friedmann model, one free function a[t] describes its varying scale. In this chapter, we consider the equation of evolution of this scale, that is, the equation of motion of the universe. Of course, this procedure explicitly incorporates the Cusanus principle. It also assumes that the inhomogeneities are negligibly small. Neglect of inhomogeneities means, in this context, that the function a[t] can really be deﬁned and isolated in the Einstein equations. To describe a homogeneously expanding universe, we take Einstein’s line element (2.21) but replace the constant radius of curvature R with a variable expansion scale a[t], and obtain dr2 2 2 2 2 + r (dθ + sin θ dϕ ) , (3.1) ds2 = c2 dt2 − a2 [t] 1 − kr2 with the comoving metric dσ 2 =

dr2 + r2 (dθ2 + sin2 θ dϕ2 ) 1 − kr2

or its equivalents obtained through (2.16), and (2.18). These latter substitutions do not change the results, but simplify the calculations in some places. The metric deﬁned by (3.1) is called a Robertson–Walker metric. Its main property is the fact that the lines {l : r, θ, ϕ = const} are geodesics, and the time t is the proper time on these lines. This is the reason why t is Dierck-Ekkehard Liebscher: Cosmology STMP 210, 53–77 (2004) c Springer-Verlag Berlin Heidelberg 2004

54

3 Expansion

Fig. 3.1. Comoving and geographical coordinates. The expansion of the universe is often depicted by a sphere of increasing size. All distances between points of constant geographical coordinates increase at the same rate. The geographical coordinates are comoving, that is, points without any motion additional to the expansion are given by constant numbers in these coordinates. The surface has no centre. — The ﬁgure is misleading, however, in suggesting that we might ask about a centre of expansion in an additional dimension and about expansion into some higher-dimensional space. All that counts here is the surface

called the cosmological time, and r, θ, ϕ are called comoving coordinates, i.e. a particle that does not move except for the expansion retains the values of its comoving coordinates for all of its history (Fig. 3.1). All distances between positions with ﬁxed comoving coordinates are proportional to a[t], i.e. L[t] = L0 a[t]/a0 . Quantities such as L0 that refer to comoving coordinates are said to be comoving or reduced for expansion. We usually obtain them by calculating their present (t = t0 ) value in ordinary units. The change in size of the universe is contained in the expansion factor a[t], the only free function in the metric (3.1). The basic task of cosmology is to determine this function, observationally and theoretically. The lines {l : r, θ, ϕ = const} are interpreted as the world lines of ideal galaxies (fundamental observers), which have virtually no peculiar motion. We call the set of these lines the fundamental congruence, and refer peculiar motion to this congruence. Of course, we cannot assume that real galaxies are individually at rest in the Robertson–Walker coordinates. However, their peculiar motion is assumed to be negligible for the basic problems of cosmology.

3.1 The Friedmann Equations

55

Only in cases where simpliﬁed considerations run into diﬃculties do we consider this kind of assumption. In general, we assume all small-scale processes to evolve in the frame of a simple Friedmann model, which itself feels the matter that it contains through only its thermodynamic properties. We now write the Ricci tensor and the curvature scalar1 for the metric (3.1): R00 = −3

g ik Rik

a ¨ , a

a ¨ a˙ 2 kc2 Rab = −gab +2 2 +2 2 a a a 2 2 a ¨ a˙ kc + = g 00 R00 + g ab Rab = −6 + 2 . a a2 a

,

Einstein’s equations become rather simple, 1 a2

da dt

2 d2 a 1 − − 2 2 a dt a

2 +

da dt

8πG kc2 Λc2 + , = a2 3 3 2 −

kc2 8πG = −Λc2 + 2 p , 2 a c

(3.2)

(3.3)

where indicates the density of mass, and p the pressure. These equations were tested by Friedmann for solutions, and they carry his name. We have derived them from the Einstein equations through substitution of the line element given by (3.1). In general, a computer program is used to manipulate formulae of this kind [7]. The expansion rate H = (da/dt)/a depends on time. This dependence is governed by the Friedmann equations if the curvature index k, the cosmological constant Λ and the matter density are given. The expansion rate H deﬁnes a length, the Hubble radius RH = c/H. When the curvature and cosmological constant can be neglected, the Hubble radius corresponds in form and size to the Einstein radius (1.2). The source of the gravitational ﬁeld is the mass distribution in the universe, given by the density , which is constant in space but varies with time because of the continuity equation d = −3( + pc−2 )

da . a

(3.4)

This continuity equation is the outcome of the condition of integrability of Einstein’s equations. It is homologous to the ﬁrst law of thermodynamics. When it is satisﬁed, the ﬁrst of Friedmann’s equations is the ﬁrst integral of 1

The scalar curvature is usually denoted by the letter R. We have tried to avoid this because R is already used for various lengths here.

56

3 Expansion

the second equation (which then must be considered only in the case of zero expansion rate in order to check stability). Equation (3.2) is kind of a balance: space-time curvature square of the + expansion rate

curvature of space

= =

matter density cosmological constant

mass + density

The expansion rate has to be counterbalanced by the curvature of space, the cosmological constant and and mass density. The value of the mass density that would by itself counterbalance the square of the present expansion rate H0 is called the critical mass density, critical =

3H02 = 1.88 × 10−26 kg m−3 h2 , 8πG

(3.5)

where h is the relative value of H0 in units of 100 km s−1 Mpc−1 . For h = 0.65 we obtain critical = 0.8 kg m−3 . One part of the mass density can be estimated by counting galaxies and using the average mass-to-light ratio of stars. This part does not reach the critical value, by far; it is only a small fraction of it (Chap. 5). This is one of the reasons to consider more complicated models. The actual variation is determined by the equation of state, i.e. the equation for the pressure p = p[]. The equation of state cannot be derived from gravitational theory. It must be obtained from a knowledge about the thermodynamic properties of the matter mixture. We shall outline a picture in which the matter is a mixture of diﬀerent components. These components are assumed either to interact eﬃciently, so that they have the same temperature as the radiation, or essentially not to interact, and evolve adiabatically. Both cases lead to the conclusion that, at least for these ‘quiet’ stages of evolution, the temperature can be eliminated from the equation of state: either we know the temperature as a function of a, as in the case of the radiation component, or we use the adiabatic case of the equation of state. With these preliminaries, we may suppose every component to have an equation of state of the form p = p[] . With such an equation of state, the continuity (3.4) can be integrated to an expression = [a] . After this has been done for every component, the Friedmann equation turns into a deﬁnition of the expansion rate H as a function of a: H2 =

8πG k [a] . 3 k

(3.6)

3.1 The Friedmann Equations

57

The short stages between the quiet epochs just sketched can be imagined, basically as discontinuous changes in the composition of the right-hand side, that is, as phase transitions. Discontinuous changes in the composition of the right-hand side are allowed by the Friedmann equations as long as the expansion rate H itself remains continuous (and therefore piecewise diﬀerentiable ¨ to give a piecewise continuous H). The dynamical action of a cosmological constant cannot be distinguished from the action of a matter component with an equation of state p = −c2 . The most prominent property of such a component is a negative pressure. It was this negative pressure that was used by Poincar´e in connection with a Lorentz-invariant (vacuum) component (T ik = pδ ik ) in order to obtain stationary solutions in the classical theory of the electron [12]. Hilbert had a corresponding term in his general variational principle, and Einstein found the cosmological constant by inserting the metric of a stationary universe of positive curvature into his equations. The expansion rate can be zero in some interval of time only in the case of Λ > 0. When Hubble found the cosmological expansion, the cosmological constant appeared to be superﬂuous. It has reappeared since 1980 in the disguise of a vacuum energy. Now it is a component of matter that can undergo phase transitions, i.e. transform into other matter components. It is generally believed that such a phase transition happened at the end of the process known as inﬂation (Chap. 7). In some respects, the Friedmann equations are a kind of bottleneck in the connection between cosmology and GRT. The reason has already been explained: with the exception of gravitational waves and perturbations, evolution proceeds like thermodynamics in an expanding closed volume, and it is solely the expansion of the volume that is determined by relativistic gravitation. The reduction of the basic problem to the determination of only one free function a[t] makes a comparison with a Newtonian cloud of particles possible. Such a cloud can be imagined as a structure that changes only its scale. The kinetic energy corresponding to the change of the scale a is of course Ekinetic = f1

M 2 a˙ , 2

where the form factor f1 depends on the internal structure, and is of order 1. M is the total mass of the cloud, and R ∝ a is its radius, or size parameter in general. The gravitational potential is, correspondingly, Epotential = −f2

GM 2 . a

The energy conservation law states that f1

M 2 GM 2 a˙ = E + f2 , 2 a

which we may rewrite in the form of the Friedmann equation

58

3 Expansion

a˙ 2 2E 2Gf2 M = + . a2 f1 M a2 f1 a3

(3.7)

Of course, the last factor is equal to the average mass density up to a third form factor f3 . The curvature corresponds to the total energy, with an opposite sign. The cosmological constant has no counterpart here. Cosmological models with a positive curvature of space appear as Newtonian systems with negative energy. They are classically bound. The size R cannot grow inﬁnitely, and we observe a recollapse after expansion. Cosmological models that appear as Newtonian systems with positive energy describe a permanent expansion (if the eﬀective mass M does not grow with expansion). Now, in a strict Newtonian sense, the mass is invariant. We may, however, eclectically use the variation of mass due to pressure in accordance with (3.4). This makes our Friedmannian interpretation of the Newtonian equations eﬀectively identical to the real Friedmann equations (without cosmological term).

3.2 The Worlds of Constant Curvature In the case of a constant, non-zero matter density, the curvature of the spacetime is also constant. This is the case for the static Einstein model (Sect. 2.4). If the universe is not static, the curvature of the world is constant if and only if the universe is devoid of mass, and the curvature of the world is given by the cosmological constant. Therefore, the cosmological constant can be understood as the basis value of the curvature of space-time. The ﬁrst case that we consider is that of a zero cosmological constant. We obtain the Milne model. It is locally isometric to the Minkowski space-time (Fig. 3.2). The simple linear expansion a[t] = a0

t , t0

which describes an ordinary explosion under force-free conditions, is a trivial solution of Friedmann’s equations for negative curvature, k = −1, with no matter added [9, 10]. The metric ds2 = c2 dt2 −

a0 t0

2

t2

dr2 + r2 (dθ2 + sin2 θ dϕ2 ) 1 + r2

is equivalent to the Minkowski metric. The space-time is the internal forward light cone of the origin; the slices corresponding to space are the spaces of constant proper time diﬀerence relative to the origin (hyperboloids); and the coordinates r, θ, ϕ the Gaussian coordinates on these hyperboloids, which have constant negative curvature. The transformation of the Minkowski coordinates, where ds2 = c2 dτ 2 − (d2 + 2 (dθ2 + sin2 θ dϕ2 )) ,

3.2 The Worlds of Constant Curvature

59

Fig. 3.2. Milne’s universe in Minkowski coordinates. The surfaces of constant cosmological time are the hyperboloids (ct)2 − x2 − y 2 − z 2 = const (all in Minkowski coordinates). These hyperboloids are surfaces of negative curvature. Special Lorentz transformations (2.3) are translations in the Milne universe. The straight lines through the vertex form the fundamental congruence that deﬁnes comoving coordinates to the Milne coordinates is given by τ =t

1 + r2 ,

= r ct

The central projection of a hyperboloid z = 1 + x2 + y 2 with the metric 2 2 2 2 ds = −dz + dx + dy yields Klein’s model of the hyperbolic plane. This projection allows us to study the non-Euclidean geometry found by Bolyai and Lobachevski [15]. When we switch to the stereographic projection (from the point z = −1, x = y = 0) we obtain an alternative representation, in which the geodesics are circular arcs (the Poincar´e model of the hyperbolic plane). Escher’s limit circles are examples of this projection in the form of tilings of negatively curved surfaces (Fig. 3.3).

The case of a positive cosmological constant is the de Sitter universe (Fig. 3.4). The sphere, 2 W 2 − T 2 + X 2 + Y 2 + Z 2 = RH

of a pseudo-Euclidean ﬁve-dimensional space with the metric

60

3 Expansion

Fig. 3.3. Pentagonal tiling of the Poincar´e plane. On a surface of constant negative curvature, there exists a regular pentagon with ﬁve right angles. This pentagon can be used to tile the surface. We see here the central projection of such a surface which maps the geodesic sides of the pentagons into circular arcs. of a The length √ side of a regular pentagon with ﬁve right angles is equal to Arsh (1 + 5)/2

ds2 = dT 2 − (dX 2 + dY 2 + dZ 2 + dW 2 ) is an empty universe: With the help of the transformation X = sin θ cos ϕ , Y = sin θ sin ϕ , Z = cos θ , 2 cosh[τ /R ] , W = RH 1 − 2 /RH H 2 sinh[τ /R ] , T = RH 1 − 2 /RH H we obtain (supposing that < RH ) 2 d2 2 2 2 2 ds2 = dτ 2 1 − 2 − 2 − (dθ + sin θ dϕ ) . RH 1 − 2 /RH

(3.8)

This is a static metric, found by de Sitter in 1915. At that time, it was puzzling to ﬁnd a time-dependent form through the transformation

3.2 The Worlds of Constant Curvature

61

Fig. 3.4. The de Sitter universe. 1. Instead of drawing the congruence of comoving objects on a plane space-time (as we did in the case of the Milne universe), we now draw on a hyperboloid that represents a sphere in a pseudo-Euclidean space-time. The lines are intersections of the hyperboloid with planes that pass the origin. Hence they are geodesics of a fundamental congruence. The planes must have a common line. Depending on this line, they generate diﬀerent congruences of geodesics and diﬀerent spatial sections orthogonal to these geodesics

τ = ct −

RH ct ct ln 1 − r2 exp 2 , = r exp 2 . 2 RH RH

This yields the Robertson–Walker form ct 2 2 2 2 ds = c dt − a exp 2 (dr2 + r2 (dθ2 + sin2 θ dϕ2 )) . RH

(3.9)

In this form, the metric (3.8) is a solution to the Friedmann equation for vanishing curvature of space, vanishing matter content but a positive cosmological constant. Metaphorically speaking, it represents a universe containing only vacuum. When we adopt the interpretation of the cosmological constant as the zero-point energy of the vacuum, the cosmological constant is a matter component. As a matter component, it may take part in phase transitions – in contrast to a ‘cosmological constant’ or to a ‘basis value of the curvature of the world’, terms that give the impression that they are constant throughout history.

62

3 Expansion

The choice (3.8) of coordinates deﬁnes the fundamental congruence so that it always expands. The expansion has no beginning, no singularity. The hypersurfaces of simultaneity turn out to be ﬂat. The generating line of Fig. 3.4 just touches the hyperboloid. When we project the ﬁgure onto a plane parallel to the axis of th hyperboloid, we obtain Fig. 3.5.

Fig. 3.5. The de Sitter universe. 2. This is the case of exponential expansion. In the full number of dimensions, the space sections turn out to be ﬂat. Any point in the sector covered by the fundamental congruence has a horizon of action. For the event A, this horizon is the region between A1 and A2

As in the Milne model, this choice is not unique, if we intend only to obtain a fundamental congruence orthogonal to simultaneity hypersurfaces. For instance, with R2 = X 2 + Y 2 + Z 2 and the transformation ∗ ∗ ∗ ct ct ct T = RH sinh , W = RH cosh cos χ∗ , R = RH cosh sin χ∗ , RH RH RH we obtain 2 ds2 = c2 dt∗ 2 − RH cosh2

ct∗ (dχ∗ 2 + sin2 χ∗ (dθ2 + sin2 θ dϕ2 )) . (3.10) RH

This is again an empty space with a cosmological constant, but now with a positive curvature. Its expansion is minimum at t∗ = 0. Before this instant,

3.2 The Worlds of Constant Curvature

63

Fig. 3.6. The de Sitter universe. 3. This is the case of contraction followed by expansion. In the full number of dimensions, the space sections turn out to be positively curved

the universe contracts; after t∗ = 0, it re-expands. The generating line of Fig. 3.4 lies inside the hyperboloid (in particular, it is its axis). When we project the ﬁgure onto a plane parallel to the axis of th hyperboloid, we obtain Fig. 3.6. A third choice is ∗ ∗ ∗ ct ct ct ∗ T = RH sinh cosh χ , W = RH cosh , R = RH sinh sinh χ∗ . RH RH RH We obtain 2

2

ds = c dt

∗2

−

2 RH

2

sinh

ct∗ (dχ∗ 2 + sinh2 χ∗ (dθ2 + sin2 θ dϕ2 )) . (3.11) RH

This is an empty space with a cosmological constant, but now with a negative curvature. It expands all the time, beginning with a singularity. The generating line of Fig. 3.4 lies intersects the hyperboloid (in particular, it is perpendicular to its axis). When we project the ﬁgure onto a plane parallel to the axis of the hyperboloid, we obtain Fig. 3.7. For curiosity, we add the metric of the anti-de Sitter universe. This is the corresponding object in a ﬁve-dimensional space with the metric

64

3 Expansion

Fig. 3.7. The de Sitter universe. 4. Projection onto the plane: The case of hyperbolic expansion. In the full number of dimensions, the space sections turn out to be negatively curved ds2 = dT 2 + dW 2 − (dX 2 + dY 2 + dZ 2 ) . The equation of a sphere in this space is now 2 − T 2 + X2 + Y 2 + Z2 , W 2 = RH

and the transformation X = sin θ cos ϕ , Y = sin θ sin ϕ , Z = cos θ ,

W = RH

1+

τ 2 cos , 2 RH RH

1+

2 τ sin 2 RH RH

T = RH produces

2

ds = dτ

2

2 1+ 2 RH

−

d2 − 2 (dθ2 + sin2 θ dϕ2 ) , 2 1 + 2 /RH

3.3 Worlds with Matter Only

65

Fig. 3.8. The anti-de Sitter universe. This is a projection of the hyperboloid onto the plane with time and position interchanged. It represents the anti-de Sitter universe. In the full number of dimensions, the space sections turn out to be positively curved analogously to (3.8). This metric is a solution of the Friedmann equation with a negative cosmological constant and negative curvature of space.

As in the case of the Milne universe, the curvature of space depends on the choice of the fundamental congruence, or comoving coordinates. The Milne universe, with negatively curved, expanding spatial sections, is isometric to the Minkowski space-time, with plane, non-expanding spatial sections. The de Sitter universe can be foliated into plane sections, positively curved sections or negatively curved sections. This degeneracy is lifted when matter that distinguishes a time-like vector of the matter stream is present.

3.3 Worlds with Matter Only After the discovery of the recession of galaxies and the insight that it would be extremely diﬃcult to measure the curvature of space and the cosmological constant, the simplest trial universe that could usefully be considered

66

3 Expansion

contained only cold, pressure-free matter,2 i.e. p = 0. With this equation of state, we obtain a density that is strictly inversely proportional to the volume indicated by the expansion parameter a: = 0 (a0 /a)3 . In general, we take p = 0 for any matter component that produces only a small pressure. The kinetic energy of the particles should be negligible in comparison with their rest energy in such a case. To a large extent, galaxies themselves are such particles. The solution of the Friedmann equations is 2 t . a3 = a30 t0 This is the Einstein–de Sitter universe [3]. The density in such a universe is, by deﬁnition, permanently equal to the critical density (3.5). The Einstein– de Sitter universe was the reference or standard model until the end of the 20th century. The curvature and cosmological constant were taken to be negligible. With the construction of inﬂationary models since 1982, arguments were found that backed this supposition. Supergravity theories even insisted on a zero vacuum energy all the time. After 1999, however, a cosmological constant has been reluctantly accepted again: the formation of structure in the universe cannot be explained without it, and the measurement of the distances of extragalactic supernovae indicates an expansion rate that is too large in comparison with the necessary age of the universe. The complement to cold matter is hot matter. We understand this matter to consist of particles that have a rest mass that can be neglected in comparison with the thermal kinetic energy, i.e. kT m0 c2 . We call them relativistic particles, of course. The equation of state of hot matter is p = c2 /3. The continuity (3.4) yields = 0 (a0 /a)4 . The most important kind of hot matter is electromagnetic radiation. As we know, there is a heat bath in the universe that has been decoupled since the formation of neutral atoms. Its temperature today is less than 3 K, and its density is 10−5 of the critical density. Therefore, this radiation dominated all processes in the universe and dominated its expansion for a < 10−5 a0 . The universe was opaque when it was radiation-dominated (z > 25 000), it was matter-dominated when it was transparent (z < 1100) and there was a short interval when it was matter-dominated but still opaque (1100 < z < 25 000). The model where radiation provides the critical mass given by (3.5) yields a2 = a20

t . t0

(3.12)

If we name the cold matter universe after Einstein and de Sitter, the hot matter universe should be named after Gamow. It provides the background 2

‘Cold’ matter is deﬁned by a relation between the pressure and the energy density only. It comprises all objects with peculiar velocities negligible with respect to the speed of light. We call the matter ‘cold’ because temperature does not matter in the equation of state.

3.4 Barotropic Components of Matter

67

and the heat reservoir for the processes that ran while the universe was opaque. These processes are considered in Chap. 6. It is tempting to formulate a power-law expansion of the form 2/n t . a(t) = a0 t0 We obtain such an expansion for any single matter term Mn /an that balances the square of the expansion rate. The cases n = 3 and n = 4 belong to this class. For these cases, the matter density is critical, and the Friedmann equation says that the Schwarzschild radius of the mass MH in a Hubble sphere is the Hubble radius RH itself: 8πG 2G 2G 4π c 3 c ⇒ R S = 2 MH = 2 = RH . H2 = = 3 c c 3 H H

3.4 Barotropic Components of Matter The main parts of the history of the universe are the radiation-dominated pre-recombination time (Gamow universe) and the matter-dominated postrecombination time (Einstein–de Sitter universe), which is apparently followed by a vacuum-dominated future. The main components of the mass distribution are radiation, for which p = c2 /3, pressure-free matter, for which p = 0, and vacuum, for which p = −c2 . We shall call a component barotropic if the pressure depends only on the density, and here we can further restrict to the canonical form of a proportionality, i.e. n − 1 c2 , p= (3.13) 3 which yields Mn , an with some integration constant Mn that represents the concentration of the component in question. The relative pressure w = p/(c2 ) is constant for the individual component; in a mixture, it depends on the expansion a. The form (3.13) of the function [a] is valid also for the curvature of space (n = 2). Therefore, we formally put it on the right-hand side, and include it formally in the function [a]. The Friedmann equation for a mixture of canonical components (3.6) is found to be =

H2 =

8πG Mn . 3 n an

(3.14)

It is important to note that a positive curvature enters the right-hand side as a negative matter component, and a negative curvature enters as a positive component.

68

3 Expansion

Locally, we may expand a[t] in powers of H0 (t − t0 ), 1 a[t] = a0 (1 + H0 (t − t0 ) − q0 (H0 (t − t0 ))2 ) , 2 to deﬁne the present deceleration parameter q0 . For our mixture of barotropic components, q0 is the present value of the expression q=−

8πG Mn a¨ a = (n − 2) n . 2 2 a˙ 6H n a

(3.15)

Phase transitions are now changes in the Mn subject to the continuity of H[a]: ∆Mn =0. an n Some interpretations of the canonical components are given in Table 3.1. Table 3.1. Canonical matter components n=6

Ultra-stiﬀ ﬂuid (Zel’dovich), 2 dp/d = vsound = c2

p = c2

n=5

Kinetic energy of non-relativistic particles

p = 2c2 /3

n=4

Hot matter (radiation)

p = c2 /3

n=3

Cold matter (the ‘dust’ of galaxies)

p=0

n=2

Gas of strings (Chap. 11), curvature

p = −c2 /3

n=1

Gas of domain walls (Chap. 11)

p = −2c2 /3

n=0

Vacuum energy, cosmological constant

p = −c2

Without phase transitions, in the earliest stage the last term in the expansion (3.14) dominates, and in the latest stage the ﬁrst term dominates. If the cosmological constant is at all positive, the universe ends in an de Sitter stage, if the cosmological constant is negative, it has to recollapse. We can plot the function H[a] in a diagram such as Fig. 3.9, and read oﬀ the qualitative consequences. More complex patterns are found in [11]. There are other simple equations of state that have both physically and mathematically interesting properties, for instance the Chaplygin gas. It is characterised by the equation of state p ∝ −1/, i.e. a negative pressure negligible for large densities. In a positively curved universe, the solutions interpolate between a pressureless high-density epoch and a vacuum-dominated low-density epoch, and the velocity of sound is real and bounded [5, 8]. A more complicated equation of state is that of a polytrope,

3.5 Friedmann’s Universe: Solving Olbers’ Paradox

69

Fig. 3.9. The evolution of the relative weight of matter components. It is often useful to refer to the density radiation of radiation, which is a measure of the entropy 3/4 density s of the universe, where s ∝ radiation . We therefore show the dependence 4 2 of the quantity a H on a. For radiation, this quantity does not change, and yields a reference for other components. All equations of state with 3p < c2 produce an increase of a4 H 2 with a (if the weight of the component is positive, of course). The weight of a positive curvature is negative, however, and we ﬁnd, in a Friedmann– Lemaˆıtre model, a stage (dominated by curvature) in which a4 H 2 decreases as a increases

p=

ρc2 kT kT . ρm c2 = 2 2 mp c mp c 1 + kT /(mp c2 (γ − 1))

When we take the radiation component as the heat bath, the temperature falls like a−1 , and we can integrate (3.4) to obtain (1 + (kT /mp

d da = −3 , 2 + kT /((γ − 1)mp c )))) a

c2 )(1/(1

to obtain the function [a] in the form a 3γ a/a + kT /((γ − 1)m c2 ) 3(γ−1) 0 0 0 p = . 0 a 1 + kT0 /((γ − 1)mp c2 )

3.5 Friedmann’s Universe: Solving Olbers’ Paradox With the Friedmann expansion, both the ﬂux of photons and their energy decrease with expansion. Integrals such as (1.1) converge. The average surface brightness of the night sky is low.

70

3 Expansion

With the expansion, and the existence of a start for star formation, Olbers’ paradox is solved. As we shall see in Chap. 4, the expansion of the universe generates a decrease in the energy Eγ = hν of the photons through the expansion-dominated increase of their wavelength Eγ [t0 ]/Eγ [te ] = a[te ]/a[t0 ], and a decrease of the photon number intensity through the corresponding dilation of the time between two photons, which yields another factor a[te ]/a[t0 ] in the energy ﬂux. The brightness of objects falls with distance very fast. In addition, in the observed universe, stars and galaxies have not been present all the time, and the corresponding start of star formation and galaxy formation can solve Olbers’ paradox, too. One can compare the two eﬀects and show that the second eﬀect dominates over the ﬁrst at present [16]. This solution of Olbers’ paradox does not depend on a particular model for the interstellar matter, the light sources or the light itself. Now, we intend to show the solutions to the Friedmann problem. At his time, the radiation bath of the universe was unknown and undetected, and the source of the gravitational ﬁeld was assumed to be cold matter and the cosmological constant. The Friedmann equation (3.2) yields

1 da a dt

2 +

8πG a0 3 kc2 Λc2 + 0 = . 2 a 3 3 a

(3.16)

We denote the fraction (a0 /a) by ζ = (1+z). We shall see in the next chapter that z is the redshift of the spectral lines of a galaxy receding through the increase of the expansion factor a of the universe. Equation (3.16) can be written in the form def

h2 [z] =

H2 Λc2 kc2 8πG0 = − 2 2 (1 + z)2 + (1 + z)3 2 2 H0 3H0 a0 H0 3H02 def

= λ0 − κ0 (1 + z)2 + Ω0 (1 + z)3 .

(3.17)

This equation deﬁnes the cosmological parameters λ0 , κ0 and Ω0 . With the substitution a0 da = −(1 + z)h[z] H0 dt (3.18) dz = − a a we obtain the integral z H0 t = H0 t 0 −

dz . (1 + z)h[z]

(3.19)

z0

This integral yields the relation between the age t of the universe and the redshift z. The latter decreases with expansion, and the former increases. At zeros of h[z], the second-order equation has to be evaluated. This procedure, found by Friedmann, was extensively discussed by Lemaˆıtre. While Friedmann was interested in the fact that consistent models existed for negative curvature, and that their existence did not depend on

3.5 Friedmann’s Universe: Solving Olbers’ Paradox

71

that of the cosmological constant, Lemaˆıtre was captured by the fact that a three-component model with positive curvature also yielded solutions with a long period of very small expansion rates (Fig. 3.12). These models can be substantially older than the Hubble age tHubble = H0−1 (Fig. 3.13). The timing of the expansion is determined by the relative weights of the terms of the right-hand side of (3.17). The sum of these terms is 1 today. It can be normalised for all times through division by h2 , and we obtain def

λ−κ+Ω =

κ0 (1 + z)2 Ω0 (1 + z)3 λ0 − + =1. 2 2 h [z] h [z] h2 [z]

(3.20)

Hence we may represent the solutions in an Ω–λ plane (Fig. 3.10). This plane is divided into several diﬀerent regions. The character of the solutions changes from one region to another (Fig. 3.11). The plane contains three fundamental points, which mark the Einstein–de Sitter universe (E in Fig. 3.10), the Milne universe (M ) and the de Sitter universe (S). The line c = ES contains the ﬂat-space universes. It divides the regions of positive curvature (I, II, III and VIII) from the regions of negative curvature. The line EM contains the class of universes without a cosmological constant, i.e. the class which is usually cited when it is argued that > critical implies positive curvature and recollapse of the universe. This conclusion is valid only for this class of solutions, and does not apply to the case of a non-zero cosmological constant. The line a in Fig. 3.10 marks the Eddington–Lemaˆıtre universe, which starts in the Einstein conﬁguration and ends in a de Sitter expansion. In region I, the expansion rate has a minimum in the past, and the age of the model can be very large. This region will be given particular emphasis in the following, and the coordinates of the minimum will be used. On the straight line b, the deceleration parameter is q equal to 1. The lines parallel to b are lines of constant deceleration parameter. The Ω–λ plane is the phase space of all Friedmann–Lemaˆıtre models. Before 1990, the cosmological constant was persona non grata, and there were, of course, reasons for this prejudice.3 Since the comeback of the cosmological constant, the observed values of λ and Ω have been presented with their conﬁdence regions in this plane, and the triangle ∆M ES has been christened the cosmic triangle [1]. The conﬁdence regions that can be drawn on the basis of various observations are believed to centre around λ0 ≈ 0.7 and Ω0 ≈ 0.3 For positive curvature, the minimum of the expansion rate is found at zmin = 3

2κ0 −1. 3Ω0

La Rochefoucauld put it this way: ‘Il y a des fausset´es d´eguis´ees qui repr´esentent si bien la v´erit´e que ce serait mal juger que de ne s’y pas laisser tromper.’ (There are disguised lies that resemble truth so well that it would seem bad judgement not to be deceived.)

72

3 Expansion

Fig. 3.10. Part of the phase plane of Friedmann models. This plane classiﬁes the solutions of the Friedmann equation with respect to the parameters Ω and λ

If this value is negative, no minimum exists; if it is smaller than one, the minimum lies ahead in the future. The square of the relative expansion rate is found to be H2 1 h2min = min = λ − κ(1 + zmin )2 . 2 H0 3 In the a–t diagram (Fig. 3.12), the various types of solutions can be compared. The formal age can be found through the integral in (3.19). Two charts of this age are given in Fig. 3.13. A particular case is the Eddington–Lemaˆıtre model. Eddington expected that the universe would evolve from a quasistationary initial state, the Einstein universe, to a ﬁnal state dominated by the cosmological constant, the de Sitter universe. The instability of the initial state would lead to expansion and a transition to the de Sitter stage. Such a model can be described as a Friedmann–Lemaˆıtre universe with h2min = 0. The beginning of expansion marks a maximal redshift z = zmax (with h2 [zmax ] = 0). Modiﬁcations of this model are still under consideration [4]. The paradox of this model is that expansion cannot start through a raise in the pressure (for instance annihilation of non-relativistic particles into radiation). A spontaneous increase in pressure that conserves the density leads

3.5 Friedmann’s Universe: Solving Olbers’ Paradox

73

Fig. 3.11. The expansion in diﬀerent Friedmann models. 1. The inﬁnite Ω–λ plane is mapped onto a circle here so that the whole plane can be seen. There are three ﬁnite special points, corresponding to the Einstein–de Sitter, Milne, and de Sitter models. The Einstein–de Sitter model corresponds to an unstable point, and the de Sitter model to a stable point. The lower right border indicates a maximal expansion with a steady turnaround, and the upper left border indicates a minimal size between contraction and expansion

to contraction, not to expansion. Only a decrease in pressure (for instance spontaneous creation of a component with n < 3 or condensation of matter that reduces the already negligible pressure) can start an expansion. This paradox of universal pressure is also the reason for Einstein’s model being unstable. With our present observation-based knowledge, the Eddington– Lemaˆıtre model is only a historical anecdote. We can observe objects with redshifts z > 5. Therefore, zmin > 5, and this includes Ω0 < 0.001, far too small for any observation. In addition, the background radiation would have to ﬁnd an unorthodox and unexpected explanation, and the universal deuterium as well. The various Friedmann models are summarised in Table 3.2.

74

3 Expansion

Fig. 3.12. The expansion in diﬀerent Friedmann models. 2. This graph shows qualitatively the behaviour of the solutions a[t] in the various regions of the parameter space. In regions V, VI, VII and VIII, the solutions have a minimal expansion in the past (a bounce). In regions III, IV and V, we obtain a maximal expansion in the future. In region V, we obtain the only true oscillating universes: these have negative λ and negative Ω as well

3.6 The Cosmological Singularity The (a4 H 2 )–a diagram shows that we can always ﬁnd a time with a = 0 whenever there is no zero of H 2 between a = a0 and a = 0. Any instant with a = 0 is a singularity, because Tki Tik diverges. We can see this happen through the full second-order Friedmann equation 4πG 1 d2 a ( + 3pc−2 ) + Λ . =− a dt2 3 For a multicomponent universe (3.14), we obtain 4πG Mn 1 d2 a =− (n − 2) n . 2 a dt 3 n a Curvature does not enter. Whenever conservative matter components (p ≥ 0) contribute positive terms to the right-hand side, there exists some a1 > 0 such that for all a < a1 , the second derivative d2 a/dt2 is negative. When da/dt > 0 at a = a1 , we ﬁnd a singularity a = 0 in the past of a = a1 . Models without a cosmological singularity can be constructed only for non-conservative matter (all Mn zero for n > 2), or for mixtures that show zeros in the function h2 [z] in the past, that is, for negative contributions to

3.6 The Cosmological Singularity

75

Fig. 3.13. The age of the universe. This map shows the relative age H0 t0 of the universe as a function of the density parameter Ω0 and the relative density of the vacuum λ0 . The isolines are spaced linearly. In the Einstein–de Sitter universe, H0 t0 = 2/3. In the Milne model, H0 t0 = 1

the density. A positive curvature could provide such a contribution. However, this contribution would not help, because it would again produce a zero in the past only if a positive cosmological constant were present. The Eddington–Lemaˆıtre model is an example. With a zero cosmological constant, any conservative Friedmann model has a singularity in the past. This cosmological singularity that is present for any conservative matter distribution also exists for general solutions of Einstein’s equations. This can be shown through consideration of a bundle of world lines that are chosen like a local fundamental congruence. These world lines represent the ﬂow of matter in the universe, which is not necessarily geodesic if the pressure is not precisely constant. Let us assume a hypersurface xi [u, v, w] with time-like normals and construct the bundle of world lines assumed to start from the various events on this hypersurface in the normal direction. We parametrise the world lines with the proper time and measure the proper time from the hypersurface itself. The bundle xi = xi [t, u, v, w] deﬁnes the unit vector ﬁeld of tangents ui = dxi /dt and the acceleration u˙ i = ui ;k uk . This vector

76

3 Expansion Table 3.2. Summary table of cosmological models Name Static Einstein universe (1917) Friedmann universe (1922), Lemaˆıtre (1930) de Sitter universe (1917) Eddington universe (1930)

λ0

1 :

κ0

Ω0

3 : 2

Remarks H0 = 0, no expansion, no redshift

Minimum of the expansion rate H[z] at λ0 − κ0 + Ω0 = 1 z = (2λ0 − Ω0 − 2/3Ω0 )

1

0

0

ξ 3 : 3ξ : 2

Exponential expansion Zero of the expansion rate at z = ξ − 1

Einstein– de Sitter universe (1931)

0

0

1

Cold matter, age < Hubble time

Milne universe (1935)

0

−1

0

Flat world, age = Hubble time

Concordance model of 2003

0.7

0

0.3

‘Concordance’ is a kind of superlative of ‘standard’

ﬁeld has divergence θ = uk;k , rotation ωik = ui;k − uk;i and shear σik = ui;k + uk;i − (1/2)gik ul;l . The deﬁnition (2.6) of the Riemann tensor yields now the Raychaudhuri equation [13, 14] 1 dθ = − θ2 + ωik ω ik − σik σ ik − Rik ui uk + u˙ k;k . c dt 3 In the case of the fundamental ﬂow, uk = [1, 0, 0, 0], u˙ k = 0 and Rik ui uk = κ(c2 + 3p). Non-vanishing rotation needs further consideration, but when c2 + 3p > 0, the bundle has a singularity in the past for θ > 0 on some open region of the hypersurface, and has a singularity in the future for θ < 0 on some open region. A general theorem has been set out by Hawking and Ellis [6], which states the following: If, in a space-time, – a simple kind of causality is observed (precisely, when one can split it into a family of Cauchy surfaces, on which a knowledge of the state of all ﬁelds suﬃces to calculate their future evolution), and

References

77

– an inequality holds for the source matter tensor (including the vacuum) of the form 1 1 R00 = g0j T0j − g00 Tjj ≥ 0 (3.21) κ 2 (in our case c2 + 3p > 0), any local contraction of the normals of a space slice yields a singularity in the future, and any local expansion of the normals yields a singularity in the past. For some particular homogeneous but anisotropic models, Belinskij, Khalatnikov and Lifshic found a kind of chaotic behaviour in the neighbourhood of the singularity [2].

References 1. Bahcall, N. A., Ostriker, J. P., Serlmutter, S., Steinhardt, P. J.: The cosmic triangle: Assessing the state of the universe, Science 284 (1999), 1481–1488. 71 2. Belinskij, V. A., Khalatnikov, I. M., Lifshic, E. M.: Oscillatory approach to a singular point in relativistic cosmology, Adv. Phys. 19 (1970), 523–573. 77 3. Einstein, A.: Zum kosmologischen Problem der allgemeinen Relativit¨ atstheorie, SBer. Preuss. Akad. Wiss. (1931), 235–237. 66 4. Ellis, G. F. R., Maartens, R.: The emergent universe: inﬂationary cosmology with no singularity, Class. Quant. Grav. 21 (2004), 223–232. 72 5. Gorini, V., Kamenshchik, A., Moschella, U., Pasquier, V.: The Chaplygin gas as a model for dark energy, SLAC-SPIRES HEP, gr-qc/0403062 (2004). 68 6. Hawking, S. W., Ellis, G. F. R.: The Large-Scale Structure of Space-Time, Cambridge University Press (1973). 76 7. Hehl, F. W., Winkelmann, V., Meyer, H.: Computer-Algebra, Springer, Berlin (1992). 55 8. Lima, J. A. S.: Alternative Dark Energy Models: An Overview, Braz. J. Phys. 34 (2004), 194–200 (astro-ph/0402109). 68 9. Milne, E. A.: Relativity, Gravitation, and World Structure, Oxford University Press (1935). 58 10. Milne, E. A.: Kinematical Relativity, Oxford University Press (1948). 58 11. Petrosian, V.: Phase transitions and dynamics of the universe, Nature 298 (1982), 805–808. 68 12. Poincar´e, H.: The dynamics of the electron, Rend. del Circ. Mat. di Palermo 21 (1906), 129–146, 166–175. 57 13. Raychaudhuri, A. K.: Relativistic cosmology. I, Phys. Rev. 98 (1955), 1123– 1126. 76 14. Raychaudhuri, A. K.: Theoretical cosmology, Clarendon Press, Oxford (1979). 76 15. Reichardt, H.: Gauss und die nichteuklidische Geometrie (1976), Teubner, Leipzig. 59 16. Wesson, P. S., Valle, K., Stabell, R.: The extragalactic background light and a deﬁnitive resolution of Olbers’ paradox, Astrophys. J. 317 (1987), 601–606. 70

4 Cosmometry

4.1 The Past Light Cone When we consider the shortness of the four thousand years of recorded astronomical observation with respect to the age of the universe and the travel time of light from its distant objects, we can understand that the observable events in the universe lie in essence on our past light cone [10]. The space at the time of the formation of neutral atoms is cut by this cone in a nearly spherical surface. This is the reason why we observe this instant, which divides the observable universe from the more distant past, as a boundary in space, an opaque ﬁreball. In an ideally isotropic universe, this boundary appears as a sphere. Loosely speaking, the universe is transparent inside the ﬁreball and opaque behind; in addition, behind the ﬁreball is before the ﬁreball. The past light cone that consists of the observable events has a small thickness that corresponds to the four thousand years of astronomical experience. Owing to this thickness, the observable quantities also comprise derivatives with respect to time, in principle. One could follow a programme aimed at discovering the state of the universe through use of the ﬁeld equations from data on this past light cone (the degenerate Cauchy problem). This programme can yield unique results only for the inner part of the cone, i.e. the past. Nevertheless, such a rigorous treatment can provide statements that are independent of the cosmological principle, i.e. that can possibly test at least parts of it [1]. This situation is relaxed by the observation of relics of prehistoric universal phase transitions such as the primordial nuclear synthesis. We have access to information about processes in the inner region of the past light cone (that depend, of course, on the model used for calculation). This information can be compared, in essence, with the results of the solution of the Cauchy problem (Fig. 4.1). To proceed along these lines, the full set of Einstein equations must be taken into account, and the gravitation-dependent evolution equations of the other quantities as well. The observable quantities are the redshifts of objects, their proper motion, apparent magnitude, size and shape; the statistics of their distribution; and their spectrum and evolutionary state in general. We deﬁne the conformal time τ by the comoving path that a light signal passes in this time interval: Dierck-Ekkehard Liebscher: Cosmology STMP 210, 79–117 (2004) c Springer-Verlag Berlin Heidelberg 2004

80

4 Cosmometry

Fig. 4.1. The ﬁeld of view. Electromagnetic radiation brings us information along the past light cone, i.e. from its surface. Investigations of fossil records yield information about the ﬁreball (local properties of cosmic structures) and the primordial nuclear synthesis (local concentrations of helium and deuterium) in our local past, i.e. about the interior of the past light cone. Of course, we observe the state of the fossils on the past light cone as well

dτ = a−1 [t]dt . We can describe space-time as a sequence of past light cones of a line of the fundamental congruence. We then use the diﬀerence between the conformal time to obtain a new time coordinate w = τ − χ, which labels the past light cones on our world line. We obtain (4.1) ds2 = a2 [w + χ] dw2 + 2 dw dχ − r2 [χ](dθ2 + sin2 θ dϕ2 ) . The coordinate χ is now a light-like, or null, coordinate. That is, the coordinate lines for which w, θ and ϕ remain ﬁxed are the world lines of photons directed towards the reference line χ = 0. If the present redshift z is a monotonic function of χ, we may substitute z for χ and obtain ds2 = a2 [z]r2 [χ[z]](dθ2 + sin2 θ dϕ2 ) on our light cone. We get rid of the unknown present value a0 by performing the integration in the following equation:

4.2 Horizons

t0 χ= t

RH c dt = a[t] a0

z

81

dz . h[z]

0

When the curvature is diﬀerent from zero, the quotient RH /a0 is just the square root of the parameter κ0 , and we can write √ RH a[z]r[χ[z]] = (1 + z)−1 √ sin [ κ0 Z[z]] , Z[z] = κ0

z

dz . h[z]

0

In this formula, only observable parameters enter.

4.2 Horizons The past light cone delineates the visual ﬁeld in every space slice t = const. The visual ﬁeld grows as we dive deeper into the past. This growth is limited absolutely by the existence of the cosmological singularity. It is limited in practice for observation with light by the ﬁreball, i.e. the intersection of the past light cone with the space of the time when the opaque state of the universe ends. The paths of photons that are arriving at present form lines on the surface of the past light cone. We choose χ0 = 0 for our position and obtain these lines from t1 χvisual

ﬁeld at t1

=−

c dt . a[t]

t0

We substitute z through (3.18) and obtain χvisual

ﬁeld at t1

RH = a0

z1

dz . h[z]

(4.2)

0

An absolute limit exists when the integral t1 χ1 − χ 0 = t0

c dt RH = a[t] a0

z1

dz h[z]

(4.3)

0

converges for t1 → tsingularity or z1 → ∞. It is our horizon of vision (technically, the particle horizon), and is sketched in Fig. 4.2. A similar situation can arise when we consider the future light cones. The intersection of the future light cone with some space t = t1 delineates the region in this space that can be reached by a signal or a visitor that starts now. This is a bounded region for every cosmological instant t1 ; it is our ﬁeld

82

4 Cosmometry

Fig. 4.2. The geometry of light rays. The parabolas are the world lines of light in a Gamow universe. An observer at B sees the region BA BB on the ﬁreball. At the instant D, this region would be only DA DB . D has been chosen so that the point O on the singularity inﬂuences just this region DA DB on the ﬁreball. Such a region is clearly smaller than the present ﬁeld of view BA BB . – The two signals that leave the comoving source at S1 and S reach the observer at B1 and B. The change in the time interval (B − B1 > S − S1 ) indicates the cosmological redshift. The comoving distance between the two signals remains constant, i.e. SS2 = BB2 . The horizon of B is the region HA HB on the singularity

of action. Again, these regions can have a common bound for any time, a horizon of action (technically, the event horizon). The ﬁeld of action is given by a formula similar to (4.2) with only the signs changed; t1 χaction

ﬁeld at t1

= t0

RH c dt =− a[t] a0

z1

dz . h[z]

(4.4)

0

A horizon of action exists when these integrals converge for t1 → ∞ or 1 + z1 → 0. In that case, the expansion of the universe can proceed so fast1 that even light cannot keep pace. In the comoving coordinates, where the milestones are the fundamental observers or galaxies themselves, that is, the 1

There is no velocity of expansion. Homogeneous expansion is characterised by a rate (the Hubble constant H).

4.3 The Cosmological Redshift

83

numerical value of the speed of light decreases with expansion when distances are measured taking the average distance between galaxies as unit. The horizon of vision, or particle horizon, increases with the time t0 , and the horizon of action, or event horizon, decreases. The older we get, the more we can see, and the less we can do. The horizon of vision at time t0 has – in the comoving measure – the same size as the region of inﬂuence of an event O on the singularity at time t0 . The radius of both regions is the comoving distance that a light ray can travel since t = 0 (Fig. 4.2). It is remarkable that in every universe which expands with some power law, i.e. a[t] = a0 ((t/t0 ))n , for n > 1 the Newtonian potential of the mass inside the horizon remains constant. When the Friedmann equation holds, we obtain Φ=

2 3 2 4π Rhorizon 1 GMhorizon = = 2 H c2 Rhorizon 8π 3 c2 2c

2 n t

ct0 t 1 − n t0

2 =

n2 . 2(1 − n)2

A cosmological model with a (particle) horizon always has a singularity, i.e. a ﬁnite age. In the Milne universe, where a[t] = V t, no horizon exists. The de Sitter universe, with ﬂat space slices, shows an event horizon only. The integral (4.2) converges for t0 → ∞: a[t] = a0 exp[Ht] , χ1 − χ0 =

RH (exp[−Ht0 ] − exp[−Ht1 ]) . a0

The static Einstein universe has no horizons. The Einstein-de Sitter universe has a particle horizon (and all models with a barotropic mixture and a positive leading component with n > 2 have a particle horizon as well). We obtain

2/3 a[t] = a0

t t0

ct0 , χ1 − χ0 = 3 a0

3

t1 −1 . t0

The Milne model has no horizon. We know that a[t] = V t here, and both integrals ((4.2) and (4.4)) diverge. In a Friedmann–Lemaˆıtre universe with both λ and Ω positive, for t → 0 the universe is approximately Einstein–de Sitter – a horizon of vision exists – and for t → ∞ the universe is approximately de Sitter – a horizon of action exists, too. In particular, the concordance model with λ = 0.7 and Ω = 0.3 has an event horizon at a distance of about 1.14 RH , and a ﬁeld of view at z = 1.8 at about the same distance (in the comoving measure). Galaxies with a redshift z larger than 1.8 cannot be reached from the earth any more (Fig. 4.3).

One remark seems necessary for safety. We have calculated the horizons that arise through the geometric properties of space-time. Our eﬀective horizon is limited further owing to the imperfectness of our telescopes, the processes in the intervening matter and the faintness of the possible sources.

4.3 The Cosmological Redshift Observation of distant objects needs light, and light needs time. Any signal from distant objects tells us about a distant past. A photon that was emitted

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4 Cosmometry

Fig. 4.3. The distance of the event horizon. The comoving distance of the (future) event horizon is represented by the redshift of the light which arrives from this distance at present. We chose a inverse scale for 1 + z

at a time temission and is detected today has travelled a distance χ, which can be calculated through integration of ds2 = 0, or t0 c dt = a[t] dχ , χ[t0 ] − χ[temission ] =

c dt . a[t]

temission

After a substitution of t by t[z] as in (3.18), we obtain RH χ[temission ] − χ[t0 ] = a0

zemission

dz ; h[z]

(4.5)

1

this distance is sketched in Fig. 4.2. This is the invariant comoving distance of a source that emitted its light at temission . The product a0 χ is the physical distance at present. The congruence χ[t, t0 ] found through (4.5) is invariant with respect to translations in χ (not to translations in t, of course). The comoving distance between two wave crests of the same photon is constant, as well as the comoving distance between two photons on the same track. The ﬁrst eﬀect leads to a constant comoving wavelength. The physical

4.3 The Cosmological Redshift

85

wavelength that enters the relation E = hc/λ changes with time through the relation λcomoving λphysical [t] = . a[t] a[t0 ] Therefore, we obtain λphysical [t0 ] λphysical [temission ] = . a[t0 ] a[temission ]

(4.6)

The quotient of the two wavelength deﬁnes a shift λphysical [t0 ] a[t0 ] = = 1 + zemission . λphysical [temission ] a[temission ] Every wavelength is shifted towards the red by the same factor because of the expansion. The solutions of the covariant wave equation g ik

D ∂Φ 1 ∂2 3a˙ ∂Φ 1 = 2 2Φ+ 2 − 2 ∇2 Φ = 0 , i k ∂x ∂x c ∂t ac ∂t a

can be expanded into a Fourier series Φ[t, xα ] = k Φk [t] exp[ik, x] with constant wave-number vectors k. The wave numbers k refer to comoving coordinates, i.e. comoving wavelengths remain constant.

The interval of time that elapses between two consecutive signals or particles on the same track at some comoving position is given by the conservation of the (inﬁnitesimal) comoving distance between them: c dt0 c dtemission − =0. a[t0 ] a[temission ] This can be written as a dilation of the detection rate: νemission a[t0 ] =1+z . = ν0 a[temission ]

(4.7)

This is a statement about frequencies in general, and not only a corollary to the statement about the wavelength in particular. The cosmological redshift (4.7) is usually interpreted as a Doppler eﬀect that arises from a recession velocity. This interpretation is problematic because the fundamental congruence represents no (peculiar) velocity at all. The indexing of an event by coordinates can be freely chosen, and the deﬁnition of relative velocities across cosmic distances is devoid of observable meaning. In the theory of general relativity, relative velocities are deﬁned only for two world lines that pass a common event.

86

4 Cosmometry We can, however, ﬁnd a kind of distance that changes through expansion. When we recall the deﬁnition (imposed by law in the International System) of distance as the travel time of light, we obtain the following: s = c(t0 − temission )

at ﬁxed values of χ .

This distance changes with the time t0 of the observer. A (formal!) recession velocity z ds =c v= dt0 1+z can be deﬁned. It never exceeds the velocity of light, and is not deﬁned for objects beyond our horizon. Using this deﬁnition of velocity, the cosmological redshift can obtained in the form 1 λ + ∆λ =1+z = . λ 1 − v/c This formula is similar but not identical to the formulae for the acoustic Doppler eﬀect (1 + z = 1 + v/c) and for the relativistic Doppler eﬀect,

1+z =

1 + v/c , 1 − v/c

but it is identical to the formula for the acoustic Doppler eﬀect for a moving observer. In spite of this discussion, the generation of the redshift by the expansion of the universe is beyond doubt.

For some time, it was argued that the redshift could be produced by some kind of ageing of photons. However, since the observation of the time dilation of supernova cooling [4, 15], we know that expansion is the correct scheme. It predicts not only the redshift, but also the dilation of the reaction times such as the cooling times of supernovae, and both by the same mechanism, as expressed by (4.7). A pure ageing of photons could not alter the kinetic rates of photon production. From the point of view of thermodynamics, there is work done by the radiation pressure when a comoving volume expands. We already know that the radiation density changes in accordance with radiation ∝ a−4 . Recalling the Stefan–Boltzmann law (radiation ∝ T 4 ), we obtain T ∝ a−1 . The number of photons in a comoving volume remains constant, so the energy of an individual photon has to vary in accordance with hν ∝ a−1 . This is consistent with our ﬁndings about frequencies. Since all frequencies show the same redshift, the form of Planck’s law does not change: the Planck exponent hν/kT is invariant. This again is consistent with our ﬁndings. The equation a[t]T [t] = constant

(4.8)

is the cosmological corollary of Wien’s law. Let us suppose that we know the function a[t]. The formula (4.5) then provides the relations between χ, r and z. We deﬁne the Hubble radius

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87

RH = c/H0 and the relative expansion rate h = H/H0 . The fundamental relation is the dependence of the comoving distance χ on the redshift, i.e. dχ RH = , dz a0 h[t[z]]

RH χ[z] = a0

z

dz , h[z]

(4.9)

0

which is the basis for all other geometric formulae. In particular, since (RH /a0 )2 is the absolute value |κ0 | of the relative curvature, we can write √ RH dz a0 r[χ[z]] = √ sin κ0 . (4.10) κ0 h[z] In the limit of zero curvature, this is

a0 r[χ[z]] = RH

dz . h[z]

The circumference of the sphere around us that is deﬁned by a redshift z is obtained in the comoving measure from U [z] = 2πr[χ[z]] .

(4.11)

The area of a sphere around us that is deﬁned by a redshift z is obtained in the comoving measure from Ωcomoving = 4πr2 [χ[z]] .

(4.12)

The volume of a sphere around us that is deﬁned by a redshift z is obtained in the comoving measure from RH dVcomoving = 4πr2 [χ[z]] . dz a0 h[z]

(4.13)

In models with matter component that obey barotropic equations of state, the Friedmann equation deﬁnes the Hubble parameter H as a function of the expansion parameter a, i.e. as a function of the redshift z itself. In a de Sitter model, we obtain χ[z] =

RH z. a0

In a Einstein-de Sitter universe, we obtain RH χ[z] = 2 a0

1 1− √ 1+z

.

Locally, we expand a[t] into a[t] = a0 (1 + H0 (t − t0 ) − (1/2)qH02 (t − t0 )2 ), and we obtain q+3 RH z 1− z . χ[z] = a0 2

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4 Cosmometry

In a closed (positively curved) space, each point has an antipode. We can construct models for which the antipode lies inside the (particle) horizon. In this case, the horizon loses its property of limiting causal action. When the antipode lies inside the horizon, every point of the space at t = tsingularity lies inside. Geometrical optics has to take the antipode into account as focal point. At the redshift of the antipode, large correlations and apparent sizes should be observed [3]. Figure 4.4 maps the redshift of the antipode of a space that is closed with κ0 > 0. The antipode lies inside the horizon for all Friedmann–Lemaˆıtre models with a minimum square of the relative expansion rate h2min < 0.83.

Fig. 4.4. The redshift of the antipode. The map shows logarithmically scaled isolines of the redshift of the antipode in minimum coordinates. The deeper and nearer the minimum of the expansion rate the nearer (in z) the antipode is

4.4 Distance Deﬁnitions in the Expanding Universe Distances are determined by triangulation with light. We can distinguish two classes of distance determination. The ﬁrst class is the use of objects of a given luminosity, where we observe the brightness, i.e. an angle subtended at

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89

the position of the source that measures the share of the luminosity that is captured by the surface of the detector. The second class is the use of objects of a given physical size, where we observe the apparent size, i.e. an angle subtendend at the position of the observer. We always refer to some kind of basis ω at one position to infer the aperture angle α spanned by ω at another position A. Let us start with objects of a given luminosity L. The brightness I of such a source indicates the angle subtended by the detector area at the position of the source. In Euclidean geometry, the energy output of the distant source is distributed over a sphere around the source that passes through the position of the observer, and a detector of aperture area ω collects a power αL, where the angle α given by ω , α= 4πD2 Hence, a comparison of brightnesses means a comparison of distances D. In an expanding universe, things are not so simple, however. First, we have to replace D by a0 r[χ[z]]. The present radius of the sphere deﬁned by χ[z] must be used. Second, we have to take account of the cosmological time dilation both in the photon frequency and in the photon rate. The energy of the individual photon decreases in accordance with hν = hνemitted

1 , 1+z

and the number per unit time decreases as well: ∆N 1 ∆N . = ∆tobserver ∆temitter 1 + z The bolometric brightness varies as I=

L (1 +

z)2 4πa20 r2 [χ[z]]

.

Comparing this result with the Euclidean result, we infer a distance Dbolometric = r[χ[z]]a0 (1 + z) . The bolometric brightness is not the end of the story if we have to account for an evolution in luminosity or for the spectral characteristics of a particular l[ν] dν, I = source or a particular detector. In the case of a spectrum L = i[ν] dν, we obtain i[ν] dν =

l[ν(1 + z)](1 + z) dν . (1 + z)2 4πa20 r2 [χ[z]]

The quotient of the above quantity with the bolometric expression, K=

l[ν(1 + z)](1 + z) , l[ν]

90

4 Cosmometry is called the K correction. If the brightness is determined with a spectral sensitivity η[ν], we obtain

η[ν]i[ν] dν =

I=

η[ν]l[ν(1 + z)](1 + z) dν . (1 + z)2 4πa20 r2 [χ[z]]

We now consider catalogues of objects with a present universal density n. These catalogues are assumed to have been compiled by brightness. We then suppose a luminosity distribution dn = Φ[L]dL. We cite the standard distribution found by Schechter: L Φ∗ L −α exp − ∗ (4.14) Φ[L] = ∗ ∗ L L L ∗ −2 3 −3 ∗ 10 −2 Φ = 10 h Mpc , L = 10 h L , α = 1 . The number of objects of redshift z down to a (bolometric) brightness Flimit is given by

∞

dN [Ilimit ] RH = 4πa20 r2 [χ[z]] dz h[z]

Φ[L] dL , Imin 4πr 2 [z]a2 (1+z)2 0

and down to a given (bolometric) surface brightness Σlimit we ﬁnd

dN [Σlimit ] RH = 4πa20 r2 [χ[z]] dz h[z]

Φ[L, ω] dL dω , L>Σlimit

ω(1+z)4

These formulae must be considered in determinations of luminosity distributions.

The other method is to use objects of a given size. We may determine the apparent size, i.e. the angle subtended by the object at the position of the observer. The apparent size is the quotient of two quantities. It compares the physical area ω of the (projection of the) object with the area of a sphere that is centred at the observer and passes through the position of the object: α=

ω 4πa2 [z]r2 [χ[z]]

=

ω(1 + z)2 . 4πa20 r2 [χ[z]]

(4.15)

The distance is obtained from the apparent size using Dsize = r[χ[z]]a[z] = r[χ[z]]

a0 . 1+z

It is important to note that the angle α has a minimum for all models with a (particle) horizon. Any object of a given size ω has a minimal apparent size α because the redshift z, but not the radial coordinate r[z], increases without limit. Figure 4.5 maps the redshift of minimal apparent size for objects of given physical size. A particularly important value is the size of the horizon or, more precisely, the size of the visual ﬁeld on the ﬁreball. We obtain the circumference

4.4 Distance Deﬁnitions in the Expanding Universe

91

Fig. 4.5. The redshift of minimal apparent size. Linearly spaced isolines of the redshift at which an object of given physical size has its smallest angular extension. Because of the expansion of the universe, the angular size increases again beyond this value. The Ω0 –λ0 plane is shown. The value for the Einstein–de Sitter universe is z[αmin ] = 1.25 z recom

Ucomoving [zrecom ] = 2πRH

dz . h[z]

0

We show this value in the Ω0 –λ0 plane (Fig. 4.6). The size is measured at present through an angular scale in the cosmic microwave background which corresponds to a physical scale that can be calculated (Sect. 8.4). We need this value to evaluate the ﬂuctuations of the microwave background radiation (Sect. 5.3). Comparing the two methods of distance deﬁnition, we obtain Dbolometric = Dsize (1 + z)2 . Both measures can be compared with the Hubble distance DHubble =

c z. H

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4 Cosmometry

Fig. 4.6. The size of the ﬁreball. The ﬁreball is the sphere that corresponds to the redshift zrecom ≈ 1100. It slightly depends, however, on the contributions of the diﬀerent matter components to the Friedmann balance. This is neglected in the ﬁgure

It is remarkable that the surface brightness depends only on the redshift and not on the cosmological model: Σ=

L I = (1 + z)−4 . α ω

The decrease with the fourth power of (1 + z) can be considered as a test of the expansion interpretation of the redshift. In a tired-light interpretation [21], the rate suppression and the expansion do not occur, and the decrease of surface brightness varies as (1 + z)−1 only. However, the rate suppression has been observed in the afterglow of supernovae.

4.5 Distance Determinations Distance determinations in the far neighbourhood of our galaxy are critical for pinning down the local expansion rate. The distances of galaxies are far less

4.5 Distance Determinations

93

precisely known than the redshifts of these galaxies, which can be determined through spectroscopy. Distances have to be measured through trigonometry. As we have seen in the preceding section, there are two ways to span a triangle across space. The ﬁrst is to identify a local reference length or area ω, to measure the angle α subtended by the object at the distant point, and to infer its distance from the quotient ω/α. The second method is to identify a reference length or area ω at the distant object, to measure the subtended angle α here and to infer the distance from the corresponding quotient. We have seen in the previous section how the cosmic expansion has to be taken into account. For both methods, absolute procedures (where we know the physical value of the basis or the angle at the distant source) are rare, but relative procedures (where we know the basis or the angle only to be constant throughout space) can be used as well – with additional uncertainty, of course. An elementary method of the ﬁrst kind is provided by the trigonometric parallax. The basis is the orbit of the earth. The apparent position of a star describes an ellipse in front of the background of more distant stars. This ellipse is a projection of the orbit of the earth and shows the apparent angle subtended by the orbit of the earth at the position of the star. When the major diameter of this ellipse is 1 arcsec, we call the distance 1 parsec. The parsec is a multiple of the astronomical unit (Fig. 4.7).

Fig. 4.7. Trigonometric parallax

94

4 Cosmometry The ﬁgure shows the orbit of the earth in the ecliptic plane and a star S above this plane. The determination of the trigonometric parallax presupposes that the sum of the angles of a triangle is equal to 180o . The two angles EAS and EBS are observed as apparent heights above the ecliptic; hence all the elements of the triangle are determined, since the size AB of the earth’s orbit is known. Bessel developed this method, and the ﬁrst star to have its distance measured was 61 Cygni. Its angle ASB is about 0.3 arcsec, and its distance is 3.4 pc ≈ 1014 km

The precision of the measurement of small angles limits the distances that can be reached by this method to less than 100 pc. The satellite HIPPARCOS and its successors are shifting this limit further out. We have already inferred the distances of the planets (and of the sun) with an earth-bound basis. The historically important example of this is the determination of the astronomical unit through observation of the transits of the planet Venus. The moving-cluster parallax provides an elementary method of the second kind. Here, the basis is the physical value of a lateral velocity that can be determined from the radial velocity and the angle subtended by the apex of a set of proper motions. The apparent size is the proper motion itself. In particular, open clusters (and, with HIPPARCOS, also globular clusters) reveal their distance through the collective motion of their members. Projected onto the sky, a collective motion with a uniform velocity deﬁnes a point towards which the motion seems to converge or from which the motion seem to come, i.e. an apex. The angle φ between this apex and the position of the cluster, the proper motion ω of its members, and their radial velocity v (measured through the frequency shift) yield the distance D (the moving-cluster parallax, Fig. 4.8): Dω . tan φ = v The observer sees the proper motion SE as a projection of the true motion SW of a stellar cluster onto the sky (perpendicular to the line of sight) and determines an apex (vanishing point) F . The angle SAF is equal to the angle RSW between the radial component (radial velocity) and the true motion. The angle RSE is a right angle. Hence SE = SR tan[ RSW ]. We now measure the proper motion as an angle, and the radial velocity as a true length per unit time. The ratio of these two provides the distance.

A similar measure is provided by the stochastic internal motion of a cluster. We can compare the velocity dispersion of the proper motion with the dispersion of the radial velocity and obtain 2 v 2 = D2 ω 2 . Both of these moving-cluster methods compare a physical size (given by the radial velocity) with an apparent size (given by the proper motion), so they belong to the second kind of distance determination. The most important object whose distance has been determined through the moving-cluster parallax

4.5 Distance Determinations

95

Fig. 4.8. Moving-cluster parallax

is the open cluster known as the Hyades, 46 ± 2 pc distant. This cluster plays a key role because it is used to calibrate the next set of distance indicators, which are all based on the comparison of luminosity with brightness. One is about to measure the apparent size of stars, in particular to compare the variation of radial velocity of the surface of Cepheids with the variation of apparent size (by direct interferometry [7] and by determination of the surface brightness [22]). This yields absolute distances to these Cepheids and absolute luminosities. With appropriate care, the maximal peculiar velocity in the radio components of a quasar can be used as a standard measuring stick. In analogy to the statistical parallax, the maximal relative proper motion is compared with the maximal relative radial velocity. The attenuation of the microwave background by the hot gas in large clusters of galaxies (Sunyayev–Zel’dovich eﬀect) provides a diameter measurement of the gas cloud. Up to a known temperature dependence H[T ], the X-ray surface brightness of the gas is proportional to its diameter times the square of its electron density, i.e. SX = n2e H[T ]d. The attenuation of the microwave background by inverse Compton scattering is only linear in the electron density, i.e. F = ne G[T ]d. Hence F 2 /SX determines d, which can be compared with the angular diameter of the cloud [14].

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4 Cosmometry

Relative methods use standard candles or standard sticks. Their being standard is a hypothesis that must be backed up by theory or other observations, and we must be aware of the resulting uncertainty. We begin with the standard candles. These are relative indicators of the ﬁrst kind. Standard candles are assumed to have a universal luminosity, or at least a universal average luminosity with a small enough variance. The apparent luminosity is proportional to the inverse square of the distance. Newton supposed the stars and the sun to have the same absolute luminosity, i.e. I D∗ = . D I∗ The apparent luminosity of a star seemed to be about 10−11 of that of the sun, and the distance 3 × 105 AU. Of course, stars have absolute luminosities that are not all the same, although they vary by only three orders of magnitude, and the method seems not to be very good. But with respect to the brightest stars in a galaxy, the brightest planetary nebulae or the brightest galaxy in a cluster of galaxies, it might work, if we only could gauge the absolute luminosity of some of these objects. For stellar clusters, we might compare the Hertzsprung–Russell diagram (Fig. 4.9) with that of the solar neighbourhood. In this case the method reaches as far as 100 000 light years, or 30 kpc. We know of more than one class of objects which may serve as standard candles because of their calculable or ﬁxed absolute luminosity. First, we mention pulsating stars, the Cepheids, with a (two-branch) period–luminosity relation, and the RR Lyrae stars. When we observe such a star, its period tells us its absolute luminosity, and the observed apparent brightness tells us its distance. This method requires galaxies to be resolved into stars by a telescope. The Hubble Space Telescope has made impressive progress here. The brightest objects in a galaxy are not stars, but large clouds of ionised hydrogen (HII regions). They can be analysed like stars, and they were indeed mistaken for stars by Hubble himself. This led him to a wrong calibration of the scale, and a inverse expansion rate of only 2 × 109 years. The method itself reaches out to some 108 light years. The most important method at present is based on the observation of supernovae. SN Ia supernovae are believed to be very uniform in luminosity and in luminosity decay. The reason is that they start at the exact Chandrasekhar point, where white dwarfs become unstable by slowly accreting mass. Supernovae of this type yielded the ﬁrst conﬁrmation of the now accepted value of the cosmological constant, or present vacuum density, λ0 = 0.7 [16, 18]. For the second kind of distance determination, we need standard sticks. We do not know the size of the observed object, we only believe that it is universal. In this case we obtain a relative distance indicator that has to be calibrated before it is useful. The ﬁrst example is the size of compact radio sources. There are reasons to assume that this size is determined solely by their internal structure, independent of the cosmic environment and its

4.5 Distance Determinations

97

Fig. 4.9. Sketch of a Hertzsprung–Russell diagram. The positions of stars in a magnitude–colour or magnitude-temperature diagram show a characteristic pattern that may be used to ﬁnd the distance to a resolved star cluster by comparing the vertical position of the pattern with that of the Hyades, for instance. While the spectroscopic measurement of temperature or colour is independent of distance, the apparent magnitude contains the distance modulus as usual

evolutionary state. If this applies, their apparent size depends on the redshift in accordance with (4.15) [6]. The second example is the graininess of homogeneous surfaces. Let us assume a galactic disc of constant surface brightness due to a constant surface density of stars. A pixellised image contains in each pixel the light from a certain number N of stars. There is an average number E[N ] and a variance D2 [N ] with respect to the individual pixels. The distribution of N in each cell is a Poisson distribution, so D2 [N ] = E[N ]. For a large enough E[N ], we can approximate the distribution by a normal one, where N ∈ N [E[N ], E[N ]]. We measure the brightness I instead of a number. Now E[I] ≈ I0 E[N ], and D2 [I] = I02 D2 [N ] = I02 E[N ]. The quotient E 2 [I]/D2 [I] = N gives the number of stars in the solid angle determined by the pixel size. The number of stars is assumed to be a measure of the physical size, and hence we can compare the physical size with the apparent size and obtain a relative measure of the distance.

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4 Cosmometry

4.6 Determinations of the Expansion Rate It was Hubble who clearly stated the law of homogeneous expansion of distant galaxies, although de Sitter had already noted the redshift in the spectral lines of the dimmer galaxies. As mentioned above, Hubble’s relation v = H0 r

(4.16)

contained a factor H0 that was too large. The Hubble age, 1/H0 , is the age itself if we suppose constant velocities v. This age, a ≈ 2 × 109 years, fell short of the age of the earth, approximately 5 × 109 years, by a factor of 2.5 and nearly one order of magnitude short of the age of globular clusters. After the error in Hubble’s interpretation had been found, there was a long debate as to whether the value promoted by Tammann and Sandage [20], H0 = 50 km/(s Mpc) , H0−1 = 20 × 109 years , or the value promoted by deVaucouleurs [2], H0 = 100 km/(s Mpc) , H0−1 = 10 × 109 years , was more reliable. The drawback of the latter value was again the age of the Einstein–de Sitter universe then accepted as standard. In order to obtain a consistent age, it was necessary to abandon the Einstein–de Sitter model and to accept a positive cosmological constant, and even a positive curvature. Since the Hubble Space Telescope came into operation, it has been possible to observe individual Cepheids and type Ia supernovae [19] in galaxies beyond a redshift of z > 0.1 and pin the value down to H0 = 65 ± 5 km/(s Mpc) , H0−1 = 15 × 109 years .

(4.17)

The precise measurement of the ﬂuctuations in the microwave background and the analysis of the formation of structure have yielded an independent, equally precise determination of H0 , together with the value of the cosmological constant, λ ≈ 0.7. The value (4.17) of the Hubble factor is consistent with all available age determinations. Basically, the value of the Hubble factor is so diﬃcult to determine because its value is less important than its relative change for the various phenomena studied. In this respect, it somewhat resembles the Cavendish constant of gravitation, which is the least precisely known fundamental constant because gravitation plays only a minor role in the structure of the constituents of matter. Distances that cannot be measured by trigonometric and quasi-trigonometric methods are now always given in terms of redshift and suppose the Hubble law, (4.18) r = RH z . The corresponding unit is the Hubble radius, RH = c/H0 . The conversion to megaparsecs contains the Hubble factor H0 = h × 100 km s−1 Mpc−1 .

4.6 Determinations of the Expansion Rate

99

With this deﬁnition of h, the Hubble radius RH is 3000 h−1 Mpc. A redshift of z = 0.1 then corresponds to a distance r = 300 h−1 Mpc. For the new standard model with λ0 = 0.7, Ω0 = 0.3, we obtain – the physical radius to χ[z] at present, given by σ = a0 χ = RH dz/h[z]; 2 – the physical surface area at present, Ω0 = 4πa20 χ2 = 4πRH ( dz/h[z])2 ; – the corresponding distance, Ω/4π; – the physical surface area at the emission time, Ω[z] = 4πa2 [z]χ2 = 2 −2 4πRH (1 + z) ( dz/h[z])2 ; – the corresponding distance, Dpresent [z] = Ω[z]/4π; – the size distance, Dsize = a0 χ[z]/(1 + z); and – the brightness distance, Dbrightness = a0 χ(1 + z). As we explained above, we observe the physical surface area in form of the apparent magnitude, the comoving volume in the form of the number of homogeneously distributed objects. The various cosmological models diﬀer in the form of the mutual dependence of these observable quantities (Figs. 4.10– 4.14)

Fig. 4.10. Number–magnitude diagram, normalised for small numbers. The three universes shown are the Einstein–de Sitter universe, the 2003 concordance model (λ = 0.7, Ω = 0.3) and the Friedmann–Lemaˆıtre model with λ = 1.09, Ω = 0.05

100

4 Cosmometry

Fig. 4.11. Number-redshift diagram, normalised at z = 1. The three universes shown are the Einstein–de Sitter universe, the 2003 concordance model (λ = 0.7, Ω = 0.3) and the Friedmann–Lemaˆıtre model with λ = 1.09, Ω = 0.05

The extrapolation of the expansion into the past leads to a formal start of the universe, which can be modiﬁed by the occurrence of phase transitions, but not necessarily removed. A Friedmann model with a positive density has a ﬁnite age of a0 ∞ 1 da dz 1 = . t= H0 ah[a] H0 (1 + z)h[z] 0

0

This age can diﬀer substantially from the Hubble time tHubble = H0−1 . For the Einstein–de Sitter universe, the age is tEdS = (2/3)tHubble . In Fig. 3.13, the dependence of the age on the Friedmann parameters was given. The consistency of this age with the ages of other objects in the universe can be checked in the vicinity of the earth. We can consider old geophysical structures (rocks) on the earth, or determine the age of meteorites, and obtain approximately 5 billion years. These ages of these objects are determined mainly by comparison of the concentrations of the various products of radioactive decay. White dwarfs cool through radiation because their nuclear energy production has ceased. Their luminosity has a lower bound (L ≈ 10−5 L ). This

4.7 Curvature Determinations

101

Fig. 4.12. Magnitude–redshift diagram. The three universes shown are the Einstein–de Sitter universe, the 2003 concordance model (λ = 0.7, Ω = 0.3) and the Friedmann–Lemaˆıtre model with λ = 1.09, Ω = 0.05

shows that there is an upper bound on their age. This age is about 1010 years and indicates the age of the galactic disc.

4.7 Curvature Determinations Owing to the universal expansion, there is no direct measurement procedure of the curvature of space that is independent of the Friedmann equation. Unlike the curvature of a static space, it has to be determined through the form of the function H 2 [z] itself. The curvature leaves a trace in the deceleration parameter (3.15), which reads q =

−Λc2 /3 + (8πG/3)((1/2)0,cold + 0,radiation ) . Λc2 /3 − kc2 /a2 + (8πG/3)(0,cold + 0,radiation )

Together with a determination of H0 , 0,cold and 0,radiation , this equation may be used to estimate the values of Λ and ka−2 0 . In contrast to the curvature, the deceleration parameter is an eﬀect of ﬁrst order in the distance or the redshift.

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Fig. 4.13. Angular-size–redshift diagram. The three universes shown are the Einstein–de Sitter universe, the 2003 concordance model (λ = 0.7, Ω = 0.3) and the Friedmann–Lemaˆıtre model with λ = 1.09, Ω = 0.05

The question of an independent measurement of the curvature of space remain open. The curvature seems to be very small compared with earth-bound sizes, but the cosmological unit of the curvature kc2 /a2 is the square of the Hubble expansion rate, H 2 . Are there direct methods to measure the curvature? As we know, the curvature reveals itself in any comparison of the surface area Ω[z] with the volume V [z] of spheres deﬁned by z. Basically, we measure V [z] by counting presumably homogeneously distributed objects, and measure the area Ω[z] from the brightness of objects of presumably identical luminosity, listed in tables of N [m]. Locally, we may expand all quantities in powers of z. The Friedmann equation becomes h2 [z] = 1 + 2αz + (2α + κ0 )z 2 , α =

3 Ω0 − κ0 = 2q + 2 . 2

We obtain the integrand 1 3 h−1 [z] = z(1 − αz − βz 2 ) , β = α + κ0 − α2 . 2 2 The Gaussian radius is given by

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103

Fig. 4.14. Spectral-magnitude–redshift diagram. This diagram shows the expected change of magnitude for a thermal source of temperature T, observed in a spectral band centred on the frequency ν. We have chosen here the case kT = hν

1 a0 1 2 r[z] = z 1 − αz − (2β + κ0 )z . RH 2 6 The area is found to be

3 2 2 1 1 − αz − Ωcomoving [z] = 2β + κ0 − α z 3 4 15 1 2 2 = 4πRH z 1 − αz − 2α + 2κ0 − α2 z 2 . 3 4 2 2 4πRH z

As regards the volume, we have to evaluate 9 1 3 Vcomoving [z] = 4πRH dz z 2 1 − αz − 2α + 2κ0 − α2 z 2 3 4 2 1 − αz − βz 4π 3 3 1 5 7 1 3 = RH z 1 − αz − α + κ0 − α 2 z 2 . 3 2 5 3 6 4 We obtain a result independent of RH in the quotient of interest, V 2 /Ω 3 . In addition, the linear terms compensate each other, and the quadratic term contains only κ0 :

104

4 Cosmometry 2 Vcomoving 4π = 3 Ωcomoving 9

9 2 3 1 + κ0 z + O[z ] . 15

The curvature can be determined from the second-order deviation from the Euclidean expectation. When we count, we measure Vcomoving directly, but when we observe the brightness, we obtain (1 + z)2 Ωcomoving . It is N –m diagrams that we have to evaluate. N –z diagrams reveal a ﬁrst-order eﬀect 3 2 z (1 − αz)dz as well as m–z diagrams such such as dVcomoving [z] = 4πRH 2 2 as Ωcomoving [z] = 4πRH z (1 − αz). This ﬁrst-order eﬀect cannot distinguish between the two parameters Ω0 and κ0 . At present, the precision of catalogues is not suﬃcient. The details of the microwave background ﬂuctuations are a better means, and indicate that the curvature is small (|κ0 | < 0.05; see Chap. 8). It should be noted that no high precision is necessary in determinations of the cosmological constant or the curvature of space in order to reach the conclusion that at early times before recombination, these two quantities do −3 not matter. If we know only that the components M0 , M2 a−2 0 and M3 a0 are of the same order of magnitude, we know that at recombination the curvature had only one-thousandth of the weight of the matter, and the cosmological constant only one-billionth. At earlier times, the relative weight of matter was overwhelming (in the case of pressure-free matter and and even more so in that of radiation). In the far past (z > 25 000) the universe not only was opaque but also was gravitationally dominated by radiation. The postrecombination universe, however, depends on the curvature of space and the cosmological constant.

4.8 Gravitational Lenses The bending of light that leads to only a small distortion of the sky in the case of the sun produces caustics, as well as multiple images with relative time delays and varying ampliﬁcation, in the case of distant stars, galaxies, and galactic clusters (Fig. 4.15). We shall demonstrate this here only for the simplest arrangement, that of a point-like source and a point-like lens (Fig. 4.16). In a Friedmann universe, the orbits of light rays are straight lines in comoving coordinates. When the source is taken to be the origin of polar coordinates, θ and ϕ are constant on a ray. The space is mapped onto the sky of the observer through an ordinary central projection. When there is a local source of gravitation in between, this projection is disturbed, the light cone is folded and the rays are bent. An observer in an appropriate position sees a signal from a source coming from more than one direction, and at more than one instant. The folding of the light cone is shown in Fig. 4.17. This ﬁgure shows the world lines of a source q, a deﬂector d, an observer b at rest in the space-time reference frame and the light cone of a ﬂash F . In the vicinity of the deﬂector, the propagation

4.8 Gravitational Lenses

105

1.5

B 1.0

ARC SEC

0.5

0.0

A2

C

-0.5

-1.0

A1 -1.5

1.5

1.0

0.5

0.0 -0.5 ARC SEC

-1.0

-1.5

Fig. 4.15. Gravitational lens. This image of MG 0414+0534 (constellation Taurus) was taken in the 22 GHz band. The radiation of the quasar in the background is bent by a galaxy which is not seen here. The galaxy was found in the visible band by the Hubble Space Telescope. Without the gravitational ﬁeld of this galaxy, the image of the quasar would be structureless and point-like. Reproduced from [5] with kind permission of J.L. Hewitt and the AAS

of light is inﬂuenced by gravitation. Independently of its exact form, this inﬂuence leads to a disturbance that necessarily folds the light cone. In the ﬁgure, the simplest kind of folding is shown. Let us ﬁrst consider a conﬁguration that is stationary in comoving coordinates. The observer is the origin, and the space R is mapped onto its sky J = {[θ, ϕ]}. This map cannot be true, because it is continuous. We may foliate the space R into a set of spheres Sz around the observer. Each individual sphere will be mapped onto the sky, and this map will no longer be unique after a folding of the light cone occurs. The sky, of course, can be mapped onto each sphere of the foliation, Pz : J → Sz , i.e. each point on the sky has a unique source point on the sphere Sz . Of course, some point of Sz may serve as image for more than one point on the sky.

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Fig. 4.16. Gravitational lensing by the bending of light. The light from the source is bent by some distribution of matter, the lens. The bending depends mainly on the gravitational potential of the mass in a cylinder, whose size is deﬁned by the point of closest approach of the photon to the lens, around its centre. This potential decreases outwards as well as inwards. Therefore, three rays will geometrically meet the observers position. The intensity along innermost ray is normally suppressed, so that the observer sees the source at two positions here

We shall apply two approximations. First, we are interested in only a small part of the sky that contains the lens. Hence we may approximate all spheres by planes. Second, the deﬂection of light is a small eﬀect in all practical cases. We approximate the eﬀect by a local change of the direction of the ray in the plane of the lens. Let us consider three planes perpendicular to the line of sight, those of the observer, the lens and the source, and use comoving Cartesian coordinates. Each bundle carried by a point ξ on the observer’s plane O maps the plane L = {η} of the lens onto the plane S = {ζ} of the source, and and each bundle carried by a point ζ on the plane of the source maps the plane L = {η} of the lens onto the plane O = {ξ} of the observer. The formula is symmetric because the deﬂection angle α depends only on the point η where the ray passes through the plane of the lens: η−ξ η−ζ + = −α[η] . ∆23 ∆12

(4.19)

The comoving distances are ∆12 = r[zL ] , ∆23 = r[zS ] − r[zL ] , where r[z] is taken from (4.10). For a point of mass M at η 0 , the deﬂection is given by 4GM η − η ∗ . α[η] = −(1 + zL ) a0 c2 (η − η ∗ )2 When we calculate in comoving coordinates, the factor is twice the Schwarzschild radius 2GM c−2 of the source in comoving coordinates, i.e. its present

4.8 Gravitational Lenses

107

Fig. 4.17. Folding of the forward light cone. The world line of the observer intersects the folded light cone of an event F three times. The observer sees the emission event at the three corresponding times, in three diﬀerent directions that correspond to the normals of the wavefront at the three events of intersection. In the case shown here, the ﬁrst image involves a deﬂection to the right, the second image involves a deﬂection to the left, and the third image is the least deﬂected. The folding always leads to an odd number of images. However, the intensity of the least deﬂected image is mostly suppressed

physical value times the factor (1 + z). If the lens has a more complicated structure, we must substitute a certain surface density µ[ξ, η] for the mass and integrate, 4Gµ[η ∗ ] η − η ∗ = −(1 + zL ) grad Φ[η] , (4.20) α = −(1 + zL ) d2 η ∗ a0 c2 (η − η ∗ )2 2Gµ[η ∗ ] Φ[η] = d2 η ∗ ln[(η − η ∗ )2 ] . a0 c2 The maps Q : L → O and P : L → S are given by the implicit (4.20). The observer at the point ξ sees in the direction of η some point ζ = Pξ [η]. This function, in the generic case, allows us to write ζ + dζ = Pξ [η + dη] and to calculate a relative ampliﬁcation of area,

108

4 Cosmometry Aarea

−1 ∂η ∂η = . ∂ζ ∂ζ 0

If ζ is seen at η from ξ, we have ξ = Qζ [η], too. An evaluation of the inﬁnitesimal neighbourhood, ξ + dξ = Qζ [η + dη], yields the relative ampliﬁcation of brightness, −1 ∂η ∂η Abrightness = . ∂ξ ∂ξ 0 We now calculate the case of the size:

∆23 − ∆23 (1 + z2 ) grad Φ , ∆ 12 ∂ζ ∂ζ = det det ∂η ∂η 0

ζ = η 1+ A−1 size

= 1−

∆12 ∆23 (1 + z2 )∆Φ + ∆13

∆12 ∆23 ∆13

2

(1 + z2 )2 det

∂2Φ ∂ηi ∂ηk

Since this formula does not vary when the source plane is replaced by the observer’s plane, the magniﬁcation in size is equal to the ampliﬁcation in ﬂux, i.e. Asize = Abrightness . The surface brightness is not changed by lensing. This is valid not only in our approximation, but also for a general projection through a gravitational ﬁeld. For a point lens at η = 0, the Laplacian of Φ vanishes, and the ampliﬁcation becomes θ4 η4 = . A= 4 η − (∆12 Einstein )4 θ4 − 4Einstein Images are√ampliﬁed more the nearer they are to the Einstein radius. Images with θ < 4 2Einstein are suppressed in brightness. This is the reason why the images seen in the well-known lenses do not have an odd number of components but have an even number: the innermost component lacks the necessary brightness. In an expanding universe, we have to be careful with the deﬁnition of areas. When any comoving distance is translated into a physical area, we must take account of the expansion parameter a at the times when the light passes through S, L and O. The plane L represents the sky, i.e. η = θr[zL ] =

1 y. a[z]

The physical coordinates on L are given by y = a[zL ]η = θr[zL ]a[z] . The physical coordinates on S are z = a[zS ]ζ .

4.8 Gravitational Lenses

109

For a simple point lens at η = 0 and an observer at ξ = 0 we obtain, in terms of the angles θ = η/∆12 and β = ζ/∆13 the equation 2 β = θ 1 − Einstein , θ2 2Einstein =

∆23 4GM (1 + z2 ) . ∆12 ∆13 a0 c2

(4.21)

A source exactly in line with the observer and the lensing mass point will be seen as a circle in the sky. The apparent radius of this circle is given by the Einstein radius of (4.21), i.e. β = 0 : θ = Einstein . In conjunction with the deﬂection of light, there is a time delay in a signal passing through a gravitational potential well, and this delay is proportional to the deﬂection: ∆t =

2Gm(1 + zL ) 2Gm 2Gm(1 + zL ) = 2 = 2 . a0 c2 η c r[zL ]a[zL ]η c RH ( dz/h[z])θ

This time delay is superposed on the geometric delay caused by the bend in the ray. Any two apparent positions have a diﬀerence in their gravitational time delay that is proportional to ∆[∆t] =

2Gm(1 + z) ∆[θ−1 ] . RH ( dz/h[z])

If we know m from the size Einstein of the apparent image, we can infer the present Hubble constant H0 (Fig. 4.18). Two other elementary lenses are the isothermal sphere with µ[r] = 1/r, which leads to a constant deviation in (4.20), η−ζ 4Gµ[η ∗ ] η − η ∗ η−ξ η + = (1 + zL ) d2 η ∗ =A , 2 ∗ 2 ∆23 ∆12 a0 c (η − η ) η and a lens with a constant surface density µ, which leads to the ordinary lens, where α = −Aη. The eﬀect of a lensing galaxy can be well described by such a constant-density plane in combination with some shear γ, i.e. µ−γ 0 α= η. 0 µ+γ In the case of a statistically distributed lens, or a lens with a small-scale variation in density, the projection onto the observer’s plane yields caustics (Fig. 4.19). The sky seen from a given position shows many images (Fig. 4.20, and, drastically, Fig. 4.21).

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Fig. 4.18. Gravitational delay between images. Diﬀerent paths through the gravitational ﬁeld yield diﬀerent time delays along the light-rays. When the source is variable it is possible to measure the diﬀerences in arrival time [9, 11]. The ﬁgure shows the ﬁt between the light-curves of two images with a shift of 417 and 420 days. Reproduced from [11] with kind permission of J. Wambsganss and the AAS

The half-space behind the lens can be mapped stereoscopically. We imagine the comoving space, with coordinates χ, and leave the lens plane ﬁxed at η3 = 0. The half-space ζ3 > 0 of sources ζ is viewed from some point ξ in the half-space of the observer, ξ3 < 0. Any one-eyed observer maps the half-space of sources onto the lens plane, where some points have more than one image. This map depends on the position ξ of the observer’s eye. Let us consider the image η of some source point ζ, seen from ξ = [0, 0, ξ3 ], ξ3 < 0: η−ζ η−ξ + = −α[η] . −ξ3 ζ3 The second eye of the observer, at ξ +dξ, sees an inﬁnitesimally diﬀerent image:

4.9 Quasar Absorption Forests

111

Fig. 4.19. Caustics of statistically distributed lenses. The intensity of the bent light is maximal at positions where the wavefronts just fold. When the lenses are statistically distributed, the caustics can be complicated. Reproduced with kind permission of J. Wambsganss from his homepage, copy in [23] dη − dξ dη ∂α η−ξ dξ3 + + =− dη . −ξ3 ζ3 ∂η ξ32 This is a linear equation for dη[dξ]:

dη =

ξ3 − ζ3 ∂α + ξ3 ζ3 ∂η

−1 ξ

dξ3 dξ − ξ3 ξ32

.

(4.22)

The stereoscopic image ζ ∗ [ζ, ξ, dξ] is found at the point of least distance between the straight lines ζ ∗ = ξ + λ(η − ξ) and ζ ∗ = ξ + dξ + µ(η + dη − ξ − dξ). We ﬁnd dξ (dη − dξ) . ζ ∗ = ξ − (η − ξ) (dη − dξ)2 For the Schwarzschild lens (4.22) can be evaluated by hand.

4.9 Quasar Absorption Forests Under the assumption that the hydrogen clouds that produce the Lyman absorption forests in quasar spectra are distributed so as to form some kind

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Fig. 4.20. The cluster lens A2218 with its conspicuously spherical lensing in the right part. Reproduced with kind permission of A. Kneib [8] and the AAS

Fig. 4.21. Droplet lensing (Credits: H. Gehlken, www.buxtehuder-fotofreunde.de)

4.9 Quasar Absorption Forests

113

of homogeneous structure in the universe, their statistics can be transformed into the relative expansion rate h[z]. These statistics can be observed in a region of comparatively large redshift (2 < z < 4). These redshifts exceed those found in galaxy counts by at least one order of magnitude. If we can make an assumption about the evolution of the structure in question, this leads us to a determination of the function a[t]. Let us calculate in comoving coordinates. The number of absorption lines is proportional to the comoving interval dχ of the path of the light ray, which can be transformed into a redshift interval by use of h[z]: N [z] dz = Nχ dχ = N [z]h[z] dχ . We use the notation – A[z] for the comoving number density of absorbers, – Q[z] for their cross-section in comoving coordinates, – S[z] for some characteristic physical size parameter in comoving coordinates, – L for a characteristic comoving size parameter, – D for some characteristic dimensional parameter, – R[z] for the density of absorbing hydrogen in the clouds, and – M [z] for the average density of absorbing hydrogen. A given physical size S[z] shrinks in comoving coordinates as a result of expansion, and this translates into an evolution of the cross-section Q[z] (Fig. 4.22). Ionisation induces an evolution of R[z] and M [z]. Condensation might change S[z] and R[z]. Merging and fragmentation will aﬀect A[z] and S[z]. The number of absorbers encountered by the light ray is again proportional to the number A of absorbers per unit volume and their cross-section Q in comoving coordinates. For nearly spherical clouds, we obtain Q ∝ S 2 [z]. For ﬁlaments of ﬁxed comoving length L, we obtain Q ∝ LS[z], and for two-dimensional comoving structures (walls of voids or bubbles; see Figs. 5.4 and 4.23), simply Q ∝ L2 . We can summarise these cases by N [z]h[z] ∝ A[z]LD S 2−D [z] .

(4.23)

The average density of absorbing hydrogen is given by M [z] ∝ A[z]LD S 3−D [z]R[z] .

(4.24)

The physical density of the absorbers may be described by R[z](1 + z)3 . This density, multiplied by the physical size S[z](1 + z)−1 in the direction of the ray, yields the column density W [z], which may be found from the equivalent width of the absorption lines: W [z] ∝ R[z]S[z](1 + z)2 .

(4.25)

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Fig. 4.22. The past light cone in comoving coordinates. The upper part shows the cut through a comoving void structure to compare with the cut through a comoving distribution of clouds of a given physical size (lower part)

The three (4.23), (4.24) and (4.25) can be combined to yield N [z]h[z]W [z] = M [z](1 + z)2 .

(4.26)

The quantities N [z] and W [z] are found by observation. This observation yields a relation between the expansion rate h[z] and the amount of absorbing hydrogen M [z]. In the approximation, N [z] ∝ (1 + z)ν , the exponent is found to be in the range 0.1 < ν < 0.4, and in the approximation W [z] ∝ (1 + z)ω , the exponent is in the range 1.8 < ω < 2.1. The result is that in an Einstein– de Sitter universe (h[z] = (1 + z)3/2 ) the absorbing mass is ionised rapidly (M [z] ∝ (1 + z)µ , with µ ≈ 1.8). If we suppose that the absorbing mass has a lifetime of the order of the Hubble time, a Friedmann–Lemaˆıtre model has to be chosen. Formally, the same holds if the walls of the voids are assumed to be universal (i.e. do not change in number) and each wall produces one line, i.e. D = 2. This yields N [z]h[z] = const. The corresponding Friedmann– Lemaˆıtre universe has a relative age H0 t0 ≈ 3, a curvature radius a0 ≈ 3.3RH and a density parameter Ω0 ≈ 0.014. When we compare these results with

References

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Fig. 4.23. Quasar absorption lines of a void structure. The diagram shows a structure that could be the source of the absorption spectrum shown in the lower part. Each absorption line could be taken as an indication of a passage through a wall of cold hydrogen clouds between two voids. Reproduced with kind permission of W. Priester [17]

the result of an analysis of the primordial nucleosynthesis (Chap. 6), we obtain a Hubble rate of 90 km/s/Mpc [12, 13, 17]. In addition, we can infer the present size Z[0] from the average separation of absorption lines. This size corresponds to the size of the voids in the local distribution of galaxies. In general, the procedures used to determine the evolution of the expansion rate combine three properties. We need assumptions about two of them in order to infer the third through theory. The theory yields only a relation between the three, which are the expansion law (h[z]), the evolution law (here M [z]) and the structure law (here D).

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5. Katz, C. A., Moore, C. B., Hewitt, J. N.: Multifrequency radio observations of the gravitational lens system MG 0414+0534, Astrophys. J. 475 (1997), 512–518. 105 6. Kellermann, K. I.: The cosmological deceleration parameter estimated from the angular-size/redshift relation for compact radio sources, Nature 361 (1993), 134–136. 97 7. Kervella, P., Nardetto, N., Bersier, D., Mourard, D., Coude du Foresto, V.: Cepheid distances from infrared long-baseline interferometry - I. VINCI/VLTI observations of seven Galactic Cepheids Astron. Astroph. (2003), astroph/0311525. 95 8. Kneib, J.-P., Ellis, R., Smail, I., Couch, W.J., Sharples, R.M.: Hubble Space Telescope observations of the lensing cluster Abell 2218, Astrophys. J. 471 (1996), 643–656. 112 9. Kochanek, C. S., Schechter, P. L.: The Hubble Constant from Gravitational Lens Time Delays, in: W. L. Freedman (ed.), Measuring and Modeling the Universe, Carnegie Observatories Astrophysics Series Vol. 2, astro-ph/0306040 (2004). 110 10. Kristian, J., Sachs, R. K.: Observations in cosmology, Astrophys. J. 143 (1966), 379–399. 79 11. Kundic, T., Turner, E. L., Colley, W. N., Gott, J. R., Rhoads, J. E., Wang, Y., Bergeron, L. E., Gloria, K. A., Long, D. C., Malhotra, S., Wambsganss, J.: A robust determination of the time delay in 0957+561A, B and a measurement of the global value of Hubble’s constant, Astrophys. J. 482 (1997), 75–82 (astroph/9610162). 110 12. Liebscher, D.-E., Priester, W., Hoell, J.: Lyman alpha forests and the evolution of the universe, Astron. Nachr. 313 (1992), 265–273. 115, 234 13. Liebscher, D.-E., Priester, W., Hoell, J.: A new method to test the model of the universe, Astron. Astroph. 261 (1992), 377–381. 115, 234 14. McHardy, I. M., Stewart, G. C., Edge, A. C., Cooke, B., Yamashita, K., Hatsukade, I.: GINGA observations of Abell 2281 - Implications for H0, Mon. Not. R. Astron. Soc. 242 (1990), 215–220. 95 15. Perlmutter, S., Pennypacker, C. R., Goldhaber, G., Goobar, A., Muller, R. A., Newberg, H. J. M., Desai, J., Kim, A. G., Kim, M. Y., Small, I. A., Boyle, B. J., Crawford, C. S., McMahon, R. G., Bunclark, P. S., Carter, D., Irwin, M. J., Terlevich, R. J., Ellis, R. S., Glazebrook, K., Couch, W. J., Mould, J. R., Small, T. A., Abraham, R. G.: A supernova at z = 0.458 and implications for measuring the cosmological deceleration, Astrophys. J. Lett. 440 (1995), L41–L44. 86 16. Perlmutter, S., Turner, M. S., White, M.: Constraining dark energy with SNe Ia and large-scale structure, Phys. Rev.Lett. 83 (1999), 670–673. 96 17. Priester, W., van de Bruck, C.: 75 Jahre Theorie des expandierenden Kosmos: Friedmann Modelle und der Einstein-Limit, Naturwissenschaften 85 (1994), 524–538. 115, 234 18. Riess, A. G., Filippenko, A. V., Challis, P., Clocchiattia, A., Diercks, A., Garnavich, P. M., Gilliland, R. L., Hogan, C. J., Jha, S., Kirshner, R. P., Leibundgut, B., Phillips, M. M., Reiss, D., Schmidt, B. P., Schommer, R. A., Smith, R. C., Spyromilio, J., Stubbs, C., Suntzeﬀ, N. B., Tonry, J.: Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998), 1009–1038. 96

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5 Matter and Radiation

In this chapter we consider how the source term in the Friedmann equation (3.2), i.e., individual terms of the right-hand side, 3kc2 Λc2 3H 2 = critical = − + cold + hot , 8πG 8πG 8πG 1 = λ 0 − κ0 + Ω0 + ω0 , can be determined through observation. Although this is not the place to acknowledge in detail the impressive amount of work that has been done, this work should be recalled in considering all the lapidary statements that will follow. The details of the astronomical and physical methods would form another book. We merely cite the literature [2, 4, 8, 24].

5.1 The Average Mass Density The direct determination of the mass M of a celestial body requires the observation of the motion of satellites in its gravitational ﬁeld. We know the masses of the sun and of the planets through observation of their natural and artiﬁcial satellites; we know the masses of double stars (the reduced mass of the two-body problem, to be more speciﬁc), the masses of some galaxies (where the rotation curves can be assumed to reﬂect Kepler motion) and the masses of clusters (where the velocity dispersion should balance the gravitational potential). We always measure the active gravitational mass, i.e. the product of the mass with the gravitational constant, GM. The mass of the Virgo cluster of galaxies can be seen from the infall of galaxies from its environment. A value of this mass can be obtained if this infall motion can be assumed to be a free fall since the recombination time (Ωinfall h2 ≈ 0.35). If we take suitable care, measurements of redshifts allow us to determine a large-scale ﬁeld of velocity that is superimposed on the overall cosmic expansion. The gravitational condensation process must be accompanied by such a velocity ﬁeld. This velocity ﬁeld generalises the infall motion onto the Virgo cluster [17, 23].

Dierck-Ekkehard Liebscher: Cosmology STMP 210, 119–142 (2004) c Springer-Verlag Berlin Heidelberg 2004

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In addition, it has been possible to evaluate the distortions of shapes caused by the light deﬂection around some of the smaller density enhancements. This distortion is called weak gravitational lensing. It is a kind of direct determination of the gravitational potential, but it requires a lot of assumptions, too. There are puzzles, however. First, the masses of individual stars can be estimated (with some uncertainties) through their relation to the luminosity and spectral type. This estimation is based on models of stellar evolution and on comparison with the determinations of the mass of binary stars. The mass of a galaxy that is condensed into stars can be estimated in the next step from the statistics of the stellar population. This mass and its distribution in the galaxy clearly disagree with the mass that is obtained from the rotation velocity. Second, we can perform analogous estimations and calculate population statistics for clusters of galaxies. Again, the mass obtained from the population statistics is smaller than the mass determined from the velocity dispersion. In order to determine the average density of the universe, the simplest way would again be to count the galaxies, correct the numbers for the identiﬁcation limits set by the minimum detectable brightness or surface brightness, and estimate their masses through their luminosity and their presumed stellar population. We call this mass the luminous mass. The ﬁrst obstacle to a precise determination is due to the large-scale structure in the distribution of galaxies which must be appropriately averaged over. The second obstacle is that the mass–luminosity relation has not yet been ﬁxed for the weak end of the distribution. The third obstacle is that the frequency of galaxies of low surface brightness is not known. The mass–luminosity relation of galaxies has been derived from the corresponding relation for stars. The question of how fast it converges for weak galaxies is open again. Nevertheless, one can give an estimate, luminous ≤ 10−28 kg m−3 . This means, for the present value of the density parameter, that Ωluminous h2 ≤ 0.005. Improvements in spectroscopy in all frequency ranges have allowed us to analyse the motion of stars in a galaxy and the motion of gaseous clouds far beyond the visual limits of a galaxy, and to use these results to derive corresponding models of the mass distribution in the galaxies. We thus obtain a direct determination of the mass. In particular, this works for rotation velocities in the discs of spiral galaxies (Fig. 5.1). Kepler’s third law yields masses that, in general, exceed the value of the luminous mass. The diﬀerence is called the dark mass, i.e. the mass of some unknown dark (transparent!) matter distributed in the halo of the galaxy [3]. When we correct the masses of galaxies correspondingly, the matter parameter grows to Ωgalactic h2 ≤ 0.015. A spherical distribution of density of the form

5.1 The Average Mass Density

121

Fig. 5.1. Rotation curves and surface luminosity. If we calculate the mass distribution by use of the observed surface luminosity and the stellar mass–luminosity relation (dashed line), the Kepler rotation velocity falls with increasing distance, as shown with the thin curved line. However, for comparatively large distances from the centre, a ﬂat rotation curve (thick line) is observed. This ﬂat rotation curve reaches further out than the visual light from the galaxy. This is taken as the mark of a dark-matter component that surrounds the galaxy in a kind of halo

[r] =

0 1 + (r/r0 )2

induces the Kepler velocity

v 2 [r] = 4πG0 r02 1 − and lim v = 0 and lim v = r→0

r→∞

r r0 arctan r r0

.

4πG0 r02 .

The statistics of the velocity distribution in a cluster of galaxies can be used to determine the mass of the cluster. If the sphericity of the cluster can be assumed, the velocity dispersion D2 [v] = (v − v)2 can be found through determination of the variance of the redshifts, i.e. the variance of the radial velocities. Assuming that the virial theorem can be applied, we obtain GM = 2D2 [v]/E[r−1 ] .

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The resulting values for the mass again exceed the masses obtained from counting. The diﬀerence is again attributed to a dark (transparent) matter distribution in which the galaxies are embedded (Fig. 5.2) [33, 41]. From this type of calculation alone, the density of the dark matter can be estimated to yield 0.1 ≤ Ωclusters h2 ≤ 0.7.

Fig. 5.2. Dark matter determines the potential of a cluster of galaxies. The gravitational potential in a galactic cluster is sketched at left as it is expected from the member galaxies, and at right as it is reconstructed from X-ray observations of lensing eﬀects. The main part of the potential is approximately spherical in spite of the asymmetric and inhomogeneous distribution of the member galaxies. It indicates a dominant and approximately spherically distributed dark-matter component

We shall see in Chap. 7 why for a long time Ω0 = 1 was expected, and in Chap. 8 why the present expectation is Ω0 = 0.3. We shall see in Chap. 6 why we have a comparatively precise value for the baryon component, which is based on the observation of the universal concentration of deuterium. As we have seen, the direct determinations show that there is dark (transparent) matter. There is, at most, some doubt about the precise contribution, but not about its existence. However, no one has found any kind of dark matter in non-gravitational detectors. When we recall history, dark matter is not without precedent. Neptune was a dark planet, observed through its gravitational action only, until its observation at the calculated position. The part of the motion of the perihelion of mercury that could not be explained by the action of the known planets was similarly attributed to an inner planet Vulcan. It was even mistakenly claimed that Vulcan had been observed before it was found out that the supposed dark matter indicated, in this case, the necessity to ﬁnd a more precise theory of gravitation. There is not a very strong belief in the latter way to solve the present dark-matter problem [1]. Besides, the uniﬁed theories of fundamental interactions seem to have many candidate

5.2 Counting Procedures

123

particles that could constitute the supposed dark matter [6]. These particles do not interact with the electromagnetic ﬁeld (that is, they are transparent), they do not interact strongly (they do not disturb our theories of nuclear structure), but they may interact weakly like neutrinos. In any case, they should decay or annihilate only slowly in the early part of the history of the universe. Two limiting cases of the behaviour of dark matter have been considered, i.e., hot dark matter (HDM) that is still relativistic when the relativistic and non-relativistic matter are of equal density, and cold dark matter (CDM) that is already non-relativistic at that time (the rest mass of the dark-matter particles is larger than the mass of the electron). Nevertheless, hot dark matter must become non-relativistic before recombination if they are expected to contribute to the evolution of structure, i.e. the rest mass of the darkmatter particles must be larger than 20 eV. Such a rest mass has often been conjectured for the ordinary neutrino, but has not yet been found deﬁnitively. Purely hot dark matter contradicts the models of structure formation (Chap. 8). The main argument is based on the theoretical spectrum of inhomogeneities that cannot be ﬁtted to observations. Dark-matter particles do not wait for the recombination to restart their condensation; they restart it at the equidensity stage (Chap. 8). Again, this must not happen too early, because of the smallness of the anisotropies in the microwave background. It is also an open question whether all dark matter is really condensed into isolated objects such as galaxies or clusters of galaxies [3, 16, 21].

5.2 Counting Procedures There are many diﬀerent counting procedures for extragalactic objects related to the question of the appropriateness of a Friedmann model and to the problem of estimating its parameters. The basic scheme is the following: We assume that we can identify a class of objects down to a redshift zlim , and can measure the apparent luminosity of these objects. In this case, the following ﬁve properties of the point set M : {zk , Ik } are not independent: 1. The absolute intensity L producing I = θ[z]L(4πr2 a20 (1 + z)2 )−1 depends on the cosmic time t in a given way: L = L[t]. 2. The comoving number density n of our objects evolves in a deﬁned way: n = n[t]. 3. The extragalactic extinction θ is known to be some function θ[z]. 4. The expansion rate H is a given function H[a] of the expansion parameter. 5. The number density n does not depend on distance for any given cosmic time (the assumption of homogeneity). If we can deﬁne a class of objects with an assumed constant number density (not depending on position and time) and an assumed constant luminosity (standard candles), not reduced by extinction in an unknown way,

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the I–z diagram should inform us about the history H[z] of the universe and its curvature parameter. Nevertheless, the actual problems are diﬃcult, because the function H[z] usually involves, eﬀectively, a diﬀerentiation of the data. Let us assume that our data set describes I[z]. First we derive the function r[z] from −1 r[z] = A I[z](1 + z)2 . The constant A has to be found from the limit condition c z + O[z 2 ] . r= a0 H0 Now we obtain

−1 c 1 − kr2 [z] dr H[z] = a0 dz

or, equivalently, dχ[z] =

c dz . a0 H[z]

(5.1)

The curvature index k has been chosen properly, if the function H[z] does not contain a quadratic term in the expansion in terms of (1 + z). We might integrate the right-hand (model) side of (5.1) and compare it with the function χ[z] (the data). However, as long as our data reach only as far as about z ≈ 0.2, we can trust only the model I = I[z], and the terms beyond H = const are very shaky. As we already noted, even statements about the order of magnitude of the various terms are quite suﬃcient to allow us to make a safe model for the early epoch of the universe. There is a simple test for the homogeneity of a distribution of objects with a given distribution of luminosity [32]. It is mostly used to test the evolution of this luminosity distribution, but it is instructive to interpret it conservatively. Let us assume that we observe a homogeneous population of objects, whose luminosity distribution is given by some dN = n[L] dL. An object will be counted if its apparent magnitude or luminosity exceeds some limiting value llim . Hence, the objects of a given absolute luminosity are seen in a comoving volume bounded by the radius rlim , for which the apparent luminosity of the chosen class of objects equals llim . We remember that there is some function f [z] that depends on the cosmological model, i.e. the curvature index k and the expansion factor a[t], such that I = Lf [z] , I = Ilim

f [z] . f [zlim ]

First, we solve this equation for zlim = zlim [L, Ilim ]; it yields rlim by the known light-ray propagation law. The corresponding comoving volume is labelled Vlim [L] = V [r[zlim [L, Ilim ]]]. Second, each object itself deﬁnes a sphere by its distance from us; this sphere has a comoving volume V which we have to

5.3 The Microwave Background Radiation

125

calculate by the formulas (2.19). Now we calculate the expectation value of this volume through the integral ES =

dL

Vlim [L]

dV (V /Vlim [L])n[L]

0

dL

Vlim [L]

=

1 . 2

dV n[L]

0

If we can determine the redshift z of our objects, the cosmological model that we have chosen tells us the volume V [r] determined by the object, and also its luminosity L and Vlim [L]. The average value of the quotient V [r[z]]/Vlim [L[l, r], llim ] has to be one-half if our assumptions about the cosmological model, homogeneity in space, the completeness of the catalogue and the absence of an evolution in time of the distribution n[L] are correct.

5.3 The Microwave Background Radiation In Chap. 1 we considered the type of argument put forward by Gamow which shows that the universe had to be opaque in some time in the past. If the moment the universe became transparent was deﬁned by a temperature of the order of magnitude of the ionisation temperature of hydrogen (diminished somewhat by the many degrees of freedom for radiation)1 and if the expansion afterwards was of the order z ≈ 1000, we expect the radiation that ﬁlls the universe homogeneously to have a temperature of a few degrees Kelvin. The reason is the frequency shift of the radiation caused by the expansion, in accordance with Wien’s law, which can be written for the universe in the form a[t] T = const . We observe the moment the universe became transparent as a ﬁreball, inside which our metagalaxy evolves. This radiation was detected in a test procedure for a microwave antenna by Penzias and Wilson in 1965 [27].2 The radiation is black, that is, in the frequency region in which it can be identiﬁed (in a gap between the galactic radio emission and the far-infrared background), its intensity follows the Planck radiation law (Fig. 5.3). The maximum lies at a wavelength of about 1 mm. Its temperature is about 2.73 K, and the most astonishing fact is its extreme isotropy, which makes it diﬃcult to see the seeds of the large-scale structure and the galaxies in this radiation. 1

2

The universe becomes transparent when the mean free path of the photons exceeds the Hubble radius. This happens with the neutralisation of most of the charged particles when atoms form. Therefore, the process is called ‘recombination’, although this word implies a previous ionisation. Such an ionisation never happened in the universe, as the initial state was an ionised one. The history of the ﬁndings about the microwave background radiation can be found in [26].

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5 Matter and Radiation

Fig. 5.3. The MBR spectrum is black. Spectrum of the Cosmic Microwave Background Radiation as measured by the FIRAS instrument on COBE and a black body curve for T = 2.7277 K. Note, the error ﬂags have been enlarged by a factor of 400. Any distortions from the Planck curve are less than 0.005 K. Reproduced from [13] with kind permission of D. J. Fixsen and the AAS

The mass density of the microwave background radiation is given by the Stefan–Boltzmann radiation law. We obtain radiation = 7.84 × 10−31 kg m−3 . We can transform this small density into the number density of the corresponding photons, and obtain nγ = 4 × 108 m−3 .

(5.2)

The number density of photons seems to be very large compared with the number density of baryons nB = baryonic /mproton = 0.1 m−3 in the metagalaxy. However, there are arguments (Chap. 6) that hint at a far larger quotient nγ /nB and show that the large (from this point of view) number of baryons requires to be explained. The present density of radiation has to be supplemented with densities of other kinds of particles that are still relativistic (neutrinos, for instance) and the present density of baryons has to be supplemented with other pressure-free components (cold dark matter, for instance). The result of the primordial nucleosynthesis puts limits on these supplements. The energy density of relativistic particles is determined by the number of species and the temperature alone. In addition to photons, electrons and positrons, at most three kinds of neutrino (νe , νµ , ντ ) would have

5.4 The Photon Bath and the Notion of the Ether

127

been present. More relativistic particles would have increased the expansion rate and, with it, the produced helium and the untouched deuterium. The alternative to the interpretation of the microwave background as a residue of the primordial heat reservoir is that it is some kind of radiation from early stars or late annihilation that has been thermalised by dust that developed early on. These models are rather complicated and far less appealing than the simple thermodynamics used in the cosmological interpretation. The background radiation is in any case not of recent origin, because we see distant objects in its foreground, in particular clusters of galaxies that show the Sunyaev–Zel’dovich eﬀect. After all, the acoustic modes in the spectrum of the angular ﬂuctuations of the background radiation strongly support the cosmological origin (Chap. 8). The high isotropy of the microwave background makes the formation of galaxies, clusters of galaxies and large-scale structures in general a diﬃcult problem of cosmology. Until 1992, only upper bounds on the anisotropy could be stated. Since 1992, through the measurements of the COBE satellite and, since 2003, through the Boomerang experiment and the WMAP mission, the ﬂuctuations of the microwave background can be mapped [5, 22, 31, 35, 37, 38]. The maximal amplitude of the ﬂuctuations is about 30 µK, i.e. 10−5 of the average value of the temperature, except for a large dipole anisotropy that indicates the motion of the earth through this heat bath. The spectrum of the angular dependence has the expected properties (Chap. 8). The ﬂuctuations of temperature have three sources. First, the radiation density and temperature may ﬂuctuate on the ﬁreball. The time of last scattering lies at locally diﬀerent redshifts. Second, at the last scattering, the matter can have a ﬂuctuating peculiar velocity. Hence, a local Doppler eﬀect will correct the redshift. Third, the integrated redshift in the timedependent gravitational ﬁeld along a light ray has to be taken into account (the Sachs–Wolfe eﬀect). This eﬀect modiﬁes the observed ﬂuctuation mainly at larger scales (>1◦ ).

5.4 The Photon Bath and the Notion of the Ether Any inhomogeneity in the sources of the radiation (i.e. in the cosmic time when the universe became transparent in the direction of observation) should produce an inhomogeneity in the temperature of the radiation. At smaller angular scales, the anisotropy is annoyingly tiny with respect to galaxy formation and structure formation in general. The quadrupole component, which would indicate a possible shear of the expansion today, is less than ∆quadrupole T ≈ 10−5 . T0 The dipole component, however, has a distinctly larger value [20]. We might interpret this as a peculiar motion of our galaxy in the Robertson–Walker

128

5 Matter and Radiation Table 5.1. Local peculiar velocities Velocity

Amount [km/s]

Direction

Sun against microwave background Sun against interstellar matter Sun in our galaxy Galaxy towards the centre of the Local group Local Group against microwave background Local Group against Hubble ﬂow Local Group against IRAS galaxies (Direction of the centre of the Virgo cluster)

370 25 400 40 610 450 510

1112–07 1700–17 1804+30 M31 1031–26 0412+35 1053+01 1053+01

coordinate system; the value we obtain for this peculiar motion is (Table 5.1; see [25]) v = 370 ± 10 km s−1 in the direction 1112–07 . The direction3 does not completely coincide with the values obtained through other determinations. This can be understood as an indication of large-scale, but local, gravitational ﬁelds [30]. The above motion diﬀers, of course, from the motion of the solar system relative to the interstellar gas, which is v = 25 km s−1

in the direction 1700–17

and from the motion within the galaxy, v = 400 km s−1

in the direction 1000+20 .

On average, the photons of the background radiation are at rest in the comoving coordinates of the Robertson–Walker system. This fact conﬁrms the appropriateness of the model and the physical foundation of cosmic time. There is a popular misunderstanding of the preferred frame of reference that is provided by the microwave background. Is it a new ether? Is it a problem for the theory of relativity? Relativity states that there is no preferred frame of rest which would allow us to determine absolute velocities without referring to other objects. Galileo demonstrated that, in a closed room, we cannot derive any velocity of this room from internal measurements, i.e. without looking out of some window. However, Newton’s mechanics got such a preferred frame when the propagation of light in form of waves was established. The frame in which light propagation was isotropic (interpreted as ether, i.e. a medium that carried the waves like the air that carries the acoustic waves) appeared to provide a preferred system of reference, i.e. a system of absolute rest. In a closed room, it appeared to be possible to deﬁne 3

We denote the direction in right ascension and declination, 1112–07 means a right ascension of 11h 12min and a declination of −7o .

5.5 The Homogeneous Large-Scale Structure

129

its velocity from the anisotropy of the propagation of light (Michelson tried to measure the anisotropy generated by the velocity of the earth). Einstein restored relativity in reconstructing mechanics such that the propagation of light was isotropic in every system of reference, i.e. there was no system of absolute rest. Now, we observe the microwave background, and it appears to provide a preferred system of reference again. However, the observation of the cosmic microwave background radiation, which comes from a region near the horizon of the metagalaxy, is a typical example of a measurement that looks out of the window. The background radiation does not contradict Galileo’s demonstration and does not question the premises of Einstein’s relativity. The deﬁnition of a special, universal system of reference by this radiation does not contradict the applicability of the theory of relativity. In a laboratory shielded from this radiation, no deviations from the predictions of relativity theory will occur. It is not the background radiation but gravitation that modiﬁes the theory of special relativity: gravitation can never be shielded because of the equivalence of inertial mass and gravitational charge. The nature of this modiﬁcation, however, is known. It is the theory of general relativity.

5.5 The Homogeneous Large-Scale Structure Structure is a prerequisite for observation, and the aim of observation is to discover this structure. Although we have to assume that the universe is homogeneous, we cannot and should not hope for that this is precisely true. We cope with inhomogeneity mostly by statistics, in particular averaging. That is, we save the theory by supposing that all inhomogeneity is a realisation of some probability density that does not depend on position, and call that density a homogeneous structure. The supposed probability density is assumed to be independent of position, just as any law of physics should be. If such a probability distribution depends on time, subtle questions about the separation of space and time arise that can be neglected only with caution [14, 36]. We shall not enter this ﬁeld here. The Hubble ﬂow of the form (4.16) has to be modiﬁed for large distances, simply because of the problems in the various distance deﬁnitions. In this sense, the formula (4.16) is a local expression. On the other hand, the velocity entering this formula is an average velocity on which peculiar motions are superposed. The average in this sense is the overall trend in the form of (4.16), which, after subtraction from the velocity ﬁeld, leaves presumably only a Maxwell distribution of velocities. Of course, this picture is too simple, because of the existence of large aggregates in our neighbourhood, i.e. in the part of the universe which can be measured in both distance and velocity. The ﬁrst of these large-scale inhomogeneities is the void structure established by deLapparent, Geller and Huchra [10]. The galaxies and clusters of

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5 Matter and Radiation

Fig. 5.4. Large-scale structure in the Las Campanas Redshift Survey. This is a sector taken from the Las Campanas Redshift Survey [34]. It is seen how the structure on smaller length scales includes voids and walls, and that the structure is kind of homogeneous although it seems to be dissolved into homogeneity at larger distances. Reproduced with kind permission of D. Tucker and the LCRS team [34] and the AAS

galaxies seem to be, in reality, condensations on the walls, edges and vertices of a basic cell structure, which reminds us a little of the crests of a wave ﬁeld (Fig. 5.4). The Great Void in our neighbourhood lies in the direction of Bootes, and has a diameter of about 50 h−1 Mpc [18]. Second, there is an apparently ﬂat region densely populated with galaxies, the Great Wall. It contains the Coma cluster (direction 1300+28), but spans more than 8 hours of right ascension across the sky between 25o and 45o declination. Its distance is about 100 Mpc, its thickness about 5 h−1 Mpc, its area about 60–170 h−2 Mpc2 . It is disc-like, with δ/ ≈ 5. Third, there is a systematic local shear in the Hubble ﬂow, producing the phenomenon of an apex of motion of the clusters in our neighbourhood, the Great Attractor [11]. It lies in the direction of ACO 3627 (1325–53), and in a distance of about 40 h−1 Mpc (Fig. 5.5). In general, galaxies and clusters of galaxies seem to be condensations on the walls, wedges and the vertices of some underlying cellular structure. This structure may be interpreted – and numerical simulations conﬁrm this – as wave crests that make smaller amplitudes of the matter density visible. On the crests, the density suﬃces to start the formation of galaxies and stars. The typical diameters of the voids or underdense regions are about 20 h−1 Mpc [12], but one should keep in mind that the distributions of the various properties of the voids have a large variance. In the approximation of uniform growth of the voids, the structure can be conjectured to be a Voronoi tessellation [43, 44]. Correlations and structure on ﬁnite scales do not contradict the assumption of large-scale homogeneity. This question becomes more diﬃcult if one takes the possibility of a fractal structure into account.

5.6 Correlations and Power Spectra

131

Fig. 5.5. Map of the sky in a Hammer projection. The map shows the sky in equatorial coordinates in order to indicate the diﬀerent positions cited in the text: C is the direction of the sun with respect to the cosmic microwave background; M is the direction of the sun with respect to the interstellar matter, G is the direction of the sun within the galaxy, L is the direction of the Local Group with respect to the cosmic microwave background; I is the direction of the Local Group with respect to the IRAS galaxies; and A is the direction to the Great Attractor

5.6 Correlations and Power Spectra The ﬁeld equations that govern the gravitational condensation relate the ﬂuctuations of the gravitational potential to the ﬂuctuations of the density. Hence we adopt the model of a probability ﬁeld of density distributions [x] that have a spatially constant function as their expectation value, E[[x]] = 0 . This is used to deﬁne the ﬁeld of overdensities, δ[x] =

[x] − 0 . 0

(5.3)

We should note that the expectation value diﬀers from any average density ¯. The latter is also a random function, but its variance is small for most practical purposes. Therefore, a kind of ergodicity assumption lets us tacitly equate to 0 , although the precise relation is again E[] = 0 . If we replace the deﬁnition (5.3) with δ[x] =

[x] − ,

we obtain exactly δ = E[δ] = 0. In the following, we shall not always distinguish between the probability ﬁeld and its observed realisation. We

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5 Matter and Radiation

shall use spatial integrals over the observed realisation as expectation values of the probability ﬁeld itself, although they are estimations of these values at best. The random variables are ﬁelds, and a pointwise evaluation would yield only a small part of the information that can be found. With the equations for the gravitational ﬁeld in mind, we decompose the overdensity ﬁeld into a Fourier integral. The structures and objects of astronomical observation are assumed to develop from and to delineate perturbations in some constant average density that can be described by the Friedmann–Lemaˆıtre models of the universe. We suppose locality in the form of periodicity over large volumes 3 ): (V > RH 1 V 3 δk = δ[x] exp(ikx) d x , δ[x] = δk exp(−ikx) d3 k . V (2π)3 V

V

where we have explicitly indicated the integration volume, and the Fourier amplitudes δk are dimensionless. It is general practice, however, to hide this volume in δk , which now acquires the dimensions of a volume and which is now interpreted directly as the variance of the mass in a volume of a size deﬁned by |k|. Hence we deﬁne 1 3 δk exp(−ikx) d3 k . δk = δ[x] exp(ikx) d x , δ[x] = (2π)3 V

V

∗ The amplitude δ[x] is real, and we obtain δk = δ−k . The amplitude of the density ﬂuctuations is deﬁned through the variance, in analogy to the ﬂuctuation amplitude of the mass itself. Again, the integration volume may be identiﬁed with that of the Fourier representation: 1 1 δ 2 [x] = E(δ 2 [x]) = d3 k |δk |2 . d3 x δ 2 [x] = V V (2π)3

For isotropic ﬂuctuation spectra where δk depends only on the modulus k = |k| and δk = δk , we can write 1 dk 3 2 2 k δk . δ [x] = V 2π 2 k We call the quantity P [k] =

1 |δk 2 | V (2π)3

(5.4)

the power spectrum of the ﬂuctuations. In the case of isotropic, Gaussian (that is, the phases are random) perturbations, it contains all the information in the distribution (Fig. 5.6). The actual power spectrum is found mainly by two methods. The ﬁrst translates the density distribution into a distribution of mass in volumes of

5.6 Correlations and Power Spectra

133

Fig. 5.6. Power. The power is proportional to k for large scales and k−3 for small scales. This ideal scaling for small scales is modiﬁed for the scales in question by transition eﬀects between the two regions. The oﬀset between a model and the data points indicates these transition eﬀects (Chap. 8). The horizontal position of COBE-data strip is taken for the case Ω0 = 1. It depends on the size of the ﬁreball, which in turn depends on the cosmological parameters (Fig. 4.6)

a given size. The mass in a given volume will vary with position because of the existence of condensed structures and objects. For a given size of this volume, we can deﬁne an expectation value E[M ] and a variation M − E[M ]. The expectation value of M − E[M ] vanishes, hence its amplitude is deﬁned by the variance δM = D2 [M ] = E[(M − E[M ])2 ]. We write

δM M

2 =

E[(M − E[M ])2 ] . (E[M ])2

This relative variance depends on the size chosen for the volumes, which is indicated through some characteristic length L = (a[t]/a0 )Lcomoving or the corresponding wave number kcomoving = a[t]/La0 . The density perturbation in a logarithmic interval 1 dk 2 3 2 2 ∆ [k] , k δ , δ = δ 2 k = ∆2 [k] = k V 2π 2 k

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5 Matter and Radiation

Fig. 5.7. Mass dispersion. The mass dispersion is nearly constant for small scales and proportional to k4 for large scales (Harrison–Zel’dovich spectrum)

is seen to be proportional to the ﬂuctuation of the mass in a volume that is deﬁned by a linear size of k −1 . Note that the symbol ∆2 is used for the logarithmic spectrum more generally, i.e. also for the perturbations of the potential δΦ. In this case we write (∆Φ)2 = (dk/k)∆2Φ [k] or (∆Φ)2 = (dk/k)(∆2 Φ)k . The relative variance of the mass in such a volume with L = k −1 is, up to form factors, equal to the amplitude of the logarithmic spectrum (Fig. 5.7): 2 δM ≈ ∆2 [k] . M L We know about the Gibbs phenomenon in Fourier transforms, which is observed for sharply bounded regions. To avoid it, we introduce a window function W [r, r0 ], which deﬁnes the volume and mass in the form

∞ W [r, r0 ]r2 dr ,

VW [r0 ] = 4π

δM M

and ﬁnally yields

2 =

3

d x

0

1 VW [r0 ]

2

3

d y δ[x + y]W [y]

,

5.6 Correlations and Power Spectra

δM M

2 =

1 2 VW [r0 ]

∆2 [k]|W [k]|2

135

dk . k

If we choose a Gaussian window with a width 2L and assume some power law |δk | = AV kn , we obtain

δM M

2

=

n+3 1 ∆2 [L−1 ] . Γ 2 2

Other window functions produce other form factors. The relative variance of the mass is proportional to ∆2 , and the precision and the form factor depend on the window function and the ﬂuctuation spectrum itself. The variance of the mass determined with a window function allows one to determine parts of the spectrum directly from galaxy counts.

The second way to determine the power spectrum is to calculate the correlation functions. In the case of Gaussian ﬂuctuations, the two-point correlation function yields the information that we need. This function is deﬁned by the expression (5.5) δ2 [x, x ] = E[δ[x]δ[x ]] . If the density [x] is realised as a set of N particles, the correlation function ξ[x] is the probability of ﬁnding another point of the set at a position x relative to a point at y, i.e. dP [x1 − x2 = x] = N 2 (1 + ξ[x]) dV1 dV2 . Homogeneity and isotropy yield δ2 [x, x ] = ξ[x − x ] and [y][x + y] ξ[x] = Ey − 1 = Ey [δ[y]δ[x + y]] 20 1 = |δk |2 exp(−ikx) d3 k . V (2π)3 It can be seen that the correlation function is the Fourier transform of the power spectrum: ∞ sin kr . ξ[r] = 4π k 2 dk P [k] kr 0

We may derive correlation functions for projected distributions (onto the sky, for instance), for other coordinates of relative position (angles, for instance), and so on. Because the power spectrum of the perturbations is the Fourier transform of the correlation function, the variance of the mass M in a sphere of radius R is its integral, i.e.

δM M

2

3 [R] ≈ 4πR3

R ξ[r] d3 r . 0

136

5 Matter and Radiation

The phases of the amplitudes δk cannot be determined simply. Their probability distribution has to be conjectured. Uniformly distributed phases and maximal entropy for given variance yield a Gaussian probability distribution of the individual values of δk . Power laws are widely used as approximations of more complicated functions that have not been determined very precisely. They characterise functions that do not contain any particular characteristics in the domain that is described by the power law. Correlation functions are often given as such power laws, i.e. r γ 0 . ξ= r The exponent γ describes the decrease of correlation with distance. Its value is approximately 1.8 both for galaxies and clusters of galaxies. The normalisation factor r0 corresponds to the scale of the existing structure and is smaller for galaxies (rgg ≈ 5 h−1 Mpc) than for clusters (rgg ≈ 24 h−1 Mpc). In a power-law spectrum |δk2 | ∝ k n , the exponent n governs the balance between large-scale and small-scale power. Because we observe condensed objects on a wide range of scales, the exponent should be approximately one. If we average in a box of linear size L, we have to put −1 L

4πk 2 k n dk ∝ L−(n+3) ∝ M −(n+3)/3

E[δ ] = 2

0

For a correlation function ξ = ((r/r0 ))−γ , we obtain 2−γ 2 π ∝ (kr0 )γ . ∆2 [k] = (kr0 )γ Γ [2 − γ] sinr π 2 This holds for n < 0; otherwise, we have to introduce a cut-oﬀ for ξ. n > −3 is necessary for large-scale homogeneity, and n < 4 because discrete matter produces the minimal spectrum, where n = 4. n = 0 corresponds to white noise, and n = 2 corresponds to a distribution obtained by random placement in regular cells. n < 0 for cosmological large-scale structure; scale invariance is deﬁned by ∆2 ∝ k 4 , P ∝ k, n = 1. The spectrum of inhomogeneities that is observed in the distribution of galaxies and in the microwave background has to be understood as a result of some evolution that starts with some plausible initial spectrum. To this end, we need an argument for the initial spectrum, the evolution equations (to be derived from the Einstein equations) and the evaluation of the observations with respect to a theoretical model. We shall follow this path in the Chaps. 7 and 8.

5.7 Fractal Structure

137

5.7 Fractal Structure At this point, the density ﬁeld is characterised by a power spectrum or correlation function. Both the power spectrum and the correlation function are approximated by power laws. The argument for power laws is the lack of a physically plausible scale; it should not necessarily be applied to the correlation function, but may be applied to the expectation value Ey [[y + x][y]] itself. In this case, we speak about self-similar structures that are evaluated by means of fractional dimensions and are called fractals. The basic model refers to a fragmentation or condensation that has identical properties on all scales, as was described in Sect. 1.7 [7]. Fractal structures can be described in several diﬀerent fashions. The simplest way is to suppose some Euclidean space and to derive the self-similarity of the hierarchy of identiﬁable objects from the form of the various correlations. The transition to the spatial structure of Einstein’s theory of gravitation requires additional considerations [29]. We adopt a model which follows Hoyle [15]. We assume a scale-invariant multiplicity. The units of each order n are composed of gn = k 3 systems of order (n − 1). If Nn is the number of elementary objects in a system of order n, we obtain Nn = k 3 Nn−1 . The linear extension must reﬂect not only the fragmentation, but also a certain contraction θ (θ < 1). Hence the typical diameters are assumed to scale as Rn−1 = k −1 θRn (Fig. 1.3). This yields Rn =

k Rn−1 , Nn = k 3 Nn−1 , θ < 1 , k > 1 , θ n k R0 , Nn = k 3n . Rn = θ

(5.6)

We directly obtain k 3n θ3n = , (θ−1 k)3n R03 R03 n n k 3m θm k 2m Φn = = , (θ−1 k)m R0 R0 m=1 m=1 n =

In =

n

n k 3m θ2m k m = . (θ−1 k)2m R02 R02 m=1 m=1

In this model, the density converges for θ < 1, the apparent brightness of the sky converges for kθ2 < 1 and the potential converges for k 2 θ < 1. Selfsimilarity is ensured because the recursion does not depend on the step number n. Although the point set is embedded in a three-dimensional space, one can deﬁne various fractional (fractal) dimensions, which would all be equal to the dimension of space if we were to consider a genuinely uniform distribution. The fractal dimensions reﬂect the particular scaling; for a genuinely uniform distribution, this would again be determined by the dimension of the embedding space.

138

5 Matter and Radiation

Our model must be constructed up to a very large order nmax . Let us consider some order n that is distinctly smaller than nmax . The total system now contains k 3(nmax −n) systems of order n, which ﬁll only k 3(nmax −n) Vn = k 3(nmax −n) R03 (k/θ)3 n of the total volume Vnmax = R03 (k/θ)3 nmax . We now cover the total system with cubes Bi [L], i = 1, . . . , N [L], of size L3 and measure each cube Bi [L] by the number of elementary objects ni [L] that it contains. We can deﬁne the frequencies pi [L] = ni [L]/ j nj [L] to obtain the R´enyi information   N [L] N [L] q 1 1   ln (5.7) pi [L] , I1 = pi [L] ln Iq [L] = 1−q p i [L] i=1 i=1 and a corresponding dimension Dq [L] = lim

Iq [L] − ln L

.

The precise deﬁnition requires the limit for L → 0. In our case (as always in evaluations of real observables) this cannot be done, because of statistical errors and physical limits on the self-similarity on small scales. One always looks for self-similarity on scales (here L) that are intermediate, in which the corresponding dimension is constant. The value of this dimension is taken to be the physical dimension of the hierarchy. These considerations hint at how the concept can be generalised mathematically, but we shall not dig as deep as that. We ﬁnd the following scalings: – – – – – – – – –

Number of elements in a system of order n: Nn = k 3n . Mass of a system of order n: Mn = M0 k 3n . Size of a system of order n: Rn = R0 (k/θ)n , k = n Rn /R0 θ. Number of cubes of linear size Rn , that cover the total: k 3(nmax −n) . Number of cubes of linear size Rn , that ﬁll the total: (Rnmax /Rn )3 . Number of cubes that are occupied by systems of order n: k 3(nmax −n) . Probability of ﬁnding an occupied cube: θ3(nmax −n) . Number of elements in an occupied cube: k 3n . pi , if not zero: k 3(n−nmax ) .

The information (5.7) can be written as 1 ln (k 3(n−nmax ) )q−1 , 1−q 1 I1 = ln 3(n−n ) max k

Iq [Rn ] =

We obtain the dimensions

References

Iq [Rn ] ln Rn 3 ln k . ≈ ln k − ln θ

Dq [Rn ] = lim −

= lim

139

3(nmax − n) ln k (nmax − n)(ln k − ln θ) − ln Rnmax

These dimensions are independent of q. In generalisations, however, they may be not equal. This case is called multifractal. D0 is called the box dimension, D1 the information dimension and D2 the correlation dimension. We expect, for the average density in a fractal in a three-dimensional space, the scaling ξ[r] ∝ rD2 −3 , and, for the decrease of the density in an element, (r) ∝ rD0 −3 . For D0 < 3, the density falls to zero; for D0 < 2, the apparent luminosity converges; and for D0 < 1, the potential converges also. To some extent, a fractal structure has been seen in the distribution of galaxies. Einasto (see, for instance [19]) obtains a dimension D = 1.3 for the clusters of galaxies, and D = 1.9 for the galaxies in the Virgo cluster. Further arguments can be found in [9, 28, 39]. In essence, it is diﬃcult to safely discriminate between a power law of the correlation function ξ ∝ r−α and a statistical law 1 + ξ ∝ r−α . In the large and in the small, there are limits to any kind of self-similarity. A comparison between diﬀerent methods to estimate fractal dimensions of point sets can be found in [42].

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7. Calzetti, D., Giavalisco, M., Ruﬃni, R., Taraglio, S., Bahcall, N. A.: Clustering of galaxies: fractal or homogeneous infrastructure? Astron. Astrophys. 245 (1991), 1–6. 137 ´ 8. Celnikier, L. M.: Basics of Cosmic Structures, Editions fronti`eres, Gif-surYvette (1989). 119 9. Coleman, P. H., Pietronero, L.: The fractal structure of the universe, Phys. Rep. 213 (1992), 311–389. 139 10. DeLapparent, V., Geller, M. J., Huchra, J. P.: The mean density and two-point correlation function for the CfA redshift survey slices, Astrophys. J. 332 (1986), 44–56. 129 11. Dressler, A.: The great attractor. Do galaxies trace the large-scale mass distribution? Nature 350 (1991), 391–397. 130 12. Dubinski, J., daCosta, L. N., Goldwirth, D. S., Lecar, M., Piran, T.: Void evolution and the large-scale structure, Astrophys. J. 410 (1993), 458–468. 130 13. Fixsen, D. J., Cheng, E. S., Galles, J. M., Mather, J. C., Shafer, R. A., Wright, E. L.: The CMB spectrum from the full COBE/FIRAS data set, Astrophys. J. 473 (1996), 576-587 (astro-ph/9605054). 126 14. Hosoya, A., Buchert, T., Morita, M.: Information entropy in cosmology, Phys. Rev. Lett. 92 (2004), 141302 (gr-qc/0402076). 129 15. Hoyle, F.: On the fragmentation of gas clouds into galaxies and stars, Astrophys. J. 118 (1953), 513–528. 137 16. Jing, Y. P., Mo, H. J., B¨ orner, G., Fang, L. Z.: The large-scale structure in a universe dominated by cold plus hot dark matter, Astron. Astroph. 284 (1994), 703–718. 123 17. Kashlinsky, A., Jones, B. J. T.: Large-scale structure in the universe, Nature 349 (1991), 753–760. 119 18. Kirshner, R. P., Oemler jr., A., Schechter, P. L., Schectman, P. A.: A survey of the bootes void, Astrophys. J. 314 (1987), 493–506. 130 19. Klypin, A. A., Einasto, J., Einasto, M., Saar, E.: Structure and formation of superclusters. X. Fractal properties of superclusters, Mon. Not. R. Astron. Soc. 237 (1989), 929–938. 139 20. Kogut, A., Lineweaver, C., Smoot, G. F., et al.: Dipole anisotropy in the COBE diﬀerential microwave radiometers ﬁrst-year sky maps, Astrophys. J. 419 (1993), 1. 127 21. Kormendy, J., Knapp, G. R.: Dark Matter in the Universe, Reidel, Dordrecht (1987). 123 22. Lange, A. E., Ade, P. A. R., Bock, J. J., Bond, J. R., Borrill, J., Boscaleri, A., Coble, K., Crill, B. P., de Bernardis, P., Farese, P., Ferreira, P., Ganga, K., Giacometti, M., Hivon, E., Hristov, V. V., Iacoangeli, A., Jaﬀe, A. H., Martinis, L., Masi, S., Mauskopf, P. D., Melchiorri, A., Montroy, T., Netterﬁeld, C. B., Pascale, E., Piacentini, F., Pogosyan, D., Prunet, S., Rao, S., Romeo, G., Ruhl, J. E., Scaramuzzi, F., Sforna, D. : First estimations of cosmological parameters from BOOMERANG, Phys. Rev. D 63 (2001), 042001. 127 23. Lauer, T., Postman, M.: The motion of the Local Group with respect to the 15000 kilometer per second Abell cluster inertial frame, Astrophys. J. 425 (1994), 418–438. 119 24. Longair, M. S.: Confrontation of Cosmological Theories with Observational Data, Kluwer Academic, Dordrecht (2002). 119

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6 Standard Synthesis

6.1 The Gamow Universe One of the two most important observations that indicate a hot past of the universe is the universal distribution of helium and deuterium. Deuterium, in particular, has no chance to escape from the deep interior of stars. It is processed to helium much faster than it is produced through two-proton processes. The presence of deuterium in the interstellar, primordial gas (Fig. 1.7) requires the hypothesis of a synthesis at a time when free neutrons were present in substantial concentration, that is, in the early universe near a temperature corresponding to the diﬀerence in rest mass, i.e. kT ≈ (mn − mp )c2 . At this temperature, the universe was necessarily dominated by radiation (more precisely, hot matter), and the radiation served not only as the source of gravitation in the Friedmann equation but also as a heat bath. This situation was predicted by Gamow. Let us consider the microscopic processes between diﬀerent matter components. Recombination is the last of this kind of process. In our simple calculations, we are bound to adopt the cosmological principle of uniformity in the physical state, i.e. the reduction of the problem to the ordinary thermodynamics of an isolated, adiabatically expanding system. In addition, we have to assume inhomogeneities to be negligibly small. In particular, the reaction rates are assumed to be everywhere the same. We tacitly assume the appropriateness of the Friedmann models, together with the possibility of calculating back into the past. We shall ﬁnd a completely natural synthesis of helium and deuterium. Furthermore, the observed abundances will give us a key to the conditions at the nucleosynthesis time. In particular, we are able to determine the expansion rate at the nucleosynthesis time from the abundance of helium, and to determine the nucleon density from the abundance of deuterium. This point of view is the conventional one, but it may be questioned [6]. It is a curious fact (but consistent with a history of the recent universe that allows carbon-based life [1]) that the recombination time more or less coincides with the point where the total energy of the photons falls below the rest energy of the small number of baryons left after the annihilation at 1013 K. That is the reason why, for the expansion of the universe in its early, pre-recombination history, we may assume a radiation-dominated model, i.e. a Gamow universe. Dierck-Ekkehard Liebscher: Cosmology STMP 210, 143–166 (2004) c Springer-Verlag Berlin Heidelberg 2004

144

6 Standard Synthesis

We recall the ﬂat metric of the Friedmann model, ds2 = c2 dt2 − a2 [t](dx2 + dy 2 + dz 2 ) . The Friedmann equation yields 2 a˙ 8πG . = a 3 The energy density of the hot component is obtained from the Stefan– Boltzmann law, 3 π ∗ kT 2 g c = kT , 30 ¯hc including the continuity equation p a d = −3 + 2 da . c We recall the expression for the dimensionless entropy density, 3 4 π ∗ kT s= g , 3 30 ¯hc which can be considered to be proportional to the number density of the photons if they are in at least partial thermodynamic equilibrium. The product M4 = radiation a4 represents the comoving number density (Sect. 3.4). If the baryon number per unit comoving volume is constant, it is proportional to the number of photons per baryon, and this is a characteristic and to some extent observable quantity in the universe. We substitute the radiation temperature T for the expansion parameter a, and ﬁnd 2 3 4 kT 1 dT 2πG 2 kT 2 2 2 = 2 f kT = c lPlanck f (6.1) T 2 dt c ¯hc ¯hc and the Hubble radius RH =

1 f lPlanck

¯c h kT

2 ,

(6.2)

where f 2 = 2π 2 g ∗ /45, g ∗ being the eﬀective number of degrees of freedom. Every particle contributes to that number according to its possible spin states, weighted for fermions with a factor 78 owing to the diﬀerence between Bose– Einstein and Fermi–Dirac statistics: 7 fermion giboson + g . g∗ = 8 j j i

6.1 The Gamow Universe

145

The particle density is given by the integral

∞

n± ∝ m0 c2

E 2 /c2 − m20 c2 E dE . exp[(E − µ)/kT ] ± 1 c2

For fermions, the sign must be + because of the Pauli principle, i.e. for any distribution of particles over the states, each state can be occupied only once. For bosons, each state can be occupied without restriction, and the normalisation factor has the minus sign. µ denotes the chemical potential.

If the various kinds i of particles have individual temperatures Ti , we obtain g∗ (T ) =

giboson

i

Ti T

4

7 fermion + gk 8 k

Tk T

4 .

(6.3)

When a component cools below the temperature Ti0 that corresponds to the rest mass, it becomes non-relativistic, its temperature starts to fall in accordance with T T Ti ≈ Ti0 and its contribution to the radiation component ends (Fig. 6.1). In a symmetric universe, we expect particles to annihilate with their corresponding antiparticles, which become non-relativistic at the same time. It makes a diﬀerence whether the annihilation occurs at this time or later. We consider a particular exothermic process that releases an energy ∆E. We use the normalised variable x deﬁned by the exponent of the canonical distribution about this energy step ∆E: x=

∆E . kT

(6.4)

Using also the Compton wavelength corresponding to this energy, l = h ¯ c/∆E, we ﬁnd for the Friedmann equation

1 dx x dt

2

2 = c2 f 2 x−4 lPlanck l−4

the solution f clPlanck 1 d t l2 d hG 2 = , lPlanck x= , t0 = , = 3 . (6.5) t0 2f clPlanck dt l2 x dx c The density of relativistic particles, given by nrel = f 2 x−3 l−3 , is conserved: nrel multiplied by any comoving volume is constant. This is the background for our standard process.

146

6 Standard Synthesis g* 100

50

20

10

5

g*s

g* 5

4

3

2

1

0

−1

−2

logT

Fig. 6.1. The relevant degrees of freedom depend on temperature. This ﬁgure shows the total number of relativistic degrees of freedom g∗ [T ] in the standard model of particle physics as a function of temperature (6.3). Most of the degrees of freedom belong to very massive particles that annihilate early. Finally, only photon and neutrino degrees of freedom remain. Reproduced from [4] with kind permission by M. Roos and the APS

6.2 The Standard Process and the Decay of Equilibrium We model the standard process as a decay of an equilibrium between a compound system D composed of two equal particles N, and its dissociated state. That is, we assume that, in a comoving volume, the number N + 2D is constant, where N is the number of particles N and D is the number of bound systems D, or N + 2D = const . ν= V nγ The kinetic equation for the system is dN N cσ =− N + ndissociation [D, T ] . dt V The change in the number N of components consists of two contributions, one given by the formation of compound systems, and the other by their dissociation. The mean free path λ of a particle N decreases with an increase in the density n and with an increase in the cross-section σ: λ = 1/nσ. The formation rate is given by the number N divided by the collision time τ = (λ/v) ≈ (V /N σv). We substitute the velocity of the particles by the

6.2 The Standard Process and the Decay of Equilibrium

147

velocity of light, assuming that we can include the resulting quotient into the estimation of the cross-section. The dissociation rate is determined from the known equilibrium: at equilibrium, this rate must be equal to the production term. After substituting the normalised variable x (see (6.4) and (6.5)), the abundance ξ = N/(N + 2D) and the photon density nγ , we obtain x2

dξ 2 = B(ξequilibrium − ξ2) , dx

where B=

νσ

1

lPlanck l ftotal

(6.6)

fγ2 .

This is a universal equation; the quantity B alone decides the evolution. The process can always be estimated by the same reaction equation, independent of the details of the particular case. The latter determine the parameter B only. The quantity σl = lPlanck l =

∆E GeV

−1

× 3.75 × 10−47 cm2

is a kind of reference cross-section for the process in question, the unit for the actual cross-section. The scenario is always the same for processes of the type considered. First, the temperature and density are so high that the process is quasi-static. The high reaction rates ensure a momentary equilibrium between the particles D, γ and N corresponding to the momentary temperature. The high concentration of γ suppresses the concentration of bound systems; this latter concentration is negligible. Not until the temperature falls below the value represented by the binding energy ∆E does one ﬁnd – still in equilibrium – a noticeable concentration of bound systems. The concentration ξ of the components N still follows the decreasing equilibrium concentration ξequilibrium (T ) with a negligible delay. With the expansion, the density of the components N falls, together with the production rate of bound systems. The latter rate becomes negligible. The concentration ξ can no longer follow the falling Boltzmann equilibrium concentration ξequilibrium (T ) and freezes in. The process leaves a relict concentration of components that do not react any longer, even when the equilibrium concentration falls to zero. The point x1 that marks this decay of equilibrium, this freezing of a relict concentration of the component N, is somewhere near x = 1. At larger values of x, the process ceases. Equilibria decay because the rapidity of the restoring processes fails to compete with that of the expansion-generated change in the equilibrium position (Fig. 6.2). For low values of x, we have a short-delay equilibrium, given approximately by a Boltzmann distribution, ξ = ξequilibrium (x) ≈ Axβ e−x .

148

6 Standard Synthesis

Fig. 6.2. Dilution competes with cooling. The Boltzmann exponent of a bound state increases for falling temperature. At some point, condensation processes begin. The reaction rates decrease by dilution, the main reason being the dependence of the mean free paths on the densities. Dilution stops the process when the mean free paths exceed the Hubble radius. The interval of eﬀective processes is limited by these two points, and the residual concentration of unprocessed components depends on the size of this interval

The delay in the equilibrium is negligible as long as the rate of decrease of the equilibrium concentration is less than the maximal consumption rate, or − x2

dξequilibrium 2 ≤ Bξequilibrium . dx

(6.7)

The equality deﬁnes the point x1 where the process ceases to be quasi-static. At the moment when x1 is reached, this value is just the quotient of the process rate nσc with the Hubble rate H. This moment is the time, when the process rate falls below the Hubble rate. For x > x1 , because of the exponential decrease of ξequilibrium , the kinetic equation is reduced to x2

dξ = −Bξ 2 . dx

6.2 The Standard Process and the Decay of Equilibrium

149

The point x1 of the decay of the equilibrium is found by evaluating (6.7). We obtain ln2 [AB] . (6.8) Bξ1 = x1 (x1 − b) → x1 ≈ ln[AB] , ξ1 ≈ B From that initial value, we integrate the equation x2 to obtain

dξ = −Bξ 2 dx

1 1 − =B ξ ξ1

1 1 − x1 x

.

From (6.8), we obtain, in the limit of x → ∞ (Fig. 6.3) B 1 B 1 1 = + = 1+ . ξ∞ x1 ξ1 x1 x1 − β A good approximation for the relict concentration is

Fig. 6.3. Quasi-static reaction and cut-oﬀ. For temperatures larger than the reaction energy, the concentration of the unprocessed component is maximal. With falling temperature, the equilibrium concentration starts to fall rapidly. Reaction sets in. For some time, the reaction is fast enough to reduce the concentration to the falling equilibrium value. At some point, the reaction ceases to be fast enough, the process is no longer quasi-static and entropy is produced until the process stops

150

6 Standard Synthesis

ξ∞ ≈

ln[AB] 1 ftotal lPlanck l x1 = ≈ ≈ . B B B fγ2 νσ

(6.9)

The larger the cross-section σ or the total concentration ν is, the smaller the relict concentration ξ∞ . The latter increases with the total number of degrees of freedom, because a faster expansion ends the short-delay process earlier. We ﬁnd – for the annihilation of protons and antiprotons (p + p ¯ → 2γ), using l = h/mp c and σ = α2 le2 , the estimate νσ/(lPlanck l) = α2 le2 /(llPlanck ) ≈ 3 × ¯ 10−14 ; ¯ → 2γ), – for the annihilation of weakly interacting massive particles (L + L 2 2 4 2 2 4 ¯ mL /(c mV ) (lV is the Compusing l = h ¯ /mL c and σ = lV /l = h ton wavelength of the intermediate boson), the estimate νσ/(lPlanck l) = m3L mPlanck /m4V , which yields the contribution ΩL =

m4V mL c2 ftotal 3 2 fγ mL mPlanck 30 eV

(6.10)

to the density parameter; – for the synthesis of deuterium (n + p → d + γ), using l = h ¯ c/∆E and −2 2 7 σ=h ¯ 2 m−2 c , the estimate νσ/(l l) = νm ∆m/m Planck Planck p p ≈ 3 × 10 −9 (with ν ≈ 10 ); ¯ c/∆E – for the synthesis of helium (symbolically d + d →4 He), using l = h −2 −2 c , the estimate νσ/(l l) = νm m and σ = h ¯ 2 m−2 Planck Planck He ∆m ≈ He 8 × 106 ; – for the recombination (p+e− → H), using l = h ¯ c/∆EI and σ = h ¯ 2/ 2 2 2 2 2 3 (α c me ), the estimate νσ/(lPlanck l) = ν¯h∆EI /(α me c lPlanck ) ≈ 5×104 . We have to remember that the formula (6.9) is to be applied only for a process that really takes place, and proves that it has taken place by a value ξ∞ < 1. If this formula yields ξ∞ > 1, it shows only that no process has taken place at all, because its conditions have never been met. For instance, in the case of the ionisation equilibrium of hydrogen, the ionised fraction is given by the Saha equation,

hc h ¯ ¯ x2 = n−1 2π 1−x me c kT

−3/2 ,

which in our case reads x2 = fγ−2 ν −1 1−x

3/2 l le

x3/2 e−x .

(6.11)

We observe that recombination sets in not at x = 1, kT = ∆E, but at about x ≈ ln A + β ln ln A, mainly because of the small value of ν. The exponential function does not cancel the large factor A before this point is reached. For the recombination of hydrogen, ln A ≈ 35.8, and x1 = 41.2. Therefore, ionisation decays not at about 160 000 K, but at about 4000 K. The remaining concentration

6.2 The Standard Process and the Decay of Equilibrium

151

of ionised hydrogen vanishes in the abundance of the hydrogen later re-ionised by quasar action. It is not observable. A similar retardation of synthesis must also be expected for deuterium, because the factor of the exponential function is nearly the same. Universes with another Friedmann equation,

2 x˙ x

= f [x] ,

can be evaluated if we substitute (6.6) by x4

f [x]

dξ 2 − ξ2) . = νcσ(ξequilibrium dx

Until the release from the equilibrium, the process is essentially quasistatic and does not produce entropy. After the release, it is irreversible and produces entropy. In the case of only a small concentration ξ, this entropy can be neglected. The interesting case is the annihilation of particle–antiparticle pairs when they still contribute to the number of relativistic degrees of free∗ be proportional to the dom. Such a process changes this number. Let g− number of relativistic degrees of freedom before kT = m0 c2 , when annihilation can start for particles with rest mass m0 . Let g0∗ be their contribution ∗ . A quasi-instantaneous (not adiabatic!) annihilation would produce an to g− increase in temperature of ∗ g− g∗ 4 aT+ = aT− = aT− 4 ∗ − ∗ g+ g− − g owing to the continuity of the expansion rate. For a quasi-static process, a4 rel ∝ f 2 T 4 changes continuously. Now the entropy, or particle number, is ﬁxed, i.e. a3 n ∝ a3 f 2 T 3 . This leads to a change in temperature of ∗ g g∗ − aT+ = aT− 3 ∗ = aT− 3 ∗ − ∗ . g+ g− − g0 If we compare this situation with the approximation of an instantaneous phase transition, we have to consider an intermediate phase of cooled particles destined to annihilate, which no longer contribute to the pressure. Therefore, ∗ ∗ /g+ they produce a slight increase in a4 , a mini-inﬂation by the factor 3 g− (see Fig. 6.4). In general, the increase in temperature is not very large, because of the ∗

g0∗ . The only exceplarge number of remaining degrees of freedom, i.e. g− tion of interest is the annihilation of electron–positron pairs. At this time, the neutrino component of the radiation is already decoupled from the heat bath of photons, so that the increase in entropy goes to the photon component only, and a temperature quotient arises between the photons and neutrinos. The electron–positron pairs contribute ge∗ = 7/2 to the pre-annihilation value

152

6 Standard Synthesis

Fig. 6.4. Entropy production through intermediate vacuum or dust phases. In a pure Gamow universe, the quantity a4 H 2 is constant. When the relativistic gas becomes non-relativistic, the quantity a4 H 2 begins to increase until some annihilation process eventually turns the non-relativistic particles into radiation again. This increase is exponential (inﬂationary) when the energy in the intermediate stage is stored in vacuum energy (Chap. 7) ∗ g− = gγ∗ +gν∗ +ge∗ . The annihilation proceeds with interaction with the photon component alone and produces a rise in temperature of ∗ 3 2 + ge 3 11 = aT− . aT+ = aT− 2 4

From now on, the photon gas has a temperature given by the constant aT+ . The neutrino gas has been left at aT− . Both temperatures fall with expansion, but their quotient Tγ 3 11 = Tν 4 remains constant. In the case of a simple decay process (D into N), we cannot hope for a switch-oﬀ of the process caused by dilution of the particles in question. All particles will decay. The question here is the virtual end of the production process, and the evaluation of the exponential decay of the concentration after that point. Here, the decay may be even faster (ξ = ξ0 exp[−x2 t0 /x20 τ ]) than the decrease of a canonical equilibrium value (ξequilibrium = e−x /(1 + e−x )). Therefore, we have to compare the characteristic production time due to some kind of particles with fw degrees of freedom in total and a cross-section σw and the decay time τ . The reaction equation is something like

6.2 The Standard Process and the Decay of Equilibrium

153

l2 dξ x =− (1 + nw σw cτ )(ξ − α(1 − ξ)) . dx f clPlanck τ The unknown coeﬃcient α is reduced to the equilibrium concentration: ξequilibrium − α(1 − ξequilibrium ) = 0 , α =

ξequilibrium = e−x . 1 − ξequilibrium

The density and cross-section of the particles that induce transitions are given by some power of x, i.e. nw σw cτ = ( xx1 )n . The process is determined by two dimensionless constants, i.e. A = l2 /f lPlanck cτ , and x1 , given by nw σw cτ = (x1 /x)n . If σw can be assumed to be constant, we have n = 3. We obtain dξ n = −Ax (1 + (x1 /x) ) (ξ − e−x (1 − ξ)) . dx

(6.12)

A solution of this equation is shown in Fig. 6.5. The abundance ξ follows the equilibrium value, i.e.

Fig. 6.5. The decay of an excited state. The diagram shows the solution of (6.12) for A = 0.02, n = 5 and x1 = 2. The broken curve shows the equilibrium abundance ξequilibrium , the dotted curve shows Ax(1 + (x1 /x)n )

154

6 Standard Synthesis

dξequilibrium ex = , dx (1 + ex )2 as long as ex l2 x(1 + nw σw cτ )ξequilibrium < x 2 (1 + e ) f lPlanck cτ x n 1 e−x , ≈ Ax 1 + x or

x n 1 1 . < Ax 1 + (1 + e−x )2 x √ The right-hand side is minimal at x = x1 n n − 1; there, its value is Ax1 n √ n n − 1. Processes with Ax1 n > 1 are always quasi-static. In the interesting case, the minimum is far lower, and the end of the quasi-static process occurs at about x n−1 1 1 ≈ Ax1 , x ≈ x1 n−1 Ax1 . x At this time, the decay is exponential in time (i.e. ∝ exp[−Ax2 ]), and is slower at the moment but faster at later times. Formally, a quasi-static process (∝ e−x ) is eventually regained, but this occurs only at a very small residual abundance.

6.3 The Primordial Nucleosynthesis The quasi-chemical evolution of the universe is caused by the overall expansion. This expansion is locally adiabatic. The local heat bath is determined by the sum of all relativistic particles. In the region of interest here, the relativistic particles have to interact electromagnetically in order to be coupled strongly enough to the particles whose processes we intend to study. We have to remember here that in the post-recombination epoch this coupling is also too to hold the other particles in thermal equilibrium with radiation. In the pre-nucleosynthesis epoch, the weak interaction is also strong enough to moderate thermal equilibrium. So when we speak about temperature in the universe, we mean primarily that of electromagnetic radiation. The other kinds of particles leave the equilibrium at characteristic times to proceed with their evolution as adiabatically isolated components. The history of the primordial nucleosynthesis begins at some time t1 below the baryon temperature, at which kT ≈ mp c2 . We ﬁnd neutrinos, muons, electrons and their antiparticles, photons, protons, and neutrons, in thermodynamic equilibrium. The relative abundance of neutrons is determined in equilibrium by the Boltzmann factor cn mn c2 − mp c2 = exp − . cp kT

6.3 The Primordial Nucleosynthesis

155

This equilibrium is ensured mainly by weak processes, and lasts for longer than the equilibrium of the neutrinos with the photons. Neutrinos decouple quietly from the photons at some earlier time t2 , they remain as abundant as the photons, and their energy distribution remains Planckian in the course of expansion just as the photon spectrum does. Now, the neutron–proton interchange is a process of the type described by (6.5). The characteristic length is given by l = 1.5×10−13 m. With a decay time τ = 886 s, we obtain A = 2 × 10−4 . The equilibrium breaks down at some time t3 ; the larger the weak cross-section is, the later this time is. The neutrons begin to decay into protons via the only channel available, n → p + e− + ν¯. The densities are already so low that we can safely extrapolate the laboratory values. A little later, at t4 , corresponding to the temperature kT = me c2 , the electron–positron pairs annihilate (without disturbing the neutron abundance, of course) and leave only a small abundance of electrons, just enough to compensate the electric charge of the protons. The plasma consists now of an adiabatically isolated neutrino component, and of photons, protons, electrons and decaying neutrons. At some time t5 , where 2 2 lnp→d 1 , t5 ≈ ln 2cf lPlanck ν the temperature falls to kT < α2 me c2 , and the deuteron formation sets in, that is, deuterons are no longer destroyed by collision with photons. The deuteron production is (compared with the stellar interior) rapid at kT ≈ 0.5 MeV; basically, all neutrons around are bound and stabilised by the nuclear binding energy. The concentration of neutrons is approximately t5 − t 3 ξn [t5 ] = ξn [t3 ] exp − . τn In order to estimate how this abundance depends on the cosmological parameters, we write the kinetic equation in the form −

2 ln→p dξn→p = xn→p dxn→p f clPlanck

ξn→p f2 + 3 ν 3 σw c(ξn→p − ξn→p τ xn→p ln→p

equilibrium )

.

The concentration of the baryons does not enter, because the equilibrium is maintained by the relativistically abundant leptons. The number f 2 of relativistic degrees of freedom is the important parameter for the neutron abundance at the point of release, xn→p 1 . It is very near one. We take x = 1 to calculate the model value of σw . The equilibrium concentration is simply ξn→p = (1 + exp[xn→p ])−1 . We obtain the switch-oﬀ condition 2 ln→p exp[xn→p ] = xn→p 1 + exp[xn→p ] f clPlanck

1 f2 + 3 ν 3 σw c τ xn→p ln→p

Its approximate solution is x2n→p = fν2 σw /2πf ln→p lPlanck , i.e.

.

156

6 Standard Synthesis 2 2ct3 ≈ x2n→p ln→p /f lPlanck .

At t5 , the synthesis of deuterium starts, and its kinetic equation is of the standard type. We obtain (6.8), i.e. xnp→d ∝ ln[Afγ2 νσnp→d /ftotal llPlanck ] (with 2 /f lPlanck . The time span for the free decay ln A ≈ 40), and 2ct5 ≈ x2np→d lnp→d of neutrons depends mainly on the number of relativistic degrees of freedom (i.e. on the expansion rate), but also on the baryon concentration. The ﬁnal abundance depends on this time span and the switch-oﬀ time t3 as well. The helium abundance after the end of the synthesis is a measure of the number of relativistic degrees of freedom at the switch-oﬀ time.

The timescale of the deuterium production, once it is possible, is short, and virtually all neutrons are used up by subsequent processes leading to 4 He: n+p → d+γ , n + d → 3H + γ , p + d → 3 He + γ , p + 3 H → 4 He + γ , n+

3

He → 4 He + γ .

The ﬁrst of these processes is the slowest. The other ones can be summarised by an eﬀective cross-section of the deuterons, and this shows that about a fraction 10−5 is left and not processed to 4 He. Of course, one can also calculate a primordially produced residual concentration of 3 He. The primordial 4 He is inert; there is no stable nucleus such as 5 He, 5 Li or 8 Be. Virtually no nuclei heavier than helium are synthesised in the primordial process. One can obtain only a tiny amount of 7 Li through the process 3 H + 4 He → 7 Li. The abundance of primordial 7 Li can be calculated too. However, its comparison with the observed values is diﬃcult because subsequent galactic and stellar processes disturb the abundance more than in the case of deuterium and helium. It is curious to note that 4 He can be processed in stars only because the unstable nucleus 8 Be has a high cross-section for collisions with 4 He because of the existence of an excited state 12 C∗ of the carbon nucleus, which is in resonance with that collision. Consequently, in the interior of a star, the unstable 8 Be nucleus, which decays via 8

Be → 2 4 He + 92 keV

with a lifetime of 10−17 s, captures a 4 He to form the excited carbon nucleus via 8 Be +4 He + 286 keV →12 C∗ →12 C + 7656 keV . A full calculation can be seen in [8]. After nucleosynthesis, we have a mixture of protons, helium nuclei, electrons and photons, with small traces of other light nuclei. Nothing happens now until the time t6 of recombination, when

6.3 The Primordial Nucleosynthesis

157

the temperature falls below the ionisation temperature of hydrogen, given by kTrec ≈ 4000 K, if the photon bath with the entropy per baryon of s = 109 . Now, basically every nucleus combines with electrons to form a neutral atom. If we take into account the fact that all stars up to the present day can have contributed to the helium content only in the per cent range, and that deuterium is used up by stars and not set free, the abundances of both helium and deuterium are traces of the primordial nucleosynthesis, and can be used to infer the conditions at that time. The deuterium abundance depends on the factor mPlanck ∆m fγ2 . r=ν m2He ftotal The helium abundance reﬂects the neutron abundance, which depends on the decoupling time t3 andthe diﬀerence t5 − t3 . Eﬀectively, we have a measurement of dT /(T dt) = 8πG/3 at that time, that is, of the total mass density . So we are able to infer an upper limit on the number of types of relativistically abundant neutrinos (not more than 3, together with their antineutrinos) and to infer the entropy per baryon (s ≈ 109 ). The details are shown in Fig. 6.6 [3]. The full calculation can be found in [8, 9]. It uses kinetic equations of the type N N N YiNi Yj j Yl l Yk k dYi Ni = [l, k]j − [i, j]k . dt Nl !Nk ! Ni !Nj ! j,k,l

The exponents Nl are the reaction numbers in the corresponding process channel. The abundances are Xi = Ai Yi ,

Xi = 1 .

i

The process rate of particles of type l with type k in the channel indicated by j is described by [l, k]j , and the reverse rate of particles of type i with particles of type j in the channel k is described by [i, j]k . For β decay, the rate is identical to the inverse of the lifetime. For reactions with photons or relativistically abundant leptons, the rates are given by [iω] = nω σv . For two-particle reactions with nuclei and nucleons, the rates are proportional to the total baryon density and inversely proportional to the mass number, i.e. [ij] = b Mu−1 σv . The network of processes up to a mass number of 8 is shown in Figs. 6.7 and 6.8. Multiple-particle reaction rates are proportional to higher powers of the baryon density. They are very small, and play no part in the primordial nucleosynthesis. The average values σv refer to the dependence of the cross-section on the energy and to the variation of the velocity distribution with the temperature [2]. The ﬁrst calculations were done by Hoyle and Tayler [5] and Wagoner, Fowler

158

6 Standard Synthesis

Fig. 6.6. Result of the primordial nuclesynthesis. Abundances of 4 He (mass fraction), D, 3 He and 7 Li (by number relative to H) as a function of the baryon over photon ratio η or Ωbaryon h2 [3]. Horizontal lines represent the primordial abundances deduced from observational data. The narrow vertical stripe represents the limits provided by the WMAP observations of the microwave background (see Sect. 8.4). Reproduced from [3] with kind permission by A. Coc and the AAS

6.3 The Primordial Nucleosynthesis 7

Be

6

3

1

H

He

2

H

n

4

He

3

H

159

Li

7

Li

n + e+ = p + ν n+p=d+γ d + n = 3H + γ d + d = 3H + p 3 H = 3 He + e− + ν 3 H + p = 4 He + γ 3 H + α = 7 Li + γ 3 He + d = 4 He + p 3 He + α = 7 Be + γ 6 Li + p = 7 Be + γ 7 Li + p = 2 α 7 Be + n = 2 α 6 Li + p = 3 He + α

p + e− = n + ν d + p = 3 He + γ d + d = 3 He + n d + α = 6 Li + γ 3 H + p = 3 He + n 3 H + d = 4 He + n 3 He + n = 4 He + γ 3 He + 3 He = 4 He + 2 p 6 Li + n = 7 Li + γ 7 Li + d = 2 α + n 7 Be + d = 2 α + p 6 Li + n = 3 H + α 7 Li + p = 7 Be + n

Fig. 6.7. Nuclear reactions up to A = 7

and Hoyle [10], and were applied to the early universe by Wagoner [9] (see also [7]). We remark that in spite of all of the uncertainties in the matter density and the present Hubble rate, the primordial abundance of helium cannot be smaller than Y = 0.22 by mass. Hence a precise explanation of the processes in objects of smaller helium content is necessary for the evaluation of the primordial synthesis. The main alternative to the model presented here is one of inhomogeneous synthesis. It assumes that the nucleons themselves are bound systems of quarks produced in a quark–gluon plasma, and that the transition from the quark– gluon plasma to the nucleon–lepton plasma was a ﬁrst-order phase transition, with temporarily coexisting quark–gluon and baryon–lepton plasmas. Such a coexistence requires constant temperature and pressure, which are maintained by production of energy during the phase transition. A constant pressure means a constant density in the quark–gluon regions; they temporarily avoid expansion and end up with a higher density than the early baryon–lepton regions. Fluctuations of the baryon density are therefore produced. Their amplitude, however, may be reduced by diﬀusion. The ﬂuctuations for protons should be frozen in the photon bath, but the ﬂuctuations for neutrons could be eliminated by the larger mean free path of the neutrons. When nucleosynthesis starts, we have regions with a smaller baryon density but higher neutron concentration, and regions with a higher baryon density but smaller neutron concentration. In total, one should expect a far higher residual concentration of light elements than in the homogeneous model.

160

6 Standard Synthesis

Fig. 6.8. Cosmic salta. Lasker invented a board game in which stones collect others by capturing them. We have interpreted this as a kind of nuclear process on the board of nuclei, where the board carries information about stable positions and processes that follow the moves eventually (like in Snakes and Ladders). Nuclei grow by capturing other nuclei (or protons and neutrons) and shifting correspondingly. When a nucleus stops on a square that indicates spontaneous decay, it shifts spontaneously unless it is moved again. The faster the spontaneous decay, the brighter the square appears in the ﬁgure

6.4 Weakly Interacting Particles The dark (transparent) matter that we observe in galaxies, clusters, and structure formation consists of particles that are massive and are, at most, weakly interacting. They are referred to as weakly interacting massive particles (WIMPs). The strong and electromagnetic interactions are excluded because the dark matter does not hamper the formation of nuclei in accordance with conventional theory. In the simplest case, we can consider these particles as a kind of lepton like neutrinos, but massive. It is a second-order question whether they are the neutrinos that we know or some other particles. They should annihilate like other particles, with a small cross-section determined by the weak interaction. This process can be estimated by use of our standard (6.6). Their residual concentration can be estimated from (6.9):

6.5 The Problem of the Baryons

ξWIMP ≈ minimum 1,

ftotal lPlanck lWIMP fγ2 σWIMP

161

.

4 −2 lWIMP . The length The weak cross-section can be described by σWIMP = lV lV characterises the virtual particle that mediates the interaction. We obtain 3 ftotal lPlanck lWIMP m4V ξWIMP ≈ minimum 1, 2 ≈ minimum 1, . 4 fγ lV mPlanck m3WIMP

For light WIMPs, the majority of the particles remain relativistic, and no annihilation occurs. We can translate the result into a density parameter, and obtain ΩWIMP = ωγ ξWIMP mWIMP c2 /kTCMB . Let us consider the consequences. If mV ≈ 250 GeV/c2 and mPlanck ≈ 10 GeV/c2 , we have to expect nearly relativistic, abundant WIMPs for masses less than mWIMP ≈ 1 keV/c2 . The average background photon has kTCMB ≈ 3 × 10−4 eV; a WIMP with 30 eV rest energy would ﬁll the density budget. Relativistic, abundant WIMPs with larger mass clearly do not exist. WIMPs as massive as this should have undergone annihilation, and ΩWIMP < 1 implies ξWIMP mWIMP < 30 eV/c2 , that is, mWIMP > 10 MeV/c2 . A more precise estimation yields mWIMP > 1 GeV/c2 . The present state of the universe excludes WIMPs in the mass range between 30 eV/c2 and 1 GeV/c2 . There is a lower limit for any residue of annihilation. This limit is set by the Hubble volume at the time when annihilation stops owing to dilution. When the Hubble volume it will host only about 1 particle, the process stops, independent of the formal cross-section. That is, the minimum relict abundance is (with x ≈ 1) 19

ξminimum ≈

H3 2πl3 (f lPlanck )3 2πf 3 m3 = = . 3 2 3 2 3 nγ c fγ c (l ) fγ2 m3Planck

(6.13)

From (3.5) and (5.2) we see that a cold component which would contribute critical/nγ ≈ 10 eV/c2 per photon would ﬁll the critical density. The maximum allowed concentration for the residue is ξmaximum ≈ 10 eV/mresidue c2 . There is no place for stable particles with masses exceeding 1012 GeV/c2 .

6.5 The Problem of the Baryons Two explanation paradigms compete with each other, and one usually switches unconsciously between the two. On the one hand, we are tempted to explain nearly but not quite symmetric states as deviations from a symmetry

162

6 Standard Synthesis

that apparently does not require explanation. On the other hand, we hope for explanation by dynamics. The consequence is that symmetry is an exception and should be an exception that has to be explained. Thermodynamics tries to do this in various ways. If we consider a universe hotter than the annihilation temperature of baryons (kT > mp c2 ) and symmetric with respact to charge (i.e. baryon number), we have to take account of some annihilation process below kTp = mp c2 . The annihilation cross-section is so large that the ﬁnal abundance of baryons is about 10−19 . In order to see this, we have only to replace the cross-section in (6.9) with the electromagnetic cross-section (the Thomson cross-section). The concentration of 10−19 is of course 10 orders of magnitude oﬀ the observed value (somewhere near 10−9 ). What to do? There must be an excess of baryons over antibaryons, small before annihilation (just 10−9 ), but dominant afterwards when the symmetric part has shrunk from 1 to a negligible 10−19 . Where does this small asymmetry come from? Asking the question means switching to the ﬁrst paradigm. With equal right, one could ask: why is the asymmetry so small? What process reduced the asymmetry, which is much more probable than exact symmetry? And, in addition: why was this process not eﬃcient enough to reduce the asymmetry to nothing? There is no right question; there is at best a productive question. Here it is the ﬁrst question that has a productive answer. First, if the asymmetry was produced by some process that began with a symmetric state, elementary processes must exist that do not conserve the baryon number. Second, if such processes exist, the proton might or should be an unstable particle. Its lifetime can be measured by the number of protons that are observed to decay. This lifetime may be estimated from our personal lifetime. The reader consists of some 1029 protons, and in his/her lifespan of 70 years not more than a few hundred should decay, otherwise damage would appear too soon. The lifetime of a proton must be longer than 1029 years. This exceeds the age of the universe by 19 orders of magnitude. There must have been a heavy unstable particle in the pre-annihilation universe that did not decay symmetrically into baryons and antibaryons. Let us call it X. We assume it to decay into a baryon with fraction r and into ¯ decays with fraction an antibaryon with fraction 1 − r. Its antiparticle, X, r¯ into an antibaryon, and with fraction 1 − r¯ into a baryon. The decay is symmetric if r = r¯. Any asymmetry would signify that = r − r¯ = 0. Let ν ¯ at the time of their decay; they produce be the abundance of the X and X an overabundance of baryons of νbaryon − νantibaryon = 2ν. We have small numbers everywhere. We need only a weakly broken symmetry. Broken symmetry is the keyword. In the realm of elementary particles, there are classiﬁcations, but no overt symmetries. It was a real task to ﬁnd out in what sense the neutron and proton were the same particle in spite of their diﬀerence in mass (indicating some kind of excitation) and their diﬀerence

6.5 The Problem of the Baryons

163

in charge. It was a real task to ﬁnd out in what sense the hyperons were the same particle, and so on. The eightfold way was the solution to this problem. For the baryon problem, it is the grand uniﬁed theory (GUT) that seems to solve the puzzle [11]. Uniﬁcation of theories is the central issue in physics: generally speaking, it is the use of a uniﬁed theory to explain one set of phenomena with the rules of another set, or with common rules. A theory itself is a uniﬁcation of phenomena under a common structure. We recall Newton’s mechanics as the uniﬁcation of celestial and terrestrial motions, Maxwell’s theory of electric and magnetic phenomena, the kinetic foundation of thermodynamics, and the uniﬁcation of mechanics and electromagnetism in the theory of special relativity. Einstein, Weyl, Kaluza, Klein, Schr¨ odinger and many others tried to construct a uniﬁed theory of electromagnetism and gravitation. Although success has not yet been achieved, they found gauge invariance, higher dimensions and many other concepts. The main success in uniﬁcation of theories was accompanied by an understanding of symmetry breakdown. This came with the uniﬁcation of the weak and electromagnetic interactions by Glashow, Salam and Weinberg. At ﬁrst glance, one would not expect that leptons with diﬀerent masses and massless photons would all have a common root. The solution is that the masses and their diﬀerences are small with respect to the characteristic mass of the interaction (mV in the preceding section). The masses of the particles turn out to be the result of a spontaneous breakdown of symmetry. The particles are quanta of ﬁelds that all interact. Although they are all massless individually, and although the action integral is symmetric, the ground state of the reference ﬁelds might be degenerate and spontaneously asymmetric. Symmetry is often restored at a high enough temperature or energy, kT = E mV c2 . The manifold of the various particles that develops through symmetry breakdown reﬂects the underlying group structure (Fig. 6.9, for instance), just as the system of chemical elements reﬂects the structure of the rotation group (Fig. 6.10). The question is whether the theory of strong interactions, which has explained all of the structure in the world of baryons and mesons, can be uniﬁed with the electroweak theory and yield a grand uniﬁed theory. As in the electroweak case, one expects symmetry at temperatures or energies that are high enough. Investigation of cross-sections at high energies shows that the characteristic energy is very high, namely EGUT ≈ 1015 GeV. The coupling constants of the strong and electroweak interactions seem to converge above this energy. In a grand uniﬁed theory, one has to take more than 200 degrees of freedom into account. Most of the corresponding particles are as heavy as 1015 GeV/c2 . Some of them may decay asymmetrically with respect to the various charges (except for the electromagnetic charge). If they are allowed to decay, that is, if they do not annihilate too fast below kT ≈ mX c2 and if they leave

164

6 Standard Synthesis

Fig. 6.9. The system of baryons. The group structure of SUc (3) yields systems of states that characterise the various baryons at left and baryon resonances at right. In some respects, the baryons are all one particle in diﬀerent states that are characterised by their values of isospin, strangeness and ﬂavour and can be seen to form these structures. The Ω− in the 20plet at right was found after the construction of the theory and was found to exist with the expected mass and spin. However, the lifetime was unexpectedly long (10−10 s) compared with the other particles, which are resonances with lifetimes of the order of 10−23 s

the quasi-static annihilation fast enough, they could produce the amount of baryon asymmetry that we observe today. In the following epoch, we have relativistic quarks and leptons in contact with the photons, that is, we again have thermodynamic equilibrium. The history of the nucleosynthesis begins at a temperature equivalent to the nucleon mass. The question of the processes that form the nucleons out of quarks is as open as the theoretical ideas about how the conﬁnement of quarks in nucleons really works. At the nucleon-mass temperature, the annihilation of nucleon–antinucleon pairs begins. Before only the asymmetric excess remains, the neutrinos lose contact with the heat bath because of their small cross-section and begin to lead an undisturbed life (except for gravitation). Their temperature is decoupled from the photon temperature, and strictly obeys Wien’s law as long as the temperature does not fall below the value corresponding to the mass of the neutrinos (which is believed to be zero or at least below 30 eV/c2 ). The temperature now drops to a value where the neutrons ‘feel’ their mass excess over the protons. This excess is no longer compensated by suﬃciently hot leptons, and the neutrons begin to decay. Soon afterwards, the electrons and positrons annihilate, up to the small asymmetry excess, and produce a considerable amount of photons, so that the photon component becomes a little hotter than the neutrino component, which has already come out of the equilibrium.

6.5 The Problem of the Baryons Ac Th

La

Ce

Pa

Pr Nd

Lr U Np

Pu

Am

Cm

Bk

Lu Pm Sm

Eu

Gd

Tb Dy

Y

Sc

Ti

V

Zr

Cf

Hf

Ho

Nb

Rf Db Sg Es Fm

Ta

Er

Mo

W

Tc

Os

Ir

Ru

Mn

Fe

Co

Rh

Ni

Pd Ag

C

N

Ge

Pb

Bi

Fr Po

At

Ra

—– n − j = 6

Rn

Au Hg Sn

Sb

Te

Cs Ba I

—– n − j = 5

Xe

Cd Rb

As

Se

Br

—– n − j = 4

Sr

Kr

Zn

Si

P

K S

Cl

Na O

Fl

Mg

—

Ne Li

H

—– n − j = 3 — – n = 3

Ca

Ar

j= 1— –

j= 2— –

B

Pt In

Cu

Al

Mt

Tl

Yb

Ga Cr

Hs

Md No

Re

Tm

Bh

165

Be

—

–

–

n

n

=

=

2

1

He

Fig. 6.10. The system of chemical elements. The three-dimensional scheme of the SUc (3) is reduced to linear multiplets in the case of the rotation group. We ﬁnd these multiplets in the periodic system of elements as multiplets with quantum numbers n and j, where they determine the chemical behaviour. For each main quantum number n there are n quantum numbers j, and for each j we obtain 2(2j + 1) states for the valence electrons. The valence electrons do not ﬁll the states of the main quantum number consecutively, as we would expect from the energy spectrum of hydrogen, but the interaction of the electrons itself leads to a consecutive ﬁlling of the states for any ﬁxed n − j, beginning with the largest j. For higher values of n − j, some of the valence electrons prefer the states of lower j before the states of higher j are ﬁlled, but this does not change the overall pattern

Just in time, before all neutrons have decayed into protons, the deuterons become stable. They form through protons capturing the neutrons and are processed into helium. This is the primordial nucleosynthesis, which is cut oﬀ by the expansion of the universe in the standard way. A small amount of deuterons is left unprocessed and traces the process for the observer today. The observed concentration of helium can be translated into the concentration of neutrons at the moment when deuterium synthesis starts. Finally, the temperature falls to the ionisation temperature of hydrogen. The atoms become neutral, and the universe transparent. The photon component no longer produces enough viscosity to suppress condensation, and gravitation takes over to form the structures that we now observe by the

166

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reheated matter that they now contain: stars, galaxies and clusters of galaxies. The late state of the universe begins.

References 1. Barrow, J. D., Tipler, F. J.: The Anthropic Cosmological Principle, Oxford University Press (1986). 143 2. Fowler, W. A., Caughlan, G. R., Zimmermann, B. A.: Thermonuclear reaction rates, Ann. Rev. Astron. Astroph. 5 (1967), 525–570. 157 3. Coc, A., Vangioni-Flam, E., Descouvemont, P., Adahchour, A., Angulo, C.: Updated big-bang nucleosynthesis compared with Wilkinson Microwave Anisotropy Probe observations and the abundance of light elements, Astrophys. J. 600 (2004), 544–552. 157, 158 4. Coleman, T. S., Roos, M.: Eﬀective degrees of freedom during the radiation era, Phys. Rev. D 68 (2003), 027702 (astro-ph/0304281). 146 5. Hoyle, F., Tayler, R. J.: The mystery of the cosmic Helium abundance, Nature 203 (1964), 1108–1110. 157 6. Rees, M. J.: Our universe and others, Q. J. R. Astron. Soc. 22 (1981), 109–124. 143 7. Schramm, D. N., Turner, M. S.: Big-bang nucleosynthesis enters the precision era, Rev. Mod. Phys. 70 (1998), 303–318. 159 8. Schramm, D. N., Wagoner, R.: Element production in the early universe, Ann. Rev. Nucl. Part. Sci. 27 (1977), 37–74. 156, 157 9. Wagoner, R. V.: Synthesis of elements within objects exploding from very high temperatures, Astrophys. J. Suppl. Ser. 18 (1969), 247–295. 157, 159 10. Wagoner, R. V., Fowler, W. A., Hoyle, F.: On the synthesis of elements at very high temperatures, Astrophys. J. 148 (1967), 3–36. 159 11. Wilczek, F.: The Universe is a Strange Place, Proceedings SpacePartII, Washington D.C., astro-ph/0401347 (2004). 163

7 Inﬂation

7.1 The Implications of the Monopole Problem The optimistic picture for the explanation of the baryon-antibaryon asymmetry that we apparently obtain with grand uniﬁed theories has one serious drawback. In generic theoretical models of a grand uniﬁed theory, with the breakdown of symmetry, there develop curious topologically stable ﬁeld structures that behave like particles with the uniﬁcation mass m ≈ 1015 GeV/c2 . These structures are called monopoles. We consider them in more detail in Sect. 11.4. Here, we shall refer only to their abundance. In contrast to the ordinary particles that are free to decay at lower temperatures, these quasiparticles are stable. They may annihilate with the corresponding antiparticles, but the minimum abundance predicted by (6.13) should survive. This minimum abundance corresponds to a mass density that is many orders of magnitude too large to ﬁt into the Friedmann balance. In addition, these monopoles would trigger many processes that are not observed. Up to now, the search for monopoles has been in vain. The predicted abundance would imply that the model cannot be applied. The solution is the supposition of some dilution through inﬂationary expansion. Inﬂation denotes an epoch in the early universe when the expansion proceeds at a constant rate and dilutes all particles to undetectably small densities. It must be invoked in order to solve the monopole problem and to save the programme of grand uniﬁed theories [3]. Expansion at a constant rate is exponential in time; it corresponds to the de Sitter universe, where H 2 = H02 =

1 2 Λc , a[t] = a0 exp[H0 t] . 3

This expansion should precede the radiation-dominated phase, which should begin at kT < EGUT . The Λ term must be of the order of magnitude of the radiation density at this temperature, that is, (TGUT /(30 K))4 ≈ 1088 times as large as it may be at present. The inﬂation period must follow the production of monopoles and precede a radiation-dominated universe at a temperature high enough for us to ﬁnd the X particles that generate the baryon asymmetry. At this point, the scheme that we intend to develop seems rather to be wishful thinking. However, we Dierck-Ekkehard Liebscher: Cosmology STMP 210, 167–188 (2004) c Springer-Verlag Berlin Heidelberg 2004

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shall see not only that the set of the properties required is consistent but that it unexpectedly generates the kind of density ﬂuctuations that we observe so precisely in the microwave background. This closure of the cosmic loop is the genuine strength of the model.

7.2 The Vacuum The most important success of the theory of elementary particles in the last 20 years was the discovery that the strong and electroweak interactions can be presented in a common theoretical frame. In this frame, the three interactions (the strong, electromagnetic and weak interactions, as diﬀerent as they are at present) have the same strength at energies larger than about 1015 GeV. We describe this state as symmetric.1 The universe cools through expansion, and the initial symmetry is reduced by various phase transitions in such a way that we observe today just the three individual interactions, embedded in the universal gravitation [7]. The symmetry reduction is a kind of property of the vacuum state. The vacuum is the state in which no real particles are present. We recall that particles are quanta of excitation of ﬁelds. Therefore we have to understand the vacuum as a state that lacks quanta that can transfer energy or charge.2 The Heisenberg uncertainty relation shows that the oscillators of the ﬁeld are nevertheless in motion. This motion can be represented by a picture of virtual particles or, better, some polarisability of the vacuum (Fig. 7.1). The vacuum is empty only in a very restricted sense. The manifold of ﬁeld oscillators in the ground state still has some energy. This residue can be described (after the ordinary renormalisation procedures) by some energy density that is accompanied by a negative pressure.3 The negative pressure of the vacuum is a consequence of local Lorentz invariance: the energy–momentum tensor of the vacuum ought not to contain the deﬁnition of a direction. It must be proportional to the Minkowski tensor, i.e. Tik ∝ ηik . This yields the equation of state p = −c2 . The vacuum energy is deﬁned by being free of preferred direction, so it is free of motion. No velocity can be attributed to this component, in particular, 1

2

3

We may call it the ‘uniﬁed phase’ and understand it as the time when the temperature was in the range 1015 GeV < kT < 1019 GeV, when quantum gravitation played no part any more, and when the electroweak and strong interactions were of the same strength and could possibly be described with a grand uniﬁed theory. The deﬁnition of such a state is not invariant with respect to accelerated motion, and requires special consideration. In comoving coordinates for a homogeneous universe, we can refer to the analogy with the Minkowski space-time. The matter term that was christened the ‘cosmological constant’ or ‘vacuum’ was introduced by Poincar´e to solve the problems of the classical theory of the electron.

7.2 The Vacuum

169

Fig. 7.1. Metaphor for the virtual pairs that form the vacuum. On a space-time plane, the ﬂuctuating height is a metaphor for the ﬂuctuating virtual-particle density (negative for the antiparticles). There is no real motion, no integral number of particles diﬀerent from zero, but we are on a ﬂuctuating surface that carries an energy, which should contribute to the gravitational ﬁeld via the equivalence principle. These ﬂuctuations have to be considered in calculating the eﬀective charges of particles that might be changed owing to the polarisability of such a ﬂuctuating vacuum

not even rest.4 It is no obstacle to any motion. Hence one cannot transport it from one region of space to another. Nevertheless, its density might depend on physical or geometrical conditions. We must remember that the vacuum energy acts like a cosmological constant in stages when no phase transition is aﬀecting the vacuum. At the end of inﬂation, such a phase transition must occur in order to restore the radiation-dominated Gamow universe. 4

Later, equations of state p = wc2 , with −1 < w < −1/3 (quintessence) and w < −1 (phantom dark energy), are discussed. Matter with such an equation of state deﬁnes a velocity and a reference object.

170

7 Inﬂation

7.3 The Inﬂaton Field The simplest ﬁeld that models the equation of state of a Lorentz-invariant vacuum is a scalar ﬁeld that does not change essentially in space and time. Such a ﬁeld not only models the vacuum, but also steers the breakdown of symmetry and represents a kind of order parameter. In our context, it is called the inﬂaton ﬁeld. We shall now consider its properties. The action integral of a scalar ﬁeld5 in a gravitational ﬁeld is given by c3 d4 x −det gmn R S=− 16πG 1 ∂φ ∂φ ik 4 + d x −det gmn ¯h g − V [φ] . (7.1) 2 ∂xi ∂xk The quantity V [φ] is the potential of the ﬁeld, which may be understood as the essence of the interaction with a heat bath made up of the other ﬁelds. If the potential is quadratic in φ, i.e. V [φ] =

1 m2 c2 2 φ , 2 ¯h2

(7.2)

the ﬁeld oscillators are purely harmonic, and all ﬁeld quanta have mass m. The model of symmetry breakdown requires a more complicated potential V [φ]. It must contain degenerate minima that allow ground states with φ = 0. In such a ground state, the mass is deﬁned for small perturbations by m2 =

¯ 2 d2 V h . c2 dφ2

The energy–momentum tensor is given by ∂φ ∂φ 1 lm ∂φ ∂φ g ¯c − gik − V [φ] . Tik = h ∂xi ∂xk 2 ∂xl ∂xm In a Friedmann universe, the energy density6 and pressure are given by the formulae, 5

6

The factors c and h ¯ are often omitted, with the argument that one could them as units. We shall not follow this misleading usage, and shall keep our formulae in a state in which they can be checked for dimensions. Setting c = 1 is not the same as using c as a unit ‘c’ of velocity. The latter would mean that we put c = 1 c. Using h and c as units gives length the dimensions mass−1 c−1 h, time −1 −2 2 3 −1 the dimensions √ mass c h, force the dimensions mass c h , and charge the dimensions hc. We choose φ such that it has the dimensions of inverse length. is already used to denote the mass density, so the energy density is denoted by ε = c2 .

7.3 The Inﬂaton Field

ε=h ¯c V + and

1 ˙2 1 1 (∇φ)2 φ + 2c2 2 a2 [t]

171

1 1 1 2 p=h ¯ c −V + 2 φ˙ 2 − (∇φ) . 2c 6 a2 [t]

A homogeneous ﬁeld that changes only negligibly realises a matter component ¯ cV [φ]). with c2 = ε = −p (= h The ﬁeld equation for the inﬂaton reads ∂2φ dV 1 3a˙ φ˙ (+3HΓ φ˙ 2 ) = 0 . − 2 ∇2 φ + 2 + 2 2 c ∂t a c a dφ

(7.3)

The term in parentheses on the left-hand side must be introduced when exchange of energy of the inﬂaton with other ﬁelds is considered. The symmetry that is present in the action but can be broken by a conﬁguration is the invariance with respect to the inversion of the sign of the inﬂaton ﬁeld here. If the potential is chosen to be harmonic (7.2), the ground state, φ = 0, is symmetric as well. In order to represent the breakdown of symmetry, we must assume a more general form (which we cannot justify in our phenomenological framework, which we do not intend to leave). We simply write for the free ﬁeld λ (7.4) Vfree [φ] = (φ2 − η 2 )2 . 4 The factors η and λ are constants. By our choice the factor λ is dimensionless, and η, like φ, has the dimensions of inverse length. The potential is still symmetric, but the ground state is unique only for η = 0. As soon as η > 0, the choice of an individual ground state breaks the symmetry. The potential (7.4) must be complemented by a contribution from the interaction with the other ﬁelds. In the simplest case, this interaction term is the product of the squares of the amplitudes of the contributing ﬁelds, and depends on φ through the square of the amplitude, for instance Lint = φ2 f 2 [Ψ, . . .]. Formally, the factor f 2 [Ψ, . . .] is proportional to the square of an eﬀective mass. Its value is determined by the temperature-dependent amplitudes of the interacting ﬁelds. Therefore, we now deﬁne an eﬀective potential that sums over all other ﬁelds and degrees of freedom, in the form 1 λ V [φ] = (φ2 − η 2 )2 + φ2 4 2

m[T ]c ¯h

2 .

(7.5)

A simple ﬁnite-temperature ﬁeld theory could write the last term as f 2 , where f 2 represents the other ﬁelds that combine to create a temperature-dependent eﬀective mass m2 [T ]. At high temperature, the interaction term dominates. The ground state φ = 0 is unique and symmetric. When the eﬀective mass falls with the temperature below the critical value

172

7 Inﬂation

Fig. 7.2. High-temperature and low-temperature potentials. The equilibrium points A and B lie at the centre of symmetry in both the high-temperature potential (left) and the low-temperature potential (right). They do not change the symmetry. In the low-temperature potential, however, B is unstable. The stable states C in the low-temperature potential lie at some distance from the centre. Their totality does not break the symmetry, but the necessary choice does. Orientation is now deﬁned by reference to that spontaneously chosen point

m[Tcritical ] =

¯√ h λη , c

the ground state becomes degenerate, and the symmetry is broken through the state of the ﬁeld φ (Fig. 7.2). This process is called spontaneous breakdown of symmetry. With falling mass m[T ], the ground state is shifted to that of the free potential (7.4). We therefore expect that a low-temperature phase follows a pre-inﬂation high-temperature phase.7 The inﬂaton ﬁeld moves to the low-temperature minimum, slowly at ﬁrst: the energy density of the hightemperature vacuum, which is still large, yields the epoch of exponential expansion. The transition ends near the new minimum in rapid oscillations (Fig. 7.3). At this point, the energy of the high-temperature vacuum is passed on to the degrees of freedom of the conventional ﬁelds, i.e. it goes into relativistic and GUT particles (Fig. 7.4). The universe is reheated. The state φ = 0 that now corresponds to a maximum of the inﬂaton potential is also called a false vacuum. 7

4 The zero-point energy E0 = ¯ hc/RH = ¯ hH is, at H 2 ≈ 8πGEGUT /(3c2 (¯ hc)3 ), about 2 8π EGUT hH ≈ ≈ 1010 GeV . E0 = ¯ 3 EPlanck

The temperature T0 ≈ 1023 K is a low temperature in this context.

7.4 Homogeneous Inﬂation

173

Fig. 7.3. Schematic illustration of the temperature-induced evolution of the potential. At high temperatures (at the very early universe), the minimum of the potential is unique (point A). With increasing age, the temperature falls, and the state is shifted to the foreground of the ﬁgure and starts (point B) to roll down into one of the side valleys (point C). The choice of one side breaks the symmetry (in this case reﬂection symmetry)

The potential (7.5) is only a simple model. In a detailed model, the potential has to be derived from a ﬁeld-theoretical construction of the part Linteraction of the Lagrangian that describes the interaction. An example is the Coleman–Weinberg potential [1], 25α2 φ2 1 V [φ, T ] = φ4 ln 2 + (φ40 − φ4 ) 16 φ0 2    4 ∞ 2 h ¯ c 18 kT 5  . + 2 dx x2 ln 1 − exp − x2 + φ2 g 2 π hc ¯ 12 kT 0

Compared with the potential sketched in Fig. 7.3, it is narrower for high temperatures and shallower for low temperatures.

7.4 Homogeneous Inﬂation The equation of motion of a homogeneous potential in the expanding universe follows from (7.3): dV [φ] . (7.6) φ¨ + 3H[t]φ˙ = −c2 dφ If we take account of the corresponding Friedmann equation,

174

7 Inﬂation

Fig. 7.4. Sketch of the history of baryons. The decay of the vacuum produces electromagnetioc radiation, massive particles with symmetric decay (Y) and par¯ The diﬀerence between baryonic and ticles that decay asymmetrically (X and X). leptonic charge may be conserved. The residue of the annihilation processes is a small amount of baryons and leptons that ﬁnally forms the atoms which constitute present matter

H2 =

h 8πG ¯ 3 c

V +

1 ˙2 φ 2c2

¯hG and H˙ = −4π 3 φ˙ 2 , c

(7.7)

the system of the two equations can be solved. It is appropriate to write the expansion rate as a function of φ. The last equation turns into a prescription of how to substitute the ﬁeld φ for the time variable,8 ¯hG dH[φ] = −4π 3 φ˙ , H = dφ c in the Friedmann equation, which now turns into a ﬁrst-order equation for H : 2 dH ¯h2 G2 ¯hG = 12π 3 H 2 − 32π 2 4 V [φ] . (7.8) dφ c c 8

In this section, a prime indicates a derivative with respect to the ﬁeld φ.

7.4 Homogeneous Inﬂation

175

During the time when the amplitude φ of the inﬂaton changes slowly, the ¨ and φ˙ 2 in (7.6) can be neglected with respect to the potential c2 V [φ]. terms φ, The evolution is governed by 8π¯hG c2 dV , H2 = V . φ˙ = − 3H dφ 3c During this time, the expansion parameter is given by (7.8);

t φ φ1 φ1 R ¯hG H H ¯hG V ln dφ ≈ 8π 3 dφ . (7.9) = H dt = dφ = 4π 3 ˙ R1 c H c V φ t1

φ1

φ

φ

The amount of inﬂation that is necessary for the required dilution of monopoles constitutes a condition for the form of the potential. In the ﬁnal part of the inﬂation, the amplitude φ oscillates about the low-temperature minimum of the potential with a frequency ω 2 ≈ c2 V . The average energy of the inﬂaton ﬁeld can be handled like the energy of an ordinary harmonic oscillator that is damped through the expansion of the universe and the excitation of the ﬁelds that are coupled to the inﬂaton. The amplitudes of these ﬁelds, which vanished through dilution during the epoch of inﬂation, are now reﬁlled nearly to their high-temperature values. They initially form a hot component of matter (ε = 3p). We take account of the transfer of energy to the matter degrees of freedom by writing 1 ˙2 3H ν d , φ + V [φ] = − 2 φ˙ 2 − dt 2c2 c ¯hc d εhot = −4Hεhot + ν , dt 1 ˙2 8πG 2 φ + V [φ] + εhot , H = ¯hc 3c2 2c2 for the phase in which the inﬂaton ﬁeld starts to change rapidly. We simplify the reaction rate by using the expression ν = Γ ¯hc−1 φ˙ 2 , take the average by h/c)φ˙ 2 = 2¯ hcV [φ], and now write use of φ = (¯ d φ = −3Hφ − Γ φ , dt d hot = −4Hhot + Γ φ , dt 8πG H2 = (φ + hot ) . 3 The inﬂaton density acts dynamically during the reheating time like conventional non-relativistic matter. Given the expansion parameter a[t], all quantities can now be calculated.

176

7 Inﬂation For a ﬁnite-temperature ﬁeld, we represent the conventional ﬁelds by ψ, and ˆ 2 ψ 2 , and obtain reheating via the coupling by (1/2)λφ

ψ¨ + 3H ψ˙ +

m2ψ

k2 ˆ 2 [t] ψ = 0 . + 2 + λφ a [t]

If φ is periodic, this is a Mathieu equation for ψ that describes parametric ampliﬁcation.

7.5 Concomitant Solutions to Fundamental Problems The universe is dominated by radiation in its early times (t ∝ R2 ). After recombination, it is an Einstein–de Sitter universe, and curvature and the cosmological constant are not important. This simple scheme implies problems for the early ages, which are solved by inﬂation too. They have the appearance of greater philosophical depth, so they now replace the monopole problem in most introductions:9 – Why do events at diﬀerent positions on the ﬁreball appear to be in thermodynamic equilibrium although they never had causal contact before? – Why is the universe homogeneous on scales that could never be spanned by a causal levelling process? – Why does the universe not consist exclusively of black holes, although the thermal density contrasts at the Planck time were far to large to avoid these black holes? – Why is the present density parameter still of order 1 although Ω = 1, λ = 0 is an unstable point in the phase space of Friedmann models? We begin the discussion with the horizon problem. It is a problem of geometry, that is, it can be formulated without referring to the microphysical processes in the universe. It arises if one tries to understand the high degree of isotropy of the microwave background radiation as the result of a causal levelling process. Such a process can be eﬃcient only if diﬀerent parts of the ﬁreball have a common past. The existence of a particle horizon has elementary consequences for the understanding of the homogeneity of the universe in general, and the ﬁreball in particular. In an Einstein–de Sitter model, the patches of the sky which can be said causally connected by belonging to a commonlight cone of one√point on the singularity have a linear size of only about a[t]/a[t0 ] = 1/ 1 + z of the sky that we observe. The ﬁreball with a redshift of about z ≈ 1000, 9

A. Einstein: ‘Wenn wir etwas suchen, steigen wir vom hohen Roß herab und schn¨ uﬀeln mit der Nase auf der Erde herum. Danach verwischen wir unsere Spuren wieder, um unsere Gott¨ ahnlichkeit zu erh¨ ohen.’ (When we search for something, we descemd from the high horse and sniﬀ with our nose on the ﬂoor. Afterwards, we spoil our traces in order to advance our likeness to God.)

7.5 Concomitant Solutions to Fundamental Problems

177

Fig. 7.5. Fields of view and horizons. This is the past light-cone in a Gamow universe. The ﬁreball has a uniform temperature up to 10−5 . Nevertheless, there are regions on the ﬁreball that never had a common past. This is indicated here by the disjoint past cones of four events on the ﬁreball

is a sphere of comoving radius a0 χ[1000] ≈ 2 RH . The singularity sphere (z = ∞) has a comoving radius a0 χ[∞] ≈ RH χ[1000] + RH /35 (Fig. 7.5). In an Einstein–de Sitter model, opposite points on the ﬁreball cannot have a common past. If the homogeneity of the ﬁreball implies that there have been events in the past that caused that homogeneity, the ﬁreball must lie in the future light cone of these events. The horizons of the points on the ﬁreball must intersect in a common region on the singularity. We need not know anything speciﬁc about this region or the processes in question. If the intersection is empty, such a region and such a process do not exist. We avoid this paradox if we accept an early phase of the universe in which the small horizon of a point at the singularity increased to a size that contains our present horizon on the ﬁreball. Here too, this is inﬂation. The inﬂation not only reduces all particle densities to negligible values, but also increases the horizon of the events on the ﬁreball by an amount suﬃcient to agree with observation (Fig. 7.6). We can use a simple pattern to show that the horizons of the points of the ﬁreball have no point in common, i.e. that the intersection of the horizons is empty. The cosmological model has the simple form

178

7 Inﬂation

Fig. 7.6. Expansion parameter, Hubble radius, horizon. The ﬁgure shows the evolution of the Hubble scale, the ﬁelds of view and inﬂuence, and an expansion parameter depending on time (the axes are logarithmically scaled, only near the zeros of t and R they are linear). In a Gamow–Einstein–de Sitter universe, the horizon on the ﬁreball of a point on the singularity is smaller than the ﬁeld of view of an observer today. This is reversed in a model with inﬂation

1/2

t ; t1 2/3 t − t2 . t1 < t < t0 : a[t] = R1 t1 − t2 0 < t < t1 : a[t] = R1

Here t2 is determined by the continuity condition for H, 3t2 = −t1 . The horizon of an observer at time t0 on the ﬁreball has a comoving radius

t0 χ0 =

c dt c =2 a[t] a0 H 0

1−

a1 a0

.

t1

The light cone of a point on the singularity covers a region on the ﬁreball of covariant radius

t1

χ1 =

c dt c = a[t] a0 H 0

a1 . a0

0

No event in the early universe, even on the singularity, can inﬂuence a region on the ﬁreball as large as our horizon. If we use z ≈ 1000 for the ﬁreball, the quotient

7.5 Concomitant Solutions to Fundamental Problems

χ0 ≈2 χ1

179

R1 ≈ 70 a0

is so large that there is no hope of obtaining a qualitatively diﬀerent result through more precise analysis. The present Hubble volume contains 105 volumes equal in size to the event horizon that a point on the singularity spans on the ﬁreball. The ﬁreball lies too near the singularity for an explanation of the observed homogeneity by some process in its past. Expanding the idea of concatenating individual expansion laws, we obtain a simple model of inﬂation. It starts with a Gamow universe that is interrupted by a de Sitter expansion, and is followed by an Einstein–de Sitter universe. At the concatenation times, the expansion rate and the expansion parameter must be continuous. We write

1/2

t1 < t < t2 : vacuum,

t ; t1 a[t] = R1 exp[H(t − t1 )] ;

t2 < t < t3 : hot,

a[t] = R2

0 < t < t1 : hot,

a[t] = R1

t3 < t < t0 : cold matter,

t − t5 1/2 ; t2 − t5 t − t6 2/3 a[t] = R3 . t3 − t6

(7.10)

The continuity conditions determine the times ti and the constant H as functions of the parameters a0 , R1 , R2 , R3 , and H0 . Let us use

L=

a0 R2 R3 , M= , N = ln R3 R2 R1

We obtain H = H0 L3/2 M 2 , t1 =

.

1 N N 1 M2 , t2 = + , t3 = + , 2H H 2H H 2H

N N N M2 2L3/2 M 2 M2 − + , t5 = , t6 = − . H 6H 3H H H 6H The ﬁreball coincides approximately with the time of equal densities of hot (i.e. relativistic) and cold (i.e. non-relativistic) matter. Our horizon on the ﬁreball has a comoving radius t0 =

t4 χﬁreball =

c dt RH (1 − L−1/2 ) . =2 a[t] a0

t3

A point on the ﬁreball has a horizon

t3 χﬁreball

horizon

=

c dt 1 RH (eN + (eN − 1) + (M − 1)) . = a[t] a0 L1/2 M

0

We may put L = 1600 and M = 1028 . We have to assume that eN ≈ 1029 in order to ﬁnd the horizon of the ﬁreball as large as our ﬁeld of view on the ﬁreball. The apparent age of the model diﬀers from the apparent age of a pure

180

7 Inﬂation Einstein–de Sitter universe. The radiation phase shortens the apparent age, and the inﬂation phase prolongs it: t0 =

2 3H0

1−

1 −3/2 3 −3/2 −2 + L M N L 4 2

.

For realistic values (L > 1000, M > 1024 and N < 100), the corrections are extremely small. More complex models have been studied by use of the same pattern by Petrosian [9] (see also [5, 10, 11]).

We now turn to the inhomogeneity problem. The isotropy of the background radiation poses a twofold riddle. The ﬁrst is the fact that a simple explanation of the large-scale structure as being formed by gravitational instability requires δ/ ≥ 10−4 , and this is certainly not observed in the temperature anisotropy of the microwave background. We therefore expect the existence of transparent matter, as indicated in Chap. 5. Second, if we calculate the density contrast of a galaxy relative to its background back to the Planck time (where we can expect or at least assume that we have density perturbations in thermodynamic equilibrium), we obtain a much smaller value than we would expect from simple thermodynamic arguments. The number of particles in the observable metagalaxy, N = 1080 , would produce √ −1 ≈ 10−40 , much too large for perturbations of the order of δ/ ≈ N what we observe. We have to answer the question of how the matter distribution becomes so homogeneous before recombination, and how it becomes so inhomogeneous after recombination. In a universe described by H 2 = H02 (λ0 − κ0 (1 + z)2 + Ω0 (1 + z)3 + ω0 (1 + z)4 ) ,

(7.11)

the mass in a Hubble volume depends on the time or, equivalently, the redshift. This mass can be obtained in the form 3 c 1 1 2 3 (1 + z)−6 H Ω0 (1 + z) M ≈ 2G 0 H0 ω03 3 3 T0 T0 Ω0 c3 −1 = = M0 h100 Ω0 , 2GH0 ω03 T T where M0 = 1.15 × 1029 M . A mass of the order of magnitude of the mass of a galaxy is not contained in a Hubble volume before T < 106 K. Where does the primary ﬂuctuation come from? If the history of the universe had known only a matter-dominated late phase and a radiation-dominated early phase, and if an estimation using linear approximations were not totally wrong, an inhomogeneity that has density contrast δ1 ≈ 1 at present would have had, at the equidensity time ttrans , the value δtrans /δ1 ≈ (ttrans /t1 )2/3 , and at the Planck time tPlanck , the value

7.5 Concomitant Solutions to Fundamental Problems

181

δPlanck /δtrans ≈ (tPlanck /ttrans ) . Therefore, we would expect δPlanck = (Ttrans /TPlanck )2 (T1 /Ttrans ) ≈ 10−52 . We now ﬁnd, in a typical galaxy of mass 1011 M , about 1068 baryons and about 1077 relativistic particles. The statistical variance is expected to be √ N −1 ≈ 10−38 , and this is much larger than δPlanck . The thermal ﬂuctuations at the Planck time are too large to ﬁt with the small ﬂuctuations on the ﬁreball. How do we solve this puzzle? We could, of course, assume that the ﬂuctuations at the Planck time were not thermal, i.e. that the universe was more homogeneous than thermodynamics suggests. This requires special initial conditions, and we intend to avoid such specialisation. It can be shown that the measure of the initial conditions that lead to an isotropic ﬁreball in a two-phase history is extremely small. We can, however, assume instead a third phase in the evolution of the universe that smoothed the thermal ﬂuctuations at the Planck time and cooled the universe. This is again inﬂation. Connected with the inhomogeneity problem, but not so serious, are the ﬂatness problem and the vacuum problem. In the present balance of the Friedmann equation described by (7.11), no term can exceed a value of the order of 1. We already know that this implies that during the time before recombination the cosmological constant, curvature and pressure-free matter can be neglected with respect to the radiation. The early universe is dominated by radiation. At GUT temperature, the share of the curvature is κGUT = (κ0 /ω0 )(a2GUT /a20 ) < 10−52 , and the share of the present vacuum λGUT = (λ0 /ω0 )(a4GUT /a40 ) < 10−108 . At ﬁrst glance, there is no physical relation that yields such small numbers without mental contortions.10 If we see a vacuum density in the cosmological constant, we accept that, on cooling below the GUT temperature, this density falls from its high-temperature 4 /(¯ hc)3 ≈ 1098 J m−3 to its present low-temperature value of value + = EGUT 2 2 − = 3c H0 /(8πG) ≈ 10−9 J m−3 . This is characteristic of a phase transition that is due to a breakdown of symmetry. The original expectation was that the small values would not be accidentally or artiﬁcially small, but would be exactly zero.11 The present analysis of structure formation and microwave background ﬂuctuations yields λ = 0.7, however (see Chap. 5). 10

11

At a second glance, we may recall that these quantities have to be compared with the radiation density and may be attributed to corresponding temperatures, for hc(Λc/8πG¯ h)−1/4 . Expressed in ratios of temperature such as instance, kTΛ = ¯ TΛ /TGUT , the numbers are still small, but not exceedingly so. The ratio between the electron mass and the GUT mass or the Planck mass is a small number too. When asked about small numbers, Zel’dovich always referred to the small numbers that are found in tunnel eﬀect calculations. One cannot do without observation. In the ﬁrst ten years, it was even claimed that the scenario would be shown to be invalid if Λ or κ proved to be diﬀerent from zero. With the same absolute certainty, of course, now the claim is κ = 0 and λ ≈ 0.7.

182

7 Inﬂation

One problem resists solution by the inﬂation scenario. This is the singularity problem. In general relativity theory (GRT), there are no models with conventional matter and global causality that are free of singularities [6]. Conventional matter means non-negative pressure: the cosmological constant ought not dominate expansion. Global causality means that global Cauchy hypersurfaces (spaces) exist that part the past from the future, do not allow closed time-like lines and forbid incoming waves. GRT and global causality require that a singularity-free model contains a cosmological constant as an essential element. This was the central issue of the hypothesis about a de Sitter phase at the beginning of the universe. Apparently, inﬂation might seem to provide such a solution. But this view is mistaken. The vacuum of the inﬂationary scenario is a high-temperature vacuum, i.e. it is accompanied by a heat bath. This heat bath is hot matter that dominates again at pre-inﬂationary times and must start from a cosmological singularity again. Inﬂation is not a solution to the singularity problem.

7.6 Inhomogeneities and Inﬂation The decisive support for the hypothesis of an inﬂationary stage in the early history of the universe comes from the conclusions about the zero-point ﬂuctuations that survive the universal suppression of inhomogeneities. These zero-point ﬂuctuations can be calculated. They can be expected to develop, via decoherence, into ordinary density ﬂuctuations with the smallness that is observed in the microwave background. This closes the cosmic loop (Fig. 1). During inﬂation, the temperature falls below any signiﬁcant value (com¯ c/RH ). That is the reason pared with the minimal zero-point energy Emin = h why the ﬂuctuations that are expected in the pre-inﬂationary times are seen neither in the gravitational ﬁeld nor in the density perturbations, and why zero-point ﬂuctuations determine the ﬂuctuation spectrum. Zero-point ﬂuctuations must be expected if we assume that ordinary quantum ﬁeld theory can be applied to perturbations of the inﬂaton ﬁeld and if the temperature falls below the value that corresponds to the zero-point energy. Comoving scales expand with inﬂation, while the Hubble radius is constant. The inﬂationary expansion shifts scales12 λ[t] = 2πa[t]/k, beginning at some time ti [k], far beyond the Hubble radius, where ﬂuctuation amplitudes evolve solely under the inﬂuence of gravitation, with a certain ampliﬁcation during the phase transition at the end of inﬂation (t = te ), where the quantity ζ remains constant (see (8.10)). After the end of inﬂation, the Hubble radius RH [t] = c/a[t] increases faster than the expansion parameter. At some time tf [k], it overtakes the scale in question, and the ﬂuctuation appears as a density perturbation or primordial gravitational wave (Fig. 7.7). For a mass 12

When it is not speciﬁcally noted otherwise, k denotes a comoving wave number k = kphysical a[t].

7.6 Inhomogeneities and Inﬂation

183

Fig. 7.7. The evolution of the scales of perturbations. The larger scales overtake the Hubble radius at an early time and fall below it again later. They measure the inﬂation at an earlier time than do the smaller scales, which overtake the Hubble radius during inﬂation later and fall below it again earlier. The region A of scales that are accessible to evaluation today corresponds to a time span B of the inﬂation and related values of the inﬂaton ﬁeld; for this time span, we can tell something – at least in principle – about the potential of the inﬂaton

of the order of magnitude of the mass of a galaxy (M ≈ 1011 M ), this time, tf [ 3 /M ], is near to the equidensity time ttrans of matter and radiation, namely at h[zf ] c 3 0 3 MH ≈ ≈ 104 . = 1 + zf H0 M M After the scale becomes smaller than the Hubble radius, the amplitude develops by approximately Newtonian rules (see Chap. 8). The largest local scale is the Hubble radius. Its zero-point energy is E0 = hc/RH = h ¯ ¯ H. The smallest amplitude δφmin of a massless (inﬂaton) ﬁeld in a de Sitter world with an expansion rate H has this energy, which corresponds to the minimal temperature13 kT0 = h ¯ H ≈ kTGUT

TGUT . TPlanck

This determines the density perturbations; 13

The temperature kT = ¯ hc/Rhorizon of an event horizon (here Rhorizon = RH ) is deﬁned for quantum processes near a black hole (Gibbons–Hawking temperature).

184

7 Inﬂation

kT04 δM δM 1 + pi /i c2 [k, ti [k]] = [k, tf [k]] ∝ . 0 M M 1 + pf /f c2 When a scale increases beyond the Hubble scale, it should carry the zeropoint amplitudes of the inﬂaton and gravitational ﬁelds. In most models, the eﬀects of gravitational excitations remain small with respect to the inﬂaton ﬂuctuations. The decisive argument is that the uniformity of the conditions at the Hubble scale produces an approximately scale-free spectrum. This uniformity must hold at least during the time interval when the scales that are important at present (galaxies, clusters, large-scale structure and the ﬁreball) freeze. During this time, the inﬂaton ﬁeld φ must roll towards the lowtemperature minimum only slowly. In any case, the behaviour of the inﬂaton ﬁeld and its potential during the time when the relevant scales pass beyond the Hubble radius is imprinted on the spectrum of perturbations, and we shall be able to infer this behaviour through the spectrum observed at the re-entry time tf [k] (Chap. 8). The evolution of the scales of perturbations is shown in Fig. 7.7.

7.7 Variations The form of the inﬂationary phase depends on the assumed microscopic mechanism for the breakdown of symmetry, and on the properties of the ﬁelds that start and steer this mechanism. The most critical part of the models is the generation of the ﬂuctuations that are observed in the large-scale structure and the microwave background [4]. The generation of ﬂuctuations is very sensitive to some properties of the inﬂation. We can make the following statements: – The inﬂationary phase should last long enough to suppress all pre-inﬂation inhomogeneities and to reduce the radiation temperature appropriately. – The inﬂation should not end in a ﬁrst-order phase transition, because the resulting foam structure could be too inhomogeneous. – The inﬂation should end rapidly, in order to enable a suﬃcient reheating of the universe. This is necessary to produce the particles that are necessary to generate the baryon excess through asymmetric decay. – The reheating should end before symmetry is restored, in order to avoid a second process of generation of GUT monopoles. Two values remain unexplained by the inﬂation paradigm, that of the curvature and that of the present vacuum density. They are exceedingly small after inﬂation. While they should be comparable to the other ﬂuctuations at the end of inﬂation, they should be below any observational limit at present. This is a good reason to use a pure matter–radiation universe after inﬂation as a reference, but not a reason to do without observation. We recall the

7.7 Variations

185

WMAP measurements, which yield Ω ≈ 0.3 and λ ≈ 0.7 as the new reference values. In particular, a present value of the curvature κ of about 0.1 decisively modiﬁes the horizon problem [2]. This is shown best with the following pattern with two phases: 0 < t < t1 :

vacuum + curvature,

h2 [z] = λ − κ(1 + z)2 ,

t1 < t < t0 : hot matter + curvature, h2 [z] = −κ(1 + z)2 + ω(1 + z)4 , λ = ω(1 + z1 )4 . We obtain the integrals

0 < t < t1 :

t1 < t < t 0 :

dz 1 = √ arcsin h κ dz 1 = √ arccos h κ

κ 1+z ω (1 + z1 )2 κ 1 ω 1+z

;

.

At the time t3 of the ﬁreball, the horizon and the ﬁeld of view have the values √ √

κ χvisual

field

κ 1 κ 1 π − 2arcsin + arcsin 2 ω 1 + z3 ω 1 + z1 κ κ 1 = arcsin − arcsin . ω ω 1 + z3

κχhorizon =

,

There is no horizon problem, even in case of no inﬂation, i.e. z1 → ∞. However, we ﬁnd a minimal expansion parameter if the curvature is not compensated by a pre-inﬂation radiation ﬁeld. The minimal expansion occurs at a redshift zmax ≈ z12

ω . κ

For κ ≈ 0.1, ω ≈ 10−5 and zf ≈ 1028 , we obtain zmax ≈ 1026 . This is too small for a model that is intended to give reasons for the observed smallness and spectrum of perturbations. The ‘big bounce’ happens at a time far from all quantum limits of the theory.

Inﬂation is not bound to a scalar ﬁeld. We recall that the scalar ﬁeld may be a mere phenomenological description of the vacuum, and not a fundamental ﬁeld itself. For instance, the renormalisation terms of the pure gravitational ﬁeld correspond to a vacuum density, through suggesting corrections to the Einstein equations in the form of higher-order terms [12]. The corrections can be written as 1 8πG Rik − gik = 4 Tik , 2 c 1 k2 2 1 1 l lm 2 Tik = Ri Rkl − RRik − gik Rlm R + gik R , hc ¯ 2880π 2 3 2 4 1 k3 1 + 2R;i;k − 2gik R;l ;l − 2RRik + gik R2 . 2880π 2 6 2

186

7 Inﬂation

These formulae can be derived by use of a kind of perturbation calculus if the condition Riklm Riklm l−4 Planck holds. Through use of the higher-order invariants Ik of the curvature tensor [8], singularity-free models of the form n √ c3 φi fi [Ii ] + V [φ1 , . . . , φn ] −g d4 x S=− R+ 16πG i=1 can be constructed. In the simplest case, the two invariants I1 = R and I2 = 4Rik Rik − R2 suﬃce. However, if the coeﬃcient λ of the potential in (7.4) is not small enough, too large an amplitude of the perturbations is generated. We obtain a problem of ﬁne-tuning: the model must yield a small enough λ for the reason explained, and a large enough λ for an inﬂation to be obtained at all. If we assume an interaction that is too weak, the end of inﬂation will not be rapid enough. We then simply live in a region of the universe that has accidentally undergone enough inﬂation. Places with a smaller initial value of φ have remained small and unimportant (chaotic inﬂation). If we have such models in mind, we can also use potentials that have a unique minimum for all temperatures. Inﬂation wipes out any traces of the pre-inﬂation times, but the singularity problem remains unsolved. This puts us in the position of having to design models for the pre-inﬂation times by taste, beauty and plausibility. We can argue for consistency, but we have only one observation at hand: the models must lead to an appropriate inﬂation. Models that use quantumtheoretical patterns can provide such a plausible prehistory. The space-time structure of the universe begin after transition from a timeless state, i.e. this space-time structure characterises a region that borders (in time) on a region without time. If we stress the metaphor that quantum ﬂuctuations may violate the energy conservation law for tiny intervals of time, we can envisage small regions that accidentally acquire energy density with negative pressure and start expanding. This expansion does not use up the energy, but ﬁlls it permanently to the GUT level through the negative work done by the negative pressure. At the end of inﬂation, the universe is ﬁlled with radiation and relativistic particles. In such a picture, however, the inhomogeneity problem appears in a new light: true homogeneity holds only for scales far beyond our present horizon, and it can never be established by observation. Our metagalaxy is accidental and not characteristic of the whole. On the one hand, quantum eﬀects can start inﬂation simultaneously only in restricted regions, and will always do so (eternal inﬂation). The pattern formed by these regions represents an inhomogeneity of large size and amplitude. On the other hand, the degenerate low-temperature vacuum can result in an equally large-sized regional structure. These regions may be characterised by diﬀerent fundamental groups:

7.7 Variations

Radiation universe

Baryon asymmetry

Grand uniﬁed theories

Inhomogeneity problem

Causality problem

Monopole problem

Primordial black holes

Flatness problem

Universe too hot

Universe too dense

Supercooling

Inﬂation

Absolute zero

Vacuum

Zero point ﬂuctuations

Inﬂaton ﬂeld

Fig. 7.8. Net of headwords of inﬂation

187

188

7 Inﬂation

The physics based on our SUc (3) × SU (2) × U (1) group may hold only accidentally. Masses and coupling constants might diﬀer regionally. These values govern not only the microscopic phenomenology of the particle zoo, but the cosmological evolution of the matter components described in Chap. 6. Our metagalaxy must be part of one such region that has been inﬂated to a size corresponding to the observation that we see no trace of its boundary. The net of headwords of inﬂation is shown in Fig. 7.8.

References 1. Coleman, S., Weinberg, E.: Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7 (1973), 1888–1910. 173 2. Durrer, R., Kovner, I.: Antipodal microwave, Astrophys. J. 356 (1990), 49–56. 185 3. Guth, A. H.: Inﬂationary universe: A possible solution to the horizon and the ﬂatness problems, Phys. Rev. D 23 (1981), 347–356. 167 4. Guth, A. H., Pi, S.-Y.: Quantum mechanics of a scalar ﬁeld in the new inﬂationary universe, Phys. Rev. D 32 (1985), 1899–1920. 184 5. Guth, A. H., Sher, M.: The impossibility of a bouncing universe, with a reply by Vah´e Petrosian, Nature 302 (1983), 505–507. 180 6. Hawking, S. W., Ellis, G. F. R.: The Large-Scale Structure of Space-Time, Cambridge University Press (1973). 182 7. Hut, P., Rees, M. J.: How stable is our vacuum? Nature 302 (1983), 508–509. 168 8. Mukhanov, V. F., Brandenberger, R.: A nonsingular universe, Phys. Rev. Lett. 68 (1992), 1969–1972. 186 9. Petrosian, V.: Phase transitions and dynamics of the universe, Nature 298 (1982), 805–808. 180 10. Rothman, T., Ellis, G. F. R.: Smolin’s natural selection hypothesis, Q. J. R. Astron. Soc. 34 (1993), 201–212. 180 11. Smolin, L.: Did the universe evolve? Class. Quant. Grav. 9 (1992), 173–191. 180 12. Starobinsky, A. A.: A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980), 99–102. 185

8 Structure Formation in the Opaque Universe

8.1 Perturbations in General In the next two chapters we shall proceed down the other side of the cosmic loop and consider the history of perturbations down to the formation of the large-scale structure in the universe. We have convinced ourselves that the ﬂuctuations of interest are residue of inﬂation. During inﬂation, all pre-inﬂationary inhomogeneities are suppressed except for the zero-point ﬂuctuations. It is the zero-point ﬂuctuations of the inﬂaton ﬁeld that generate density perturbations, and the zero-point ﬂuctuations of the metric ﬁeld that generate primordial gravitational waves. In verifying that the structures seeded during inﬂation determine the ﬂuctuations in the microwave background and the correlations in the distribution of galaxies and clusters of galaxies, we close the cosmic loop, and the story of inﬂation acquires a second leg to stand on. Structures, in general, are witnesses to evolution. The notion of a structure is associated with some more or less stably constructed object in space, such as a crystal or a building. Structures are determined partly by internal laws, and partly by inheritance from earlier structures. The less they represent a stable ﬁnal state, the more they tell us about their history. Structures such as crystals are determined by laws that act internally. They are a kind of ground state. If the ground state is not degenerate, and a system reaches such a state, the system becomes independent of its history. Atoms, thermodynamic states and attractors (chaotic as well as ordinary) belong to this class. They do not witness evolution; they cannot. But this stability has another side, too. In nuclear physics, for instance, the timescales of evolution are rather short. While individual atoms and molecules do not tell us about their history,1 their stability is the reason why their relative concentrations are frozen or develop comparatively slowly, and the concentrations now tell us about history. This is well known in the case of stellar production of the various nuclei. The mass distribution of the nuclei is a frozen structure that is out of equilibrium. These kinds of structures are 1

This property was used by Einstein in order to criticise Weyl’s uniﬁed ﬁeld theory, which involved history-dependent masses of particles [12].

Dierck-Ekkehard Liebscher: Cosmology STMP 210, 189–213 (2004) c Springer-Verlag Berlin Heidelberg 2004

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8 Structure Formation in the Opaque Universe

genuine fossils. They more or less conserve a metastable state created in the past. But we do not use only fossils. Structures in full evolution will depend on their history as well, and tell us about it. As long as the system is still on its path through its conﬁguration space, it characterises this path to some extent, and is a witness of its own evolution. Any structure witnesses its evolution as long as it is not in a (unique) ﬁnal equilibrium state. Now for gravitation. Gravitation is peculiar. First, it is weaker than any other force, and the timescales for the resulting evolution are long. Second, the notion of equilibrium is a particularly intricate question for a system of gravitationally interacting objects. In a strict sense, such a system is not a thermodynamic system, because of the inﬁnite amount of free energy that it formally contains, and the virial state is only a limited substitute (e.g. [17]). Third, for an unbound system, the expansion of the universe slows down evolution and may even stop it. Short-range interactions are frozen in any case when the dilution of the reactants makes the reaction rate smaller than the expansion rate. It is an open question whether gravitational clustering leads to universal distributions, i.e. distributions that no longer witness the past. For larger gravitationally interacting systems, the expansion of the universe produces slow motion, with a timescale that becomes longer the larger the system is. This makes the recently formed large-scale structure in the universe an ideal witness of the history of the universe, i.e. an ideal witness for cosmology. How does the evolution of the universe work? As we shall see, condensation would proceed steadily if there were no pressure in the matter and if the Hubble scale did not increase too fast. The density contrast would increase with the expansion parameter until the point where non-linear evolution begins. Pressure has two eﬀects. First, it changes the evolution of the expansion rate, and second, it resists condensation, as we know from stellar structure, for instance. It is a good approximation to say – provided that the contributions from the curvature of space and the vacuum are not dominant – that the amplitudes of the inhomogeneities of the gravitational potential are constant. This statement resembles the theorem that the outer gravitational ﬁeld of a star is independent of its composition and structure provided that its mass does not vary. The evolution of the density will depend on the Hubble radius and the Jeans radius, which both depend on time. Calculating the evolution starting from small inhomogeneities includes the averaging of Einstein’s equations. In addition, it requires a reason for the existence of such inhomogeneities. For large scales,2 there is the problem of 2

We use the concept of scale as a substitute for comoving distance or comoving size. The physical measure of a scale grows at the same rate as the universe, i.e. at the Hubble rate H[t]. The scales of the matter distribution are constant, by the deﬁnition of the cosmic expansion. In contrast, the Hubble scale a−1 RH

8.1 Perturbations in General

191

gauge invariance, which mu be solved by the construction of gauge-invariant potentials [6]. Beyond the Hubble radius, the theory yields a proportional evolution of all components that conserves inhomogeneities in the potential. Outside the Hubble radius, the short-range pressure eﬀects are not eﬃcient. Ampliﬁcation proceeds undisturbed. When a scale is overtaken by the Hubble radius, the situation might be diﬀerent. The evolution depends on the relation of the instant tf at which this occurs to the time ttrans of equal densities of the pressureless component and of the radiation component of matter, when the expansion rate changes from H = (2t)−1 to H = 2(3t)−1 , and to the time trecom of recombination, when the universe becomes transparent. We obtain two critical scales. The ﬁrst is determined by the Hubble scale Ltrans = c/H[ttrans ]a[ttrans ] at equidensity of radiation and matter, that is, the transition from the Gamow to the Einstein–de Sitter universe. The second is determined by the Hubble scale Lrecom = c/H[trecom ]a[trecom ] at recombination, when the universe becomes transparent. Now, the distinction between dark (transparent) and baryonic3 matter becomes important. Dark matter decouples from the heat bath before nucleosynthesis. It starts a life of its own against the background of a still radiation-dominated universe. Cold dark matter (CDM) is non-relativistic at the end of the radiation-dominated epoch (kTtrans ≈ 10 eV). If the scale in question is smaller than Ltrans , it is overtaken by the Hubble radius when the expansion is still dominated by radiation (tf < ttrans ). Then the ampliﬁcation is suppressed until ttrans . With a Hubble scale increasing as RH /a ∝ 1/(1+z), the theory still yields condensation for dark matter, but only marginally. The situation does not change before the universe becomes matter-dominated, √ at matter–radiation equidensity. The Hubble scale then increases as 1/ 1 + z only, and the small-scale DM inhomogeneities resume growth in a way similar to the growth of the large-scale inhomogeneities, which is proportional to 1/(1 + z) (Fig. 8.1). Baryonic matter is tightly coupled to radiation until recombination, and although it is non-relativistic, its temperature is determined by interaction with photons. In addition, pressure can build up and resists condensation. In a cloud ready for condensation, pressure ‘acts faster’ than gravitation when the crossing-time√for the sound tsound = vsound /R is smaller than the free-fall√time tfree fall = 1/ G. The critical size is the Jeans radius RJeans = vsound / G. On scales smaller than this, condensation is suppressed. Before recombination, the Jeans radius for baryonic matter is determined by the velocity of light. It is of a size similar to the Hubble radius. For scales smaller Ltrans , the inhomogeneities in the baryon–photon plasma begin to oscillate. No further

3

depends on time. We shall later use Fourier components without explicitly noting this fact. This will be the case when the wave number vector k or its modulus k ˜ for the curvature appears somewhere in the equation. In this chapter, we use k index. Here, we mean matter that is coupled to the electromagnetic ﬁeld.

8 Structure Formation in the Opaque Universe

variation in mass

192

tim e

R

spectrum

t en ec

Hubbl e radiu s

wave number [k]

10−4 10

−6

H ub bl

cr iti e

sc al

e

ca ls ca l

e

m

tru

ee -fr e l ca ls a i d or im r P

c pe

s

Fig. 8.1. Sketch of the linear evolution. The surface shows the linear evolution with time over the observable scales. The horizontal axis corresponds to the comoving wave number, the depth corresponds to time and the vertical axis corresponds to the variation in mass (i.e. the power times k3 ). For a long time, before the Hubble radius is crossed and before matter–radiation equidensity, the spectrum is a nearly featureless Harrison–Zel’dovich spectrum (∆2 ∝ k4 ) and increases as (1 + z)−4 . We see the region of suppression, and the free increase as (1 + z)−2 in the matterdominated epoch. After the Hubble radius is crossed, the increase is equal again for all scales, but the form of the spectrum has changed

condensation occurs until photons cool and the baryons are diluted to the point where the electromagnetic interaction rate falls below the expansion rate. This is the time of recombination of charged particles (which were never combined before). The velocity of sound sharply falls to the thermal velocities of the atoms. Radiation pressure does not protect baryons from gravitational condensation any more. During inﬂation, the Hubble radius is constant, and the corresponding scale falls to a very small value. For the scales λ[k] = 2π/k of interest, there is an instant ti [k] when the scale grows beyond the Hubble radius, i.e. a[ti [k]]λ[k] = RH , during inﬂation. Depending on k, this scale is surpassed by

8.1 Perturbations in General

193

Fig. 8.2. All scales lose against the Hubble radius. At left, physical distances and physical sizes are shown. Logarithmic scales are used. The Silk scale is proportional to a3 . The Hubble radius becomes larger than all scales which are smaller than that of the future horizon. The scale is large, if this happens after trecom , it is medium, if this happens between ttrans and trecom , and it is small if this happens already before ttrans . Compare with Fig. 7.7. At right, comoving distances and comoving scales are shown. In comoving measure, scales are constant, of course

the Hubble radius again at some later time tf [k]. In the time between ti [k] and tf [k], the perturbations are ruled solely by gravitational laws, and we obtain a particularly simple result. There are small, medium, and large scales deﬁned by the position of tf [k] with respect to ttrans and trecom (Fig. 8.2). Large scales are overtaken by the Hubble radius later than trecom . The gravitational condensation proceeds all the time. Medium scales are overtaken by the Hubble radius between ttrans and trecom . The condensation of DM proceeds but the density of the baryon matter oscillates between tf [k] and trecom . Small scales are overtaken by the Hubble radius before than ttrans . Between tf [k] and ttrans , the condensation is suppressed. This suppression yields the change of the slope of the power spectrum for small scales. The perturbation of the baryon component oscillates again between ttrans and trecom . The calculation of the spectrum of inhomogeneities consists of three parts. First, we have to ﬁnd the spectrum at time ti [k]. Before this time, the wave number k carries a short-wave perturbation (kc Ha[t]). The calculation of the generation and evolution of structure thus begins with an estimation of the ﬂuctuations during inﬂation, at time ti [k]. Hence we calculate the amplitudes by quantising short-wave ﬂuctuations in a de Sitter space-time [4]. Next we have to consider the phase when the scale in question is larger than the Hubble radius. This phase ends at the time tf [k], when the Hubble radius overtakes the scale in question. In the interval ti [k] < t < tf [k] these perturbations are long-wave, i.e. the scale is larger than the Hubble radius (kc H[t]a[t]). We consider them as classical

194

8 Structure Formation in the Opaque Universe

perturbations in a relativistic context (Sect. 8.4). The phase transition at the end of inﬂation (t = te ) ampliﬁes the perturbations without changing the spectrum. That is, the spectrum of perturbations at tf [k] can be related to that for the time ti [k]. If the universe is still opaque, the recombination transition will heavily aﬀect the perturbation through all the local processes at that transition. If the universe is already transparent at tf [k], the scale will not be aﬀected by recombination-time processes. After the time tf [k] and the end of the radiation-dominated universe at ttrans , the spectrum will have changed, because the ampliﬁcation was suppressed on small scales for some time interval that depends on k.

8.2 The Relativistic Approach and the Evolution on Large Scales We describe the undisturbed, homogeneous ﬁeld using the conformal time η that is deﬁned by a[t] dη = c dt. In the background metric, the expansion parameter appears as a conformal factor, i.e. ds = a [η](dη − γab dx dx ) , γab = δab 2

2

2

a

b

k˜ 1 + δcd xc xd 4

−2 .

In what follows, derivatives with respect to η are denoted by a prime. In general, f = (a[t]/c)f˙. The Friedmann equations now read Hη2 =

8πG 2 ˜ , H = − 4πG a2 + 3 p , = −3Hη + p , a − k η 3c2 3c2 c2 c2

where Hη = a /a = a/c, ˙ and k˜ denotes the curvature index. The complete metric is decomposed into a background and a perturbation: ds2 = a2 [η]((1 + 2ϕ) dη 2 − 2B,a dxa dη + 2Sb dxb dη (8.1) − ((1 − 2ψ)γab + 2E|ab ) dxa dxb + 2Fa|b dxa dxb + hab dxa dxb ) . With respect to substitutions of other space coordinates, the perturbation potentials ϕ, ψ, E and B are scalars, Fa and Sa are vectors, and hab is a tensor. The vertical bar in E|ab denotes a derivative that is covariant with respect to coordinate substitutions in spaces of ﬁxed time. Scalar perturbations deﬁne density perturbations, and tensorial perturbations deﬁne gravitational waves. Vectorial perturbations correspond to velocity ﬁelds. These are of importance only in the case where shear tensions in the source are signiﬁcant; we have nowhere invoked such tensions. In a linear approximation, the three types are decoupled and can be considered separately. In evaluating the Einstein equations, it is necessary to take account of the freedom in the local choice of coordinates. This is a kind of gauge invariance.

8.2 The Relativistic Approach and the Evolution on Large Scales

195

Since the metric tensor is covariantly constant (2.9), we obtain S a |a = 0, F |a = 0, ha a = 0, hab |b = 0 [1]. We now have to look for relations that do not change under permitted coordinate substitutions. The form (8.1) is invariant with respect to several substitutions a

x∗i = xi + ξ i [x]

(8.2)

that are analogous to gauge transformations if we consider the perturbation potentials as ﬁelds in a given background. What about potentials that are invariant individually? The tensorial perturbation is itself invariant with respect to the substitution (8.2). If we omit the gauge transformations that generate vectorial components from scalar ones, the invariant combinations are Φ=ϕ+

((B − E )a) a , Ψ = ψ − (B − E ) . a a

The simplest gauge is called the longitudinal gauge and puts B = E = 0. We shall use it in the following. The gauge that puts Φ = B = 0 is called the synchronous gauge.

The metric with scalar perturbations is used in the form4 ds2 = a2 [η](dη 2 (1 + 2Φ) − γab dxa dxb (1 − 2Ψ )) .

(8.3)

When Φ and Ψ are kept apart, Φ is called the perturbation of the gravitational potential, and Ψ the perturbation of the curvature. The tensorial perturbations can be analysed with a metric tensor such as ds2 = a2 [η](dη 2 − (δab + 2hab ) dxa dxb ) .

(8.4)

The Einstein equations yield 4πG 2 a 1 a δTb = (Φ − Ψ )|cb γ ac c4 2 −((2Hη + Hη2 )Φ + Hη Φ ˜ + 1 ∇2 (Φ − Ψ ))δ a , +Ψ + 2Hη Ψ − kΨ b 2 4πG 2 0 a δTb = (Hη Φ + Ψ )|b , c4 4πG 2 0 ˜ . a δT0 = −3Hη (Hη Φ + Ψ ) + ∇2 Ψ + 3kΨ c4 The Laplacian refers to comoving coordinates. If the spatial part of the energy–momentum tensor is diagonal, i.e. if the anisotropic stresses vanish, 4

This notation was chosen by Bardeen and Brandenberger with the intention of using it in the longitudinal gauge. In the synchronous gauge, Ψ is the remaining potential, and Φ is replaced by Ψ , and Ψ by −Φ. We shall stick to the original notation.

196

8 Structure Formation in the Opaque Universe

the diﬀerence Φ − Ψ depends only on time. It hence must vanish, because it is a perturbation that vanishes on average, and therefore Φ=Ψ .

(8.5)

In this case, we need to take account of only one potential Φ. The diﬀerence is important only when shear stresses exist, i.e. when the approximation of a laminar ideal ﬂuid cannot be applied. This happens during the time when the photons acquire a large mean free path during recombination. We do not intend to present the calculation in such detail. The qualitative features and their dependence on the main parameters can be seen in the approximation (or case) of Φ = Ψ . We shall consider Φ and Ψ separately only when it is necessary for the interpretation of their eﬀects. We now consider perturbations of the energy–momentum tensor. The scalar perturbations of the energy–momentum tensor have the form δε −(ε + p)γbc δuc i . (8.6) δT k = (ε + p)δuc −δp δba + σcb γ ac Here, ε denotes the energy density c2 . The velocities δuc are to be understood as peculiar motions per conformal time (otherwise a factor a[t] would appear). Because vorticity always decays and may be neglected here, the shear and velocity should be derivatives of corresponding potentials, i.e. σcb = σ|cb , δub = V|b . The velocity potential V and the shear potential σ are ﬁrst-order quantities similar to δε or δp. Except at the time of recombination, we can neglect dissipation and viscosity, and the shear vanishes, i.e. σ = 0. The pressure can be assumed to depend on the energy density ε = c2 and on the 3/4 speciﬁc entropy s ∝ εrad /εcold as well, i.e. p = p[ε, s]. We implicitly write 3/4

ε = εcold + εrad , 3p = εrad , s =

εrad . εcold

Fluctuations are described again by δp =

2 vsound δε + τ δs , c2

(8.7)

Entropy perturbations become important in multicomponent mixtures when the components are converted into each other far from equilibrium. Here, we consider only the mixture of radiation and non-relativistic particles that describes the transition from an opaque to a transparent universe. We obtain p = prad =

1 εrad , 3

1 δεrad , 3 3 δεrad δεcold δs = − . s 4 εrad εcold δp =

8.2 The Relativistic Approach and the Evolution on Large Scales

197

When we substitute this in (8.7), we obtain β2 =

2 1 εcold vsound 1 . = , τ = β2 2 c 3 1 + 3εcold /4εrad s

We now proceed by using the simpliﬁcation of vanishing shear, (8.6), and use Ψ = Φ. The Einstein equations are reduced to ˜ = ∇2 Φ − 3Hη Φ − 3(Hη2 − k)Φ

4πG 2 a δε , c4

4πG 3 a (ε + p) δub , c4 ˜ = 4πG a2 δp . Φ + 3Hη Φ + (2Hη + Hη2 − k)Φ c4 We now substitute the ﬁrst and the last of these equations in (8.7), and ﬁnd (aΦ)|b =

4πG 2 a τ δs = Φ + 3Hη (1 + β 2 )Φ − β 2 ∇2 Φ c4 ˜ + (2Hη + (1 + 3β 2 )(Hη2 − k))Φ .

(8.8)

In these four equations, k˜ is the curvature index of the three-dimensional space. We can further condense the equations through use of a variable u, i.e. 4πG Φ= (ε + p) u = Hη2 − Hη + k˜ a−1 u , c4 and a potential Z, i.e. Hη Z= a

−1/2 2 2 ε −1 ˜ (Hη − Hη + k) =a 3 ε+p

With the abbreviation 4πG a3 N= 4 τ δs = c Hη2 − Hη + k˜

1−

3k˜ . 8πGa2 ε

4πG a2 √ τ δs , ε+p c4

we ﬁnally obtain

2 vsound Z 2 u=N . (8.9) ∇ u − c2 Z This is the central equation [13]. It describes a parametric ampliﬁer. The other two equations of the initial system read

u −

δε 2 1 ˜ = (∇2 Φ − 3Hη Φ − 3(Hη2 − k)Φ) , ε 3 Hη2 + k˜

198

8 Structure Formation in the Opaque Universe

˜ −1 (aΦ) ,b . δub = a−1 (Hη2 − Hη + k) In the case of large-scale perturbations in a negligibly curved universe, (8.8) can be written in the form of a conservation law. The quantity ζ =Φ+

2 Φ + a(dΦ/da) 3 1 + p/ε

(8.10)

obeys, for adiabatic perturbations, the equation & % 2 ˜ 2 vsound dζ kc = Hζ O . +O 2 2 dt L2 H 2 a H

(8.11)

The symbol k˜ again represents the curvature index. With some care, we can take ζ ≈ δε/(ε + p). We now consider cosmological models with one-component matter. In the Einstein–de Sitter universe, the variables of (8.9) turn into Z = a−1 , u =

Φ √ . 4πG ε

2 An adiabatic perturbation (δs = 0) with a small velocity of sound (vsound −2 2 c ∇ Φ = 0 or negligible) must satisfy the equation

∞

u Z = uZ → u = C0 [x ]Z + C1 [x ]Z a

a

dη . Z2

Z

When we substitute this result into the deﬁnitions of the gauge-invariant quantities a ∝ η 2 , Z = a−1 ∝ η −2 , Φ ∝ a−3/2 u , we obtain u = C0 [xa ]η −2 + C1 [xa ]η 3 , Φ = C0 [xa ]η −5 + C1 [xa ] and, most importantly, 1 δε = ((η 2 ∇2 C1 − 12C1 ) + (η 2 ∇2 C0 + 18C0 )η −5 ) . ε 6 For small scales (where ∇2 C1 is more important than C1 ), the perturbations of the gravitational potential Φ increase in proportion to η 2 ; for large scales (C1 more important than ∇2 C1 ), the perturbations do not change. The corresponding density perturbations do not increase faster than η 2 ∝ 1/(1 + z). In a Gamow universe, the velocity of sound is of the order of the velocity of light. The term proportional to the sound velocity can be neglected only for large-scale perturbations. Again, potential perturbations on scales larger

8.3 Inhomogeneities and Inﬂation

199

than the Hubble radius do not increase. In the case of the de Sitter universe, the denominator in (8.10) vanishes, ε + p = 0. Hence, the analysis must take account in more detail of how the perturbations in ε and p depend on the perturbations in the scalar ﬁeld (Chap. 7). We have already derived that adiabatic perturbations beyond the Hubble radius are subject to a quasiconservation law for the quantity ζ (8.10). In this quantity, the perturbations do not increase.

8.3 Inhomogeneities and Inﬂation Inhomogeneities are residue of the inﬂation period. Inﬂation can wipe out any perturbation, with the exception of the zero-point ﬂuctuations. The latter transform into perturbations in the potential and the density. For a pure combination of the gravitational and the inﬂaton ﬁeld, we start with consideration of the spatially plane metric (8.3), or ds2 = a2 [η] (dη 2 (1 + 2Φ) − δab dxa dxb (1 − 2Ψ )) . In what follows, the derivative with respect to time is indicated by a dot, and the derivative with respect to η (conformal time) by a prime. The ﬁelds that generate inﬂation and breakdown of symmetry are condensed into a one-component scalar inﬂaton ﬁeld φ, subject to the action S given by (7.1). The ﬂuctuations of this ﬁeld are subject to the equation δφ + 2Hη δφ − ∇2 δφ + Vφφ a2 δφ = 4φ Φ + 2Vφ a2 Φ . The background metric is a solution to 3H 2 = 8πG, and H˙ = −4πG( + p/c2 ). With the scalar ﬁeld in place of matter, we obtain instead of (8.8) the equation ¨ ¨ φ φ 1 Φ˙ + 2H˙ − 2 Φ¨ + H − 2 Φ − 2 ∇2 Φ = 0 . (8.12) a φ˙ φ˙ This can be transformed into 3˙ 1 ζH(1 + w) − 2 ∇2 Φ = 0 , 2 a where ζ is deﬁned by (8.10) and the relation 1 + w = φ˙ 20 /ε is used, so that w = p/ε. This is the known quasi-conservation law [6] in the case of pure inﬂaton matter. The quantity ζ is constant to the precision that remains after we neglect the Hubble radius compared with the scale of the perturbation and the curvature radius. This conservation law relates the ﬂuctuation spectrum at inﬂation to the perturbation spectrum at the time tf when the Hubble radius overtakes the scale of the perturbation again.

200

8 Structure Formation in the Opaque Universe One of the many ways to shorten the argument reminds us of the short-scale relation δε 3 −k 2 Ψ = 4πGa2 δε = a2 H 2 ; 2 ε it uses the Hawking temperature ¯ hH = T in order to estimate the energy density of the perturbations at the time ti the Hubble radius is crossed, and we obtain δε ≈ (¯ h/c3 )H 4 . Now Ψ is assumed to vary like ζ, and we obtain at the reentry time tf

(¯ h/c3 )H 4 (¯ h/c3 )H 4 δε δε ≈ ≈ =O 2 2 /c)2 ε + p f ε + p i (¯ h /c)(H (¯ h/c)φ˙

= O[1]

This is only four orders of magnitude larger than what is observed, i.e. it is better than the thermodynamic estimate of Sect. 7.5, but it needs further consideration, and probably ﬁxing of parameters for the inﬂation.

The essential point of the calculation is again the transformation of the evolution equation into the known form (8.9), i.e. u − ∇2 u −

Z u=0. Z

(8.13)

The function u is proportional to the perturbation in question, and Z is a potential that depends on the type of perturbation. For a scalar perturbation, φ aφ u = a[t] δφ + Φ ; (8.14) , Z= Hη Hη for gravitational waves, u = aΨ , Z = a .

(8.15)

Equation (8.13) is the Euler–Lagrange equation of the action integral [23] Z 2 1 2 2 4 def u d4 x L . d x = S= u − (∇u) + 2 Z On the one hand, an equation of the form (8.13) admits formal integration. On the other hand, it provides a basis for a quantisation in analogy to quantum ﬁeld theory in a Minkowski space-time. The quantum variance of the perturbation operators u ˆ will be interpreted classically. We replace the classical ﬁeld u with an operator u ˆ, which is expanded into the integral u ˆ=

1 1 2 (2π)3/2

∗ − d3 k (uk [η]a+ k exp[−ik, x] + uk [η]ak exp[+ik, x]) .

The mode functions are subject to the condition

uk +

Z k − Z 2

uk = 0 .

8.3 Inhomogeneities and Inﬂation

201

The a± k are the ordinary decrement and increment operators of the ﬁeld oscillators. The commutation relation between the ﬁeld u and the corresponding momentum ﬁeld πu = ∂L/∂u determines the normalisation of the mode functions: uk [η]u∗k [η] − u∗ k [η]uk [η] = 2i . The vacuum expectation values can be recast in terms of the expectation values of classical quantities, and support the estimation that follows.

Let us consider a simple scalar ﬁeld v such as the inﬂaton. Its mass ought to play no part. It is representative of the perturbations of both the gravitational ﬁeld and the scalar ﬁeld (as long as its potential V [v] does not vary sigiﬁcantly with v). This is the case at times t ≈ ti [k]. The Hubble radius at ti deﬁnes the characteristic length La[ti ] = cH −1 [ti ] of the perturbation in question. We can calculate with an average Hubble radius because it does not vary much in the interesting interval of t. The ﬁeld v ﬂuctuates ﬁrst with a still unknown amplitude δv[L] at LHa[ti ] = c. The gradient of the ﬂuctuation then has a value of about δv H/c, and the total energy of the ﬂuctuation is

δE ≈ ¯ hc

d3 x

H δv c

2 ≈

¯ c2 h (δv)2 , H

3 where we take RH for the integration volume. The characteristic time interval −1 is δt = H , and the Heisenberg uncertainty relation corresponds to

δE δt ≈ ¯h , |δv| ≈

H . c

For ﬂuctuations of the gravitational ﬁeld, we now have simply v = c3/16π¯hGΦ; for the inﬂaton ﬁeld, we can put v = φ. The ﬂuctuations of the gravitational ﬁeld hence lead to the amplitude given in (8.21) Φ ≈ lPlanck

H[ti ] , c

which we shall call ∆T . The transition from the ﬂuctuations of the scalar ﬁeld to the density perturbations of the subsequent Gamow universe can be estimated by elementary methods. We merely integrate the equation for the evolution of the ﬂuctuation δφ[t] formally, through comparing it with the derivative φ˙ 0 of the average potential. The two quantities are subject to the same equation, y¨ + 3H y˙ = −

d2 V y, dφ2

if the term k 2 δφ can be neglected. This is the case for the interval ti [k] < t < te . Hence, (8.16) δφ[x, t] = −δτ [x]φ˙ 0 [t] .

202

8 Structure Formation in the Opaque Universe

The factor δτ [x] may be understood as a variation of the instant te when the inﬂation locally ends. This instant can be understood as being characterised by a certain value of the inﬂaton ﬁeld φ = φe . At this characteristic value, 2 /(8πG) is handed over to the vacuum energy density inﬂation = 3c2 Hinﬂation ordinary matter that is largely relativistic and is diluted with expansion like radiation, i.e. t2 [x] (8.17) radiation ≈ inﬂation e 2 . t We now write φ[x, t] = φ0 [t − δτ [x]] , δte [x] = δτ [x] . Equation (8.17) yields the relation δ δτ ≈2 . te We take into account the fact that in all inﬂationary models, the time te −1 is determined by the expansion rate during inﬂation; te ∝ HInﬂation . From (8.16) and δφ ≈ HInﬂation /c, we ﬁnally obtain (8.20) in the form δ H 2 ∆S = ∝ . after t cφ˙ 0 e

t=ti

We now consider the ﬁrst order in the small parameters5 1 V −H˙ d ln H = 2 =− 2 H d ln a 8lPlanck V 2

(8.18)

1 d ln φ˙ V φ¨ =− 2 . = d ln a 4lPlanck V H φ˙

(8.19)

2

= and δ=

The expansion of Z yields Z = 2a2 H 2 Z

and η=

1 1 ˙ + δ˙ 1 3 1 + δ + + δ 2 + δ + 2 2 2 2 H

c dt = a

c da c =− + a2 H aH

c da . a2 H

In order to obtain the standard results, and δ are neglected. If we take account of them to a ﬁrst approximation, we ﬁnd for scalar perturbations the more precise formula 5

and δ are generally used, although these symbols have also other meanings.

8.3 Inhomogeneities and Inﬂation

203

1 H 2 √ |∆S | = (1 + (2 − ln 2 − b)(2 + δ) − ) , 4π 2 cφ˙ t=ti and for tensorial perturbations (gravitational waves) ¯hG 1 √ H[ti ] . |∆T | = (1 − (ln 2 + b − 1)) c5 2 2π If the dependence of and δ on time can be neglected, the spectral indices are d ln PS ≈ 1 − 4 − 2δ nS [k] = 1 + d ln a and d ln PT ≈ 1 − 2 . nT [k] = 1 + d ln a Further cases are considered in [23]. In the limit of ‘slow rolling’, i.e. 0 and δ 0, the roughest approximation yields the scalar ﬂuctuations 1 H 2 √ |∆S |f = , (8.20) 4π 2 cφ˙ at ti and the tensorial gravitational waves 1 |∆T |f = √ 2 2π

¯G h H[ti ] . c5

(8.21)

The relations (8.20) and (8.21) allow us to start a programme to infer the potential of the inﬂaton. This programme shall be sketched here. The evolution (7.7) of the expansion rate of the inﬂationary universe is interpreted in (7.8) as a dependence of the expansion rate on the inﬂaton ﬁeld, 2 dH ¯h2 G2 ¯hG = 12π 3 H 2 − 32π 2 4 V [φ] . dφ c c This equation maps the evolution H[φ] of the expansion rate straight onto the potential V [φ] [10, 11, 18, 24]. In the case of slow roll, H 2 remains small with respect to V . The expansion parameter is governed by (7.9). The essential part of the integral corresponds to the time between ti and tf , when the scale of the perturbation exceeds the Hubble radius:   φi a[t0 ] ¯hG H  = exp N∗ + 4π 3 dφ  . a[t1 ] c H φf

204

8 Structure Formation in the Opaque Universe

This expression, when multiplied by the Hubble radius at the time ti , yields the scale of the perturbation at the time ti . We now consider the end of inﬂation at te : t φ φe a ¯hG H H dφ . = Hdt = dφ = 4π 3 N [t] = ln ˙ ae c H φ

te

φe

φ

The present scale λ is given by λ=

a[t0 ] c a[t0 ] N [tf ] c e = . a[tf ] Hf a[te ] Hf

It depends via tf on the value φf of the inﬂation ﬁeld. We obtain d ln λ dN [φf ] 1 dH[φf ] ¯hG H H . = − = 4π 3 − dφf dφf H dφf c H H

(8.22)

At this point, we use the relations (8.20) and (8.21) in order to express the quantities H and H as functions of the amplitudes ∆S and ∆T . First, we ﬁnd d ln ∆T 1 |H | 1 ∆T . = √ = √ ∆S 2 πlPlanck H 2 πlPlanck dφ When we now substitute the expression √ ∆T H /H = 2 πlPlanck ∆S into (8.22), we obtain the condition ∆2 d ∆T = 2 T 2 . d ln λ ∆S − ∆T This relation does not depend on the particular potential that produces the inﬂation, and states the essential connection between the inﬂaton perturbations and the gravitational waves that are generated by inﬂation. In detail, it is certainly not precise enough. However, in all cases we obtain such a condition in the form d ∆T /d ln λ = F [∆S , ∆T ], and its validity for the observed amplitudes will test the viability of the inﬂationary history. If we could know the spectrum of the gravitational waves generated by inﬂation, we would be able to infer the inﬂaton perturbation from the potential. The other way round, an integration constant remains free, and we obtain ∆2T [λ] −4 2 V [φ[λ]] = lPlanck ∆T [λ] 3 − 2 , ∆S [λ] λ dλ ∆S [λ ]∆T [λ ] 1 φ[λ] = ± √ λ ∆2S [λ ] − ∆2T [λ ] 2 πlPlanck ∆T 1 ∆S [∆T ] √ =± . d∆T ∆2 2 πlPlanck T

8.4 The Evolution of Small Scales and the Microwave Background

205

This states a connection that allows us in principle to determine the dependence of the potential on the inﬂaton ﬁeld from the spectra of the scalar and tensorial perturbations. At present, we know only small parts of these spectra. In particular, the observation of ∆T and its separation from ∆S is still an unsolved problem. The solution is expected to be found from polarisation measurements [8, 9].

8.4 The Evolution of Small Scales and the Microwave Background We now know the origin of perturbations during inﬂation and their rather simple evolution as long as their scale exceeds the Hubble radius. Sooner or later, however, any particular scale will fall below the Hubble radius, which grows in comoving coordinates, where the perturbation scales are constant by deﬁnition. When the scale is smaller than the Hubble radius, the equation for its evolution has an ordinary Newtonian interpretation. In addition, the isotropy of the microwave background tells us that the evolution will be well approximated by a linear approximation before recombination and also for some time after. We hence start our considerations with small density perturbations, retaining only the overall expansion of the background. We may decompose all perturbed quantities into eigenfunctions of the comoving Laplacian, ∇2 Q = γ ij Q|ij = −k2 Q. In the simple case of a ﬂat background, this is kind of a Fourier decomposition, and we may deﬁne vector and tensor modes through Qj = −k−1 Q|j and Qij = k−2 Q|ij + (1/3)γij Q, respectively. This leads to the simple rules Qi|j γ ij ∇ 2 Qj Qi|j Qij γ ij Qij|l γ jl

= = = = =

kQ , ˜ j , −(k2 − 3k)Q −k(Qij − (1/3)γij Q) , 0, ˜ i. (2/3)k−1 (k2 − 3k)Q

This procedure compensates all terms proportional to H(r grad) through the mere use of comoving wave numbers k. That is, we use the fact that wave numbers and wavelengths are constant in comoving coordinates, i.e. the law of the cosmological redshift.

The line element (8.3), which to ﬁrst order is ds2 = a2 [η]((1 + ΦQ) − (1 − 2Ψ Q)γij dxi dxj ) , yields the Einstein tensor E i j = Ri j + (1/2)δji Rk k in the form Eji

=

3 2 a2 (Hη 0 δE j

˜ + δE 0 0 + k)

δE 0 j ˜ i + δE i j (1/a2 )(2Hηη − Hη2 + k)δ j

,

206

8 Structure Formation in the Opaque Universe

where a2 δE 0 0 a2 δE 0 j a2 δE i 0 a2 δE i j

˜ ), = −2Q(3Hη2 Φ + 3Hη Ψ˙ + (k 2 − 3k)Ψ ˙ = −2Qj (Hη kΦ + k Ψ ) , = 2Qi (Hη kΦ + k Ψ˙ ) , = −2δji Q((2Hηη − Hη2 )Φ + Hη (Φ˙ + Ψ˙ ) ˜ ) + k 2 Qi (Φ − Ψ ) , −(k 2 /3)Φ + Ψ˙ + Hη Ψ + (1/3)(k 2 − 3k)Ψ j

together with the energy–momentum tensor of a component of the quasiFourier expansion, (1 + δQ)ε −(ε + p)V Qj i Tj = . (ε + p)V Qj p(δji (1 + (δp/p)Q) + ΠQij ) To zeroth order, we obtain the Friedmann equations, and to ﬁrst order, d δ = −(1 + w)(kV − 3Ψ ) − 3Hη δw . dη Alternatively, we can write 2 δp vsound δ d − δ. = −(kV − 3Ψ ) − 3Hη wΓ , wΓ = dη 1 + w δε c2 This equation holds for all isolated components of Q individually; only Ψ does not carry the number index of the component. The term Ψ represents the eﬀect of a local stretching of space in addition to the Hubble expansion. In a barotropic ﬂuid, δ/(1+w) is the ﬂuctuation of the number density (Sect. 3.4). If we take only the scalar component of Q, part of the Euler equation T αk ;k = 0 can be cast in the form 3k˜ V kδ δp 2 w dV + Hη (1 − 3w)V = − w + − (1 − 2 )kΠ + kΦ . dη 1+w 1 + w δε 3 1 + w k Alternatively, 2 dV vsound + Hη 1 − 3 2 V dη c 2 2 w w kδ vsound kΓ − + = 1 + w c2 1+w 31+w

3k˜ 1− 2 k

kΠ + kΦ .

We have to include, of course, the Einstein equations for the metric, E i j = (8πG/c4 )T i j , i.e. ˜ − 3Hη (Ψ + Hη Φ) = 4πG a2 δε . ∇2 Ψ + 3kΨ c4

8.4 The Evolution of Small Scales and the Microwave Background

207

Both equations are separately valid for diﬀerent matter components as long as those components do not interact microscopically. In the case of tight coupling, the components can be taken as a single component with an appropriately constructed equation of state. In other cases, in particular in the transition times, we must refer to the Boltzmann equation for distributions in phase space. A curious circumstance allows us to observe a characteristic structure in the spectrum of the anisotropy of the microwave background. This structure is generated in the interval between ttrans and trecom through the oscillation of amplitudes on the medium and small scales. First, the oscillation starts with a phase determined by that of the growing mode in the time before tf . It ends with a certain phase at the recombination time trecom . This phase now depends on the scale k as does the duration of the oscillation. However, when the density amplitude goes through zero, the velocity amplitude has its maximum. Hence, the impression of the phase on the amplitude of the observed temperature ﬂuctuations should be expected to vanish in this approximation. The inertia of the baryons, however, leads to a reduction in the velocity amplitude, and the cancellation is lifted. We observe an eﬀect that shows the scale of the ﬁreball, as well as the matter and baryon concentrations during recombination. Because this eﬀect is – with respect to homogeneity of the plasma – purely second order, it is extremely sensitive to the ingredients that determine it (the concentrations of components and the expansion rate). That makes this eﬀect a tool for precision measurements of just these ingredients [21, 22]. We shall merely sketch the decisive arguments. In the background radiation, we observe a temperature ﬂuctuation. The temperature ﬂuctuates with the radiation density (the higher the density, the later the radiation is decoupled and the less it is redshifted), with the gravitational potential (the deeper the potential the more the radiation is redshifted) and with the radial velocity (through the Doppler eﬀect on the frequency). The temperature ﬂuctuation at the last scattering in the local rest frame is corrected for the gravitational potential (the ﬁrst classical eﬀect of GRT, here Sachs–Wolfe eﬀect) and the velocity of the plasma. The observed temperature ﬂuctuation is given by ∆T = (Φ + vradial + Θ)|at last scattering . (8.23) T observed Θ denotes the relative ﬂuctuation in the temperature in the local rest frame, vradial the ﬂuctuation in radial velocity of this frame and Φ the ﬂuctuation of the gravitational potential. The Einstein equations for a homogeneous plasma must be supplemented by or replaced with Boltzmann equations for the phase-space distributions of the individual components with interaction terms, all embedded in the expanding space of the Friedmann universe. We only sketch the result, following the exposition given by Bunn [7] (see also [15, 16]). For tight coupling,

208

8 Structure Formation in the Opaque Universe

the distribution of photons is isotropic in the rest frame of the baryons, and depends only on the temperature, nbaryonic ∝ nγ ∝ T 3 . Hence, 1 ∆T [x, t] = δ[x, t] . T 3

Θ[x, t] =

(8.24)

The equation for the isotropic part of the ﬂuctuations is derived from the Boltzmann equation and reads d k2 ((1 + R)Θ ) + Θ = F [η] , dη 3 where F [η] = −

k2 (1 + R)Φ + ((1 + R)Ψ ) 3

and R=

εb + p b 3εb = . εγ + p γ 4εγ

If we use Φ = Ψ , and Ψ ≈ 0, we obtain (1 + R)Θ +

k2 k2 Θ = − (1 + R)Φ . 3 3

This is the equation for a simple harmonic oscillator, with the solution6 Θ[η] = −(1 + R)Φ + K1 cos[kvsound η] + K2 sin[kvsound η] , 1 . vsound = 3(1 + R) With Θ [0] = 0 and Θ[0] = −(2/3)Φ, we obtain 1 Θ[η] = −Φ + Φ cos[kvsound η] . 3 This is the intrinsic temperature variation. We have to add the Sachs–Wolfe eﬀect and the Doppler eﬀect (8.23) ∆T 1 = vradial + Φ cos[kvsound η] . T observed 3 at last scattering If there would be no Doppler term, we would obtain pure oscillations and the so-called Doppler peaks, which have nothing to do with the Doppler eﬀect. The physical scale of the ﬁrst peak is given by 6

When as a wavenumber k appears in a formula, all variables are understood as Fourier components. This convention saves us from introducing too many letters or indices. Where it is necessary to explicitly remind the reader of the Fourier decomposition, a component is denoted by a subscript k.

8.4 The Evolution of Small Scales and the Microwave Background

209

Fig. 8.3. Draft of the evolution of perturbations in the oscillatory regime. The upper part shows the path of a perturbation in the phase space of its components in potential and velocity. In the lower part, the resulting amplitude (of temperature ﬂuctuations) is sketched. The amplitudes of perturbations on all scales start at some point on the vertical axis. The time left until the end of acoustic oscillations (the time of recombination) depends on the scale. The point in phase space reached at this instant marks the amplitude for that scale in the microwave background. Without baryon inertia and damping, the path is a circle centred on the origin, and the amplitude is constant (dotted lines). Including baryon inertia, we obtain a shifted circle and a varying amplitude (broken line). Including damping, we obtain the qualitative picture of what we observe (Fig. 8.4)

kvsound ηls = π , λ = k −1 = vsound ηlast

scattering /π

≈ 30 Mpc ,

D = η0 − ηls ≈ 6000 Mpc and α ≈ 0.25o . If we put back the Doppler eﬀect, the peaks vanish; they cancel if R is set to zero, i.e. if the baryon contribution is negligible. We can check this for δ = 3Θ (8.24). In this case v=

3i Θk, k2

and the corresponding contribution is

210

8 Structure Formation in the Opaque Universe

Fig. 8.4. Angular power of the microwave background. With WMAP [3], the acoustic ﬁne structure of the power spectrum has been measured precisely enough to constrain the main cosmological parameters (H0 , Λ, Ω, Ωbaryon and κ). Reproduced from [3] with kind permission of J. Weiland (NASA/WMAP Science Team) and the AAS

∆T i |Doppler = sin[kvsound η] . T 3 For R = 0, the combined eﬀect vanishes. We ﬁnd that any eﬀect of the oscillations must be proportional to the baryon concentration. The dynamical eﬀect of the baryons cannot be ignored. Baryons contribute to the mass, but not to the pressure. When R = 0, the speed of sound becomes smaller, and the driving term becomes larger. The result is 1 Θ[η] = −(1 + R)Φ + (1 + 3R)Φ cos[kvsound η] . 3 In this approximation, we see that the spacing of the extrema in the anisotropy spectrum depends on the expansion rate at recombination, and the

8.4 The Evolution of Small Scales and the Microwave Background

Expansion

Expansion is decelerated by matter The Hubble radius is permanently increasing All scales start large and end small

Radiation

Decelerated ampliﬁcation while scale is large Damped oscillation while scale is small Jeans radius is approximately Hubble radius Full set of initial conditions known at the time when the Hubble radius is crossed

Baryon mix

Decoupling through dilution, completed by neutralisation Oscillation in radiation frozen in the microwave background Decelerated condensation of baryons afterwards

Dark matter

Baryon mass

Pure baryon condensation too slow, too late (z ≈ 1100) Dark matter restarts ampliﬁcation at equidensity, and provides the potential wells for the baryon–photon plasma

Breaks the equality between amplitudes of temperature and velocity, thus producing the structure in the spatial spectrum of the microwave background.

Fig. 8.5. The ﬁve steps of perturbation analysis

211

212

8 Structure Formation in the Opaque Universe

amplitudes depend on the baryon content. The phase plane of the acoustic oscillations is shown in Fig. 8.3 an the result in the angular spectrum of anisotropy in Fig. 8.4. More precise calculations take the details of the interaction term into account. The ﬁrst of the eﬀects taken into account is the Silk damping through the weakening of the interaction during recombination. This damping reduces the amplitudes on scales that are smaller than or comparable to the mean free path of the photons at that time. This scale can be estimated from the expression ctrecom . (8.25) (∆x)2comoving = (1 + zrecom )νnγ0 σThomson At the end of recombination, this scale has the order of magnitude of the size of a galaxy. Amplitudes on smaller scales are damped out. We resume the chain of arguments for the analysis of the perturbations in the microwave background in Fig. 8.5.

References 1. Bardeen, J. M.: Gauge-invariant cosmological perturbations, Phys. Rev. D 22 (1980), 1882–1905. 195 2. Bardeen, J. M., Bond, J. R., Kaiser, N., Szalay, A. S.: The statistics of peaks of Gaussian random ﬁelds, Astrophys. J. 304 (1986), 15–61. 221 3. Bennett, C. L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S. S., Page, L., Spergel, D. N., Tucker, G. S., Wollack, E., Wright, E. L., Barnes, C., Greason, M. R., Hill, R. S., Komatsu, E., Nolta, M. R., Odegard, N., Peirs, H. V., Verde, L., Weiland, J. L.: First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: preliminary maps and basic results, Astrophys. J. Suppl. 148 (2003), 1–27 (astro-ph/0302207). 210 4. Birrel, N. D., Davies, P. C. W.: Quantum Fields in Curved Space, Cambridge University Press (1982). 193 5. Blanchard, A., Bartlett, J. G., Douspis, M.: Cosmological implications from the observed properties of CMB, C. R. Phys. 4 (2004), 909 (astro-ph/0402297). 6. Brandenberger, R., Feldman, H., Mukhanov, V.: Classical and quantum theory of perturbations in inﬂationary universe models, in K. Sato (ed.): Evolution of the Universe and Its Observational Quest, Frontiers Science, Tokyo, astroph/9307016 (1993). 191, 199 7. Bunn, E. F.: Calculating cosmic background anisotropies and implications (four invited lectures), in The Cosmic Background Radiation, NATO ASI Series C 502, 135 (astro-ph/9607088) (1997). 207 8. Bunn, E. F.: Detectability of microwave background polarization, Phys. Rev. D 65 (2002), 043003 (erratum in 66, 069902). 205 9. Cooray, A. R.: Polarized CMB: reionization and primordial tensor modes, in M. Colless, L. Staveley-Smith (eds.): Maps of the Cosmos, Proceedings IAU Symposium 216, ASP Conference Series, Sydney, astro–ph/0311059 (2003). 205 10. Copeland, E. J., Kolb, E. W., Liddle, A. R., Lidsey, J. E.: Observing the inﬂaton potential, Phys. Rev. Lett. 71 (1993), 219–222. 203

References

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11. Copeland, E. J., Kolb, E. W., Liddle, A. R., Lidsey, J. E.: Reconstructing the inﬂaton potential – perturbation reconstruction to 2nd order, Phys. Rev. D 49 (1994), 1840–1844. 203 12. Einstein, A.: in Schulman, R., Kox, A. J., Janssen, M., and Illy, J. (eds.): Einstein Collected Papers, p. 803, Doc. 565 (1918). 189 13. Grishchuk, L. P.: Ampliﬁcation of gravitational waves in an isotropic universe, Zh. Eksp. Teor. Fiz. 67 (1974), 825–838, J. Exp. Theor. Phys. 40 (1975) 409– 415 . 197 14. Heath, D. J.: The growth of density perturbations in zero pressure Friedmann– Lemaˆıtre universes, Mon. Not. R. Astron. Soc. 179 (1977), 351–358. 217 15. Hu, W., White, M.: Acoustic Signatures in the Cosmic Microwave Background, Astrophys. J. 471 (1996), 30–51. 207 16. Hu, W., Zaldarriaga, M., White, M., Seljak, U.: Complete treatment of CMB anisotropies in a FRW universe, Phys. Rev. D 57 (1998), 3290–3301. 207 17. Kiessling, J.: On the equilibrium statistical mechanics of isothermal classical self-gravitating matter, J. Stat. Phys. 55 (1989), 203. 190 18. Liddle, A. R.: The inﬂationary energy scale, Phys. Rev. D 49 (1994), 739–747. 203 19. Lucchin, F.: Introduzione alla cosmologia, Zanichelli, Bologna (1990). 221 20. Peacock, J. A.: Clustering of mass and galaxies, in Robert G. C., N. G. Turok (eds.): Structure Formation in the Universe Proceedings NATO ASI, Kluwer Academic, Dordrecht (2000). 222 21. Schwarz, D. J., Terrero-Escalante, C. A.: Primordial ﬂuctuations and cosmological inﬂation after WMAP 1.0, J. Cosmol. Astropart. Phys. 08 (2004), 003 (hep-ph/0403129). 207 22. Shaﬁeloo, A., Souradeep, T.: Primordial power spectrum from WMAP, Phys. Rev. D 70 (2004), 043523 (astro-ph/0312174). 207 23. Stewart, E. D., Lyth, D. H.: A more accurate analytic calculation of the spectrum of cosmological perturbations produced during inﬂation, Phys. Lett. B 302 (1993), 171–175. 200, 203 24. Turner, M. S.: Recovering the inﬂationary potential, Phys. Rev. D 48 (1993), 5539–5545. 203

9 Structure Formation in the Transparent Universe

After the recombination of hydrogen, the universe is transparent, i.e. the interaction between the massive component and the homogeneous (background) radiation is negligible for the overall picture. The production of additional radiation through the condensation of matter and the nuclear processes in the stars is negligible, even though it seems imposing. We see the evolution of condensations in the basically cold, gravitationdominated dust of atoms, and in stars, galaxies and clusters of galaxies. The dust of atoms is primarily neutral and free of pressure, although it might be reheated and reionized in very dilute clouds. The evolution of condensations begins in a linear regime, taking into account the fact that the anisotropy in the background radiation is less than 10−4 . The morphology of the condensations, in particular that of galaxies, is dominated by their angular momentum, their magnetic ﬁeld and their history of encounters with other galaxies. This will not be our subject here. The large-scale structures consisting of regularities in the positions of galaxies in the form of ﬁlaments, walls and clusters are dominated by gravitation. We ﬁnd that the correlations we observe have to evolve before recombination time. This requires the presence of dark matter, because interaction with the radiation component in the pre-recombination epoch holds matter quasifrozen in the radiation, which is homogeneous. Most of the questions about the post-recombination evolution are tackled today with the help of numerical simulations. These simulations are being used to explain the structure that is represented by the voids of the distribution of galaxies, the relation between the distributions of dark and baryon matter, the distribution of satellites of galaxies, the eﬀect and history of merging, and other features.

9.1 The Newtonian Interpretation In order to identify the relativistic corrections, let us compare the above results with a quasi-Newton approximation for perturbations of a homogeneously expanding gas. The density can be written as ε := ε[t](1 + δ), the velocity as v := H[t]r + δv, the dimensionless potential as Φ := Dierck-Ekkehard Liebscher: Cosmology STMP 210, 215–242 (2004) c Springer-Verlag Berlin Heidelberg 2004

216

9 Structure Formation in the Transparent Universe

2 (2πG/6c4 )ε[t]r 2 + Φ and the pressure as p = p[ε[t](1 + δ)] = p[ε[t]] + vsound ε[t]δ. We use the known continuity equation

dε = −(ε + p) div v , dt and the Euler equation ε

dv = −εc2 grad Φ − grad p , dt

(9.1)

with the hydrodynamic derivatives. The rotation mode must decay purely because of the dilution, and we can restrict the discussion to the divergence of the perturbation of the velocity. The zeroth-order equations are dε = −3H(ε + p) dt dH[t] 4πG r + H 2 r = −c2 4 (ε + 3p)r . dt 3c These are the Friedmann equations. We obtain them by completing the source of the Newton potential with a pressure term. This is a relativistic correction to the active gravitational mass due to the internal state of the source distribution. To ﬁrst order, we obtain the equations dδ ∂εδ + H[t]( r·grad )(εδ) = ε − 3H(ε + p)δ ∂t dt 2 = −(εδ + vsound εδ)3H − (ε + p) div (δv) ∂δv 2 + H( r·grad ) δv + H δv = −c2 grad Φ − vsound grad δ . ∂t First, we take the rotation of the last equation, and obtain d rot δv ∂ rot δv + 2H rot δv = + H(r grad ) rot δv + 2H rot δv = 0 . dt ∂t This result shows that vorticity decays continuously. One can see that viscosity only accelerates this decay. Hence we are left with a mere potential ﬂow. The divergence yields d div δv 2 + 2H div δv = −c2 ∇2 Φ − vsound ∇2 δ . dt We now diﬀerentiate the continuity equation again to obtain p dδ p 2 =+ − vsound δ 3H − 1 + div (δv) , dt ε ε

(9.2)

9.1 The Newtonian Interpretation

217

and insert the expansion (9.2) to obtain a second-order equation for the density perturbation δ: 3 2 d2 δ dδ 2 2 2 − (1 + w) H δ(1 + 3vsound ) + vsound ∇ δ + S = 0 , + 2H dt2 dt 2 S = ω3Hδ ˙ −

w˙ 2 (δ˙ + 3ωHδ) + 3(vsound − w)(H δ˙ + (H˙ + 2H 2 )δ) . 1+w

2 Here, we have omitted some terms due to variation of w = p/ε and vsound with time. For the Poisson equation, we write

∇2 Φ =

4πG 3 2 2 εδ(1 + 3vsound ) = 2 H 2 δ(1 + 3vsound ). 4 c 2c

We here thus obtained a homogeneous linear equation of second order in the perturbation δ. There exist two diﬀerently behaving modes. In the interesting cases, there is an increasing and a decreasing mode. If the two are of the same order of magnitude of some initial time, the increasing mode becomes the only interesting mode. 2 = The ﬁrst case to consider is that of cold matter. We put w = ω = vsound 0 to ﬁnd d2 δ 3 dδ − H 2δ = 0 . + 2H (9.3) 2 dt dt 2 In an Einstein–de Sitter universe, H = 2/3t, and the solution is δ = δincr

t t0

2/3

+ δdecr

t t0

−1 .

The increasing mode of the perturbations is a product δ[x, t] = C[x]b[t] .

(9.4)

In a Friedmann–Lemaˆıtre universe, the evolution with time obeys the equation ∞ (1 + z) dz → δEdS = C[x](1 + z)−1 . b[t] = h[z] (9.5) h3 [z] z Figure 9.1 shows the linear ampliﬁcation δz=0 /δz=1000 of perturbations calculated from this formula, normalised by the value for the Einstein–de Sitter universe. Equation (9.3) can be integrated exactly [14] if the form h2 = 1 + κ − Ω − κ(1 + z)2 + Ω(1 + z)3 can be assumed and ε ∝ a−3 . First, this particular form of h2 yields

d2 δ dh + h(1 + z) (1 + z)h − h2 dz 2 dz 2

dδ 1 − dz 2

d2 h dh (1 + z) − dz dz

This equation can be recast into the frequently cited form

δ=0.

218

9 Structure Formation in the Transparent Universe

Fig. 9.1. Linear ampliﬁcation since recombination. This map of the Ω0 –λ0 plane shows the linear ampliﬁcation of inhomogeneities between recombination and now in relation to the Einstein–de Sitter values. The isolines are logarithmically spaced. It is evident that only for Friedmann–Lemaˆıtre models can larger values be expected

1 d2 Z 1 d2 y − = 0 → y[x] = Z[x] C1 + C2 2 y dx Z dx2

dx Z 2 [x]

.

(9.6)

To this end, we have to make the substitutions y = hδ,

Z = h2

and

dx = (1 + z)h dz .

In the case of cold matter, we obtain

δ = h C0 + C1 z

∞

(1 + z) dz h3

.

C0 stands for a mode that decays with h, at least in the ﬁrst part of the late epoch. C1 stand for the increasing mode. In an Einstein–de Sitter universe, we ﬁnd δ0 [0] δ1 [0] = (1 + z1 )−3/2 , = 1 + z1 . δ0 [z1 ] δ1 [z1 ] In a de Sitter universe with a small contribution from matter, h2 = 1 − (1/(1 + ze ))3 + ((1 + z)/(1 + ze ))3 , ze > z1 , the zeroth component is constant and the ﬁrst component increases only weakly:

9.1 The Newtonian Interpretation δ0 [0] δ1 [0] = h[z1 ]−1 , ≈1+O δ0 [z1 ] δ1 [z1 ]

1 + z1 1 + ze

219

2 .

If the universe is almost empty (Milne model, h ≈ (1 + z), but with a small contribution from matter, h2 = (1 + z)2 (1 − 1/(1 + ze ) + (1 + z)/(1 + ze ))), a perturbation cannot grow in the same way, and we obtain δ1 [0] δ0 [0] = (1 + z1 )−1 , ≈1+O δ0 [z1 ] δ1 [z1 ]

1 + z1 1 + ze

2 .

We compare these results with the case of hot matter, i.e. matter with an 2 = 1/3 and leave extremely relativistic equation of state. We put w = vsound ω = 0. We obtain dδ 4 2 2 d2 δ 2 − 4H c + 2H δ + ∇ δ =0. dt2 dt 9 There is a characteristic time-dependent scale, k −1 = (1/3)RH , that separates large scales, with an increasing and a decreasing mode, from small scales, with two damped oscillating modes. In a pure Gamow universe (H = 1/2t), thermalized and opaque through enough ‘Kohlest¨ aubchen’ (coal dust particle), increasing modes exist as long as their scales are larger than the Hubble radius. The latter increases beyond any scale, the scales become small with respect to the Hubble radius, and from this moment the perturbations begin to oscillate until they are damped away. For very large scales, ∇2 δ 3δ/RH , and we obtain −1 t t δ = δincr + δdecr . t0 t0 This is not the last word, however, because relativistic corrections are important here.1 Finally, we consider a warm gas, i.e. a strongly coupled mixture of radiation and cold matter with the equation of state atrans atrans 1 , p = εcold , ε = εcold + 3p , ε = εcold 1 + a 3 a 2 vsound =w=

atrans 1 , ω=0. 3 a[t] + atrans

We have to take account of the time derivative 1

In addition, relativistic or weakly interacting particles are not simple ﬂuids, because their streaming does not conserve the coherence of ﬂuid volumes. We have to use a Boltzmann equation for the distribution f [xi , pk , t] of the particles in phase space,

∂f ∂f dxi ∂f dpk ∂f . + + = ∂t ∂xi dt ∂pk dt ∂t collisions

220

9 Structure Formation in the Transparent Universe

dw = wH(1 − 3w) , dt and now obtain Hw(1 − 3w) dδ d2 δ dδ − − (1 + w) + 2H dt2 dt 1+w dt

3 2 H δ(1 + 3w) + w ∇2 δ 2

=0.

There is an increasing mode when 4πG0 >

2 vsound

a0 a[t]

2 k2 .

On small scales (large k), the amplitude of a perturbation oscillates with some damping caused by the expansion. The Jeans length 2 vsound (9.7) RJeans = G indicates the minimum scale of an unstable perturbation. The corresponding mass is given by π 3 MJeans = RJeans . 6 In a radiation-dominated universe, the condensation of dark (transparent) matter is suppressed. The two equations d2 δ dδ − 4πG¯ matter δ − 2H dt2 dt and H2 =

8πG (¯ matter + ¯rad ) 3

yield the Meszaros eﬀect, i.e. t < ttrans , ¯matter negligible → δ ∝ ln

ttrans . t

Cold dark matter is non-relativistic at the end of the radiation-dominated epoch (kTtransition ≈ 10 eV). As soon as the Hubble radius passes the scale in question, the radiation pressure begins to resist condensation, but it aﬀects only the heat bath and the baryons coupled to it. The dark matter is not aﬀected, but it is not yet gravitationally dominant. The growth of its perturbations is far smaller than before. The dark matter resumes its former rate when it becomes gravitationally dominant, i.e. when the universe enters the second epoch. Now, the baryons are still coupled to the heat bath, and the Jeans radius is determined by the high pressure of photons (3p ≈ c2 ). Hence, the baryonic and heat bath perturbations are still in an oscillatory regime. In the third epoch, after recombination, baryon perturbations not only resume

9.1 The Newtonian Interpretation

221

Fig. 9.2. Temporarily acoustic modes. The diagram shows the behaviour of baryons and photon in the evolving dark-matter background for large scales. Before recombination, the perturbations in baryons and photons are ﬂuctuating. After recombination, photon perturbations decay, and baryon perturbation keep up with the DM perturbations. Reproduced from [19] with kind permission of Zanichelli Editore

their η 2 growth, but also fall into the potential wells prepared by the DM perturbations that have already grown (Fig. 9.2). These non-linear processes give rise to a rich structure in the background radiation, and contribute to the bias between mass and luminosity for the evolving structures. The suppression of condensation for some time on scales smaller than the critical scale produces formally a break, with ∆n = 4, in the power spectrum. This limit, however, is reached for scales so small that they cannot be checked today. The scales that are actually observable feel only an incomplete reduction in growth near the ﬁrst critical scale, where the DM perturbations do not really stop condensating, but assume some linear law with logarithmic outcome (Meszaros eﬀect). This produces the observed slope in the power spectrum. Damping by diﬀusion (the diﬀusion length is the geometric mean of the horizon scale and the mean free path of the photons) and oscillation caused by pressure modify the expected state of the ﬁreball. Accurate results require a solution of the Boltzmann equation to follow the evolution in detail. But we can ﬁnally obtain the transfer functions [2]. In a DM model, baryons now fall into the potential wells prepared by the DM, and begin to keep pace with the condensation of the DM. The onset process makes its impression on the microwave background as well as on the phenomenology of the condensation process, but overall condensations appear in the late universe as if they started at matter–radiation equidensity. If DM were not present, the baryonic matter would have had to prepare its

222

9 Structure Formation in the Transparent Universe

own potential wells for condensation, the time available would be too short for simple condensation, and the characteristic point in the power spectrum would indicate a scale larger than what is observed. As long as the linear approximation of scales that do not interfere is valid, analytical computation may be performed (the diﬀerential equations are solved mainly numerically, of course). The non-linear stage is characterised by interference between the evolution on diﬀerent scales, and could be followed analytically only by a full-sized theory of gravitation and kinetics. Therefore, simulation has been tried. Here, particles are subject to Newtonian gravitation and their behaviour is followed. If a large box (containing the system) is considered, 100 particles must suﬃce for a galaxy. This number is increasing with improvements in hardware and software. In particular, successive reﬁnement procedures have been developed. The primary evolution is that of the DM. Presumably, in halos of DM, protogalactic blocks form and merge into galaxies. Swimming in DM halos that provide potential wells, galaxies are self-determined structures that are deﬁned by the comparability of the free-fall and cooling times. Inside most of the large galaxies, black holes are believed to exist, which are responsible for a characteristic velocity proﬁle and produce virialisation by the chaoticity of orbits in their vicinity, in cooperation with or in contrast to violent relaxation. One ﬁnds many small, rather compact galaxies that merge between z = 3 and z = 1. Later than z = 1, one ﬁnds only a marginal evolution in number. All evolution is aﬀected by environment. There are only few properties that seem to be independent of the environment on a large scale, for instance the Tully–Fisher relation, and the fundamental-plane relations: at z = 1, the bulk of high-surface-density ellipticals have formed and settled down to the fundamental plane [20]. Evolution can yield transmission of an initial spectrum, but cannot explain the initial spectrum. The initial spectrum stems from very early times, and the theory of an inﬂationary universe can make predictions for the spectral distributions. The suppression of evolution for small scales during the radiation-dominated interval between tf and ttrans , which produces the small-scale part of the spectrum shown in Fig. 5.6, and the subsequent evolution in the transparent universe can be resumed by empirical approximation formulae, called transfer functions [13]. The power spectrum at times before tf is approximately P = P0 k/k0 , and evolves into k P = T 2 [k]P0 . k0 An adequate approximation for adiabatic perturbations is given by −1/ν T [k] = 1 + (ak + (bk)3/2 + (ck)2 )ν ,

9.2 Decoupling of Condensations from the Cosmic Expansion

223

where a = 6.4(Ω0 h2 )−1 Mpc, b = 3.0(Ω0 h2 )−1 Mpc, and c = 1.7(Ω0 h2 )−1 Mpc, ν = 1.13 [3]. For entropy perturbations, the corresponding approximation is δ[k, t0 ] = TS [k, t0 , ti ] δS[k, t0 ] −1/ν , T [k] = (kτ∗ )2 1 + (ak + (bk)3/2 + (ck)2 )ν where a = 15.6(Ω0 h2 )−1 Mpc, b = 0.9(Ω0 h2 )−1 Mpc, c = 5.8(Ω0 h2 )−1 Mpc and ν = 1.24 [4]. The details of the evolution of structure are becoming more and more important for discrimination between the various models [10, 11, 14, 15, 16].

9.2 Decoupling of Condensations from the Cosmic Expansion The universal expansion does not inﬂuence the internal dynamics of either the solar system or the galaxy. In some way, bound systems are decoupled from the cosmic evolution. To estimate this process, we consider a simple model (a reﬁnement can be found in [9]). We start with a spherical region of small and uniform overdensity and consider it as a small universe where the overdensity is compensated by an appropriate spatial curvature. We compare this universe with an undisturbed Einstein–de Sitter universe, i.e. we compare H2 =

8πG 8πG k with H 2 = (0 + δ) − 2 or 3 3 R

a30 a20 a30 2 with h = Ω + κ , Ω0 + κ 0 = 1 . 0 0 R3 a3 a2 The curvature leads to a relative deceleration of the overdense region, which reaches its maximal extension at a time given by h2 =

H0 tmax =

π Ω0 . 2 κ30

This is the formal time of decoupling. The density contrast at this time is found to be 2 Ω0 κ30 tmax 9π 2 − 1 ≈ 4.5 , −1= 3 −1= 1 Ω0 t0 16 If the constants a0 and t0 in the solution a Ω0 Ω0 , H0 t = (θ − sin θ) √ = (1 − cos θ) 3 a0 2(Ω0 − 1) 2 Ω0 − 1

224

9 Structure Formation in the Transparent Universe

are ﬁtted to the reference solution (a/a0 )3 = (t/t0 )2 . The quantity H0 t0 = √ 2/(3 Ω0 ) yields the comparison

t0 tmax

2

16 κ30 16 = ≈ 9π 2 Ω03 9π 2

δ

3

16 ≈ 9π 2

δM M

3 .

Hence, there is a relation between the amplitude of the mass ﬂuctuation and the time of decoupling from the universal expansion: δM = M

3π t0 4 tmax

2/3 .

(9.8)

If we accept that a range of masses of more than 12 orders of magnitude has entered the condensed phase in less than three orders of magnitude of time between recombination and now, the relative dispersion of mass cannot depend in any important way on the extension of the region in question. This corresponds to the simplest hypothesis about the spectrum of inhomogeneities, the Harrison–Zel’dovich spectrum [7, 20]: 2 δM ∝ k 4 , δk2 ∝ k . M at t=t0

This consideration is complemented by the existence of an exact solution of the Einstein equations that describes an ordinary Schwarzschild ﬁeld (as known for a static spherical source in an asymptotically ﬂat space-time) surrounded by an expanding Friedmann universe with a similarly expanding transition sphere (Fig. 9.3), the Einstein–Straus solution [5, 6]. The cosmological constant Λ, however, inﬂuences the local gravitational ﬁeld. For Λ = 0, the Schwarzschild solution must be replaced by the Weyl– Treﬀz solution, 2

ds =

2m Λr2 − 1− r 3

2m Λr2 − dt − 1 − r 3 2

−1 dr2 − r2 dω 2 .

For m = 0, this solution becomes the de Sitter universe, written in coordinates where it appears as a static solution.

9.3 Jeans’ Contraction The equations for the evolution of perturbations are, in general, equations of second order. They contain a ﬁrst-order term that produces a kind of dissipation, and a force term that combines gravitation (∝ H 2 Φ) and pressure 2 (∝ ∇2 Φ = −k 2 vsound Φ) with opposite eﬀects. The balance depends on the scale k. The critical scale is the Jeans scale

9.3 Jeans’ Contraction

225

Fig. 9.3. The Einstein-Straus solution. This sketch illustrates the existence of an exact solution for the condensation of a spherical region inside an expanding universe, collecting all matter inside a comoving spherical volume

kJeans −1 ≈ vsound tfree

fall

.

For larger scales, gravitation outweighs pressure, and ampliﬁcation of density contrasts can occur; for smaller scales, perturbations can oscillate, but not increase. In particular, non-relativistic gas can contract through its own gravitation, which outweighs the internal pressure if its extension is at least kT vsound . ≈ RJeans ≈ √ µG G The Jeans radius can be understood as an equilibrium of the gravitational acceleration f and buoyancy a at this distance: GM p 2 ≈ vsound . ≈ GR2 , a = − grad p ≈ 2 R R R √ If we compare the Jeans length vsound / G with the instantaneous Hubble radius 3c2 /8πG, two things must be remarked on. First, during the time of eﬀective interaction with a dominating radiation (that is, before recombination), perturbations in the baryon distribution cannot increase, and second, during recombination, the Jeans length falls by many orders of magnitude, and the Jeans contraction can start at this very moment. The dark matter is electromagnetically inert. Its particles interact only weakly, and this interaction produces no thermodynamic state. Nevertheless, f=

226

9 Structure Formation in the Transparent Universe

there is an equivalent of the sound velocity. As the Boltzmann equation for the particle density n[x, v, t] tells us, the mean velocity is the equivalent of the sound velocity. This velocity falls with expansion in proportion to 1 + z. A set of particles at rest is always unstable: there is no pressure and no sound propagation. The velocity of sound varies with time as a result of the universal expansion (Fig. 9.4).

Fig. 9.4. Jeans’ criteria. The Jeans radius is of the order of the product of the ratio of the velocities of sound and light and the Hubble radius. In the Gamow universe (before equidensity), the Hubble radius (in the comoving measure) increases linearly with the expansion; in the Einstein–de Sitter universe, it is proportional to the square root of the expansion. The DM enters its non-relativistic stage early, but this does not aﬀect the sound velocity, which is determined by the baryon–photon mixture and begins to decrease after the baryon–photon equilibrium in inverse proportion to the expansion. At recombination, the universe clears and the sound velocity falls abruptly to the DM particle velocity

9.4 The Virial Theorem

227

For DM particles of mass M , there is a critical temperature kTcrit ≈ M c2 and a corresponding redshift 1 + zcrit = Tcrit /T0 . For T > Tcrit , the velocity vDM in the DM is about c, and for temperatures below Tcrit , we estimate vDM ≈ c(1 + z)/(1 + zcrit ). For baryons, the sound velocity remains about c until baryon–radiation equilibrium, and then begins slowly to decrease in accordance with 2 = vsound

radiation 1+z ≈ b + radiation 1 + zequilibrium

until recombination. After recombination, the baryon pressure can be estimated through ordinary gas dynamics, i.e. 2 vsound ≈

kT Mbaryon

.

These estimates are shown in Fig. 9.4. Before the equidensity time ttrans , i.e. the transition to the Einstein– de Sitter phase at ztrans ≈ 42000 Ωh2 , the sound velocity is proportional to the velocity of light: dp c2 = . d 3 When the scale of a perturbation falls below the increasing Hubble radius, it falls below the Jeans radius too. It cannot grow, but it starts to oscillate. The Jeans mass increases with the Hubble mass. In a universe without dark matter, after the baryon–radiation equilibrium, the Jeans mass ceases to grow, and remains at its maximum MJeans

1

= 9 × 1016 (Ωh2 )−2 M .

This is the order of magnitude of the mass of a supercluster. After recombination, the sound velocity falls to the value for a monatomic gas: 2 vsound =

5kT . 3mp

At the recombination temperature, the Jeans mass is MJeans

2

= 1.3 × 106 (Ωh2 )−1/2 M .

This is the order of magnitude of the mass of a globular cluster. With falling 2 ,) the Jeans mass falls, too. The reheating of the gas by temperature (∝ Trad the evolving quasars and galaxies causes this decrease to be halted.

9.4 The Virial Theorem In classical mechanics, for bound (quasi-periodic) systems of particles, a theorem about averages over time, the virial theorem, can be derived.

228

9 Structure Formation in the Transparent Universe If the solution to a mechanical problem with the Lagrangian function 1 mik q˙i q˙k − V [q 1 , . . . , q n ] 2

L=T −V =

remains bound in all generalised coordinates, total time derivatives must vanish in the long-term average; for instance,

d dt

i

∂L q ∂ q˙i i

=

i ∂L q˙

i

∂ q˙i

+ qi

∂L ∂q i

=0.

In the ordinary case of a constant mass matrix and a potential that is homogeneous and of degree n (i.e. L = T [q] ˙ − V [q], V [λq] = λn V [q]), we obtain for the long-term average 2T − nV = 0 . In a gravitational ﬁeld (and in an electrostatic ﬁeld as well), we have n = −1, and 2T + V = 0. The virial theorem can be generalised to other quantities. For instance, in the case of a tensorial potential, we obtain, through use of the equation ∂ ∂φ ∂ (va vb ) = − , (¯ va ) + ∂t ∂xb ∂xa the tensor of inertia and its derivatives, i.e.

d3 x xa xb

Iab =

=⇒ d ∂ d3 x Iab = xa xb dt ∂t

= − =

1 2

d3 x

∂¯ vc xa xb ∂xc

d3 x ¯ vc (xb δac + xa δbc )

=⇒ 1 d2 3 I = d x ¯ v v ¯ + d3 x (va − v¯a )(vb − v¯b ) a b ab 2 dt2 −

1 2

d3 x d3 x [x][x ]

(xa − xa )(xb − xb ) |x − x |3

= 2Tab + Πab + Wab . The three terms on the left-hand side can be identiﬁed with the tensor of average motion Tab , the covariance matrix of the small-scale velocities (tensorial pressure) Πab and the tensorial potential Wab . In a quasi-periodic system, integrals of derivatives with respect to time must vanish in the long-term average. Hence we obtain 2Tab + Πab + Wab = 0 . A corollary can be found in the relation between the velocity dispersion and the correlation function for a system of mass points. With the velocity dispersion

9.4 The Virial Theorem

229

m v2 1 A A K= 2

and the integral 1 W = − Ga2 2

mA

d3 x ξ[x] x

over the correlation function (where x denotes comoving coordinates), we obtain d (K + W ) + H(2K + W ) = 0 . dt For a virialised state, the equation 2K + W = 0 can be stated.

When a cloud of particles with radius a0 , potential energy V ≈ −GM 2 a0 −1 and a small kinetic energy (1/2)M v 2 −V , begins to contract through its own gravitational ﬁeld, the kinetic energy rises until it reaches the value that is allowed by the virial theorem. Even in the case of a zero initial kinetic energy, the cloud cannot contract below half of its initial radius: the energy is conserved, i.e. T + V = V0 , and the virial theorem requires 2T + V = 0. Combining the two equations, we obtain V = 2V0 for the virialised state. This means that the condensation is halted at this value. Pure Jeans contraction cannot lead to a fully condensed state. In order to reach such a state, the settling into a virialised state must be prevented by cooling, either by radiation or by expulsion of particles. These two eﬀects make the hypothesis of a system bound √ in a volume invalid. In a virialised cloud, the mixing time tmixing = 2a/ < v 2 >, is equal to the free-fall time. In other words, a virialised cloud is at the verge of a Jeans instability. The virial temperature that marks equilibrium in a cloud of particles of mass µ is given by kT = G

N Gµ2 µ2 N = α¯hc , where α = . R R ¯hc

The dimensionless number α is a kind of gravitational ﬁne-structure constant. The conceptual diﬀerence between the virial equilibrium that presumes boundedness in a volume and the thermal equilibrium of a gas cloud that leads to a density distribution reaching to inﬁnity indicates the diﬃculty of applying ordinary thermodynamics to gravitating systems. Although we have derived the virial theorem for the gravitational potential of the cloud itself, it is valid also for motion in the external gravitational ﬁeld of a central source. The factor in front of the potential may depend on the distribution of the particles and the form of the potential, but the order of magnitude of the virialised kinetic energy is the same. The virialised kinetic energy of gas particles represents a temperature, and the observation of a temperature distribution can be used to infer the value of the potential. This has been done with the X-ray emission of the intergalactic gas in clusters of galaxies. A potential of Φ ≈ 10−6 corresponds to a baryon temperature of kT ≈ mp c2 Φ ≈ 1 keV.

230

9 Structure Formation in the Transparent Universe

The contraction of a cloud is halted at the virial equilibrium. Further contraction requires cooling by radiation or by expulsion of particles (Fig. 9.5). The expulsion of particles is usually very slow and ineﬃcient. Hence we have to consider cooling by radiation. Cooling by radiation can start at temperatures beyond the excitation scale. The radiating atom or molecule hWas to be excited by collisions with other particles, and has to radiate this energy before it returns it to the gas in the next collision. In a primordial hydrogen– helium cloud, below the temperature that corresponds to the Lyman α line (11 eV), cooling is ineﬃcient. The minimum temperature for cooling is lowered to about 1 eV when metals (which have a nuclear charge larger than 2) are present, and to about 10−1 eV in the presence of molecules such as water or carbon monoxide. Dust lowers the minimum temperature to a negligible

Fig. 9.5. The size of galaxies is determined by cooling eﬃciency. The curves show the mass scales that virialise at a given time. This time depends on the size Q of the initial perturbation that is assumed to virialise. For smaller Q, virialisation occurs later; for Q of order 1, it occurs immediately after recombination for small scales and immediately after the Hubble scale is crossed for large scales. The virial temperature increases with mass, and with virialisation time as well. The virial temperature has to reach the Lyman α limit for eﬀective cooling to occur. After a long time of virialisation, cooling (now by bremsstrahlung) becomes ineﬃcient. Reproduced from [18] with kind permission by M.Tegmark and the AAS

9.5 Peculiar and Bulk Velocities

231

value. When the cooling is very eﬃcient, the condensing cloud will fragment into smaller cloudlets because the Jeans radius falls during cooling to a correspondingly small value. The system of cloudlets will behave like a system of particles again. This system will lose its kinetic energy much more slowly than the individual cloudlets lose their internal energy. Hence fragmentation stops the condensation of the primary cloud again. A large cloud can condense eﬀectively only when a cooling mechanism that is not too fast is in eﬀect. The formation of massive black holes, for instance, requires such a careful cooling. Fast cooling induces fragmentation and keeps the temperature at the lower limit of what can be achieved by cooling. If the cooling does not aﬀect the reaction equilibrium, i.e. when the temperature rises beyond the ionisation energies, the gas can become opaque, and the cooling is reduced. The fragmentation is stopped, and the condensation as well, because the radiation pressure keeps the gaseous cloud apart. This happens in stars when hydrogen fusion starts in the core and produces more energy than the cooling mechanism can transport away. Further contraction to a cold ﬁnal state must wait for the fusion fuel to be exhausted. There is another obstacle to condensation. This is the conservation of angular momentum. In a contracting cloud of particles, it leads to an increasing rotation, which is not so easily coupled to radiation. The rotation velocities are of the order which is prescribed by the virial theorem. One observes the formation of disks, in an external potential accetion disks, which can lose their angular momentum by some transport to the outer regions. In the end, the system loses particles.

9.5 Peculiar and Bulk Velocities The universal expansion deﬁnes a homogeneous velocity ﬁeld and comoving coordinates. The expansion does not produce changes in these coordinates. Any motion through this comoving coordinate system is a peculiar motion, its velocity the peculiar velocity. Proper motion can only be produced by peculiar motion. In the simplest (force-free) case, peculiar motion is subject to the geodesic equation, i.e. the Euler–Lagrange equation of the action integral 2 dχ . S = m0 c dt c2 − a2 [t] dt The integrand does not depend on χ, and hence we ﬁnd a ﬁrst integral, d dχ d ∂L = m0 c a2 [t] =0, dt ∂ χ˙ dt ds The spatial components pχ = m0 ca2 (dχ/ds) of the (covariant) momentum vector are constant. The energy E = cp0 then obeys the equation

232

9 Structure Formation in the Transparent Universe 2 E 2 = c2 (m20 c2 + (1 + z)2 a−2 0 pχ ) .

In the case of photons (m0 = 0), this is the law of the cosmological redshift. In the case of slow (non-relativistic) particles, we obtain Ekinetic = cp0 − m0 c2 = (1 + z)2 a−2 0

p2χ . 2m0

The kinetic energy of a free particle (and hence the temperature of a homogeneous monatomic gas) fall as (1 + z)2 , i.e. faster than the radiation temperature (Trad ∝ (1 + z)). The physical peculiar velocity is given by v = ca

pχ dχ = ds m0 a

→ av = const .

Peculiar velocities decrease with the universal expansion when local gravitational ﬁelds do not enter the picture. We can argue that the contribution of primordial peculiar velocities can be neglected today, and that the observed peculiar velocities are the eﬀect of local gravitational action. It is highly instructive to look directly at the velocity perturbations. The matter deﬁnes rest through its average motion, which corresponds to the expansion, and (peculiar) velocities can be considered to be of ﬁrst order, i.e. as perturbations of the velocity ﬁeld. The evolution equations of pressure-free matter, 1 1 1 div v = 0 , v˙ + Hv + grad Φ = 0 , 2 ∇2 Φ = 4πGδ , δ˙ + a[t] a a [t] can be read as equations for perturbations of the velocity ﬁeld and may provide a clearer picture of the perturbations. If we assume the velocity ﬁeld to be free of rotation, it is the gradient of some velocity potential ψ, i.e. v = grad ψ. Neglecting the decreasing mode, we substitute δ = C[x]b[t], corresponding to (9.4). This yields ˙ ba 2 ˙ δ div v + a[t]δ = 0 → ∇ ψ = −Ha ba˙ ˙ 1 ba y−x → v[x, t] = Ha d3 y δ[y, t] 4π ba˙ |y − x|3 on the one hand, and on the other hand 1 dv =− grad Φ = G[t]a[t] dt a[t]

d3 y

y−x δ[y, t] . |y − x|3

The velocity ﬁeld is proportional to the acceleration ﬁeld: ˙ H ba v= g. 4πG ba˙

(9.9)

9.6 Models for Non-Linear Evolution

233

This is the reason why a perturbation in the gravitational potential can be determined through a measurement of the velocity ﬁeld. We have to assume, however, an Einstein–de Sitter universe. In other models, the proportionality factor diﬀers from the one calculated here. We are now ready to compare diﬀerent Friedmann models with respect to the spectrum of inhomogeneities, linear ampliﬁcation and bulk velocities (Fig. 9.6). The expected spectrum has a break at the scale of the Hubble radius on the ﬁreball (i.e. at recombination) or, in the case of dominant CDM, at equidensity. The ampliﬁcation is described by (9.4), and the factor ˙ a˙ of (9.9) is taken as velocity parameter. ab/b

9.6 Models for Non-Linear Evolution We now consider an inertial motion of the particles that constitute the density, together with the inhomogeneities that develop in the velocity ﬁeld at least transiently (Fig. 9.7) [21]. We assume a velocity ﬁeld that is composed of the Hubble expansion r = a[t]x and a peculiar motion r = a[t]x + r local [t, x]. The decisive point is the substitution of the evolution law for inhomogeneities by the dependence of the peculiar velocity-ﬁeld on time. We write r = a[t] (x + b[t]ξ[x]) .

(9.10)

The function b[t] is chosen to be b ∝ t2/3 for the Einstein–de Sitter universe. Primordial velocities can be neglected and local gravitational ﬁelds can produce only rotation-free velocity ﬁelds. Hence we assume rotation-free ﬁelds, i.e. rot dr dt = 0 or rot ξ[x] = 0 . Equation (9.10) describes a ﬂow as a transformation of comoving into physical coordinates. The density evolves in the same way as the Jacobi determinant: −1 ∂x ∂r . [r, t] = x = x ∂r ∂x To ﬁrst order, this gives us δ = −b[t] divx ξ[x] . We shall follow the evolution a step further. The Jacobi matrix ∂ξi Jik = a[t] δik + b[t] ∂xk is symmetric because the ξ are free of rotation. At each point, we ﬁnd principal axes, and we set the principal values equal to (1−b[t]λi ). In this form, the

234

9 Structure Formation in the Transparent Universe

Fig. 9.6. Velocity parameter and ampliﬁcation. In this combined diagram, Friedmann–Lemaˆıtre models are compared with an Einstein–de Sitter universe. The Einstein-de Sitter universe yields the broken lines, the Friedmann–Lemaˆıtre universe of [12, 13, 17] the full lines. In the upper right part, the course of the Hubble radius is shown in comoving coordinates. The Hubble radius of the Friedmann–Lemaˆıtre model is always larger than that of the Einstein–de Sitter model, while the ﬁeld of view does not diﬀer as much. Hence the break in the spectrum occurs for larger sizes in the Friedmann–Lemaˆıtre case. In addition, because of the lack of dark matter ion this case, it is shifted from the equidensity to the recombination time (upper left part). The recent velocity parameter is too small in the Friedmann–Lemaˆıtre case (lower right part), The ampliﬁcation is only marginally enhanced in spite of the longer time interval for galaxy formation. In Friedmann–Lemaˆıtre models, the ampliﬁcation may exceed the Einstein–de Sitter value, but at the cost of an earlier formation and smaller bulk velocities today

λi depend only on the perturbation ﬁeld ξ[x]. The evolution of the density is found to be [r] = x a−3

1 . (1 − b[t]λ1 )(1 − b[t]λ2 )(1 − b[t]λ3 )

(9.11)

Let λ1 be the largest of the three principal values and let it be positive. In this case, the density diverges at time t1 (b[t1 ]λ1 = 1). Because of the continuity of the displacement ﬁeld ξ, the direction that corresponds to λ1 is the direction

9.6 Models for Non-Linear Evolution

235

Fig. 9.7. The Zel’dovich scenario. The primary increase of perturbation is due to simple velocity inhomogeneity. The picture shows the light of the sun reﬂected on the weakly disturbed surface of the Mosel at Piesport. The perturbation of the ﬁeld of directions of the reﬂected light on the surface of the river leads to a caustic structure that resembles both the void structure considered here and the caustics of gravitational lensing (Fig. 4.19)

of fastest contraction for the adjacent points. The result is that in this simple model of non-linear evolution, ﬂat two-dimensional objects (called ‘bliny’ by Zel’dovich2 ) are formed, in which the density attains large values. These ‘bliny’ are due to the structure in the ﬁeld of peculiar velocities. One is tempted to see in these structures the places of eﬃcient galaxy formation and the walls of the large-scale void structure in the universe. They are caustics, however, and intrinsically transient. Permanency can be attained only through interaction of substructures during their formation. Such an interaction can be modelled through the viscosity that has been neglected here, by use of the Navier–Stokes equations. Viscosity is now introduced not in the form that we know in the kinetic theory of gases, but in a form designed to model gravitational interaction in the caustics. We remain in the Einstein–de Sitter model (a/a0 )3 = (t/t0 )2 and use the ampliﬁcation factor b[t]/b[0] = (1+z)−1 = (t/t0 )2/3 as the time variable. The linear left-hand side 2

‘Bliny’ is usually translated as pancakes, but it means crˆepes.

236

9 Structure Formation in the Transparent Universe

of the Navier–Stokes equations (9.1) is complemented by a formal viscosity term on the right-hand side [19]. We obtain, for the velocity ﬁeld u=

dx , db

a Burgers equation, du + (u·grad )u = ν ∇2 u . db The viscosity has the eﬀect that the originally transient caustics become permanent, and the particles are caught in the regions of high density. In the case of the existence of a velocity potential, where u = grad Φ = grad (2ν ln[1/U ]) = −2ν

grad U , U

the Navier–Stokes equation obtains the form dU = ν ∇2 U db and has the formal solution −Φ0 [x ] −(x − x )2 U [x, b] = exp exp d3 x . 2ν 4bν Although a potential ﬂow remains a potential ﬂow all the time, the velocity potential Φ is only proportional to the gravitational potential at the beginning. For very small values of the eﬀective viscosity ν, the main contribution to the potential U comes from the minimum of the function G[x, x , b] = Φ0 [x ] +

(x − x )2 . 2b

This yields

x − xmin [x, b] . b This formula for the velocity ﬁeld is a corollary to the considerations concerning the Jacobi determinant of the displacement ﬁeld, that led to (9.11). We remind the reader at this point of the parallelism of the velocity ﬁeld and the acceleration ﬁeld (9.9). u[x, b] ≈

9.7 Numerical Simulation Most of the problems of post-recombination evolution can be tackled only by numerical simulation. Numerical simulation, however, has only a limited

9.7 Numerical Simulation

237

power for proving relations, because large sets of non-linear equations sometimes are badly behaved and produce numerical solutions that seem to converge as the step size is reduced but do not converge to an exact solution of the system of equations. Similarity of a numerical solution to a corresponding observation is a good sign, but it is no proof either of the validity of the assumptions or of the appropriateness of the method. But we have nothing better, and therefore use the numerical simulation as a guiding tool that seems to prevent us from going astray. The amount of computer power that is necessary to model the large-scale evolution is rather impressive. The large-scale many-body problem is a task in its own right, and one can learn many things from it that are useful for tackling topics beyond cosmology. Numerical simulations of the evolution of DM condensations use a large set of points that represent the density and the velocity distribution [2, 8, 12]. The power function of the perturbations is realised by this set of points, and it need no further explanation that the best simulation is a simulation with as many particles in the volume as there are DM particles (≈ 1080 ). In 2004, we are happy to follow 109 particles. The evolution has been calculated with several diﬀerent codes. In general, one observes the evolution of a cellular net structure (Fig. 9.8). The discretisation of the initial probability distribution and the representation of complicated structures (galaxies or parts

z=25

z=10

z=5

z=2

z=1

z=0

Fig. 9.8. The evolution of the large-scale structure in a numerical simulation. The diagram shows six instants in a simulation. It is evident that the large-scale structure forms a comoving pattern and changes its position during evolution only marginally

238

9 Structure Formation in the Transparent Universe

of galaxies) in one formal particle requires particular attention (see [1], for example). The density contrast increases rapidly and permanently only at some vertices of the structure that are interpreted as clusters of galaxies. We also see in the simulation the evolution of substructures that merge in the end. If we check the initial positions of the particles that end in a large concentration, we ﬁnd regions that are astonishingly spherical in spite of the fact that the initial evolution leads to ﬂat ‘bliny’. The comoving equation of motion is given by a˙ 1 d u = −2 u − 2 grad Φ . dt a a We substitute a for t: x − xA G d a (a2 u) = g = mA . d ln a H aH |x − xA |3 A

With the help of a Fourier transformation, δ= δk exp[−ikx] , k

we obtain −k 2 Φk = 4πGa2 ¯δk , grad Φk = −ikΦk . With the expression ¯ = Ω(3H02 /8πG), we obtain 3ΩHa2 d (a2 u) = Fk exp[−ikx] , Fk = −ikδk . d ln a 2k 2 k

For N particles of mass m in a box of size L, x = x/L , u =

u Nm δv = , = , HLa HL (aL)3

and we ﬁnd d 3 1 x − xA (f [a]u) = Ω[a]f [a] . d ln a 8π N |x − xA |3 A

The function f [a] ∝ a2 H[a] has an arbitrary normalisation. If the evaluation is performed on a grid, we can write K = kL, and d (f [a]u) = F K exp[−iKX] , d ln a K

3Ω[a]f [a] , 2K 2 dX = U d ln a .

F K = −iKδK

The initial displacements are taken to be ∆X = U .

9.8 The Nails of the Universe

239

It is a problem of its own to discuss the relation between the local overdensity of dark matter and the concentration of ordinary matter. Only the latter is observed, and it does not map the distribution of dark matter in a constant proportion. Numerically, this problem is tackled with codes that supplement the code for the particle cloud with hydrodynamics (see for instance [17]). We shall not consider this topic here.

9.8 The Nails of the Universe We now come to the point where we can resume the state between theory and observation in cosmology. We have learned that the central function is the expansion parameter, which gauges the pysical distances with respect to the comoving distances. This expansion yields a continuous dilution of matter and a continuous cooling (with the exception of some phase transitions). Dilution and cooling give rise to the various universal processes. Most of what happens after recombination can be observed (in principle) in the electromagnetic radiation, and even the dark matter is analysed through the comparison of the large-scale structure and the structure of galaxies and clusters of galaxies with numerical simulations. In the early history of the universe, it is opaque, and detailed structure cannot be observed. However, there are instants in the opaque universe that left observable traces or fossils, which can be used to pin down also the early history. This is condensed into Fig. 9.9. The ﬁgure sketches qualitatively the history of the universe and indicates the times that leave traces for observation. The dark full line indicates the expansion of the scale of the present Hubble radius. The Hubble radius is shown in the light full line. It divides the small from the temporarily large scales. The long dashed line shows the temperature. The broken lines represent the ﬁeld of view and the event horizon of a point on the singularity. The dotted lines show the expansion of scales smaller than the scale of the present Hubble radius. If the universe started its classical evolution before inﬂation, it must have been a radiation-dominated universe. We assume that it run into an inﬂationary vacuum-dominated stage. This stage ended with a reheating to a second radiation-dominated phase. This phase hosts the time span of the primordial nucleosynthesis (the original ‘big bang’) and ends at equidensity of matter and radiation, which occurs shortly before the universe becomes transparent. The expansion parameter is represented by the thick full line. It is chosen to be the size of the scale of the present Hubble radius, which is determined at present and yields the point N . The observation of the temperature of the microwave background is represented by the point T . The Hubble radius itself has a scale that varies with time. Each scale smaller than the present Hubble scale (dotted lines in the ﬁgure) deﬁnes two instants (ti and tf ) in the early universe when they coincide with the scale of the Hubble radius at that time. These scales can be subdivided in large scales (reentering the Hubble

240

9 Structure Formation in the Transparent Universe

Fig. 9.9. Nails of the Universe

References

241

radius on the interval EN ) and small scales (passing the dark triangle of suppressed evolution). A representant of the large scales is indicated by the dotted line that passes the points A and D, a representant of the small scales is shown by the dotted line that passes the points C and F . The critical scale where the perturbation spectrum changes its exponent is shown by the dotted line that passes the points D and E. The observation of the structure of the microwave background is represented by the point M . We know the temperature of the start of the primordial nucleosynthesis (point I), and we know it in certain limits after reheating (point R). We know the expansion which ends the primordial nucleosynthesis (point O). The observation of the evolution of structure yields informations at the points D, E, F . These informations can be used to infer the conditions at the points A, B, C. Finally, we expect that our ﬁeld of view on the ﬁreball (point G) is smaller than the ﬁeld of action (point H) of a point on the singularity.

References 1. Binney, J.: Discreteness eﬀects in cosmological N-body simulations, Mon. Not. R. Astron. Soc. 350 (2004), 939 (astro-ph/0311155). 238 2. B¨ orner, G.: The Early Universe, Springer, Berlin (1992). 237 3. Bond, J. R., Efstathiou, G.: The statistics of cosmic background radiation ﬂuctuations, Mon. Not. R. Astron. Soc. 226 (1987), 655–687. 223 4. Efstathiou, G., Bond, J. R.: Microwave anisotropy constraints on isocurvature baryon models, Mon. Not. R. Astron. Soc. 227 (1987), 33p–38p. 223 5. Einstein, A., Strauss, E. G.: The inﬂuence of the expansion of space on the gravitation ﬁelds surrounding the individual stars, Rev. Mod. Phys. 17 (1945), 120–124. 224 6. Einstein, A., Strauss, E. G.: Corrections and additional remarks to our paper: the inﬂuence of the expansion of space on the gravitation ﬁelds surrounding the individual stars, Rev. Mod. Phys. 18 (1946), 148–149. 224 7. Harrison, E. R.: Fluctuations at the threshold of classical cosmology, Phys. Rev. D 1 (1970), 2726–2730. 224 8. Hockney, R. W., Eastwood, J. W.: Computer Simulation Using Particles, Adam Hilger, Bristol (1988). 237 9. Hwang, J.-C., Hyun, J. J.: Perturbing Friedmann equations: an introduction to Lifshitz instability, Astrophys. J. 420 (1994), 512–524. 223 10. Jing, Y. P., Mo, H. J., B¨ orner, G., Fang, L. Z.: The large-scale structure in a universe dominated by cold plus hot dark matter, Astron. Astroph. 284 (1994), 703–718. 223 11. Loeb, A., Zaldarriaga, M.: Measuring the small-scale power spectrum of cosmic density ﬂuctuations through 21 cm tomography prior to the epoch of structure formation, Phys. Rev. Lett. 92 (2004), 211301 (astro-ph/0312134). 223 12. Peacock, J. A.: Inﬂationary cosmology and structure formation, Lectures given at the EADN summer school, The structure of the universe, Leiden, July 1995, astro-ph/9601135 (1995). 237 13. Peacock, J. A., Heavens, A. F., Davies, A. T. (eds.): Physics of the Early Universe, Edinburgh University Press (1990). 222

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9 Structure Formation in the Transparent Universe

14. Peebles, P. J. E., Silk, J.: A cosmic book of phenomena, Nature 346 (1990), 233–239. 223 15. Peebles, P. J. E., Schramm, D. N., Turner, E. L., Kron, R. G.: The case for the relativistic hot big bang cosmology, Nature 352 (1991), 769–776. 223 16. Pope, A. C., Matsubara, T., Szalay, A. S., Blanton, M. R., Eisenstein, D. J., Gray, J., Jain, B., Bahcall, N. A., Brinkmann, J., Budavari, T., Connolly, A. J., Frieman, J. A., Gunn, J. E., Johnston, D., Kent, S. M., Lupton, R. H., Meiksin, A., Nichol, R. C., Schneider, D. P., Scranton, R., Strauss, M. A., Szapudi, I., Tegmark, M., Vogeley, M. S., Weinberg, D. H., Zehavi, I.: Cosmological parameters from eigenmode analysis of Sloan digital sky survey galaxy redshifts, Astrophys. J. 607 (2004), 655–660 (astro-ph/0401249). 223 17. Abadi, M. G., Navarro, J. F., Steinmetz, M., Eke, V. R.: Simulations of Galaxy Formation in a Lambda CDM Universe I: Dynamical and Photometric Properties of a Simulated Disk Galaxy, Astroph. J. 591 (2003), 499–514; II: The Fine Structure of Simulated Galactic Disks, Astroph. J. 597 (2003), 21–34; III: The Dissipative Formation of an Elliptical Galaxy, Astroph. J. 590 (2003), 619–635; 239 Authors: Mario G. Abadi, Julio F. Navarro, Matthias Steinmetz, Vincent R. Eke Comments: 15 pages, 13 ﬁgures, submitted to ApJ Journal-ref: Astrophys. J. 591 (2003) 499–514. 18. Tegmark, M., Rees, M. J.: Why is the cosmic microwave background ﬂuctuation level 10−5 ? Astrophys. J. 499 (1998), 526–532. 230 19. Weinberg, D. H., Gunn, J. E.: Large-scale structure and the adhesion approximation, Mon. Not. R. Astron. Soc. 247 (1990), 260–286. 236 20. Zeldovich, Ya. B.: A hypothesis, unifying the structure and the entropy of the universe, Mon. Not. R. Astron. Soc. 160 (1972), 1p–3p. 224 21. Zeldovich, Ya. B., Novikov, I. D.: The Structure and Evolution of the Universe, Chicago University Press (1983). 233

10 Higher Dimensions

10.1 Why Three Dimensions of Space? The invariance group of mechanics and ﬁeld theory tells us that space has three dimensions, and space-time four. Nearly all physical theory uses these numbers without consideration. There are various approaches to deﬁning a dimension, as we discussed in Sect. 5.7, but for physics, the ordinary four space-time dimensions are anchored in the form of the ﬁeld equations. We shall ﬁrst answer the question of the reasons that indicate that there are just four dimensions for space [7]. Three reasons are fundamental: – The ﬁrst reason is provided by the conservation theorems of thermodynamics. These conservation theorems calculate with a three-dimensional volume. Their applicability shows that there are no variations of the volume that correspond to a variation of a measure in any supplementary dimensions. The variation dU of the internal energy of a homogeneous system is the diﬀerence between the added heat and the work done by the system: dU = δQ − p dV . The known number of dimensions of the volume used here is three. If there were more (undetected) dimensions, it should happen at least sometimes that sizes in these dimensions would also vary, in an uncontrolled way. This would be observed as a deviation from the ﬁrst law of thermodynamics, but nothing like that is observed. – Energy conservation is the foundation of the relation between the brightness I and luminosity L that we considered in Chap. 4. Locally, we observe I[r] =

L , Ω[r]

and the validity of a 1/r2 law tells us that the surface area of a sphere is Ω[r] ∝ r2 . Hence, the dimension of space is 3. – Locally, gravitation is determined through Newton’s law as a solution of the Poisson equation. We can interpret the Poisson equation here as being analogous to the relation between luminosity and brightness. The ﬂux of the force density through a closed surface is proportional to the enclosed Dierck-Ekkehard Liebscher: Cosmology STMP 210, 243–252 (2004) c Springer-Verlag Berlin Heidelberg 2004

244

10 Higher Dimensions

mass, but independent of the form of the surface. The basic solution in an n-dimensional space is given by Φ ∝ r−(n−2) . Hence, the potential of the gravitational ﬁeld (Φ ∝ 1/r) indicates n = 3. Only such a potential Φ ∝ 1/r allows closed Kepler orbits. A potential Φ ∝ r−(n−2) with n ≥ 4 does not allow quasi-stable bound orbits any more as one can see in the energy conservation: Ekinetic + Epotential =

1 mr˙ 2 − GmM Φ[r] = const , 2

1 2 L2 mr˙ + − GmM Φ[r] = const , 2 2mr2 where L2 is the square of the angular momentum. Only in the case of n < 4 does the radial eﬀective potential L2 /2mr2 − GmM Φ[r] has a minimum at a ﬁnite distance (0 < rmin < ∞). Also, if the electrostatic potential were not inversely proportional to the distance, the form of atoms and the structure of spectral lines would change drastically. However, the general theory of relativity weakens these arguments by modiﬁcation of the Poisson equation in such a way that the asymptotic form of the metric is g00 = 1 − 2m/r + O[r−2 ] in any case. – Planck’s law for black-body radiation contains a count of the oscillators of the electromagnetic ﬁeld in a given volume. The result depends on and therefore shows the number n of dimensions of space: u[ν] dν ∝ dν ν n−1

hν . exp[hν/kT ] − 1

The determination of the black-body spectrum is a determination of the number of dimensions of space. There are other peculiarities of three-dimensional space that are rather abstract: – The propagation of signals is free of distortion only in three dimensions. Only in three dimensions does the wave equation have a radial solution of the form φk [r, t] = u[r] exp[ik(r − ct)] , in which the amplitude does not depend on the wave number. For an outgoing signal, the above relation yields u[r] r − r0 φ[r, t] = dk A[k]u[r] exp[ik(r − ct)] = φ r0 , t + . u[r0 ] c – Propagation of a pulse is composed solely of retardation and amplitude decrease, and produces no change in the form of the pulse. Huygens’ principle excludes even numbers for the dimension of space. If the number of dimensions of space were even, the various components of a spherical wave would propagate with diﬀerent velocities, and we would always ﬁnd retarded parts of the pulse.

10.2 Why More Than Three Dimensions of Space?

245

– Only for three dimensions of space is there a correspondence between rotational and translational degrees of freedom, i.e. between axial and polar vectors. – The dimensions of the fundamental constants depend on the number of dimensions of space. We obtain the following dimensions of the constants: ¯ has dimensions ML2 T−1 , h c has dimensions LT−1 , G has dimensions M−1 LD T−2 , e2 has dimensions MLD T−2 , where D is the number of dimensions of space. Hence, the dimensionless combinations are powers of α = e2D−2 G3−D ¯h4−2D c2D−8 . Only for D = 3 does the gravitational constant not enter this combination. For D = 3, the Sommerfeld ﬁne-structure constant (which should not depend on G, because gravitation does not enter) must contain constants other than e, h and c, for instance masses of particles or the cosmological constant. – Only for D = 3 do several diﬀerential structures of space-time exist; the physical importance of these is not yet clear [3]. – Only a potential Φ ∝ 1/r yields a force-free inner region of a spherical source shell.

10.2 Why More Than Three Dimensions of Space? In spite of all arguments for three dimensions of space, there are arguments for constructing theories with higher dimensions. Of course, such theories always have the task of explaining why the supplementary dimensions have succeeded in hiding their existence. The usual excuse is to assume some spontaneous reduction, closing or splitting of these dimension like the spontaneous symmetry breakdown that we considered in Chap. 7. The success of the theory of general relativity suggests that we should try an analogous geometrisation of the electromagnetic ﬁeld, which should end in some uniﬁed theory of gravitation and electromagnetism. One could try to follow this programme by adding a ﬁfth dimension of space-time. With respect to the usual four-dimensional coordinate transformations, the new non-diagonal elements gi4 of the metric tensor could correspond to the electromagnetic vector potential, and the new diagonal element g44 to a not yet interpreted scalar ﬁeld that modiﬁes the eﬀective gravitational constant. In its simplest variant [11, 13], this programme has had no decisive success.

246

10 Higher Dimensions

In contrast, the eﬀorts to unify the electromagnetic and weak interactions by a diﬀerent method has been more successful than the work on a uniﬁed gravo-electromagnetic theory. From the point of the theory of elementary interactions, it seems natural to interpret certain degrees of freedom as virtual dimensions of space. It turns out that particular numbers for the dimension of space far larger than three yield particularly welcome properties of the theory that concern renormalisability and freedom from anomalies. The numbers 10, 11 and 26 play a distinguished role. One could also try to take the dimension as a dynamical variable [10]. If one takes a higher-dimensional theory seriously, a reason must exist why the supplementary dimensions are not found in the observations and experiments at hand [8]. The simplest supposition is that admissible solutions of the ﬁeld equations should not depend on the supplementary dimensions. If in this case the additional scalar is constant, the Einstein equations of the ﬁve-dimensional theory split into a combination of the Einstein equations of four-dimensional space-time and Maxwell’s equations. The supplementary dimensions have no eﬀect, cannot be found and cannot be found to be false, i.e. falsiﬁed. They are an artiﬁcial construct without physical content. In this interpretation, the recent higher-dimensional theories would also lose their expected importance. These theories are guided by the expectation that at least at high energies (recall the scale of uniﬁcation, i.e. 1015 GeV), the supplementary dimensions cannot be distinguished from the usual ones. The symmetry breakdown below the Curie temperature of the vacuum should remove the similarity of the known and the supplementary dimensions. A second possibility is a reduction to such solutions that are periodic in the new coordinates with a very small period (the Planck length or GUT length). This is not necessarily an additional requirement, since one can argue that the new dimensions form (after the symmetry breakdown) one or more highly curved internal factor spaces with curvature radii that correspond to the lengths of the periods in question. Therefore, this interpretation is generally accepted. A pure periodicity would be equivalent to a torus topology of the factor spaces. In this case, they need not be curved [5]. The simplest physical consequences of higher dimensions of this kind are found in the wave equation ∂2 g µν µ ν Φ = sources . ∂x ∂x The Greek indices count all dimensions. When the supplementary dimensions are hidden from observation, we separate the wave equation to obtain the form g ik

∂2 ∂2 ∂2 Φ = −2g iA i A Φ − g AB A B Φ + sources . i k ∂x ∂x ∂x ∂x ∂x ∂x

Lower-case Latin indices relate to the usual space-time, and capitals to the factor spaces of the supplementary dimensions. For a Fourier component in the factor space,

10.3 Multidimensional Cosmological Models

247

∂Φ = ikA Φ , ∂xA the ﬁrst term on the right-hand side turns into kind of a gauge-ﬁeld coupling with e[k]Ai = 2g iB kB , and the second turns into an eﬀective mass, m2 [k]c2 = −g AB kA kB . ¯h2

(10.1)

The quantisation of the wave numbers due to the periodicity condition, kA = 2πnA /Lperiod , yields immediately a quantisation of the mass and the charge. However, even the smallest masses have a size comparable to the GUT mass, and hence they cannot explain the masses of the elementary particles [14]. The particle-like excitations in the supplementary dimensions are called pyrgons. If we require that the masses of these pyrgons are real, i.e. that they are not tachyons, the supplementary dimensions must be space-like. The additional mixed metric degrees of freedom are a kind of massive gravitational modes and lead to contributions to dark matter [2, 9]. Finally, one can try to use a construction consisting of a four-dimensional hyperspace (brane) in a higher-dimensional embedding (bulk space). These constructions require an appropriate deﬁnition of wave propagation that supports the three-dimensional appearance of the familiar space [15, 16, 17].

10.3 Multidimensional Cosmological Models A factorised space is illustrated in Fig. 10.1. It contains the option to explain the inﬂation in the ordinary space as accompanied by a spontaneous contraction of the supplementary factor spaces. If the D-dimensional space splits into individual dj -dimensional factor spaces (where D = j dj ), a homogeneously isotropic universe need be internally isotropic only within the factor spaces, and not between them. Instead of a metric such as dr2 2 2 2 2 2 2 + r (dσ )(D−1)-dimensional unit sphere , ds = c dt − a [t] 1 − kr2 we assume the line element ds = c dt − 2

2

2

α j=0

dj 2

(aj [t])

(1

i 2 i=1 (dx ) + kj rj2 /4)2

.

(10.2)

This is the metric of a space consisting of α + 1 factors that are homogeneous and internally isotropic, and have individual expansion parameters aj [t] and

248

10 Higher Dimensions

Fig. 10.1. The torus metaphor of a factorised space. The two radii of the torus shall visualise the expansion parameters of the conventional and the supplementary factor space. One expects that the contraction of the supplementary space accompanies the inﬂationary expansion of the conventional one

individual curvature indices kj . Each factor space contributes one term. The ﬁrst of these factor spaces is assumed to be the ordinary space (d0 = 3). From one point of view, these models generalise simple homogeneous but anisotropic solutions of the Einstein equations in four-dimensional spacetime. The factorisation corresponds to an energy–momentum tensor T ab of an ideal ﬂuid. This tensor is diagonal, but its space components (i.e. the pressure) need be equal only internally within the factors, not between the factors. We obtain diﬀerent factor pressures pj . Microscopically, if the energy in the centre-of-mass frame cannot be redistributed between the diﬀerent factor spaces by collisions, we obtain a constrained equilibrium with pressures of such a kind. The continuity equation now reads dε = −

α j=0

dj

daj (ε + pj ) . aj

(10.3)

The energy density ε is a density in a D-dimensional space. The density in the three-dimensional interpretation is the volume integral over the supplementary dimensions. The ordinary continuity equation (which corresponds to the ﬁrst law of thermodynamics) is valid only in the case of α j=1

dj

daj pj = 0 . aj

10.3 Multidimensional Cosmological Models

249

This can be read as a condition on the equation of state in the supplementary dimensions or as a condition on their expansion rate. In a fundamental theory, conditions on a (macroscopic!) equation of state are diﬃcult to justify. Hence, the ﬁrst law should be seen as a conﬁrmation that the small expansion rates of the supplementary factor spaces are extremely small during the evolution of the universe since the time of nucleosynthesis. This interpretation is supported by the observation that a change in the extension of the supplementary factor spaces would result in an eﬀective dependence on time of the fundamental constants, which is extremely small if not zero [12, 19, 20, 21]. In analogy to the standard model (Sect. 3.4) we consider barotropic equations of state in the form mj −1 ε , (10.4) pj = dj which can be integrated through use of the ﬁrst law (10.3) to yield ε = Mm0 m1 ...mα

m1 mα 0 am 0 [0]a1 [0]...aα [0] . m0 m1 mα a0 a1 ...aα

(10.5)

Such barotropic equations of state describe cold matter in the sense that the pressure depends on the density only, without additional explicit dependence on temperature. Any matter component is now characterised by a list of indices [m0 , m1 , . . . , mα ]. Just as in the four-dimensional space-time, the curvature and the cosmological constant are also characterised by such a family. For the cosmological constant, [0, 0, . . . , 0] is found, and for the curvature of the kth factor, mi = 0 for i = k, and mk = 2. The important case of vanishing trace of the energy– momentum tensor (we may call this case superradiation, in analogy to the conventional space-time) is obtained from α i=0

mi = 1 +

α

di .

i=0

Examples for the case of only one supplementary factor space are given in Table 10.1. At this point it must be remarked that the various models for the vacuum yield contributions that, in general, are no longer dynamically equivalent to the cosmological constant. The main advantage of the assumption of matter components with equations of state of the form (10.4) is that we can map the cosmological evolution onto the motion of a particle in a formally (α + 1)dimensional space that is endowed with a scalar potential [1, 2, 23]. The Einstein equations obtain the form 2  α α a˙ 2j a ˙ k j j  dj  − dj − (dj − 1) 2 = 2κε , aj a2j aj j=0 j=0

250

10 Higher Dimensions

Table 10.1. Matter components in multidimensional cosmology with one supplementary factor space Indices m, n

Interpretation

Source

m+n=d+4

Superradiation

[1]

m = 3, n = d + 1

Dust-like superradiation

[1]

m = 4, n = d

Radiation-like superradiation,

m = 0, n = d + 4

Candelas–Weinberg vacuum,

radiation at low temperatures

m = d + 4, n = 0

[1]

contribution of Casimir energies

[4]

Vacuum following Moss

[18]

m = 3(d + 4)/(d + 3), Vacuum following Sahdev, n = d(d + 4)/(d + 3)

radiation at high temperatures

m = 6, n = 2d

Ultrastiﬀ ﬂuids following Zel’dovich [1]

m = 0, n = 2d

Variant of the string vacuum, eﬀect of monopoles

[6]

m = 6, n = 0

Variant of the string vacuum

[6]

m = 2, n = 0

Curvature of ordinary space

m = 0, n = 2

Curvature of supplementary factor

m = 0, n = 0

Cosmological constant

d dt

a˙ i ai

[22]

α ε − j d j pj ki a˙ i a˙ j dj − (di − 1) 2 = κ pi + + , ai j=0 aj ai D−1

where pj and ε are obtained from the (10.3) and (10.5). If the density consists of cold components only, we obtain mij

d2 ξ j ∂Φ = i . 2 dτ ∂ξ

(10.6)

The ﬁrst integral of this equation is simply mij

dξ i dξ j = 2Φ . dτ dτ

Here, the formal coordinates ξj are determined from aj j ξ = ln , aj0 the formal time τ from dτ = dt exp[−dj ξ j ] ,

(10.7)

References

251

the potential from 8πG ε exp[2dj ξ j ] c4 and the formal mass matrix from Φ=

mik = di dk − di δik , mik =

1 1 − δ ik . D − 1 dk

Let us now determine the conditions which ensure that the expansion rate of the supplementary factor spaces is extremely small. Normalising by the current values of the expansion parameters, we put ξ j [t0 ] = 0 (j = 0, . . . , α). By the deﬁnition of the Hubble constant, ξ˙0 [t0 ] = H0 . The deceleration parameter is of the order ξ¨0 [t0 ] = O[H02 ]. In order not to disturb the tested properties of the universe such as the relation between the background radiation and primordial nucleosynthesis, the internal expansion parameters must be much smaller than the above value, for instance ξ˙j [t0 ] = o[H0 ], ξ¨j [t0 ] = o[H02 ] . When we insert this into the evolution (10.6) and (10.7), we obtain Φ[ξ[t0 ]] = O[H02 ] ,

∂Φ [ξ[t0 ]] = O[H02 ] , ∂ξ j

and this means that 8πG 3c2 8πG 3c2

Mm0 ,...,mα = O[H02 ] ,

all components

mi Mm0 ,...,mα = O[H02 ] , i = 0, . . . , α .

all components

We obtain α + 2 conditions for the matter components, curvatures and cosmological constant included. The number of internal factor spaces is α. If we hope for very large curvatures of these internal factors (because we intend to argue that they are not observable in macroscopic experiments), there have to exist at least two other matter components that are as large as the curvatures and are able to compensate them in the α + 2 conditions.

References 1. Bleyer, U., Liebscher, D.-E.: Kaluza-Klein cosmology: Phenomenology and exact solutions with three-component matter, Gen. Rel. Grav. 17 (1985), 989– 1000. 249, 250 2. Bleyer, U., Liebscher, D.-E., Polnaryov, A. G.: Mixed metric perturbations in Kaluza-Klein cosmoloy, Nuovo Cimento B 106 (1991), 107–122. 247, 249

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3. Brans, C. H., Randall, D.: Exotic diﬀerentiable structures and general relativity, Gen. Rel. Grav. 25 (1993), 205–221. 245 4. Candelas, P., Weinberg, S.: Calculation of gauge couplings and compact circumferences from self-consistent dimensional reduction, Nucl. Phys. B 237 (1984), 397–441. 250 5. Chodos, A., Detweiler, S.: Where has the ﬁfth dimension gone? Phys. Rev. D 21 (1980), 2167–2170. 246 6. Demianski, M., Szydlowski, M., Szczesny, J.: Dynamical dimensional reduction, Gen. Rel. Grav. 22 (1990), 1217–1227. 250 7. Ehrenfest, P.: Welche Rolle spielt die Dreidimensionalit¨ at des Raumes in den Grundgesetzen der Physik? Ann. Phys. (Leipzig) 61 (1920), 440–446. 243 8. Frank, A., van Isacker, P., Gomez–Camacho, J.: Probing additional dimensions in the universe with neutron experiments, Phys. Lett. B 582 (2004), 15–20. 246 9. G¨ unther, U., Starobinsky, A., Zhuk, A.: Multidimensional cosmological models: cosmological and astrophysical implications and constraints, Phys. Rev. D 69 (2004), 044003 (hep-ph/030619132). 247 10. Hochberg, D., Wheeler, J. T.: Spacetime dimensions from a variational principle, Phys. Rev. D 43 (1991), 2617–2621. 246 11. Kaluza, T. von: Zum Unit¨ atsproblem der Physik, SBer. Preuss. Akad. Wiss. (1921), 966–972. 245 12. Kimberly, D., Magueijo, J.: Varying alpha and the electroweak model, Phys. Lett. B 584 (2004), 8–15 (hep-ph/0310030). 249 13. Klein, O.: Quantentheorie und f¨ unfdimensionale Relativit¨ atstheorie, Z. Phys. 37 (1926), 895–906. 245 14. Kolb, E. W., Slansky, R.: Dimensional reduction in the early universe: Where have the massive particles gone? Phys. Lett. B 135 (1984), 378–382. 247 15. Maartens, R.: Brane-world gravity, Living Reviews Relativity 7 (2004), grqc/0312059. 247 16. Maartens, R.: Brane-world cosmological perturbations, Proceedings RESCEU 6, astro-ph/0402485 (2004). 247 17. Maia, M. D.: Brane-worlds and Cosmology, in J. M. Salim, S. P. Bergliaﬀa, L. A. Oliveira and V. De Lorenci (eds.): Inquiring the Universe, Frontiers Group, gr-qc/0401032 (2004). 247 18. Moss, I. G.: The dynamics of dimensional reduction, Phys. Lett. B 140 (1984), 29–32. 250 19. Mota, D. F.: Variations of the Fine Structure Constant in Space and Time, Ph.D. Thesis; DAMTP, University of Cambridge, astro-ph/0401631 (2004). 249 20. Potekhin, A. Y., Varshalovich, D. A.: Non-variability of the ﬁne-structure constant over cosmological time-scales, Astron. Astrophys. Suppl. Ser. 104 (1994), 89–98. 249 21. Sisterna, P., Vucetich, H.: Time variation of fundamental constants: bounds from geophysical and astronomical data, Phys. Rev. D 41 (1990), 1034–1046. 249 22. Sahdev, D.: Perfect-ﬂuid higher-dimensional cosmologies, Phys. Rev. D 30 (1984), 2495–2507. 250 23. Szydlowski, M.: Geometrized dynamics of multidimensional cosmological models, Int. J. Theor. Phys. 33 (1994), 715–734. 249

11 Topological Quasi-Particles

11.1 The Topology of a Field Distribution The subject of topology is the set of properties of relative positions that can be deﬁned without the use of metric properties. A metric induces a particular topology only locally, and a metric is not necessary to deﬁne a topology. In the familiar metric geometry, equivalence is constructed by maps that conserve distances. In topology, equivalence is constructed by maps that conserve neighbourhood or continuity. Even in the simplest spatially ﬂat case, there exist topologically inequivalent cosmological models that do not diﬀer in their metrical properties [2, 3]. The unexpectedly low amplitudes of the quadrupole and octupole components of the cosmic microwave background have led to the proposal of a dodecahedral topology of space [6, 7, 12]. We learned about requirements of global periodicity in Chap. 10. Periodicity makes points that are widely separated in our familiar understanding into neighbours. Any topologically non-trivial properties of the space-time or of the ﬁeld conﬁguration indicate that global relations in addition to local metric properties are important. In some cases, global topological peculiarities are an expression of identiﬁable local peculiarities (analogous to knots, for instance) in the conﬁguration of the ﬁelds. In this case, we can speak of ‘quasi-particles’. They are our subject in this chapter. Topological peculiarities can exist from the beginning, that is, with an expression in the past sky; they also can come into being at phase transitions at the Curie point of the vacuum, if the symmetry is appropriately reduced. Hence, the peculiarities that might be formed depend on the model of uniﬁcation used. Here we discuss only the general variants. Phase transitions that yield the development of structure in an initially structureless (symmetric or isotropic) medium are quite common in physics. As we learned in Chap. 7, some phase transitions in elementary-particle physics are related to structures in the vacuum. We have learned that these phase transitions can be described with the help of a scalar inﬂaton ﬁeld. If its expectation value is diﬀerent from zero, the ground state, the vacuum, loses its symmetry. We shall avoid the question of the algebraic properties of the ﬁelds and use simply some multicomponent ﬁeld φA , A = 1, . . . , N [9]. Its components can be considered as real scalars, because we do not consider algebraic properties. Dierck-Ekkehard Liebscher: Cosmology STMP 210, 253–263 (2004) c Springer-Verlag Berlin Heidelberg 2004

254

11 Topological Quasi-Particles

The action integral is a generalisation of that in Chap. 7, (7.1): 1 ik 4 S = d x¯ h g φA,i φA,k − V [φ] . 2 A

The symmetry breakdown should proceed over a very short time span, so that the time dependence of the metric, which is mow not important, can be neglected. The ﬁelds have the dimensions of reciprocal length. For Minkowski coordinates (which always exist locally), we obtain 11 4 2 2 ˙ h φ − (∇φA ) − V [φ] . S = d x¯ 2 c2 A A

The ﬁeld equations are ∂V 1 ¨ , φA − ∆φA = − 2 c ∂φA and the energy–momentum tensor is obtained as 1 Tik = h g mn φA,m φA,n − V [φ] ¯c φA,i φA,k − gik . 2 A

A

The energy density follows: 11 ¯c εφ = h φ˙ 2 + (∇φA )2 + V [φ] . 2 c2 A A

The potential is modelled by 2 1 m2 [T ]c2 2 λ 2 2 φA − η + φA . V [φ] = 4 2 ¯h2 A

(11.1)

A

At low temperatures, the classical ground state of the N -dimensional ﬁeld is degenerate, and there are a variety of topological peculiarities that we intend to consider from the point of cosmology. The eﬀective Compton length of a perturbation near the minimum of the potential is given by 1 lδφ = √ . η λ In the centre of a topological peculiarity, a region of false (i.e. high-temperature) vacuum is included. Owing to topological (i.e. continuity) conditions, it cannot decay. From the Lorentz invariance of the vacuum, we can draw conclusions about the equation of state for a gas of such objects. In the familiar vacuum, Lorentz invariance implies

11.2 Domain Walls

T ik = εδ ik ,

255

(11.2)

i.e. p = −ε. For a wall, (11.2) remains valid except for the normal component, which now vanishes. A statistical average over all directions yields the equation of state p = −2ε/3. For a string, only the tangential component does not vanish, and an average yields p = −ε/3. Topological quasi-particles are stable classically because of the continuity conditions. According to the rules of quantum theory, however, we should not be astonished if they decay through a kind of a tunnel eﬀect [10].

11.2 Domain Walls A ferromagnet consists of a number, usually large, of so-called Weiss domains, in each of which the ground state of the magnetisation is homogeneous. On the walls between these domains, the direction of the magnetisation is discontinuous. Analogous domains and domain walls may develop in the case of grand uniﬁed ﬁeld theories [5, 17]. In general, domain walls develop if the set of ground states of the vacuum (here φ = η and φ = −η) is not connected. For the simple example of a one-component ﬁeld, the potential has two minima, φ = ±η. The minimum that attracts the ﬁeld can vary with position. The space is divided into diﬀerent domains that diﬀer in the sign of φ. The domain walls are regions where the ﬁeld cannot fall into a ground state. We describe these walls as inﬁnitesimally thin surfaces z = 0, and obtain the surface density of the energy from the ﬁrst integral of the ﬁeld equation, 2 dφ = 2V [φ] , dz where φ = −η for z negative and φ = +η for z positive. We integrate the energy–momentum tensor perpendicular to the wall, and obtain the surface densities dz T00 = − dz Txx = − dz Tyy ≈ 2 h ¯ cV [0] ∆z , dz Tzz ≈ 0 , where ∆z is the eﬀective thickness of the wall. The surface density of the active gravitational mass is negative: dz (2T00 − g ik Tik ) = −2 ¯hcV [0] ∆z . The formal equation of state of a gas of such domain walls is 2 p=− ε. 3 The eﬀective thickness of the wall can be estimated from (φ/∆z)2 ≈ (η/∆z)2 ≈ λη 4 , which gives

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1 ∆z ≈ √ . η λ The surface density of the energy is √ Σ ≈ ¯hcV ∆z ≈ ¯hcη 3 λ . The simplest estimation of the possible density of these peculiarities assumes that they will ﬁnd and react with partners for annihilation inside their horizon, during the formation stage at best. Therefore, we have to take account of one quasi-particle per horizon volume at the symmetry-breakdown stage. This corresponds to 1080 particles in the present Hubble sphere, and 100–1000 h3 m−3 at symmetry-breakdown time. The energy density of the domain walls must not be too large, because we have to expect at least one in our horizon, and the isotropy of the background radiation must not be disturbed beyond a level of 10−3 . We expect Σc2 H0−2 ≤ 10−4

c2 H02 c3 2¯ hcG 1 √ ≥ 104 3 , . 3 2G H03 c H0 η λ

Considering only orders of magnitude, this gives us 2 RH , lφ3 > lPlanck

lφ > 10−13 cm .

The ﬁeld mass should be smaller than the electron mass, i.e. far below the grand uniﬁcation scale of 1015 GeV. Domain walls that form at the end of inﬂation would be excluded by the present state of the universe. If the energy scale was that of the electron mass, we would expect the formation of domain walls at the end of the radiation-dominated stage [8]. These domain walls could be the precursors of the present cell structure of the distribution of galaxies. The hypothesis of such an energy scale, however, interferes deeply with the standard model of elementary-particle physics. Nevertheless, the question of whether the cell structure of the distribution of galaxies is connected to some cause beyond gravitation is still discussed as alternative to the scenario described in Chaps. 8 and 9.

11.3 Strings Vortex ﬁlaments (strings) develop in our model theory in the simplest way in a two-component ﬁeld φA = [φ1 , φ2 ] that has a set of ground states of the form φA [θ] = η [sin θ, cos θ] . Let us assume that the formal angle θ of the ground state depends on position. The ﬁeld may now be ‘smoothly combed’, or it may contain vortices. In the cylindric conﬁguration,

11.3 Strings

257

x y , r = x2 + y 2 , φA = ϕ[r] , r r where the amplitude ϕ[r] must have a zero on the z axis (i.e. at r = 0) because of continuity. The local ground state, ϕ = η, cannot be reached in the inﬁnitesimal neighbourhood of the z axis. We calculate with an eﬀective radius ∆r of the string and obtain, analogously to our estimations in the case of a domain wall, the following integrals across the string: (11.3) dx dy T00 = − dx dy Tzz ≈ 2 ¯hcV [0](∆r)2 , dx dy Tyy = dx dy Txx ≈ 0 . The line density of the active gravitational mass is zero, and the formal equation of state of a gas of strings is 1 p=− ε. 3 The eﬀective diameter of the strings can be estimated from (φ/∆r)2 ≈ (η/∆r)2 ≈ λη 4 as 1 ∆r ≈ √ . η λ The line density of energy yields µ ≈ ¯hcV (∆r)2 ≈ ¯hcη 2 . Because of (11.3), this line density is identical to the tension that determines the oscillations and contraction of closed strings. Depending on the details of the model, the line density of the energy varies over a wide interval from 1022 N to 1038 N. The upper value corresponds to a line density of approximately 1 M /R . Strings appear when the set of ground states of the vacuum (here φ21 + φ22 = η 2 ) are not simply connected, i.e. when it contains closed curves that cannot be contracted continuously to one point without the state leaving the set. Because the active gravitational mass of our string is zero, the space-time around the string remains ﬂat; only the string is a singularity of curvature like the vertex of a cone. A spatial section orthogonal to the string corresponds to the surface of a cone, and the angle deﬁcit is proportional to the line density of mass: = 8πGµ ≈ 8π

2 lPlanck ≈ 10−6 . lφ2

The case > 0 yields a kind of attraction that is represented best as a deﬂection of light (Fig. 11.1). The surface of the cone is ﬂat and and can

258

11 Topological Quasi-Particles

Fig. 11.1. Light deﬂection around a string. The observer sees the source from two diﬀerent directions. Owing to the procedure for the identiﬁcation of the boundaries of the cut, the continuation of a ray intersects the second boundary of the cut at the same angle as the primary ray intersects the ﬁrst boundary of the cut, while the intersection points have the the same distance from the vertex (γ + δ = π). Consequently, two points (source and observer) may have two straight rays connecting them. The sum of the angles α and β is independent of the position of the two points and equal to the excess of the sum of angles in a triangle that contains the string (the vertex)

be rolled oﬀ. A plane is formed that contains a cut that eliminates1 a sector with an angle > 0. The strings have a precursor in a solution of the Einstein equations known as the Weyl–Curzon solution. This ﬁrst string was found when the two-body problem was considered in GRT. An attempt to ﬁnd a static solution to Einstein’s equations led to the use of cylindrical coordinates. The ﬁeld equations show that one can put the metric in the form ds2 = e2(ν−λ) (dr2 + dz 2 ) + r2 e−2λ dϕ2 − e2λ dt2 . The equation R33 − R44 = 0 turns out to be the potential equation for λ in cylindrical coordinates, given by λ,rr + λ,zz + λ,r /r = 0 .

(11.4)

As far as ν is concerned, we obtain two equations, ν,r = r(λ2,r − λ2,z ) , ν,z = 2rλ,r λ,z , which do not contradict each other, because of (11.4); we also obtain the integral

B νB − νA =

r((λ2,r − λ2,z ) dr + 2λ,r λ,z dz) . A

1

We can also imagine that we insert a sector, and then calculate for < 0. This would imply a negative line density, and a repulsion of passing rays

11.3 Strings

259

If we choose the z axis as the connection between the two sources, we obtain the Curzon solution, λ=−

m2 m1 − , 1 2

m2 r 2 m2 r 2 2m1 m2 ν = − 14 − 24 + (z1 − z2 )2 21 22

r2 + (z − z1 )(z − z2 ) −1 1 2

,

where 2i = r2 + (z − zi )2 . This metric seems to describe the gravitational ﬁeld of two point masses. Of course, this is an error. A theory with this metric would allow two particles to be at rest in spite of their gravitational attraction, which the theory claims to describe. However, the line between the two sources is a string, i.e. a source itself. A section perpendicular to the z axis is like the surface of a cone. On the z axis, between the point sources, we have ν = −4m1 m2 (z1 − z2 )−2 . If both masses are positive, ν is negative. The circumference of a circle with radius = eν−λ r is now diﬀerent from U [] = 2π + o[], namely U [] = 2πe−ν + o[] . The space around the connecting line contains a conical singularity distributed along this line that apparently compensates the attracting force. However, the line density is negative, and yields a pressure along the string, not a tension. So we have a rather unstable object, diﬀerent from strings with positive line density and tension.

In the theory of elementary particles, strings are used to generalise the concept of particles as point masses. Macroscopic analogies have been found experimentally in the phase transition of helium to the superﬂuid state [4]. Here, we shall consider only cosmic, i.e. macroscopic, strings [15, 17]. We have to distinguish between inﬁnitely long open strings and ﬁnite closed strings. Open strings should be stretched into nearly straight lines that form a kind of net in the universe [1]. One might expect to ﬁnd them from binary images of galaxies that cannot be modelled by ordinary gravitational lenses. Closed strings have a wider theoretical appeal. First, they are expected to contract owing to their internal tension and to decay by some quantum process into ordinary particles when the radius of the loop decreases to the diameter of the string. These particles are formed from the energy of the false vacuum in the core of the string and the kinetic energy of the contraction. Lots of science ﬁction can be devised. Loops should oscillate because of their tension. The corresponding quantum numbers might model the generalised charges of the particles that the loop is assumed to model. A general interaction between two strings is illustrated in Fig. 11.2. A string can cross itself and decay into smaller strings on such an occasion. Oscillating loops can emit gravitational radiation owing to their varying quadrupole moment. A string that moves through the intergalactic medium might form caustic phenomena in its wake, which might

260

11 Topological Quasi-Particles

Fig. 11.2. Metaphor of two closed strings interacting in space-time. The history of the strings is indicated by a kind of spaghetti in space-time. The wobbling indicates the quanta of the string that are interchanged during the interaction

be important for the formation of structure in the metagalaxy. In addition, cosmic strings might be superconducting [18]. They could emit short, highenergy pulses of radiation, or even charged particles that contribute to cosmic rays. They could heat the surrounding plasma by radio emission, or empty their neighbourhood to contribute to the formation of voids.

11.4 Monopoles and Textures If the inﬂaton ﬁeld has three components, the degenerate, low-temperature ground state is topologically a sphere: φA [θ1 , θ2 ] = η [sin θ1 sin θ2 , sin θ1 cos θ2 , cos θ1 ] . Again, it can have nontrivial conﬁgurations, x y z , r = x2 + y 2 + z 2 , φA = ϕ[r] , , r r r for instance. The ground state ϕ = η cannot be attained at r = 0 because of the continuity condition. The conﬁguration around r = 0 is called a monopole. Again, the eﬀective radius is given by 1 ∆r ≈ √ , η λ

11.4 Monopoles and Textures

261

and the mass is approximately η M ≈ V (∆r)3 ≈ √ . λ Monopoles form when the set of ground states (here φ21 + φ22 + φ23 = η 2 ) contains closed surfaces, that cannot be continuously contracted to a point.2 Monopoles are spatially concentrated. One is forced to assume that they are sources of a magnetic ﬁeld [14], in contrast to the explicit statement of classical electrodynamics that magnetic monopoles do not exist. Monopoles can trigger the decay of baryons [16]. The mass of the monopoles and their creation rate posed the problem that initiated the development of the concept of inﬂation (Chap. 7). Without the dilution of the monopoles induced by inﬂation, they would dominate the present matter density. The same argument that restricted the existence of domain walls can be used to estimate the abundance of monopoles (6.13). GUT monopoles with mass MM ≈ 1016 GeV are formed at kTM ≈ 1014 GeV. The minimum abundance is obtained for fast annihilation, that is, at least one monopole per Hubble volume survives. From (6.2), we obtain the Hubble volume at the time of T ≈ TM : VHubble [TM ] ≈

1 (f lPlanck )3

¯c h kTM

6

and a present density of about M [t0 ] ≈

T0 TM

3

3

kTM (f lPlanck )

kTM ¯hc

6 ≈ 1011 critical .

This would be an absurd universe [19]. Structures in ﬁelds with more than three components are stationary no more. Let us assume that a four-component ﬁeld φA [θ1 , θ2 , θ3 ] with a ground state [η sin θ1 sin θ2 sin θ3 , η sin θ1 sin θ2 cos θ3 , η sin θ1 cos θ2 , η cos θ1 ] can have a non-trivial conﬁguration around r = 0, for instance x y z sin θ1 , sin θ1 , sin θ1 , cos θ1 , r = x2 + y 2 + z 2 . φA = ϕ r r r Even such a conﬁguration can smoothly approach the state ϕ = η at r = 0 if the angle θ1 [0] is a multiple of π at r = 0. With an appropriate function θ1 [r, t], a non-trivial structure can be described that cannot be removed in four-dimensional space-time but appears transient in three-dimensional 2

One must distinguish between short-range local monopoles and long-range global monopoles. The latter may annihilate over a longer period of time and do not necessarily exceed the critical density.

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11 Topological Quasi-Particles

space. Such a structure is called a texture. These structures collapse, that is, they condense more and more with time [11], disturb the metric with the transiently bound energy of the false vacuum, and induce gravitational ﬂuctuations in the matter ﬁelds. If more than four ﬁeld components are involved, textures can disappear completely. Such an event releases energy in a kind of small explosion and may induces structure formation. Numerical simulations [13] lead us to expect that the number of these events per unit comoving volume and conformal time should be scale-invariant. Textures are interesting for structure formation because their lifetime could be so large that their abundance could be important even after recombination, when they would start structures in disappearing that are completely invisible in the background radiation. The expected spectrum diﬀers from the form that is expected in an inﬂationary generation scenario only for small numbers of ﬁeld components. The WMAP observations, however, rule out these ‘active perturbations’, at least for the simpler evolution models.

References 1. Albrecht, A., Turok, N.: Evolution of cosmic string networks, Phys. Rev. D 40 (1989), 973–1001. 259 2. Fang, L. Z., Mo, H. J.: Topology of the universe, in Hewitt, A., Burbidge, G. R., Fang, L.-Z. (eds.): Observational Cosmology, IAU-Seminar 124, Kluwer Academic, Dordrecht 461–475 (1987). 253 3. Heidmann, J.: Introduction a ` la cosmologie, Presses universitaires de France, Paris (1973). 253 4. Hendry, P. C., Lawson, N. S., Lee, R. A. M., McClintrock, P. V. E., Williams, C. D. H.: Generation of defects in superﬂuid 4 He as an analogue of the formation of cosmic strings, Nature 368 (1994), 315–317. 259 5. Kolb, E. W., Wang Yun: Domain-wall formation in late-time phase transitions, Phys. Rev. D 45 (1992), 4421–4427. 255 6. Luminet, J.-P., Weeks, J., Riazuelo, A., Lehoucq, R., Uzan, J.-P.: Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background, Nature 425 (2003), 593–595 (astro-ph/0310253). 253 7. McInnes, B.: String Theory and the Shape of the Universe, SPIRES HEP, hepth/0401035 (2004). 253 8. Nambu, Y., Ishihara, H., Gouda, N., Sugiyama, N.: Anisotropies of the cosmic background radiation by domain wall networks, Astrophys. J. Lett. 373 (1991), L35–L37. 256 9. Peebles, P. J. E.: Principles of physical cosmology, Princeton University Press (1993). 253 10. Preskill, J., Vilenkin, A. A.: Decay of metastable topological defects, Phys. Rev. D 47 (1993), 2324–2342. 255 11. Prokopec, T., Sornborger, A., Brandenberger, R. H.: Texture collapse, Phys. Rev. D 45 (1992), 1971–1981. 262 12. Schwarz, D. J., Starkman, G. D., Huterer, D., Copi, C. J.: Is the low-l microwave background cosmic? CERN-PH-TH/2004-052, astro-ph/0403353 (2004). 253

References

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13. Spergel, D. N., Turok, N., Press, W. H., Ryden, B. S.: Global texture as the origin of large-scale structure: numerical simulations of evolution, Phys. Rev. D 43 (1991), 1038–1046. 262 14. t’Hooft, G.: Magnetic monopoles in uniﬁed gauge theories, Nucl. Phys. B 79 (1974), 276–284. 261 15. Tseytlin, A. A., Vafa, C.: Elements of string cosmology, Nucl. Phys. B 372 (1992), 443–466. 259 16. Turner, M. S.: The end may be hastened by magnetic monopoles, Nature 306 (1983), 161–162. 261 17. Vilenkin, A.: Cosmic strings and domain walls, Phys. Rep. 121 (1985), 263–315. 255, 259 18. Witten, E. L.: Superconducting cosmic strings, Nucl. Phys. B 249 (1985), 557– 592. 260 19. Zeldovich, Ya. B., Khlopov, M. Yu.: On the concentration of relic magnetic monopoles in the universe, Phys. Lett. B 79 (1978), 239–241. 261

12 Quantum Cosmology

12.1 Why Quantum Cosmology Up to this point, the cosmological model has been a kind of classical framework for processes that are mainly microscopic. These in turn have determined the evolution of the classical framework through the history of the matter content. Such a division between frame and content presupposes that the orders of magnitude of the characteristic lengths, times or energies are distinctly diﬀerent. Since the time of inﬂation, such a supposition may be justiﬁed. However, what happened before inﬂation began? Should we ever expect an answer? After all, we recall that the goal of explaining the universe through its dynamics requires that the conclusions about its early history become more and more fuzzy and shaky the further we dig into the past, and ﬁnally must become empty. From this point of view, a particularly welcome property of the models with inﬂation is the fact that the universe in a sense forgets about its history before inﬂation. Only a small fraction of its large-scale properties must be assumed in order to ﬁnd a model that develops via an inﬂationary epoch. These properties not only allow us to ﬁnd some evidence of the inﬂation itself (Sect. 5.6), but also allow us to look even deeper into the past, and give us reason to consider the history before inﬂation. We observe no details from that time, but we know that the preinﬂationary phase must be modelled in such a way that it can lead to inﬂation. It is still possible to construct phenomenological models for the preinﬂationary phase. Their absolute limit is reached when the sizes of the universe (the curvature radius and Hubble radius) become comparable to the Planck length. At that time and before, quantum laws must have governed what we understand as the history of the universe. These laws must have concerned not only the microscopic matter content, but the universe itself. If we accept this reasoning, the ﬁrst paradox that arises is the fact that quantum theory requires the dichotomy between the quantum-determined object of observation and the classically interpretable measurement apparatus (or observer). The universe, however, is literally everything together, by deﬁnition, and does not contain any reference to an external observer: there is nothing external by deﬁnition. The familiar concepts of a measurement or of an ensemble of identical systems cannot be applied. One can manage this Dierck-Ekkehard Liebscher: Cosmology STMP 210, 265–272 (2004) c Springer-Verlag Berlin Heidelberg 2004

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paradox in various ways. In the end, when it comes to formal calculations, it is mostly disregarded. Instead, the familiar equations are formally translated into the language of quantum theory. An interpretation is constructed ad hoc afterwards. The consequence is that we have to check whether this interpretation is compatible with the standard interpretation when applied to familiar quantum mechanics [9]. The formal problems are diﬃcult enough. Nevertheless, they seem easier than those of the subsequent interpretation. We can, however, identify a minimum set of statements that should be obeyed by a quantum picture of the pre-inﬂationary universe [8, 14]. Quantum cosmology should – determine the conditions that must be met to allow us to describe the universe as a classical system, – identify the initial conditions for the classical evolution, and – determine the particular state of the vacuum that allows us to use quantum ﬁeld theory in a curved background. Loosely speaking, quantum cosmology has to remove thew singularity in the curvature (and density) at t = 0 for the Gamow universe, where ds2 = c2 dt2 + a20

t (dx2 + dy 2 + dz 2 ) , t0

and where space-time appears to be connected to a four-dimensional Riemannian space [12].

12.2 The Canonical Formulation of General Relativity Theory If we intend to construct the quantum equivalent of the Friedmann equations, we have to recall that the Friedmann equations are a particular case of the Einstein equations, and that the reduction procedure itself may be modiﬁed by the quantisation procedure. It may not suﬃce to take the Friedmann equations as equations of motion of a formal mechanical system and to use the quantisation rules of quantum mechanics without further ado. It is necessary to settle the formal aspects of the quantisation of the Einstein equations themselves [13, 15]. The general theory of relativity is not a ﬁeld theory in a given spacetime frame, but is the theory of space-time itself. The Einstein equations have no evident canonical structure. The ﬁrst task in quantisation is to ﬁnd a canonical formulation. Of course, such a formulation is important for the initial-value problem and the numerical solution of the Einstein equations as well. It is now necessary to leave the strictly four-dimensional path and to choose a function t = t[xi ], which is used as a speciﬁc cosmological time.1 The 1

We give the time t the dimensions of a length in this chapter. A dot indicates a derivative with respect to this time.

12.2 The Canonical Formulation of General Relativity Theory

267

hypersurfaces t[xi ] = const must not intersect, so that space-time appears as a sequence (foliation) of spaces ordered by the time t. The gravitational ﬁeld is given by the internal geometry of the spaces and their relative arrangement. Each particular hypersurface is a three-dimensional space. Its state is given by the metric tensor hab of the corresponding line element, ds2 = hab dxa dxb . In this way, a model of a superspace is constructed. The points of the superspace are the three-dimensional spaces, and the coordinates are represented by the components of the metric ﬁeld hab [x]. The gravitational ﬁeld of the four-dimensional space-time, i.e. the history of the universe, is a curve in this superspace. The inﬂuence of the choice of coordinates must be considered separately. A point in superspace is a metric tensor hab [x] modulo the coordinate transformations. For the moment, its velocity is ∂hab /∂t, the coordinate-independent part of which is Kij = ∂hab /∂t − Na|b − Nb|a , with functions N a that have not yet been determined. Together with the deﬁnition of some kind of ephemeris time, these velocities mediate the association between space-time and superspace. The quantity that represents the potential energy is the curvature of the three-dimensional space, 3R = hab 3Rc abc .

In analogy to the familiar type of mechanics, we deﬁne a metric in superspace. Let us return to the construction of the hypersurfaces t[xk ] = const. The metric gik of space-time induces the internal metric hab in the hypersurfaces and splits into the terms ds2 = −(N 2 − Nc N c ) dt2 + 2Na dxa dt + hab dxa dxb = −N 2 dt2 + hab (N a dt + dxa )(N b dt + dxb ) .

(12.1)

We choose the notations hab hbc = δac , N a = hab Nb . The indices a, b, . . . run from 1 to 3. In order to explain the functions Na and N , we consider two adjacent hypersurfaces t = 0 and t = . The vector connecting points with equal internal coordinates has the coordinates [, 0, 0, 0]. It is orthogonal to the hypersurface [0, δxa ] only in the case of vanishing components Na . Hence the Na describe a kind of shift of the adjacent hypersurface, t = , with respect to t = 0. Orthogonal to a hypersurface (t = const) are the vectors mi = g ik mk , where mk = t,k = [1, 0,√0, 0]. Normalisation yields the normal vectors ni = [−1/N, N a /N ], nk = [ N , 0, 0, 0]. Thus the components Na (shift) and N (lapse) determine the relative position of points in adjacent hypersurfaces. Even in the case of ﬁxed internal coordinates, transformations of the form xa∗ = xa + tf a [x] can still be chosen. With these transformations, we may

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reach the particular form N∗ a = 0 by the choice f a = N a . A transformation that yields N = 1 must include the choice of a diﬀerent (speciﬁc) set of adjacent hypersurfaces.2 In any case, however, the equations for the metric must leave the lapse and shift open. The Einstein-Hilbert action integral with a surface term is given by   3 √ √ c  S= d4 x −gR + 2 d3 x hK  + Smatter . 16πG M

∂M

The surface term does not inﬂuence the Euler-Lagrange equations, but it corrects the value of the integral. Here, the letter R denotes the scalar curvature of the space-time, h denotes the determinant of the inner metric of the surface, and K denotes the trace of its external curvature K ab . We obtain the expansion R = K 2 − Kab K ab − 3R + total divergence . For our metric (12.1), the Lagrangian can be transformed to yield Lgrav = where Kab

1 = 2N

√ c3 N (Gabcd Kab Kcd + h 3R) , 16πG

∂hab + 2N(a|b) − ∂t

, Gabcd =

1√ h(hac hbd + had hbc − 2hab hcd ) . 2

The tensor Gabcd will yield the coeﬃcients of the diﬀerential operator of the Schr¨ odinger equation for our problem and is therefore the metric for the tensors Kab . It has a pseudo-Euclidean signature, with one time-like and ﬁve space-like directions. In the classical picture, the closed empty universe is only a particular case of the de Sitter model, and hence we should also have sources in an elementary picture. The simplest representative of such sources is the same scalar ﬁeld as usual, which makes a contribution √ 1 ik ab a 2 g [h , N , N ]φ,i φ,k − V [φ] , ¯ N h Lφ = h 2 to the Lagrangian density. The components hab of the internal metric, and the ﬁeld φ serve as observables. The canonical momentum densities are √ √ h ˙ ∂L c3 h ab (K − hab K) , πφ = h (φ − N a φ,a ) . =− ¯ π ab = 16πG N ∂ h˙ ab 2

With respect to purely internal coordinate transformations xa∗ = xa∗ [xb ], the quantity N a is a vector and N is a scalar.

12.3 The Wheeler–deWitt Equation

269

The action integral can be written as S = d3 x dt (π ab h˙ ab − N H0 − Na Ha ) . For the Hamiltonian function, we obtain H = d3 x (N H0 + Na Ha ) , where H0 =

16πG 1 c3 √ 3 Gabcd π ab π cd − h R 3 c h 16πG √ πφ2 ¯h h ab + √ + (h φ,a φ,b + 2V [φ]) , 2 2¯ h h

Ha = −2π ab|b + hab φ,b πφ .

12.3 The Wheeler–deWitt Equation Following the general pattern of ﬁeld theory, we replace the momenta with partial derivatives with respect to the coordinates and thus satisfy the commutation rules of position and momentum, π ab :=

¯ δ h ¯h δ . , πφ := i δhab i δφ

The equation Ha = 0 turns out to be nothing else than the condition for covariance in the hypersurfaces t = const. After the above replacement, it yields a formal diﬀerential equation of second order, called the Wheeler– de Witt equation: √

√ δ δ ¯h h 3R δ 2 00 + + hT φ, 16π¯ hlPlanck Gabcd Ψ =0. 2 δhab δhcd 16πlPlanck δφ This equation expresses the expectation that the total energy of a closed universe vanishes. A particular problem, which is not considered here, is the ordering of factors because the metric tensor that describes the ﬁnal causal order is not prescribed (as in special-relativistic ﬁeld theories) but is expected to be the result of the calculations. The state of the universe is described by odinger wave function. some functional Ψ [hab , φ] that plays the role of the Schr¨ Because the classical condition on the Hamilton density is H = 0, the time does not appear in the Wheeler–de Witt equation. The wave function Ψ [h, φ] does not depend on the time t because t does not enter the action integral explicitly. The latter statement is a consequence of invariance with respect

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to gauging the quantities N a and N . It must be possible to determine the relative position of hypersurfaces in space-time, and with it the evolution in time, simply through analysis of their internal geometries. The Wheeler–de Witt equation is usually treated by a particular method of integration that can be interpreted as taking a weighted average of all paths3 through the superspace of the metrics gµν with given end points [hab 1 , φ1 ] and [hab 2 , φ2 ]. These end points are metric tensors and ﬁeld distributions on hypersurfaces that play the role of the initial and the ﬁnal instant. The action integral again attributes values to the paths. Its minimum indicates the path that contributes most to the amplitude [5, 7]. Formally, we write

i S . (12.2) P[hab1 , φ1 , hab2 , φ2 ] = Dgµν Dφ exp ¯h The time does not appear in the probability amplitudes P, This picture yields a kind of description of the internal relations of the superspace.4 How can such a picture describe the early, i.e. pre-inﬂationary, universe? At best, we obtain a wave function with an amplitude that represents a probability density in superspace. The regions of superspace with large amplitude are interpreted as the states of the universe that are observed. If the wave function is oscillatory in some sense (i.e. if the real part of the function S is important), we may ﬁnd analogies with classical solutions. The time can be introduced as an ephemeris time if one uses the analogy of the action in superspace to the action in Maupertuis’s or Jacobi’s principle (Chap. 13). The transition to a classical universe is obtained through a kind of Wentzel–Kramers–Brillouin approximation. In such an approximation, time is bound to expansion [3, 9, 10, 11, 18]. It turns out that the amplitude is large on the total classical trajectory when it can be shown that it is large on one point of the trajectory. With 3 The action integral S = dt L[x, x ¨, t] attributes a value to any path between two end points. The classical motion proceeds along a path with stationary action (δS = 0). The probability amplitude Ψ [x, t] is given by

Ψ [x, t] = N

D[ξ[τ ]] exp

i S[ξ[τ ]] , h ¯

up to a normalisation factor, if the integration runs over the set of all paths ξ[τ ] that pass through the state ξ[t] = x and start at the prepared initial state. The variation of the ﬁnal state shows that Ψ solves the Schr¨ odinger equation, −

4

¯ ∂ h Ψ [x, t] = HΨ [x, t] . i ∂t

The method of path integrals was the ﬁrst approach to quantum mechanics. It was found by Wentzel in 1924 and developed twenty years later by Feynman. We note that in the integral on the right-hand side, complex rotations into imaginary time (i.e. Euclidean four-dimensional spaces) are often used in order to ensure regularity of the results. Of course, this does not mean that we interpret time as being an imaginary coordinate.

12.4 Minisuperspaces

271

this property, the wave function corresponds to a probability distribution on the set of classical trajectories.

12.4 Minisuperspaces A signiﬁcant simpliﬁcation can be obtained when the superspace can be restricted to some subspace of a few dimensions. In fact, only in these cases can a detailed treatment be performed. For instance, the full metric gik [xl ] may be restricted to homogeneous, isotropic models. Such a small part of the superspace is called a minisuperspace. We have met this construction already in Chap. 10. The restriction to a minisuperspace assumes implicitly that the degrees of freedom left are really separable from the others. This, however, is hard to decide. One example is the familiar cosmological model ds2 = c2 dt2 − R2 [t]d2 ω3 , endowed with a homogeneous scalar ﬁeld φ[t] that obeys the familiar equations R˙ dV =0 φ¨ + 3 φ˙ + c2 R dφ and

1 dR R dt

2

kc2 8π¯hG + 2 = R 3c3

1 ˙2 2 φ + c V [φ] . 2

The Wheeler–de Witt equation, ∂ 1 ∂2 ∂ kR4 2 6 − + R V [φ] Ψ (R, φ) = 0 , 16πlPlanck R R − 2 ∂R ∂R 16πlPlanck 2 ∂φ2 is expected to determine the form of the wave function Ψ [R, φ]. Near classical conﬁgurations that correspond to a positive kinetic energy in mechanics, the wave function is expected to oscillate, and its amplitude is expected to be large. In the classically forbidden regions, the amplitude should be small and can be considered to behave in analogy to the tunnel eﬀect [2, 16, 17]. One can construct models in which the universe tunnels from a qualitatively different and not necessarily deﬁned state into the familiar expanding universe. It is desirable to obtain probabilities that this expanding universe develops via an inﬂationary stage, or that a symmetry breakdown leads to the coupling constants and particle masses, etc. now observed. The approach of Hartle and Hawking [10] uses the construction of Euclidean worlds in order to eliminate the problem of the initial values for the expanding universe. The formal identiﬁcation of the initial state is then replaced with a restriction on the paths in the path integral [6].

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If the problem is expressed in terms of Ashtekhar’s variables, which here led to a new approach to the initial-value problem and to the quantisation of gravitation as well, the method of loop quantisation is expected to lead to a kind of singularity-free description [1, 4].

References 1. Ashtekar, A., Bojowald, M., Lewandowski, J.: Mathematical structure of loop quantum cosmology, Adv. Theor. Math. Phys. 7 (2003), 233–268. 272 2. Atkatz, D., Pagels, H.: Origin of the universe as a quantum tunneling event, Phys. Rev. D 25 (1982), 2065–2073. 271 3. Barbour, J. B.: Time and complex numbers in canonical quantum gravity, Phys. Rev. D 47 (1993), 5422–5429. 270 4. Bojowald, M.: Loop Quantum Cosmology: Recent Progress, Plenary talk at ICGC 04, Cochin, India AEI-2004-017, gr-qc/0402053 (2004). 272 5. Brown, J. D., Martinez, E. A.: Lorentzian path integral for minisuperspace cosmology, Phys. Rev. D 42 (1990), 1931–1943. 270 ¨ 6. Dereli, T., Onder, M., Tucker, R. W.: Signature transitions in quantum cosmology, Class. Quant. Grav. 10 (1993), 1425–1434. 271 7. Feynman, R. P., Hibbs, A. R.: Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965). 270 8. Grishchuk, L. P.: Quantum eﬀects in cosmology, Class. Quant. Grav. 10 (1993), 2449–2477. 266 9. Halliwell, J. J.: The interpretation of quantum cosmological models, CTP preprint 2130, gr-qc/9208001 (1992). 266, 270 10. Hartle, J. B., Hawking, S. W.: Wave function of the universe, Phys. Rev. D 28 (1983), 2960–2975. 270, 271 11. Hawking, S. W., Laﬂamme, R., Lyons, G. W.: Origin of time asymmetry, Phys. Rev. D 47 (1993), 5342–5356. 270 12. Kim, S. P.: Quantum cosmology for tunneling universes, Proceedings VI APCTP International Conference of Gravitation and Astrophysics (ICGA6), Seoul, gr-qc/0403015 (2004). 266 13. Kuchaˇr, K. V.: Canonical quantum gravity, in Gleiser, R. J., Kozameh, C. N., Moreschi, O. M. (eds.): General Relativity and Gravitation 1992, Proceedings, Institute of Physics, Bristol, 119–150 (1993). 266 14. Linde, A. D.: Inﬂation and quantum cosmology, Physica Scr. T 36 (1991), 30–54. 266 15. Smolin, L.: Three Roads to Quantum Gravity: A New Understanding of Space, Time and the Universe, Weidenfeld, London (2001). 266 16. Vilenkin, A.: Quantum cosmology and the initial state of the universe, Phys. Rev. D 37 (1988), 888–897. 271 17. Vilenkin, A.: Interpretation of the wave function of the universe, Phys. Rev. D 39 (1989), 1116–1121. 271 18. Zeh, H. D.: The Physical Basis of the Direction of Time, 2nd edn., Springer, Berlin (1992). 270

13 Machian Aspects

13.1 Mach’s Paradox The question of a possible inﬂuence of the global structure of the universe on local physical laws is one of the most fundamental problems of natural science. The topicality of this question arises from the geometrisation of all interactions by modern gauge ﬁeld theories following Einstein’s theory of gravitation, as well as the consideration of energy regions in elementary-particle physics which could be reached only in the early stages of the evolution of the universe. In correspondence to Poincar´e’s epistemological sum, which states that the physical content of a theory is deﬁned by geometry plus dynamics, we might handle the interconnection between physics at small and large distances in two diﬀerent ways. In uniﬁed ﬁeld theories, dynamics is based on a geometry of the space-time manifold in which the global existence of a causal structure is assumed a priori, and the local laws determine everything else. In the opposite case, the connection between local motion and global structure is given by the Mach–Einstein postulate of the induction of the inertial properties of matter by the joint gravitational inﬂuence of cosmic masses. The notion of absolute, that is independent of its content, space should be eliminated from physics and the inertia of mass should be related to the universe [23]. This demand, Mach’s principle, is the ﬁrst manifest formulation of the view that the state of the universe – at least in this respect – is responsible for the form of the laws of physics that we ﬁnd at present. This view is the opposite of the familiar procedure that we use to design a model of the universe. Normally, we construct a cosmos from the ‘earthly’ laws of physics and understand the universe to be its subject. To some extent, this is also necessary to interpret the cosmologically relevant observations. For instance, we interpret, with every right to do so, the light of the stars as radiation emitted by atoms and ions that do not diﬀer from atoms and ions on earth. However, if the model of the universe constructed with local physics, does it not force us to interpret the observations in such a way that they support the model, at least in its foundations? That is, can we really obtain observations that contradict this procedure? Can the assumption of viability be falsiﬁed? Mach could not make speciﬁc his demand for a cosmic reference. A constructive theory was not presented, and subsequently various theoretical Dierck-Ekkehard Liebscher: Cosmology STMP 210, 273–285 (2004) c Springer-Verlag Berlin Heidelberg 2004

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schemes have been constructed, but a satisfactory answer has not been obtained. Mach’s principle is the last important principle of classical physics that lacks such an answer [4, 33]. Although Mach’s principle played a documented role in the history of the general theory of relativity, it is not yet uniquely deﬁned and has not yet been uniquely theoretically realised [8, 28]. The reason is, not least, that Mach’s principle is justiﬁed by a paradox that requires a solution that turns out not to be easy to obtain [34, 35]. The paradox is easily explained: even without explicit reference to the stars, the rotation of the plane of oscillation of a Foucault pendulum tells us that the earth rotates. When we compare this dynamically observed rotation with the kinematical rotation with respect to the sky, we ﬁnd that the two values are equal. Newton postulated an absolute space, observable only in this context, with the sole aim of deﬁning rotationfree motion. The surface of the water in Newton’s rotating bucket curves into a paraboloid not because of its rotation with respect to the bucket, but because of its rotation with respect to absolute space. The sky of the ﬁxed stars cannot rotate relative to absolute space, because of the expected centrifugal forces. Mach insisted upon the non-detectability of absolute space and upon the ﬁxed stars as the reference and source of inertia. He argued that absolute space cannot be independently identiﬁed. The inertial reference systems of Newton should be ﬁxed only to real objects or to the collective motion of a system of masses. There is no motion with respect to absolute space, there is only motion of objects with respect to each other. The collectively rotating sky of stars does not feel any centrifugal force, because the relative positions of the stars do not change during a rigid rotation. Mach’s demand was that mechanics should be formulated in such a way that only relative motions are of physical interest and of physical impact. Inertial forces then arise not through rotation with respect to some ﬁcticious absolute space but through rotation with respect to the totality of cosmic masses. Mach wrote: ‘Niemand kann sagen, wie der (Eimer-)Versuch verlaufen w¨ urde, wenn die Gef¨ aßw¨ande immer dicker und massiger und zuletzt mehrere Meilen dick w¨ urden’.1 This is his judgement of Newton’s bucket experiment. The general theory of relativity was able easily to underline this with numbers. It can describe without further ado how Newton’s bucket inﬂuences the conﬁguration of inertial reference frames and the surface of the ﬂuid as the mass of the bucket (more speciﬁcally, its gravitational potential) increases. One can compare solutions obtained with diﬀerently rotating sources and compare their eﬀects on a freely falling gyroscope (the Lense–Thirring eﬀect [19, 27]); however, this eﬀect must then be separated from the eﬀects of spatial curvature. We may call a certain component of the excess in the sum of the angles of a triangle (Fig. 2.9) 1

No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick.

13.2 Machian Mechanics

275

in the ﬁeld of a rotating source a Mach eﬀect. The decisively unsatisfactory feature lies in the fact that the limit of the rotating universe is beyond reach, and this is apparently due to the diﬃculty of deﬁne rotation in a simple way without boundary conditions. It is the place here to recall also a microscopic feature of rotation: rotation has discrete quantum numbers, but the spectrum of translational motions is continuous.

13.2 Machian Mechanics In the framework of mechanics, a consistent realisation of the Mach–Einstein principle was obtained in an analytical description of inertia-free mechanics by Schr¨ odinger [29] the help of the Weber potential, by Treder with the help of the Riemann potential [32, 33] and by Barbour in his forceless, or relational, mechanics [3, 7]. In these constructions, inertia is replaced by an interaction in that the kinetic terms are replaced by (velocity-dependent) potentials. The integral of these interactions over the surrounding universe leads, for small subsystems, to kinetic terms, which can be interpreted as induced inertia. The simplest solution is to write the expression for the kinetic energy as a sum over distance variations. The kinetic energy, which is in Newtonian mechanics 1 mA r˙ 2A , (13.1) Ekinetic = 2 A

turns into a sum over all pairs of point masses of the kind that we are used to in the potential-energy term. A characteristic example is Schr¨odinger’s mechanics, which uses Weber’s potential [29], Ekinetic =

2 1 mA mB r˙AB G . 2 rAB c2

(13.2)

AB

Here, only distances rAB and variations of distances are accepted as physical quantities. The simplest consequences of such a mechanics are: – For a lone particle in an otherwise empty universe there is no law of motion at all: there is no reference for position and velocity. – For two particles in an otherwise empty universe, there is only a law for the variation of their distance, and not for the revolution around each other described by Kepler: there is no reference for orientation. – Independently of how many particles are considered, a common rotation and common velocity cannot be determined, and do not enter the equations of motion. The universe cannot rotate by lack of deﬁnition; only parts of it can rotate with respect to the whole. – The Kepler problem is part of the motion of a subsystem in the universe of the other particles, which in reality are overwhelmingly many.

276

13 Machian Aspects

The action integral that determines the motion, S = dt (Ekinetic − Epotential ) , is now invariant with respect to the total kinematical group of Euclidean space, i.e. not only with respect to position, orientation and translation, but also with respect to rotation. In contrast to this mechanics, Newton’s mechanics is only invariant with respect to the Galileo group. Accelerated translation and rotation are determined by the equation of motion and are measureable absolutely, i.e. independent of the existence and conﬁguration of surrounding particles. Accelerated translation and rotation produce inertial forces absolutely. Newton felt this to be a proof of the absoluteness of space. Inertial forces, however, arise in relational mechanics, too. They are absolute in an absolute space, and relational in a relational one. We can reduce the problem of motion to that of a small subsystem (for instance the solar system) in a large cloud of particles that represents the universe (Fig. 13.1). The relational kinetic energy splits into three parts. The ﬁrst part is the sum over all pairs of particles in the cloud. This does not change with the variables of the subsystem and falls out of the equations of motion of the latter. The second part is the sum over all pairs of particles in the subsystem. This yields a small, velocity-dependent correction to Newton’s potential energy (which may correspond to the motion of the perihelion of Mercury). The decisive third part contains the pairs consisting of one particle

Fig. 13.1. The universe given by a Machian cloud. The local subsystem is sketched as a planetary system in a cloud of stars

13.2 Machian Mechanics

277

in the cloud and another in the subsystem. For an isotropic cloud, this part yields the familiar kinetic energy, EkineticE,A Ekinetic = E,A

mE β β γ γ = mA nEAβ nEAγ x˙ E − x˙ A x˙ E − x˙ A rEA E

M mA r˙ 2A . = R The factor M/R indicates that we have to calculate with a quantity of this size. Its precise value depends on the distribution of the matter in the universe. The inertial mass is proportional to the potential of the universe, i.e. mA

inertial

= mA

f

B∈universe

GmB , c2 rAB

(13.3)

with some form factor f . The invariance of the subsystem is no longer the full (telescopic) invariance of the universe; it is broken down to some (approximate) Galilean invariance. The universe plays the role of the vacuum for the subsystem in question. It has a structure that reduces the telescopic invariance of the total system. The subsystem obtains position and orientation from this kind of vacuum. This example models Mach’s principle in mechanics. However, it is far too simple to correspond to the facts. The most important of these facts are: – The induced mass (13.3) varies with the expansion of the universe. This would correspond to an increase in the eﬀective gravitational constant. – The induced mass is anisotropic (the leading term in this anisotropy stems from our home galaxy, which has a potential Φ = GM/rc2 of the order of 10−6 ). – The (special) theory of relativity is not taken intoe account. The gravitational ﬁeld varies instantaneously. An attempt to construct a Riemannian geometry through a bi-metric procedure [20] fails if we stick to an absolute time (instantaneous propagation of gravitation) in the background. For any bi-metric construction of the kind mentioned, the absolute simultaneity in the background shows up in the post-Newtonian coeﬃcient [17] α2 =

vlight vgravitation

2 −1.

(13.4)

Form this point of view, the breakdown of the telescopic group to the Galileo group must be replaced with a breakdown to the Lorentz group.

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13.3 Time without Time One way out of this situation is the argument of Barbour [5] that we have gone only half the way if we give up absolute position and orientation in space but not in time. Barbour returns to a concept of motion that takes the time as a pure variation in the conﬁguration of a system and measures the time by this variation only. This concept is familiar through Maupertuis’s, Fermat’s and Jacobi’s principles: the orbit of a motion is a kind of shortest path, and the measure of the path reﬂects the potential of gravitation and the inertia, but does not contain the time explicitly. In classical mechanics, the motion through conﬁguration space is determined by the minimum of the integral S = dt E − Epotential 2Ekinetic . This integral does not depend on time; t is a freely exchangeable parameter. The kinetic energy yields a metric for the conﬁguration space, ds2Maupertuis = 2(E − Epotential )Ekinetic dt2 . When a solution has been found, the condition Ekinetic + Epotential = E yields the ﬂow of time. Ephemeris time is determined explicitly in this way. The discovery is that this procedure can be applied to the theory of general relativity. Here, we deﬁne a superspace. The elements of this superspace are the states of a space, that are represented by a metric ds2 = hik dxi dxk of the space. The gravitational ﬁeld in fourdimensional space-time has the form ds2 = −(N 2 − Ni N i )dt2 + 2Ni dxi dt + hij dxi dxj . It is a curve in superspace (Chap. 12), which represents the history of the universe. In analogy to the familiar type of mechanics, we deﬁne the Maupertuis metric in the superspace [2, 6]. The corresponding action integral is obtained as ∂gab ∂gcd 3 S = dλ d x 3R Gabcd − N(a|b) − N(c|d) . ∂λ ∂λ This yields the complete solution of the problem of how the metrics of space are constructed exclusively relative to each other, and of how they vary along the path in superspace and determine the time ﬂow like an ephemeris time as well. A particular point is that one can possibly use this integral to interpret a

13.5 The Assumed and the Explained

279

canonical quantum ﬁeld theory in which the Wheeler–de Witt equation does not contain the time but yields only probability amplitudes in superspace. From this point of view, Mach’s principle supports the three-dimensional concept of the canonical quantisation of the gravitational ﬁeld, and does not reach beyond the theory of general relativity.

13.4 Measure by Mass In view of the initial problem, one may ask for more: after all, the metric ds2 = gik dxi dxk of GRT gives an expression for the orientation of an inertial system, which should be determined not with respect to itself but with respect to matter, i.e. the distribution of the energy-momentum tensor. The Einstein equations state this immediately, 1 Rik − gik R = κTik . 2 From these equations, the metric, i.e. the conﬁguration of an inertial reference system, is not determined solely by the matter distribution. We have to choose boundary conditions to ﬁx the geometry for a given matter distribution. This is the back door through which an equivalent to absolute space comes back in. Hence, it is important to look for conditions that allow us to proceed without choosing boundary conditions, i.e. to look for an integral formulation of Einstein’s equations. This can be done for space-times with closed space sections, i.e. for space-times that are related to the Einstein model. Independent of other ﬁndings along this route, however, the result can only be a general selection rule for solutions of the Einstein equations. This concept is strange in the context of other ﬁeld theories [2, 15, 16, 24].

13.5 The Assumed and the Explained The historical development from Newton’s mechanics to the GRT can be understood as a repeated withdrawal of absolute elements from physics. Special relativity removed absolute simultaneity and replaced the formal product of space and time with an inseparable union. The determination of absolute velocities that appeared to be feasible through a combination of mechanics and electrodynamics was found to be impossible, and the relativity of velocity was restored. The GRT explained the local relativity of accelerations as a consequence of the local equivalence of inertia and gravitation. The fact that all objects fall with the same acceleration allows the determination of relative accelerations only, that is, of tidal forces [12]. If Mach’s principle is expected to lead beyond the GRT, it must be understood as a programme to explain absolute features of the GRT as the result of some generalised

280

13 Machian Aspects

dynamics. The absolute element in the GRT is the (local) Lorentz invariance of inertial reference frames, the existence (not the actual form) of a unique metric of space-time that ﬁxes the validity of the special relativity theory on an inﬁnitesimal scale. From this point of view, Mach’s principle is a programme to replace the local Lorentz invariance with a more general invariance, and to understand the local Lorentz invariance as a result of a classical breakdown of symmetry that is observed in small regions of space-time (on a cosmic scale) only. Such a programme corresponds to what we have learned from relational mechanics and has a ﬁeld-theoretical background now. The simplest generalisation of Lorentz invariance is conformal invariance. This expects that all ﬁeld equations and equations of motion not only are also valid irrespective of Lorentz transformations, but do not change with rescaling, in particular with a rescaling of mass. The mass is immediately dependent on the conﬁguration of the universe. We start with elementary arguments. We know that the natural unit of mass, the Planck mass ¯hc ≈ 10−5 g , MPlanck = G is a combination that contains the velocity of light, even though the notion of mass was born in Newtonian mechanics and does not depend on the special-relativistic geometry of space-time. If, instead, we construct the natural unit of mass as a combination of the gravitational constant, the Planck constant and the cosmological constant2 Λ ≈ Λcrit ≈ 10−52 m−2 , we obtain the Eddington mass [31, 36] 4 6 h Λ ≈ 4 × 10−25 g . MEddington = G2 This value is an astonishingly good approximation to the mass scale of nucleons. Are we allowed to interpret this result as an indication that the size of the universe represented by Λ is reﬂected in the masses of the stable particles, which are nearly zero in comparison with the characteristic masses of uniﬁed quantum ﬁeld theories (mp ≈ 10−15 MGUT ≈ 10−19 MPlanck )? With a vanishing cosmological constant, the Eddington mass vanishes too. A constructive theory for such a relation, however, does not exist, and it remains an anecdote only. A theory that contains the reduction of an a priori conformal invariance to Lorentz invariance usually requires a scalar ﬁeld that describes the scaling that is obtained through the breakdown of a symmetry. Such a theory is a scalar–tensor theory for the gravitational ﬁeld, in contrast to the GRT, which is a pure tensor theory. The Brans–Dicke theory [10] and the Hoyle–Narlikar theory [13, 14] are the main representatives. 2

The cosmological constant can be determined today by various methods. Our −2 . general argument may use Λcrit = 3RH

13.5 The Assumed and the Explained

281

Irrespective of the observational limits, however, a conformal theory is too short an extrapolation of the GRT; it does not change anything in relation to the existence of a causal structure, and needs purely conventional calculations only. The next step is to determine not only the local scale but also the very existence of the light cone when a symmetry breakdown is induced by the universe. This may be achieved by a theory that concerns a priori only the aﬃne connection, and assumes the relation between two space-time volumes to be the simplest dynamical element [9, 18, 21, 22, 30]. In such a theory, there is no causality beyond the limits of a region that is small on a cosmic scale, and the existence of time and of causal order is a consequence of the existence and symmetry of the universe. The metric ﬁeld does not exist without the matter in the universe that surrounds us.3 Even without a particular theory of this type, characteristic eﬀects can be identiﬁed. If the universe diﬀers from the ideal state that leads to local Lorentz invariance, if it is not ideally isotropic (and our galaxy disturbs the ideal universe with its potential of 10−6 ), ﬁeld components must exist that propagate with velocities slightly diﬀerent from the abstract absolute velocity that characterises Lorentz invariance. We can test for component-dependent signal propagation [1]. The general Euler–Lagrange equations of second order for a set of ﬁelds ΦA , CAB ik ΦB , ik = −

∂2L ∂L ∂L ΦB , i − , g, i + ∂ΦA , i ∂ΦB ∂ΦA , i ∂g ∂ΦA

(13.5)

determine the propagation properties of a signal through the coeﬃcients of the second-order derivatives CAB ik =

∂2L . ∂ΦA ,i ∂ΦB ,k

Let us suppose a situation where the ﬁeld is discontinuous on a wavefront z(x0 , . . . , x3 ) = const , for instance ΦA (z > 0) = ΦA (z < 0) + ϕA z 2 . We obtain, through the ﬁeld equations, the condition CAB ik z,i z,k φB = 0 . 3

‘Es w¨ are meiner Meinung nach unbefriedigend, wenn es eine denkbare Welt ohne Materie g¨ abe. Das g µν -Feld soll vielmehr durch die Materie bedingt sein, ohne dieselbe nicht bestehen k¨ onnen. Das ist der Kern dessen, was ich unter der Forderung von der Relativit¨ at der Tr¨ agheit verstehe.’ (In my opinion it would be dissatisfying, if there were a conceivable world without matter. The g µν -ﬁeld should rather be determined by the matter, and not be able to exist without it. This is the heart of what I understand by the demand of the relativity of inertia.) Einstein writes this in a letter to de Sitter (24.3.1917). However, he then believes that this in an argument for the inclusion of the cosmological constant.

282

13 Machian Aspects This is a linear homogeneous equation for the discontinuity amplitudes φB . Non-trivial solutions exist on hypersurfaces with normals z,k that obey det(CAB ik z,i z,k ) = 0 . In a Lorentz-invariant theory, this condition is simpliﬁed to yield g ik z,i z,k = 0 . The system (13.5) of equations is Lorentz-invariant in the shock-wave approximation if the set of coeﬃcients C factorises into a product of the form CAB ik = aAB g ik . If the Lorentz invariance is weakly disturbed, we expect CAB ik = aAB g ik + AB ik . The determinant condition for the existence of a non-trivial shock amplitude yields, to ﬁrst order in , the equation (g ik z,i z,k )(N −1) (g lm + aAB AB lm )z,l z,m = 0 . To this order, all ﬁeld components except one propagate on the common light cone, but the one exception deﬁnes its own cone. In every subsequent order, a further component leaves the common light cone. The propagation becomes dependent on a kind of generalised isospin (Fig. 13.2).

Only a real theory can decide which ﬁeld components yield the maximal deviation and whether one should observe an anisotropy of the mass (which has been shown for some components to be smaller than 10−24 ). Hence, the task would be to identify the ﬁeld components that show anomalous propagation in lower orders and are observable. The concept of a spontaneous breakdown of symmetry was ﬁrst introduced in the form of the Higgs mechanism (Chap. 7). Can this mechanism be made responsible for the symmetry breakdown considered in this chapter? A theory that uses a Higgs mechanism [11, 25, 26] refrains from using the universe to deﬁne orientation. It remains in this sense local, and it has to explain the longrange coherence of the symmetry-breaking ﬁeld. This ﬁeld then represents the former absolute space in a new form. The concept of ﬁelds that determine the current physical laws with universe-wide coherence leads to the inﬂationary universe, in all its forms. From this point of view, the inﬂationary universe is the anti-Mach universe per se.

References

283

Fig. 13.2. Metaphor of the splitting of the light cone with increasing perturbation

References 1. Audretsch, J., Bleyer, U., L¨ ammerzahl, C.: Testing Lorentz invariance with atomic beam interferometry, Phys. Rev. A 47 (1993), 4632–4640. 281 2. Baierlein, R. F., Sharp, D. H., Wheeler, J. A.: Three-dimensional geometry as carrier of information about time, Phys. Rev. 126 (1962), 1864–1865. 278, 279 3. Barbour, J. B.: Forceless Machian dynamics, Nuovo Cimento B 26 (1975), 16–22. 275 4. Barbour, J. B.: Absolute or Relative Motion? Vol. 1, The Discovery of Dynamics, Cambridge University Press (1990). 274 5. Barbour, J. B.: The End of Time, Weidenfeld & Nicholson, London (1999). 278 6. Barbour, J. B.: The emergence of time and its arrow from timelessness, in J. J. Halliwell, J. P`erez-Mercader, W. H. Zurek (eds.): Physical origins of time asymmetry, Proceedings, Cambridge University Press (1992). 278 7. Barbour, J. B., Bertotti, B.: Gravity and inertia in a Machian framework, Nuovo Cimento B 38 (1977), 1–27. 275 8. Barbour, J. B., Pﬁster, H. (eds.): Mach’s Principle. From Newton’s Bucket to Quantum Gravity, Birkh¨ auser, Boston (1994). 274, 283, 284 9. Bleyer, U., Liebscher, D.-E.: Mach’s principle and local causal structure, in [8], 293-307 (1994). 281

284

13 Machian Aspects

10. Brans, C. H., Dicke, R. H.: Mach’s principle and a relativistic theory of gravitation, Phys. Rev. 124 (1961), 925–935. 280 11. Carlini, A., Greensite, J.: Why is spacetime Lorentzian? Phys. Rev. D 49 (1994), 866–878. 282 12. Ehlers, J.: Machian ideas and general relativity, in [8], 458-473 (1994). 279 13. Hoyle, F., Narlikar, J. V.: Action at a Distance in Physics and Cosmology, Freeman, New York (1974). 280 14. Hoyle, F., Narlikar, J. V.: Mach’s principle and the creation of matter, Proc. R. Soc. London A 273 (1963), 1–11. 280 15. Isenberg, J., Wheeler, J. A.: Inertia here is ﬁxed by mass-energy there in every W model universe, in Pantaleo, M., DeFinis, F. (eds.): Relativity, Quanta and Cosmology in the Development of the Scientiﬁc Thought of Albert Einstein, Johnson Reprint Corp., New York, 267–293 (1979). 279 16. Isenberg, J. A.: Wheeler–Einstein–Mach space-times, Phys. Rev. D 24 (1981), 251–256. 279 17. Kasper, U., Liebscher, D.-E.: On the post-newtonian approximation of theories of gravity, Astron. Nachr. 295 (1974), 11–17. 277 18. Lanczos, C.: Signal propagation in a positive deﬁnite Riemannian space, Phys. Rev. 134 (1964), 475–480. 281 ¨ 19. Lense, J., Thirring, H.: Uber den Einﬂuß der Eigenrotation der Zentralk¨ orper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie, Phys. Z. 19 (1918), 156–163. 274 20. Liebscher, D.-E.: Inertia-free mechanics and the bi-metric procedure, Astron. Nachr. 302 (1981), 137–142. 277 21. Liebscher, D.-E.: Purely aﬃne ﬁeld theories, Ann. Phys. (Leipzig) 45 (1988), 200–204. 281 22. Liebscher, D.-E., Yourgrau, W.: Classical spontaneous breakdown of symmetry and the induction of inertia, Ann. Phys. (Leipzig) 36 (1979), 20–24. 281 23. Mach, E.: Die Mechanik in ihrer Entwicklung, Leipzig (1883). 273 24. Maltsev, V. K., Markov, M. A.: On the integral formulation of the Mach’s principle in a conformally ﬂat space, Trudy FIANa (1976), 9–23. 279 25. Ne’eman, Y., Sijacki, Dj.: Gravity from symmetry breakdown of a gauge aﬃne theory, Phys. Lett. B 200 (1988), 489–494. 282 26. Peldan, P.: Gravity coupled to matter without the metric, Phys. Lett. B 248 (1990), 62–66. 282 27. Pﬁster, H., Braun, K. H.: Induction of correct centrifugal force in a rotating mass shell, Class. Quant. Grav. 2 (1985), 909–918. 274 28. Reinhardt, M.: Mach’s principle – a critical review, Z. Naturforsch. 28a (1973), 529–537. 274 29. Schr¨ odinger, E.: Die Erf¨ ullbarkeit der Relativit¨ atsforderungen der klassischen Mechanik, Ann. Phys. (Leipzig) 77 (1925), 325–336. 275 30. Schr¨ odinger, E.: Space-Time Structure, Cambridge University Press (1950). 281 31. Treder, H.-J.: Elementare Kosmologie, Akademie-Verlag, Berlin (1975). 280 32. Treder, H.-J.: Die Relativit¨ at der Tr¨ agheit, Akademie-Verlag, Berlin (1972). 275 ¨ 33. Treder, H.-J.: Uber die Prinzipien der Dynamik von Einstein, Hertz, Mach und Poincar´e, Akademie-Verlag, Berlin (1974). 274, 275 34. Treder, H.-J.: Philosophische Probleme des physikalischen Raums, AkademieVerlag, Berlin (1974). 274

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35. Yourgrau. W., van der Merwe, A.: Did Ernst Mach ‘miss the target’ ? Synthese 18 (1968), 234–250. 274 36. Zeldovich, Ya. B.: The cosmological constant and the theory of elementary particles, Sov. Phys. Uspekhi 11 (1968), 381–393 (Uspekhi Fiz. Nauk 95 (1968), 209–230). 280

14 Anthropic Aspects

14.1 Cosmological Principle and Cosmological Result Cosmology attempts to understand the observable metagalaxy as a representative part of the universe and as a necessary consequence of the laws that govern physical processes in this universe. How far have we got? Just through the claim that the metagalaxy represents the universe, one characteristic degree of freedom can be isolated – unexpectedly at ﬁrst. This is the expansion parameter. With some success, cosmological models have been constructed that describe appropriately the large-scale behaviour beyond the observed structures and individual celestial objects. The stability considerations of the next step show, however, that in simple models the evolution of the expansion parameter is too unstable to yield a large-scale homogeneous picture. The characteristic indication is the stability of calculations into the past, which seems always to yield a cosmological singularity. It remains to be discussed whether a reference to the necessity for a satisfactory quantum theory of gravity really defuses the problem [5], or whether the cosmological singularity is intertwined with the inﬁnity problems of elementary-particle physics [7, 8]. If we consider only phenomena in the post-quantum universe, it is the reasoning about the inﬂationary stage that appears to be the solution to the problem of explaining the history of the universe through dynamics – irrespective of the initial conditions. The stage of inﬂationary expansion provides an argument for why we observe large-scale homogeneity today, for why an intermediate state of extremely low temperature can be considered as the initial state of the universe in spite of the extremely large temperatures before, and for why we can forget about the details of the time before that. This picture, however, requires us to accept large negative values for the pressure. When large negative pressures are accepted, the evolution of the universe is determined and restricted by the subtleties of microscopic physics. Just because of these restrictions, the state of the universe and the objects in it is a kind of observation and measurement of microphysical quantities in the early universe that may be beyond our reach in earthbound laboratories. Of course, there are also observations that can be compared with more precise microphysical measurements. Such a comparison simply states the Dierck-Ekkehard Liebscher: Cosmology STMP 210, 287–291 (2004) c Springer-Verlag Berlin Heidelberg 2004

288

14 Anthropic Aspects

consistency of the evolution of the universe with our familiar physics. This was the goal of constructing a cosmos. Our success in answering the question of the necessity of evolution into the presently observed state has been paid for by giving up the concept that the metagalaxy is representative of the real universe. From the interplay of quantum gravitation and uniﬁed ﬁeld theories, it is obvious that inﬂation did not start everywhere and simultaneously; that inﬂation did not reach the same order of magnitude everywhere; and that the symmetry breakdown did not everywhere produce the same group (at present SU (3) × SU (2) × U (1)), the same particle masses, or the same coupling constants for the lighter particles (particles with m MGUT ). An attempt to calm ourselves down with the conception that the metagalaxy lies wholly in a quasi-homogeneous region that is suﬃciently inﬂated can only be a cold comfort. In the end this conception admits that far beyond the horizon, equally important regions exist where not only the evolution of cosmic objects has proceeded at diﬀerent rates to diﬀerent states but also the objects of the local physics are made up diﬀerently. Even if one does not doubt the universal validity of quantum theory, the properties of atoms and molecules may depend so sensitively on the values of the elementary masses and coupling constants that the qualitative phenomenology of nature is sensitive to these constants, i.e. that this phenomenology measures the constants. On scales so much larger than the horizon, the universe will be inhomogeneous, not only in terms of small variations of density, peculiar velocity or structure, but in a grossly phenomenological way in the qualitative composition of its bound systems, from atoms to stars. On these scales, in an extreme case, the universe may have even diﬀerent dimensions. What does this mean? If quantum gravitation, inﬂation and symmetry breakdown are the ﬁnal way to solve the problems related to the expanding universe, we might have to abandon the Cusanus principle. Thus the whole chain of conclusions that leads to the homogeneous model, inﬂation and a quantum stage of the universe falls to pieces. This may be a reason to look for another way out, for instance by relating the assumption of homogeneity and the initial-value problem to a quasi-static stage at a density below the Planck density.1

14.2 Life as an Observational Datum Why does the observation that there are observers have such far-reaching consequences? The existence of carbon-based life2 in a late enough stage of 1

2

The Eddington-LeMaˆıtre universe (Sect. 3.5), with interplays by inﬂation, is the simplest background of such a programme. Stanislaw Lem describes, in his Star Diaries, the visit of the protagonist to creatures living in an environment of boiling sulphur who demonstrate to each

14.2 Life as an Observational Datum

289

the evolution of the universe – often called the ‘existence of intelligent life’3 – puts limits on the existence, age and lifetime of main-sequence stars and supernovae, and on the possibility of the production of heavy elements in spite of the bottleneck described in Chap. 6. Even the start of the primordial nuclear synthesis depends sensitively on the masses of the proton and neutron and the diﬀerence between those masses. Deuteron synthesis must be able to start, but not too early, in order to have helium, but not too much helium. The mass of the electron and the Sommerfeld constant determine the structure of atoms. Virtual variations of these constants would make atoms too loose or too rigid for the stability of the many molecules we know. The strength of the strong coupling is just as important. Only in a narrow region of its value can the wealth of chemical elements that are used by the architecture of life coexist. The most important point here is the position of the resonances of carbon and oxygen, as we have seen in Chap. 6. Also, the strength of gravitation decides the age and lifetime of the stars, and in particular the question of the coexistence of a long age with recycling of stellar material that releases heavy elements into free space [9]. If we consider our own existence in the framework of the evolution of the universe, we live beyond doubt in a peculiar stage. The universe had to cool down to allow large-scale structure and to allow stars to form and to explode, before stars such as our sun and planetary systems such as the solar system could form from the debris, with planets that could host life [3]. The universe must not be too old. In the far future, when stellar fuel is exhausted, life will not be able to develop in the familiar form. We can live only in a particular stage of the evolution of the universe and observe it. This is the content of an interpretation that is called the weak anthropic principle [1, 4, 6]: the values of all observed cosmological and physical parameters are bounded by the observation that we exist as observers. In this context, the well-known large Dirac numbers (for instance the 1080 protons within the horizon) can also be included. The point is that the popular misunderstanding of probability makes the observed state of the universe feel so improbable that there should be a strong anthropic principle stating that the universe has to have just those properties that admit the development of life during a certain stage of its history [6]. This is, of course, a concept that cannot possibly be falsiﬁed, so it cannot contribute to scientiﬁc explanation.

3

other that the universe has to admit their life because life in another form would be obviously unthinkable. One could also refer also to The Black Cloud by Fred Hoyle or Out of the sun by Arthur C. Clarke, which discuss this topic, as do many other books. A professor is asked whether he believes that there is intelligent life in the universe. ‘There is an optimistic and a pessimistic answer,’ he states. ‘The optimistic answer is ‘Yes, have a look at me’, the pessimistic is ‘Why should we expect intelligent life in the universe? We would be happy to ﬁnd some on earth.’’

290

14 Anthropic Aspects

The basic question all the time is, given that the state of the universe is what is observed, with life and observers and all that, has this happened by chance (more precisely, is this due to particular initial conditions for the evolution of the universe), or is it a necessary consequence of the evolution of any cosmological model (see Chap. 1). The inﬂationary models of the universe propose that the observable metagalaxy developed from a far smaller part of the universe than one would expect if one were to follow a simple Einstein–Gamow model into the past.4 For a long time, proponents of inﬂation insisted that inﬂation implies that the density of matter in the universe has exactly the critical value that can be calculated from the observed recession velocity of galaxies. They know better now, and try to cover up their mistake by christening the cosmological constant dark energy. The inﬂationary models provide arguments for how the observed large-scale structure has evolved from zero-point quantum ﬂuctuations. They also claim that the assumption of homogeneity and isotropy is a viable approximation only because the metagalaxy evolved from an extremely small part of the pre-inﬂationary universe, while the universe could be rather inhomogeneous again on scales far beyond the horizon. We do not get information from these regions, of course. The present values of the elementary masses and coupling constants therefore obtain a truly accidental character. One is tempted to construct models where these values are state variables and evolve with the universe [10, 11]. One may compare the weak anthropic principle with an aspect of quantum theory. When a measurement is interpreted in quantum theory, one has to take the conditions that the measuring apparatus imposes into account. In our case, the imposed condition is our own existence. However, humanity is not a measuring apparatus in the classical sense but – in this context – is again subject to observation. The consistency of processes in nature implies that our existence cannot contradict the cosmological parameters. We are an observational statement; the amazing fact is that this qualitative observation leads to so many quantitative conclusions. The weak anthropic principle is of methodical use; we should not forget, however, that there are many observational questions that are irrelevant to life. If we accept the concept that the universe might be structured in a rather complicated way and that it is not necessarily homogeneous and isotropic on a large scale in order to be so in the metagalaxy, the conditions for the evolution of life need to be met only in the latter small part. If the overall state is homogeneously chaotic, there are again so many diﬀerent paths of evolution on the small (i.e. metagalaxy-sized) scale that our metagalaxy is not a preferred one. All other evolutionary scenarios exist with the same 4

At T = TPlanck , the Einstein–Gamow model expects R[TPlanck ] ≈ 10−5 m. In an inﬂationary model, this temperature is attained only before inﬂation. Therefore we have to take inﬂation into account, and this yields R[TPlanck ] lPlanck . For a model with positive spatial curvature, the temperature TPlanck may never be reached [2].

References

291

right, but they exist far beyond our horizon. From this point of view, the strong anthropic principle loses its content totally.

References 1. Barrow, J. D., Tipler, F. J.: The Anthropic Cosmological Principle, Oxford University Press (1986). 289 2. Blome, H. J., Priester, W.: Big bounce in the very early universe, Astron. Astrophys. 250 (1991), 43–49. 290 3. Baez, C.: Is life improbable? Found. Phys. 19 (1989), 91–95. 289 4. Balashov, Yu. V.: Multifaced anthropic principle, Comments on Astrophys. 15 (1990), 19–28. 289 5. Bondi, H.: Cosmology, 2nd edn., Cambridge University Press (1960). 287 6. Carter, B.: Large number coincidences and the anthropic principle in cosmology, in M. S. Longair (ed.): Confrontation of Cosmological Theories with Observational Data, Proceedings, IAU Symposium No. 63, Reidel, Dordrecht (1974). 289 7. Einstein, A.: Spielen Gravitationsfelder im Aufbau der materiellen Elementarteilchen eine wesentliche Rolle? SBer. Preuss. Akad. Wiss. (1919), 349–356. 287 8. Einstein, A.: Bietet die Feldtheorie M¨ oglichkeiten f¨ ur die L¨ osung des Quantenproblems? SBer. Preuss. Akad. Wiss. (1923), 359–364. 287 9. Rosental, I. L.: Big Bang – Big Bounce: How Particles and Fields Drive Cosmic Evolution, Springer, Heidelberg (1988). 289 10. Rothman, T., Ellis, G. F. R.: Smolin’s natural selection hypothesis, Q. J. R. Astron. Soc. 34 (1993), 201–212. 290 11. Smolin, L.: Did the universe evolve? Class. Quant. Grav. 9 (1992), 173–191. 290

Index

inﬂation 167 inhomogeneities

aberration 26 absorber statistics 113 anthropic principle 289

Jeans barotropic matter 67 baryon asymmetry 161 breakdown of symmetry 170 canonical GRT 266 cosmological principle 5, 287 cosmological singularity 74 cosmos 1 curvature 46 decoupling 223 distance 88 domain walls 255 Einstein 11 Einstein’s equations 43 equivalence principle 34 ether 127 explanation 3 ﬂuctuations 182 fractal structure 137 Fresnel 27 Friedmann 12 Friedmann’s equations Galileo 26 Gamow 16, 143 gravitational lenses higher dimensions horizon 81 Hubble 12

53

104 243

182, 199

18, 224

kinetic equation

146

Lambert 10 Lemaˆıtre 19 light cone 80, 108, 177 Mach 273 metagalaxy 1 microwave background minisuperspace 271 monopoles 260 neutrinos 162 Newton 8, 215 non-linear evolution Olbers

125

233

9

peculiar velocities 231 perturbations 132, 189 power spectra 131 primordial synthesis 154 proper time 32 quantum cosmology 265 quasi-particles 253 recombination redshift 83

125, 205

standard model 70, 176 strings 250 structure formation 189

294

Index

tensor calculus textures 260 topology 253 universe

1

38

vacuum

168

virial theorem

227

Wheeler-deWitt equation

269

Cosmology (Springer Tracts in Modern Physics)

Neutral Kaons (Springer Tracts in Modern Physics)

Semiconductor Junctions (Springer Tracts in Modern Physics)

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Semiconductor Cavity Quantum Electrodynamics (Springer Tracts in Modern Physics)

High Power Laser-Matter Interaction (Springer Tracts in Modern Physics)

Jets of Hadrons (Springer Tracts in Modern Physics 90)

Handbook of Feynman Path Integrals (Springer Tracts in Modern Physics)

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Neutral Kaons (Springer Tracts in Modern Physics 153)

Coverings of Discrete Quasiperiodic Sets (Springer Tracts in Modern Physics)

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Flavor Physics and the TeV Scale (Springer Tracts in Modern Physics, 233)

Flavor Physics and the TeV Scale (Springer Tracts in Modern Physics, 233)

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Advances in Modern Cosmology

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Precision Electroweak Physics at Electron-Positron Colliders (Springer Tracts in Modern Physics)

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Modern Cosmology

Cosmology (Springer Tracts in Modern Physics)

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Heavy Quark Effective Theory (Springer Tracts in Modern Physics)

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Heavy Quark Effective Theory (Springer Tracts in Modern Physics)

Semiconductor Cavity Quantum Electrodynamics (Springer Tracts in Modern Physics)

High Power Laser-Matter Interaction (Springer Tracts in Modern Physics)

Jets of Hadrons (Springer Tracts in Modern Physics 90)

Handbook of Feynman Path Integrals (Springer Tracts in Modern Physics)

High Power Laser-Matter Interaction (Springer Tracts in Modern Physics)

Neutral Kaons (Springer Tracts in Modern Physics 153)

Coverings of Discrete Quasiperiodic Sets (Springer Tracts in Modern Physics)

Modern Cosmology (Series in High Energy Physics, Cosmology and Gravitation)

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Flavor Physics and the TeV Scale (Springer Tracts in Modern Physics, 233)

High Energy Polarized Proton Beams: A Modern View (Springer Tracts in Modern Physics)

Advances in Modern Cosmology

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Modern cosmology

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Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)

Capillary Surfaces: Shape - Stability - Dynamics, in Particular Under Weightlessness (Springer Tracts in Modern Physics)

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