NEW WORLDS IN >
H
Proceedings otthe Fourth International Workshop editors
Alexander Krasnitz Ana M. Mourao Mario Pimenta Robertus Potting
World Scientific
NEW WORLDS IN
n°
C J 1-3
n Proceedings of the Fourth International Workshop
This page is intentionally left blank
NEW WORLDS IN
)—I
n m Proceedings of the Fourth International Workshop
editors
Alexander Krasnitz CENTRA & Universidade do Algarve, Faro, Portugal
Ana M. Mourao CENTRA & Instituto Superior Tecnico, Lisbon, Portugal
Mario Pimenta UP & Instituto Superior Tecnico, Lisbon, Portugal
Robertus Potting CENTRA & Universidade do Algarve, Faro, Portugal
V f e World Scientific wB
New Jersey • London • Singapore Si. • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Cover design is based on graphics material from the New Worlds in Astroparticle Physics—2002 poster by Dimensao 6.
NEW WORLDS IN ASTROPARTICLE PHYSICS Proceedings of the Fourth International Workshop Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-238-584-3
Printed in Singapore by World Scientific Printers (S) Pte Ltd
This page is intentionally left blank
PREFACE
The Fourth International Workshop on New Worlds in Astroparticle Physics took place from the 5th through the 7th of September 2002 on the Campus of Gambelas, at the University of the Algarve, near Faro. For three days we had talks and discussions, with presentation of recent results on distant supernovae and cosmological background radiation, of developments in new projects on gravitational wave searches and extremely high energy cosmic ray detection based on the International Space Station, and of theoretical insights into extremely dense matter and the possibility of recreating Big Bang at the CERN Large Hadron Collider. Astroparticle physics is growing and nourishing - even if we still do not know where the dark mass and the dark energy come from... The symposium was organised by the University of the Algarve, Instituto Superior Tecnico, CENTRA (Multidisciplinary Center for Astro physics) and CFIF (Center for Physics of Fundamental Interactions). Fi nancial support from FCT (Foundation for Science and Technology) under the Programa Operacional Ciencia, Tecnologia, Inovagao do Quadro Comunitario de Apoio II, from FLAD (Portuguese American Foundation) and from Gulbenkian Foundation is gratefully acknowledged.
Jorge Dias de Deus
VII
This page is intentionally left blank
CONTENTS Preface PART 1
vii OVERVIEWS IN ASTROPARTICLE PHYSICS
Dense Matter N. K. Glendenning
3
Extreme Energy Cosmic Rays and Fundamental Physics H. J. De Vega
20
Gravitational Waves: Probing the Extremes of Physics N. Andersson
28
Cosmological Parameters: Fashion and Facts A. Blanchard
50
PART 2
CONTRIBUTIONS
Astroparticle Physics Beyond the Standard Model Physical Implications of Quintessential Brane Cosmologies K. E. Kunze and M. A. Vdzquez-Mozo
75
Dark Radiation and Localization of Gravity on the Brane R. Neves and C. Vaz
82
Astrophysical Tests of Lorentz Symmetry in Electrodynamics M. Mewes
89
Apparent Lorentz Violation through Spacetime-Varying Couplings R. Lehnert
96
The CERN Axion Solar Telescope
103
M. D. Hasinoff for the CAST experiment Matter Under Extreme Conditions Properties of Dense and Cold QCD H. J. De Vega
113
IX
Stability of Quark Matter and Quark Stars M. Fiolhais. M. Malheiro and A. R. Taurines
123
Heavy Quarks or Compactified Extra Dimensions in the Core of Hybrid Stars G. G. Barnafoldi, P. Levai and B. Lukdcs
133
Ratios of Antibaryon/Baryon Yields in Heavy Ion Collisions
143
Yu. M. Shabelski Cosmic Rays EUSO: Basic Parameters M. C. Espirito Santo
151
The Radio Technique 40 Years Later: Where Do We Stand? E. Zas
161
GPS Synchronization in Cosmic Ray Experiments P. Assis
171
Results from the AMS01 1998 Shuttle Flight M. Steuer
178
Electric Charge Reconstruction with RICH Detector of the AMS Experiment L. Arruda. F. Barao, J. Borges, F. Carmo, P. Goncalves, A. Keating, M. Pimenta and I. Perez Velocity Reconstruction with the RICH Detector of the AMS Experiment L. Arruda, F. Barao, Joao P. Borges. M. Pimenta and I. Perez
191
202
Neutrino Physics and Astrophysics Results from the Sudbury Neutrino Observatory /. Maneira for the SNO Collaboration
217
Status Report on Borexino A. de Bellefon
227
Testing Neutrino Parameters at Future Accelerators J. C. Romdo
233
XI
Gravitational Waves and Tests of General Relativity Relativistic R-Modes in Slowly Rotating Neutron Stars S. Yoshida and U. Lee
245
Collision of Highly Relativistic Particles with Black Holes: The Gravitational Radiation Generated V. Cardoso and J. P. S. Lemos
252
Pair of Accelerated Black Holes in an Anti-de Sitter Background: The AdS C-Metric O. J. Dias and J. P. S. Lemos
260
Timing the PSR J2016+1947 Binary System: Testing the Fundamental Assumption of General Relativity P. C. Freire. J. A. Navarro and S. B. Anderson
270
Supernovae and Dark Matter Supernovae and Dark Energy A. Goobar Current Status of Type IA Supernovae Theory and Their Role in Cosmology S. Blinnikov
281
291
Intensive Supernovae Searches K. Schahmaneche
298
Probing the Dark Matter within the Solar Interior /. P. Lopes
306
List of participants
319
This page is intentionally left blank
PART 1
OVERVIEWS IN ASTROPARTICLE PHYSICS
This page is intentionally left blank
DENSE MATTER
NORMAN K. GLENDENNING Nuclear Science Division and Institute for Nuclear & Particle Astrophysics Lawrence Berkeley National Laboratory Berkeley, California 94720 USA Neutron stars are the densest objects in the universe today in which matter in adiabatic equilibrium can be found. Various high-density phases, both geometric and constitutional are spatially spread out by the pressure gradient in the star. Boundaries between phases slowly move, appear, or disappear as the density profile of the star is changed by the centrifugal force due to spindown caused by the magnetic torque of a pulsar, or the spinup of an x-ray neutron star due to the torque applied by mass accreted from a companion star. Phase transitions in turn produce their own imprint on the spin behavior through changes in the moment of inertia as one phase replaces another, in some cases on single stars, and in others on populations. These are the clues that we elucidate after first reviewing high-density phases.
1. A Brief History • 1054 Chinese astronomer "observed the apparition of a guest star ...its color an iridescent yellow". • 1933 Baade and Zwicky—binding energy of "closely packed neutrons" powers supernova. • 1939 Oppenheimer, Volkoff and Tolman—neutron fermi gas. • 1967 Pacini predicted magnetic dipole radiation. • 1967 Hewish & Bell's serendipitous discovery of neutron stars producing a radio pulse once every revolution from beamed radiation along the magnetic axis which is fixed in the star. They are believed to be the direct product of core collapse a mature massive star and its the subsequent supernova. • 1974 Hulse and Taylor binary neutron star pair in close orbit. • 1984 Bacher's discovery of first Millisecond pulsar. They are believed to be very old supernova products that have been spun up by mass
3
4
accretion from a low-mass companion star. • 1992 Wolszczan & Frail, discovery of 3 planets around a neutron star.
2. Gross Features of Neutron Stars • Surface gravity M/R of Black hole =0.5, Neutron star =0.2, Sun =10~ 6 • Gravitational binding / Nuclear binding ~ 10 • Radius = 10 - 12km, Mass> 1.44MQ • Spin periods from seconds to milliseconds • Neutron stars are degenerate objects (/i «T). • Stars are electrically neutral. (Z^et/A ~ (m/e) 2 < 10 - 3 6 ) • Baryon number and charge are conserved. • Strangeness not conserved (beyond 10~ 10 seconds). • Millisecond pulsars have remarkably stable pulses: P = 1.55780644887275 ± 0.00000000000003 ms (measured for PSR 1937+21 on 29 Nov 1982 at 1903 UT)
120< z> L_ CO CO ■D
100-
Q.
80-
n
r-lI
H—
o l_
i
U
CD
+
60-
CD -Q
E
fT 1
40-
CM
i
+
<
I
♦ W
\
r^ CO
CD
0
^LEI
10'
H
CJ
20_
10
.Mm4 "
10 -1
1
.%, _
1
10
Period in seconds Figure 1. There are two classes of pulsars. The great bulk of known ones are the canonical pulsars with periods centered at about 0.7 seconds. The millisecond pulsars are believed to be an evolutionarily different class. They are harder to detect, and were first discovered in 1982.
5
3. Hyperonization Free neutrons are unstable, but in a star the size and mass of a neutron star, gravitational binding energy is about ten times greater than nuclear binding so that neutrons are a stable component of dense stars. What about protons? The repulsive Coulomb force is so much stronger than the gravitational, that the net electric charge on a star must be very small (Znet/(N + Z) < (m/e) 2 ~ 10~ 36 ). We can say that it is charge neutral. Since mp + me > mn, neutrons are the preferred baryon species. However, being Fermions, with increasing density of neutron matter, the Fermi level of neutrons will exceed the mass of proton and electron at some, not too high a density. Therefore, protons and electrons will also occupy neutron star matter. Because strangeness is conserved only on a weak interaction time-scale, this quantum number is not conserved in an equilibrium state. So with increasing density, the Pauli principle assures us that baryons of many species will be ingredients of dense neutral matter. 1 ' 2 Generally, it suffices to take the baryon octet into account together with electrons and muons. In Figure 2 we see that the A is most strongly popu lated in the center of a typical neutron star if quarks have not become deconfined at those densities. Notice that the lepton populations decrease as the populations of negatively charged hyperons increase. This is in accord with conservation of baryon number in the star. The number of electrons and muons are not by themselves conserved. The equation of state is softened in comparison with a neutron matter equation of state. The softening means that the Fermi pressure is reduced so that hyperon matter cannot support as large a mass against gravitational collapse than would be the case otherwise. The hyperon transition is second order; particle populations vary continuously with density in a uniform medium. However, the densities reached in neutron star cores, 5 to 10 times nuclear matter density, are in all likelihood too high for baryons to exist as separate entities—quarks are likely to become deconfined at lower density than that. This is likely to be a first order phase transition. 4. First Order Transitions in Stars Generally, physicists think of a phase transition such as from water to vapor as being typical of a first order transition. In the real world it is far from typical. Its characteristics are: if heated at constant pressure, the temper ature of water and vapor will rise to 100 C and remain there until all the water has been evaporated before the temperature of the steam rises. This
6
0
1
0.5
0
1.5
2
Baryon density (frrf3)
4
6
8
10
r (km)
Figure 2. Particle populations as a function of baryon density in dense matter, and as a function of radial coordinate in a neutron star.
is true of substances having one independent component (like H 2 0 ) . The situation can, and usually is much more interesting for substances with two or more independent components. Neutron stars are an example. The in dependent components are the conserved baryon and electric charge. Until about 1990, stellar models were forced into the mold of single-component substances by imposing a condition of local charge neutrality and ignor ing the discontinuity in electron chemical potential at the interface of two phases in equilibrium. Let us see how Gibbs criteria can be satisfied in complex systems and what new physics is introduced. 3
4 . 1 . Degrees of freedom and driving forces Two features can come into play in phase transition of complex substances that are absent in simple substances. The degree(s) of freedom can be seen in the following way. Imagine assembling a star in a pure phase (say ordinary nuclear matter) with B baryons and Q electric charges, either positive, negative or zero. (Of course, more precisely, we consider a typical local inertial region.) The concentration is said to be c = Q/B. Now consider another local region deeper in the star and at higher pressure with the same number of baryons and charges, but with conditions such that part of the volume is in the first phase and another part in the second phase. Suppose the baryons and charges in the two phases are distributed such that concentrations in the two phases are Qi/Bi
= c\
and
Q2/B2 = C2 .
7
The conservation laws are still satisfied if Qi+Q2
= Q,
BX+B2
= B.
Why might the concentrations in the two phases be different from each other and from the concentration in the other local volumes at different pressure? Because the degree of freedom of redistributing the concentration may be exploited by internal forces of the substance so as to achieve a lower free energy. In a single-component substance there was no such degree of freedom, and in an n-component substance there are n — 1 degrees of freedom. In deeper regions of the star, still different concentrations may be favored in the two phases in equilibrium at these higher-pressure locations. So you see that the each phase in equilibrium with the other, may have continuously changing properties from one region of the star to another. (This is unlike the simple substance whose properties remain unchanged in each equilibrium phase, until only one phase remains.) The key recognition is that conserved quantities (or independent com ponents) of a substance are conserved globally, but need not be con served locally. Otherwise, Gibbs conditions for phase equilibrium cannot be satisfied.3 Let us see how this is done.
Kaon phase -
Maxwell
5 a
ioo-
Normal phase
500
1000
£ (MeV/fm 3 )
Figure 3. Solid line: equation of state for neutron star matter with a kaon condensed phase. Regions of the normal nuclear matter phase, the mixed phase, and the pure kaon condensed phase are marked. Notice that the pressure changes monotonically through the mixed phase. Dashed line: The Maxwell construction with the typical constant pressure region does not satisfy equality of the electron chemical potential in the two phases.
8
Gibbs condition for phase equilibrium in the case of two conserved quan tities is We have introduced the neutron and electron chemical potentials by which baryon and electric charge conservation are to be enforced. In contrast to the case of a simple substance, for which Gibbs condition— Pi((i,T) = p2(ii,T)—can be solved for fi, the phase equilibrium condition cannot be satisfied for substance of more than one independent compo nent without additional conservation constraints. Clearly, local charge conservation (q(r) = 0) must be abandoned in favor of global conserva tion (J q(r)q(r) = 0), which is after all what is required by physics. For a uniform distribution global neutrality reads, (1 - X)qi (l*n,Ve,T)
+ Xq2(Hn,Ve,T)
where x = V2/V, V = V\ + V?. Given T and x Thus the solutions are of the form IJ-n = Hn(X,T),
= 0,
we can
Ve = Ve(X,T)
solve for fj,nandfie.
.
Because of the dependance on x, we learn that all properties of the phases in equilibrium change with proportion of the phases. This contrasts with simple substances.
U"1—
1
1
1
0
500
1000
1500
e (MeV/fm3)
U.UUI-1—r-T—1—1—1—r—1—1—i—]—i—I—i—i—|
0
0.5
1
1,5
Baryon density (fm" 3 )
Figure 4. Equation of state for mater compositions as shown. H stands for hyperons. Hybrid denotes a low density nuclear phase, high-density quark phase, and intermediate mixed phase. Figure 5. populations as a function of density from charge neutral nuclear matter, mixed nuclear and quark matter to pure quark matter.
9
4.2. Isospin symmetry energy as a driving force A well known feature of nuclear systematics is the valley of beta stability which, aside from the Coulomb repulsion, endows nuclei with N = Z the greatest binding among isotones (N + Z = const). Empirically, the form of the symmetry energy is ^N-sym = -e[(N - Z)/(N + Z)f . Physically, this arises in about equal parts from the difference in energies of neutron and proton Fermi energies and the coupling of the p meson to nucleon isospin current. Consider now a neutron star. While containing many nucleon species, neutron star matter is still very isospin asymmetric— it sits high up from the valley floor of beta stability—and must do so because of the asymmetry imposed by the strength of the Coulomb force compared to the gravitational. Let us examine sample volumes of matter at ever-deeper depth in a star until we arrive at a local inertial volume where the pressure is high enough that some of the quarks have become deconfmed; that both phases are present in the local volume. According to what has been said above, the highly unfavorable isospin of the nuclear phase can lower its repulsive asymmetry energy if some neutrons exchange one of their d quarks with a u quark in the quark phase in equilibrium with it. In this way the nuclear matter will become positively charged and the quark matter will carry a compensating negative charge, and the overall energy will be lowered. The degree to which the exchange will take place will vary according to the proportion of the phases—clearly a region with a small proportion of quark matter cannot as effectively relieve the isospin asymmetry of a large proportion of neutron star matter of its excess isospin as can a volume of the star where the two phases are in more equal proportion. We see this quantitatively in Figure 6 where the charge densities on hadronic and quark matter are shown as a function of proportion of the phases. 4.3. Geometrical phases In equilibrium, the isospin driving force tends to concentrate positive charge on nuclear matter and compensating negative charge on quark matter. The Coulomb force will tend to break up regions of like charge while the sur face interface energy will resist this tendency. The same competition is in play in the crust of the star where ionized atoms sit at lattice sites in an electron sea. For the idealized geometries of spheres, rods, or sheets of the rare phase immersed in the dominant one, and employing the Wigner-Seitz
10
X
r (km)
Figure 6. Charge densities on Hadronic and Quark matter as a function of proportion. Note that overall the mixture is neutral. Figure 7. Diameter (bottom) and spacing (top) of geometrical phases as a function of position r in the star of indicated mass, (see also Figure 8)
approximation (in which each cell has zero total charge, and does not in teract with other cells), closed form solutions exist for the diameter D, and spacing S of the Coulomb lattice. The Coulomb and surface energy for drops, rods or slabs (d = 3,2,1) have the form: ec = Cd(X)D2,
es =
Sd(x)/D,
where Cd abd Sd are simple algebraic functions of x- The sum is minimized by es — 2ec- Hence, the diameter of the objects at the lattice sites is D =
[Sd(x)/2Cd(X)]1/s,
where their spacing is S — D/x1^ if the hadronic phase is the background or S = D/(\ - x) 1 / / d if the quark phase is background. Figure 7 shows the computed diameter and spacing of the various geometric phases of quark and hadronic matter as a function of radial coordinate in a hybrid neutron star. 4.4. Color-flavor locked quark-matter phase (CFL) Rajagopal and Wilczek have argued that the Fermi surface of the quark deconfined phase is unstable to correlations of quarks of opposite momen tum and unlike flavor and form BCS pairs 4 . They estimate a pairing gap of A ~ 100 MeV. The greatest energy benefit is achieved if the Fermi sur faces of all flavors are equal in radius. This links color and flavor by an
11
E
?
E
i
Figure 8. Pie sections showing geometric phases in two stars of different mass
invariance to simultaneous rotations of color and flavor. The approximate energy density corresponding to the gap is CA-CFL ~ -C(kF
A) 2 ~ 50 • C MeV/fm 3 ,
where C is an unknown constant. This is another "driving force" as spoken of above in addition to the nuclear symmetry energy e s y m . It acts, not to restore isospin symmetry in nuclear matter, but color-flavor symmetry in the quark phase. Alford, Rajagopal, Reddy, and Wilczek have argued that the CFL phase, which is identically charge neutral and has this large pair ing gap may preempt the possibility of phase equilibrium between confined hadronic matter and the quark phase; that any amount of quark matter would go into the charge neutral CFL phase (with equal numbers of u, d and s quarks, irrespective of mass) and that the mixed phase spoken of above would be absent. 5 That the nuclear symmetry driving force would be overcome by the color-flavor locking of the quark phase leaving the de gree of freedom possessed by the two-component system unexploited. The discontinuity of the electron chemical potential in the two phases, hadronic and quark matter would be patched by a spherical interface separating a core of CFL phase in the star from the surrounding hadronic phase. For that conclusion to be true, a rather large surface interface coefficient was chosen by dimensional arguments. However, my opinion is that nature will make a choice of surface inter face properties between hadronic and quark matter such that the degree of freedom of exchanging charge can be exploited by the driving forces (here
12
two in number as discussed below). This is usually the case. Physical sys tems generally have their free energy lowered when a degree of freedom (as spoken of above) becomes available. With two possible phases of quark matter, the uniform uncorrelated one discussed first, and the CFL phase as discussed by Rajagopal and Wilczek, there is now a competition between the CFL pairing and the nuclear symmetry-energy densities, and these energy densities are weighted by the volume proportion \ of quark matter in comparison with hadronic matter in locally inertial regions of the star. That is to say, CCFL and e sym are not directly in competition, but rather they are weighted by the relevant volume proportions. It is not a question of "either, or" but "one, then the other". The magnitude of the nuclear symmetry energy density at a typical phase transition density of p ~ 1/fm3 is eN-Sym = -35[(iV - Z)/(N + Z)}2 MeV/fm 3 . To gain this energy a certain price is exacted from the disturbance of the symmetry of the uniform quark matter phase in equilibrium with it; eQ_ sym . As can be inferred from Figure 6, the price is small compared to the gain. On the other side, the energy gained by the quark matter entering the CFL phase was written above and is offset by the energy not gained by the nuclear matter because the CFL preempts an improvement in its isospin asymmetry. So we need to compare -sym
XeQ-sym
with X e A-CFL - (1 - X) e N-sym •
The behavior of these two lines as a function of proportion of quark phase X in a local volume in the star is as follows:a The first expression for the net gain in energy due to the formation of a mixed phase of nuclear and uniform quark matter monotonically decreases from its maximum value at X = 0 while the second expression, the net energy gain in forming the CFL phase monotonically increases from zero at x = 0. Therefore as a function of x or equivalently depth in the star measured from the depth at which the first quarks become deconfined, nuclear symmetry energy is the dominating driving force, while at some value of x in the range 0 < x < 1 the CFL pairing becomes the dominating driving force. In terms of Figure 8, several of the outermost geometric phases in which a
The behavior of the quantity in square brackets can be viewed in Figure 9.14 of reference [2, 2'nd ed.]
13
quark matter occupies lattice sites in a background of nuclear matter are undisturbed. But the sequence of geometric phases is terminated before the series is complete, and the inner core is entirely in the CFL phase. In summary, when the interior density of a neutron star is sufficiently high as to deconfine quarks, a charge neutral color-flavor locked phase with no electrons will form the inner core. This will be surrounded by one or more shells of mixed phase of quark matter in a uniform phase in phase equilibrium with confined hadronic matter, the two arranged in a Coulomb lattice which differs in dimensionality from one shell to another. As seen in Figure 6, the density of electrons is very low to essentially vanishing, because overall charge neutrality can be achieved more economically among the conserved baryon charge carrying particles. Finally, All this will be surrounded by uniform charge neutral nuclear matter with varying particle composition according to depth (pressure), (cf. Figure 5.)
5. Rotation and Phase Transitions Except for the first few seconds in the life of a neutron star, at which time they radiate the vast bulk of their binding energy in the form of neutrinos, we think of them as rather static objects. However the spin evolution at millisecond periods of rotation brings about centrifugally induced changes in the density profile of the star, and hence also in the thresholds and densities of various hyperons, dense phases such as kaon condensed phase and inevitably quark matter. We shall assume that the central density of the more massive millisecond pulsars—being centrifugally diluted—lies below the critical density for pure quark matter, while the central density of canonical pulsars, like the Crab, and more slowly rotating ones, lies above. We explore the consequences of such assumptions. Because of the different compressibility of low and high-density phases, conversion from one phase to another as the phase boundary slowly moves with changing stellar spin (Fig. 9) results in a considerable redistribution of mass (Fig. 10) and hence change in moment of inertia over time. The time scale is of the order of 107 to 109 yr. The behavior of the moment of inertia while successive shells in the star are changing phase is analogous to the so-called backbending behavior of the moment of inertia of deformed rotating nuclei brought about by a change of phase from one in which the coriolis force breaks nucleon spin pairing to one in which spins are paired. Compare Figs. 11 and 12. Elsewhere we have discussed the possible effect of a phase transition on isolated millisecond radio pulsars. 6 Here we discuss x-ray neutron stars
14
I 10 I 0
r (km)
. ■ 1 5
10
15
r (km)
Figure 9. Radial boundaries at various rotational frequencies separating various phases. The frequencies of two pulsars, the Crab and PSR 1937+21 are marked for reference. Figure 10. Mass profiles as a function of equatorial radius of a star rotating at three different frequencies. At low frequency the star is very dense in its core, having a 4 km central region of highly compressible pure quark matter. Inflections at e m 220 and 950 are the boundaries of the mixed phase.
that have a low-mass non-degenerate companion. Beginning at a late stage in the evolution of the companion it evolves toward its red-giant stage and mass overflows the gravitational barrier between the donor and neutron star. The neutron star is spun up by angular momentum conservation of the accreted matter. The heated surface of the neutron star and its rotation may be detected by emitted x-rays. These are the objects found in observations made on the Rossi X-ray Timing Explorer (RXTE). In either case—neutron star accretors or millisecond pulsars—the radial thresholds of particle types and phase boundaries will move—either out ward or inward—depending on whether the star is being spun up or down. The critical density separating phases moves slowly so that the conversion from one phase to another occurs little by little at the moving boundary. In a rapidly rotating pulsar that is spinning down, the matter density ini tially is centrifugally diluted, but the density rises above the critical phase transition density as the star spins down. Relatively stiff nuclear matter is converted to highly compressible quark matter. The overlaying layer of nuclear matter squeezes the quark matter causing the interior density to rise, while the greater concentration of mass at the center acts further to concentrate the mass of the star. Therefore, its moment of inertia decreases
15
over and above what would occur in an immutable rotating gravitating fluid that is spinning down. If this occurs, the moment of inertia as a function of spin exhibits a backbend as in Figure 11. The opposite evolution of the moment of inertia may occur in x-ray neutron stars that are spinning up when the spin change spans the critical region of phase transition. As a result of the backbend in moment of inertia, an isolated ms pulsar may cease its spindown and actually spin up for a time, even though loosing angular momentum to radiation as was discussed in a previous work.6 An x-ray neutron star with a companion may pause in its accretion driven spinup until quark matter is driven out of the star, after which it will resume spinup. Spinup or spin down occurs very slowly, being controlled by the mass accretion rate or the magnitude of the magnetic dipole field, respectively. So, the spin anomaly that might be produced by a conversion of matter from one phase to another will endure for many millions of years. If it were fleeting it would be unobservable. But enduring for a long epoch— if the phenomenon occurs at all—it has a good chance of being observed. A very interesting work by Spyrou and Stergioulas has recently ap peared in the above connection.7 They perform a more accurate numerical calculation for a rotating relativistic star, as compared to our perturbative solution. They find that the backbend in our particular example occurs very close to, or at the maximum (non-rotating) star, but that it is generic for stars that are conditionally stabilized by their spin. This is possibly the situation for some or eventually all accretors. In fact, we expect a phase transition to leave a permanent imprint on the distribution in spins of x-ray accretors. Because of the increase of moment of inertia during the epoch in which the quark core is driven out of a neutron star as it is spun up by mass accretion, spinup is—during this epoch—hindered. Therefore we expect the population of accretors to be clustered in the spin-range corresponding to the expulsion of the quark phase from the stellar core. Spin clustering is actually observed in the population of x-ray neutron stars in binaries that have been discovered by use of the Rossi X-ray Timing Explorer. 8
6. Calculation The theory and parameters used to describe our model neutron star are pre cisely those used in previous publications. Its initial mass is M = 1.42MQ, close to the mass limit. The confined hadronic phase is described by a generalization of a relativistic nuclear field theory solved at the mean field
16
i6
n ocleatpWf!
-
^-^
120-
H-^time
>
c
V J=14
CD
7B
c
P^i^ e d
V j = 10
2
CO
80-
\l2
*-* 40-
10' yr
0-
1—
1325
13BO
9*
| Pulsar |
1375
n (rad/s)
1400
1425
I 0.04
156&
! 0.08
(hu)
2
1 0.12
—. 0.16
0.2
(MeV 2 )
Figure 11. Development of moment of inertia of a model neutron star as a function of angular velocity. The backbend in this case is similar to what is observed in some rotating nuclei. (Adapted from Ref. 6 .) Figure 12. Backbending in the rotating Er nucleus and an number of others was dis covered in the 1970s.
level in which members of the baryon octet are coupled to scalar, vec tor and vector-isovector mesons. 9 ' 2 The parameters 6 ' 1 0 of the nuclear Lagrangian were chosen so that symmetric nuclear matter has the follow ing properties: binding energy B/A — -16.3 MeV, saturation density p = 0.153 fm - 3 , compression modulus K = 300 MeV, symmetry en ergy coefficient asym = 32.5 MeV, nucleon effective mass at saturation m*at = 0.7m. These together with the ratio of hyperon to nucleon cou plings of the three mesons, xa = 0.6, xw = 0.653 = xp yield the correct A binding in nuclear matter. 10 Quark matter is treated in a version of the MIT bag model with the three light flavor quarks (mu = mj, = 0, ms = 150 MeV) as described. n A value of the bag constant B 1 / 4 = 180 MeV is employed.6 The transition between these two phases of a medium with two independent conserved charges (baryon and electric) has been described elsewhere.3 We use a sim ple schematic model of accretion. 12>13>14 All details of our calculation can be found elsewhere. 15 ' 16 ' 17 7. Results The spin evolution of accreting neutron stars as determined by the changing moment of inertia and the evolution equation 15 is shown in Fig. 13. We assume that up to O.4M0 is accreted. Otherwise the maximum frequency
17
attained is less than shown. Three average accretion rates are assumed, M_io = 1, 10 and 100 (where M_ 1 0 is in units of l O _ l o M 0 / y ) .
1000
200 10°
10'
10
10"
time (years)
10'"
400 600 v (s-1)
800
1000
Figure 13. Evolution of spin frequencies of accreting neutron stars with (solid curves) and without (dashed curves) quark deconfinement if QAMQ is accreted. The spin plateau around 200 Hz signals the ongoing process of quark confinement in the stellar centers. Spin equilibrium is eventually reached. (From Ref. 1 5 .) Figure 14. Calculated spin distribution of the underlying population of x-ray neutron stars for one accretion rate (open histogram) is normalized to the number of observed objects (18) at the peak. Data on neutron stars in low-mass X-ray binaries (shaded histogram) is from Ref. 8 . The spike in the calculated distribution corresponds to the spinout of the quark matter phase. Otherwise the spike would be absent. (From Ref. 15
0
We compute a frequency distribution of x-ray stars in low-mass binaries (LMXBs) from Fig. 13, for one accretion rate, by assuming that neutron stars begin their accretion evolution at the average rate of one per million years. A different rate will only shift some neutron stars from one bin to an adjacent one. The donor masses in the binaries are believed to range between 0.1 and 0.4M© and we assume a uniform distribution in this range and repeat the calculation shown in Fig. 13 at intervals of 0.1M Q . The resulting frequency distribution of x-ray neutron stars is shown in Fig. 14; it is striking. A spike in the distribution signals spinout of the quark matter core as the neutron star spins up. This feature would be absent if there were no phase transition in our model of the neutron star.
18
8. Discussion The observed frequency clustering of x-ray neutron stars is about 100 Hz higher than what we calculate. This discrepancy should not be surprising in view of our ignorance of the equation of state above saturation density of nuclear matter and the fact that we employed a stellar model that was previously used, without change. Spyrou and Stergioulas are currently studying a selection of models to learn whether the position of the clustering of spin frequencies can be used to discriminate among them. What is clear is that however crude any model of hadronic matter may be, the physics underlying the effect of a phase transition on spin rate is robust, although not inevitable. We have cited an analogous phenomenon discovered in rotating nuclei. 18 ' 19 - 20 The data in Fig. 14 is gathered from Tables 2-4 of the review article of van der Klis concerning discoveries made with the Rossi X-ray Timing Explorer. 8
9. Conclusion The apparent clustering in rotation frequency of accreting x-ray neutron stars in low-mass binaries may be caused by the progressive conversion of quark matter in the core to confined hadronic matter, paced by the slow spinup due to mass accretion. When conversion is completed, normal accre tion driven spinup resumes. To distinguish this conjecture from others, one would have to discover the inverse phenomenon—a spin anomaly near the same frequency in an isolated ms pulsar. 6 If such a discovery were made, and the apparent clustering of x-ray accretors is confirmed, we would have some degree of confidence in the hypothesis that a dense matter phase, most plausibly quark matter, exists from birth in the cores of canonical neutron stars, is spun out if the star has a companion from which it ac cretes matter, and later, having consumed its companion, resumes life as a millisecond radio pulsar and spins down.
Acknowledgments This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S. Department of Energy under Contract DE-AC03-76SF00098.
19
References 1. N. K. Glendenning, Phys. Lett. 114B (1982) 392; Astrophys. J. 293 (1985) 470. 2. N. K. Glendenning, Compact Stars (Springer-Verlag New York, l'st ed. 1996, 2'nd ed. 2000). 3. N. K. Glendenning, Phys. Rev. D 46 (1992) 1274. 4. K. Rajagopal and F. Wilczek, Phys.Rev.Lett. 86 (2001) 3492. 5. M. Alford, K. Rajagopal, S. Reddy, and F. Wilczek, Phys.Rev. D 64 (2001) 074017. 6. N. K. Glendenning, S. Pei and F. Weber, Phys. Rev. Lett. 79 (1997) 1603. 7. N.K. Spyrou and N. Sterigoulas, Astron. & Astrophys., 395 (2002) 151. 8. M. van der Klis, Ann. Rev. Astron. Astrophys, 38 717 (2000). 9. N. K. Glendenning, Astrophys. J. 293 (1985) 470. 10. N. K. Glendenning and S. A. Moszkowski, Phys. Rev. Lett. 67 (1991) 2414. 11. E. Farhi and R. L. Jaffe, Phys. Rev. D 30 (1984) 2379. 12. R. F. Eisner and F. K. Lamb, Astrophys. J. 215 (1977) 897. 13. P. Ghosh, F. K. Lamb and C. J. Pethick, Astrophys. J. 217 (1977) 578. 14. V. M. Lupinov, Astrophysics of Neutron Stars, (Springer-Verlag, New York, 1992. 15. N. K. Glendenning and F. Weber, Astrophys. J. Lett 559 (2001) L119. 16. N. K. Glendenning and F. Weber, astro-ph/0010336 (2000). 17. N. K. Glendenning and F. Weber, Signal of quark deconfinement in millisec ond pulsars and reconGnement in accreting x-ray neutron stars, in Physics of Neutron Star Interiors, Ed. by Blaschke, Glendenning and Sedrakian (Springer-Verlag, Lecture Notes Series, 2001). 18. B. R. Mottelson and J. G. Valatin, Phys. Rev. Lett. 5 (1960) 511. 19. A. Johnson, H. Ryde and S. A. Hjorth, Nucl. Phys. A179 (1972) 753. 20. F. S. Stephens and R. S. Simon, Nucl. Phys. A183 (1972) 257.
E X T R E M E E N E R G Y COSMIC RAYS AND FUNDAMENTAL PHYSICS
H. J. DE VEGA LPTHE, Universite Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), UMR 7589 CNRS, Tour 16, ler. etage, 4, Place Jussieu, 75252 Paris, Cedex 05, France This is an overview on the physics behind cosmic rays and extreme energy cos mic rays. We discuss the acceleration mechanisms of cosmic rays, their possible astrophysical sources and the main open physical problems and difficulties. The top-down and bottom-up scenarii are contrasted.
Cosmic rays are one of the rare systems in the Universe which are not thermalized. Their energy spectrum follows approximately a power law over at least thirteen orders of magnitude [see fig. 1]. The understanding of the cosmic ray spectrum involves several branches of physics and astronomy. First, to explain how cosmic rays get energies up to 1020eV according to observations 1 ' 2 ' 3 and with such power spectrum. Then, to study the effect of galactic and extragalactic magnetic fields and explain the knee and the ankle effects. Furthermore, reconciliate the GZK effect 4 with the current observations of cosmic rays events beyond 1020eV. Last and not least to understand the interaction of the cosmic rays with the atmosphere, the extended air shower formation, especially at extreme energies and the fluorescence effects. At the same time, one has to identify the astronomical sources of cosmic rays and relate the observed events, composition and spectra with the properties and structure of the sources. The standard physical acceleration mechanism goes back to the ideas proposed by E. Fermi in the fifties. That is, charged particles can be effi ciently accelerated by electric fields in astrophysical shock waves6'7. This is the so-called diffusive shock acceleration mechanism yielding a power spectrum with n(E) ~ E~a ,
20
(1)
21
with 2.3 < a < 2.5 - 2.7. Such spectrum is well verified over more than ten orders of magnitude in energy. In short, electric fields accelerate charged particles in the wavefront of the shock. Then, magnetic fields deviate and diffuse the particles. Charged particles take energy from the wavefront of macroscopic (astronomic!) size. Particles trapped for long enough can acquire gigantic energies. This mechanism can accelerate particles till arbitrary high energies. The upper limit in energy is given by the time while the particle stays in the wavefront which depends on the size of the source and on the magnetic field strength. Particles gain energy by bouncing off hydromagnetic disturbances near the shock wave. Particles are both in the downstream and the upstream flow regions. If a particle in the upstream crosses the shock-front, reaches the downstream region and then crosses back to the upstream region This cross boosts its energy by a factor proportional to the Lorentz factor of the shock squared ~ T 2 6 . Such factor could be very large. Indeed, the larger is the acceleration energy the less probable is the process yielding a power spectrum of the type of eq.(l). Accelerated particles are focalized inside a cone of angle 9 < p. To rotate the particle momentump this angle takes a time At ~ rg 9 = Ze^ r where rg = zfB stand for the relativistic gyration radius of the particle in the magnetic field. It must be At < Rs where Rs is the radius of the spherical wave. Otherwise the particle is gone of the shock front. Therefore, E < Z e B± T. That is, one finds for the maximal available acceleration energy 6 ' 7 for a particle of charge Z e, Emax = ZeB±T
(2)
There are other estimates but all have the same structure. The maximal energy is proportional to the the magnetic field strength, to the particle charge, to Rs which is of the order of the size of the source and to a big numerical factor as T. Further important effects are the radiation losses from the accelerated charged particles and the back-reaction of the particles on the plasma. That is, the non-linear effects on the shock wave. Particle acceleration in shock-waves can be described at different levels. The simplest one is the test particle description where the propagation of charged particles in shock-waves is studied. A better description is obtained with transport equations. In addition, non-linear effects can be introduced in such a Fokker-Planck treatment 6 ' 7 . The distribution function for the particles in the plasma f(x,p, t) obeys
22
Figure 1. Cosmic ray spectrum[5].
the Fokker-Planck equation, ^
+ tf-V/ + | ^ d i v t f - d i v ( K V / ) = Q .
(3)
Here, u stands for the velocity field, the third term describes the adiabatic compression and it follows from the collision terms in the transport equation for small momentum transfer, K describes the spatial diffusion and Q is a injection or source term. The energy spectrum follows from this equation irrespective of the de tails of the diffusion. It must be recalled that the coefficients in this Fokker-
23
Planck equation are only sketchily known for relevant astrophysical plas mas. A microscopic derivation of the Fokker-Planck equation including reliable computation of its coefficients will be important to understand the acceleration of extreme energy cosmic rays (EECR). Let us consider stationary solutions of the Fokker-Planck equation (3) for a simple one dimensional geometry. Let us consider a step function as velocity field6. That is, v(x) = vi upstream, for x > 0 and v(x) = vi downstream, for x < 0 (4)
Eq.(3) then takes the form,
, , df d n(x,p) 9f] dx Yx = Yx
for x ^ 0 .
V{X)
Integrating upon x taking into account eq.(4) yields,
df v(x) f(x,p) = n(x,p) -^ + A(p) , where A(p) is an integration constant. Integrating again upon x gives the solution h(p) + 9i(p) e~Vl
" (I ' ,P> ,upstream, x > 0
f(x,P) = {
(5) f2(p) , downstream, x < 0 .
Matching the solutions at the shock-wave front at x = 0 yields, (r-l)p^=3r(/i-/
2
)
,
Op
andrEE^.
(6)
V2
Notice that the solution is independent of the diffusion coefficient Eq.(6) has the homogeneous solution (no incoming particles), f2(p)=Ap-a
,
a=
n(x,p).
3r r-1
For ultrarelativistic shock-waves we have r = 3 and a = 9/2 1 3 . Therefore, the cosmic ray (CR) energy flux follows the law n(E)=p2f2(p)=Apl, in close agreement with the observations 1. It must be stressed that this example is quite oversimplified. The ge ometry is not one-dimensional in astrophysical plasmas, the velocity field is not uniform, neither constant. However, the results obtained show the robustness of the approach 6,7 .
24
The particles back-reaction on the shock-wave can be neglected pro vided: (a) the shock thickness is much smaller than the CR mean free path and (b) the CR energy is much smaller than the shock energy. Otherwise, one has to take into account the back-reaction by coupling the transport equation for the CR with the hydrodynamic equations for the plasma in cluding the CR pressure. The parameter that measures this non-linear effect is the so called injection parameter: v = UCRI'Ni(upstream). For v < vc ~ 0.001 the linear theory can be used. For v > vc intrinsically nonlinear phenomena can show up as solitons and perhaps self-organized criticality. The non-linear effects predict a maximum CR energy and a hardening of the energy spectrum near this maximum 6 . Let us now discuss the possible astrophysical sources of EECR. Accord ing to eq.(2) sources of large size and/or large magnetic fields are needed. For big sources natural candidates are active galactic nuclei (AGN). These are supermassive black holes (mass ~ 106 — 109 solar masses) in the core of quasars. They are powered by the matter accreting onto the core black hole. It has been found from X and 7 ray detection that they are Kerr black holes near their critical limit. [They should have formed by rotat ing collapsing matter. Their angular momentum is large just because it is conserved during collapse. The excess should be radiated by gravitational radiation and what remains is a critical Kerr hole]. They usually exhibit powerful jets identified through radio emission where shock-waves can accel erate CR. For example, in radio lobes of radio galaxies 6 ' 9 . The interaction regions may extend as much as a Mpc. The jet energy is dissipated into a bow shock inside the inter galactic medium and in shocks within the jet plasma itself. The characteristic size of the shock is about Rs ~ lOkpc and B ~ 10 - lOO^uG which yields7 E m a x ~ 1021eV. Blazars (quasars, with their jets pointing toward us) are good candi dates like BL Lacertae. They have lower matter density that facilitates propagation of the CR. There are a few known radio jets inside the GZK sphere. We have Centaurus A at about 3 Mpc away and M87 at 18 Mpc from us. Other highly luminous sources as Cygnus A are too far ~ 200Mpc. In order to explain the apparently isotropic distribution of detected events one needs an abundant and uniform distribution of sources or intergalactic magnetic fields able to isotropize the CR flux. Extragalactic neutron stars, in particular magnetars (fastly rotating young neutron stars with B ~ 10 15 Gauss and Q, ~ 104 s _ 1 ) may be small size sources of EECR. Galactic neutron stars would give a CR spectrum
25
concentrated in the galactic plane unless strong galactic magnetic fields could isotropize the CR flux7'10. Sources of gamma ray bursts (GRB) may also accelerate CR since they have strong ultrarelativistic shock-waves. However, the extragalactic dis tribution of these sources is well beyond the GZK sphere. It is difficult to have the same origin for EECR and nearby originated GRB.
Figure 2. Diagram illustrating current ideas concerning microquasars, quasars and gamma-ray burst sources (not to scale) extracted from ref.[13]. It is proposed in ref.[13] (see also ref.[14]) that a universal mechanism may be at work in all sources of relativistic jets in the universe.
The present data from HiRes and AGASA seem incompatible above 1020eV where AGASA has eight events beyond the GZK bound. Besides these AGASA events the available EECR spectrum today seems compatible with the GZK effect showing up as predicted 2,3 ' 8 . It is the task of the forthcoming experiments like Auger and EUSO to clarify the situation and show whether the GZK effect is present or not. In particular, the measurement of the CR spectrum at GZK energies will provide a strong check of Lorentz invariance (or the discovery of its violation) since the Lorentz factor involved is extremely large 8 ~ 10 11 . Furthermore, Auger and EUSO should be able to see or reject the pileup effect. The pileup effect is an enhancement of the CR spectrum just below
26
the GZK cutoff due to CR starting out at higher energies and crowding up at or below the GZK energy8 Let us make a short comment about the so-called top-down scenarios. There, 11 ' 12 EECR are assumed to be originated from unstable heavy relics from the early universe or from topological defects also originated in the early universe. In the first case, both the relic mass and its lifetime must be fine tuned. The relic mass to the observed EECR energies and the lifetime to the age of universe. Indeed, a manifold of models have been built to fulfil such requirements. As stressed in 12 , the top-down scenarios are just tailored to explain the observed events. There is absolutely no physical reason to assume that relics have such a mass (and not any other value) and such a lifetime. The natural lifetimes of such heavy objects are microscopic times associated to the GUT energy scale (~ 10~28sec. or shorter). It is at this energy scale (by the end of inflation) where they could have been abundantly formed in the early universe and it seems natural that they decayed shortly after being formed. The second type of models rely on the existence if a network of topo logical defects formed during phase transitions in the early universe. Such topological defects should survive till nowadays to produce the observed EECR. In case they decay in the early universe we go back to the previous case. It must be first noted that only some grand unified field theories support topological defects12. Moreover, recent CMB anisotropy measure ments from Boomerang, Maxima, Dasi and Archeops have seen no evidence of topological defects strongly disfavoring their eventual presence in the present universe. A wonderfully interesting physics is involved in EECR. (i) Non-linear phenomena in relativistic plasmas, shock waves and tur bulence in astrophysical fluids. (ii) Particle interactions in the standard model of particle physics and beyond (10 19 eV= 10 10 GeV). Microscopic derivation of the trans port equations computing their coefficients. Energy losses compu tation. (iii) Identification of the astrophysical sources. Big sources with small magnetic fields (as AGN's) or small compact sources with big mag netic fields (neutron stars)? Structure and properties of the sources should be understood from the observed EECR.
27 In summary, t h e standard model of cosmic ray acceleration (diffusive shock acceleration) based in Fermi ideas explains the non-thermal power energy spectrum of CR over at least thirteen orders of magnitude. It is r e a s o n a b l e to extend such spectrum to E E C R and this seems p l a u s i ble. However, stimulating physical and astronomical problems remain to understand and explain the CR spectrum well below extreme energies!
References 1. J. Linsley, Phys. Rev. Lett. 10, 146 (1963). M. A. Lawrence et al. J. Phys. G 17, 773 (1991). D. J. Bird et al. Ap J, 441, 144 (1995). D. Kieda et al. Proc. 26th ICRC, Salt Lake City, Utah, USA. 2. N. Hayashida et al. astro-ph/0008102 and http://www-akeno.icrr.u-tokyo.ac.Jp/AGASA/results.html#100EeV. 3. T. Abu-Zayyad et al. astro-ph/0208243. 4. K. Greisen, Phys. Rev. Lett. 16, 748 (1966). G. T. Zatsepin and V. A. Kuzmin, JETP Letters, 4, 144 (1966). 5. S. Swordy, The data represent published results of the LEAP, Proton, Akeno, AGASA, Fly's Eye, Haverah Park, and Yakutsk experiments. 6. Y. A. Gallant and A. Achterberg, MNRAS, 305 L6, 1998. M. A. Malkov and P. H. Diamond, astro-ph/0102373. P. L. Biermann and P. Strittmatter, Astropart. Phys. 322, 643 (1987). J. A. Rachen and P. L. Biermann, A&A, 272, 161 (1993). E. V. Berezhko and D. C. Ellison, ApJ, 526, 385 (1999). M. A. Malkov, P. H. Diamond and H. J. Volk, ApJ, 533, L171 (2000). M. Ostrowski, astro-ph/0101053. A. Achterberg, Y. A. Gallant, J. G. Kirk and A. W. Guthmann, astro-ph/0107530. 7. T. W. Jones, astro-ph/020677, Lectures at the 9th. Chalonge School, Palermo September 2002, and references therein. 8. F. Stecker, astro-ph/0208507, Lectures at the 9th. Chalonge School, Palermo September 2002, and references therein. 9. F. Halzen, Lectures at the 9th. Chalonge School, Palermo September 2002. 10. J. Arons, astro-ph/0208444. M. Vietri and L. Stella, ApJ, 527, 443 (1998). 11. See, for example P. Bhattacharjee and G. Sigl, Phys. Rep. 327, 109 (2000) 12. H. J. de Vega and N. Sanchez, hep-ph/0202249. 13. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, vol. 6 of Course in The oretical Physics, Pergamon, Oxford, 1963. 14. I. F. Mirabel, astro-ph/0211085 and references therein. 15. M. A. Abramowicz, W. Kluzniak, J-P. Lasota, astro-ph/0207270 and refer ences therein.
GRAVITATIONAL WAVES: P R O B I N G T H E E X T R E M E S OF PHYSICS
N. ANDERSSON Department of Mathematics University of Southampton Southampton SO 17 1BJ, UK email: N.AnderssonOmaths.soton.ac.
ufe
This article provides a brief overview of gravitational-wave physics. The main focus is on the challenges and opportunities associated with gravitational-wave observa tions of the most extreme inhabitants of the Universe: neutron stars and black holes. The various ways in which these compact objects can radiate gravitationally and the physics that one can hope to probe by observing the associated waves are discussed.
1. Einstein's elusive waves With the formulation of his theories of relativity Einstein prompted a com plete change in our views of space and time. Relativity is now well estab lished and we know that we live in a wonky Universe where time and space may play rather confusing roles. The existence of exotic, highly relativistic, objects like neutron stars and black holes is beyond any serious doubt. Another intriguing consequence of the General theory of Relativity is the existence of gravitational waves. These "ripples in the spacetime curva ture" are generated by the acceleration of matter and travel away from the source at the speed of light. At the time of writing, we are still waiting for the first undisputed direct detection of gravitational radiation. Yet we are confident that our theoreti cal predictions are reasonable and that these elusive waves will be detected in the next few years. Is this confidence justified? Well, we know that grav itational radiation plays a key role in the dynamics of cataclysmic variables and binary neutron star systems. In fact, the observations of the shrinking orbit of the binary pulsar PSR1913+16 (which earned Hulse and Taylor the 1993 Nobel Prize in physics) agree with the predictions of Einstein's theory to within 1%. Furthermore, a new generation of gravitational-wave
28
29
detectors are now beginning to take data. This means that we will — for the first time — have detectors that should be sensitive enough to see gravitational waves from events in distant galaxies. As the detectors are improved over the next decade, our chances of detecting these weak waves will be much improved and with the launch of the space-based interferometer LISA (planned for around 2011) detection will be virtually guaranteed. At a previous Faro meeting Bernard Schutz succinctly described the current situation: "Gravitational radiation is no longer a ghost, but it is not quite concrete either..." This article is an attempt to demystify this ghost. It provides an overview of the very exciting physics that "gravitational-wave astronomy" should be able to probe, and is intended as an introduction to a currently vibrant field of research. In modeling gravitational waves it is often sufficient to work at the level of linear perturbations of the spacetime metric, i.e. use
9n* = 5°„ + K"
C1)
where |/iM„| is suitably small. If we take the background metric g^v to be that of flat space, then Einstein's equations lead to the wave equation □ft M W = 0
(2)
which (obviously) permits outgoing-wave solutions /iM„(£ — x/c). According to General Relativity gravitational waves are transverse and have two possible polarization states a , customarily referred to as "plus" and "cross", and the general wave-form can thus be written h^v = h+(t-
x/c) + hx(t - x/c)
(3)
This nomenclature is inspired by the way that gravitational waves affect a ring of test-particles (see Fig. 1 in Ref. 1). Since gravitational waves act like a tidal force they change all proper distances by the same ratio. If we consider two "free masses" placed a distance L apart, then the gravitationalwave induced strain will lead to a change AL in the proper distance such that
"Alternative theories (like Brans-Dicke theory) may allow also for longitudinal components and/or a weak scalar component of gravity. Hence future observations may be able to constrain the parameters of such theories by studying the gravitational-wave polarization.
30
It is often said that there is a close analogy between electromagnetic waves and gravitational ones. Despite this being useful to a certain extent, one must be careful not to push the analogy too far. A key conceptual difference between the two cases concerns the fact that electromagnetic radiation corresponds to oscillations of electric and magnetic fields propa gating in a given spacetime, while gravitational waves are oscillations of the spacetime itself. This crucial distinction has led to several controver sies concerning the "reality" of gravitational waves, in particular whether gravitational waves carry energy away from the source. This is a delicate point since one cannot localize the radiation energy to regions smaller than a wavelength or so in General Relativity, and the issue was not settled until the 1960s with the work of Bondi and colleagues2. It is easy to appreciate the confusion. After all, it is not trivial to define (in a precise way) what a gravitational wave is. In order to single out a gravitational wave one must identify an oscillating contribution to the spacetime that varies on a lengthscale much shorter than that of the "background" curvature. Other differences between gravitational waves and electromagnetic ones illustrate both the promises and challenges of this research area: — While electromagnetic waves are radiated when individual particles are accelerated, gravitational waves are due to non-spherical bulk motion of matter. In effect, the incoherent electromagnetic radiation generated by many particles carry information about the thermodynamics of the source. Gravitational radiation tell us about the source dynamics. — The electromagnetic waves that reach our detectors will have been scat tered many times since their generation. In contrast, gravitational waves couple weakly to matter and arrive at the Earth in pristine condition. They carry key information about violent processes that remain hidden from our electromagnetic "eyes", eg. supernova core collapse or merging black holes. Of course, the waves also interact very weakly with our detectors, making their observation a serious experimental challenge. — Standard astronomy is based on deep imaging of small fields of view, while gravitational-wave detectors cover virtually the entire sky. A con sequence of this is that their ability to pinpoint a source on the sky is not particularly good. On the other hand, any source on the sky will be detectable, not just sources towards which we aim the detector. — Electromagnetic radiation typically has a wavelength smaller than the size of the emitter, while the wavelength of a gravitational wave is usu ally comparable to or larger than the size of the radiating source. Hence, gravitational waves cannot be used to form a meaningful "image" of the source.
31
Morale: Gravitational waves carry information about the Universe's most violent phenomena. Information that would be very difficult to glean from electromagnetic data. Once the "gravitational-wave window" has been opened we can hope to learn much about the most exotic inhabitants of the skies — especially neutron stars and black holes — and perhaps also test our theories of the very early Universe. These observations should help unveil the "dark side" of the Universe, and we must be prepared for surprises since many strong sources of gravitational radiation may not have been predicted. 2. Gravitational-wave estimates and detection In order to estimate the strength of a gravitational-wave signal, we can use the well-known flux formula which relates the luminosity to the gravitational-wave strain h; c3 16nGW
-2_
1
dE
~ 47T7-2 dt
[b}
where r is the distance to the source. This relation is exact for the weak waves that bathe the Earth. If we characterize a given event by a timescale r and assume that the signal is essentially monochromatic (with frequency / ) , we can use dE/dt « E/T and h s=s 2nfh and readily deduce that
^ 5 x l 0 " 2 2 ( l a w ) 1 / 2 ( i ^ ) " 1 / 2 ( l 4 ) ' ( i s f c ) '(«) Here we have taken the distance to be that of a source in the Virgo cluster. This is necessary to ensure a reasonable event rate for most astrophysical scenarios. At that distance one would expect to see many supernovae per year, which means that one can hope to see a few neutron stars/black holes being born during one year of observation. We can proceed further and estimate an "effective amplitude" that re flects the fact that detailed knowledge of the signal can be used to dig deeper into the detector noise. A typical example is based on the use of matched filtering, for which the effective amplitude improves roughly as the square root of the number of observed cycles n. Using n w fr we arrive at
From this relation we see that the "detector sensitivity" essentially depends only on the radiated energy and the characteristic frequency. Hence, an
32
estimate of the total energy radiated and the frequency of the signal may be sufficient to assess the relevance of a gravitational-wave source. Example: Assume that a supernova collapse in the Virgo Cluster releases 52 10 erg of energy as gravitational waves. If the characteristic frequency of these waves is about 1 kHz we find that the associated gravitational wave amplitude is hc « 10~ 21 . This estimate illustrates the sensitivity required to detect gravitational waves from astrophysical sources. It shows that we need an instrument sensitive enough that it can "resolve the width of a hair at the distance of the nearest star" in order to detect the weak gravitational waves that bathe the Earth. This is an immense experimental challenge, and at first sight it would seem more or less impossible. Yet today's gravitational-wave detectors have almost reached the target sensitivity. How is this possible? A part of the answer is obvious from Eq. (4): The problem is alleviated if we make the detector large. For the interferometric detectors that are now coming into operation L = 3 — 4 km. Furthermore, the optical path length can be made much longer than the physical distance between the mirrors of these interferometers. By allowing the light multiple bounces in each interferometer arm the effective optical path can be increased by perhaps a factor of a hundred. However, even with L m 100 km we need to be able to measure AL s=a 1 0 - 1 6 m, i.e. resolve distances below nuclear dimensions. In fact, the problem is even harder since we must have a signal-to-noise ratio much larger than unity in order to have confidence in any detected signal. In reality we will need to measure something like AL » 10~ 18 m. The key things that allow experimenters to overcome this daunting challenge are 3 : i) A sophisticated servo system and clever use of the light in the interferometer arms which facilitate a comparison of the arm lengths at the level of 1 0 - 1 0 rad, and ii) the fact that in an interferometer beam of size a few centimeters one is dealing with ~ 10 17 atoms in each mirror. This means that the unavoidable thermal oscillations of individual atoms are irrelevant. All that matters is the coherent motion of the mirror surface, so the real limitation is provided by the lowest normal modes of vibration of the mirror. Gravitational-wave detection was pioneered by Weber in the late 1960s. The initial efforts made use of resonant-mass detectors (bars). Since then, this technology has been much improved and today several (narrow band) cryogenic bar detectors, sensitive enough to see unique events from within our Galaxy, are in continuous operation. In parallel, the idea of using laserinterferometry has been developed and after decades of R&D, a generation
33
of large-scale interferometers are now near reaching their target sensitiv ity. The interferometers use laser light to measure changes in the differ ence between the lengths of two (essentially perpendicular) arms. When combined with the bar-detectors, these interferometers — the two LIGO facities in the USA, the French-Italian Virgo detector, and the slightly smaller German-British GEO600 and Japanese TAMA300 interferometers — provide a global network of broad-band detectors sensitive in the range 10 - 103 Hz. The main noise sources that limit the ground based interferometers are 4 : i) unavoidable seismic noise below a few Hz, ii) photon shot noise at high frequencies (arising because the photons are quantized which leads to random fluctuations in the light intensity), iii) thermal noise due to vibrations of the mirrors and the suspension system in the intermediate range near ~ 100 Hz. These features are clear from the noise-curves shown in Fig. 1. These are, of course, not the only noise sources that one need to worry about. Many other kinds of noise, which must be considered for the development of future generations of interferometers, lurk underneath. Gravity-gradient noise prohibits ground based observations below 1 Hz or so. At lower frequencies fluctuations in the local tidal field, which can not be screened, are much larger than the expected gravitational-wave amplitudes. However, this noise falls off as 1/r 3 as one moves away from the Earth. Hence it would seem natural to attempt to place a gravitational-wave detector in space. Indeed, such a project is currently being planned. The space-based interferometer LISA, which will have a baseline of 5 x 106 km, is due to be launched as a joint ESA-NASA mission around 2011. LISA is expected to be sensitive in the range 1 0 - 4 — 0.1 Hz, cf. Fig. 1. From, for example, Eq. (7) we see that the gravitational-wave strain h falls off with distance as 1/r. This means that even modest improvements in detector sensitivity can have a large impact on the achievable science. If we (say) want to increase the number of detectable sources, eg. the volume of the Universe within which such sources can be seen, by one order of magnitude then we need to roughly double the sensitivity of the detector. In other words, while factors of two are often considered irrelevant in standard astrophysics they are precious to the gravitational-wave physicist. Example: The planned upgrades of the LIGO detectors (due to take place around 2007) will improve the sensitivity by roughly a factor of 15, cf. Fig. 1. This would correspond to a factor of 300 in the detectable volume, eg. the number of observable events. Since this is the difference
34
between seeing one event per year and one per day, it represents a serious enhancement of the detectors capability! 3. Overview of gravitational-wave sources Before discussing the detailed nature of various gravitational-wave sources it is useful to estimate what the most important ones may be. We can do this in a relatively straightforward way if we note that the dynamical frequency of any self-bound system with mass M and radius R can be approximated by x
/w
l
G
M
(8)
SFV«r
Given this, it is easy to show that the natural frequency of a (non-rotating) black hole should be
*(£)
/BH«104(^J
HZ
(9)
This immediately shows that medium sized black holes, with masses in the range 10 — 100M Q , will be prime sources for ground-based interferometers, while supermassive black holes with masses ~ 10 6 M Q expected to exist in the cores of many (most?) galaxies should radiate in the LISA bandwidth. We also find that neutron stars, with a canonical mass of 1.4M© compressed inside a radius of 10 km or so, would be expected to radiate at / N S M 2 kHz
(10)
It is also highly relevant to consider binary systems. In that case we can (roughly) take R to represent the separation between the two objects. If we assume that the final plunge sets in at something like R m QM we find that a binary system comprising two 10M© black holes will radiate at frequencies below 1.5 kHz or so. In other words, one would expect such systems to be promising sources for ground based detectors. Meanwhile, supermassive black hole binaries — likely resulting from galaxy mergers — would radiate in the LISA frequency band. In comparison with the ground-based detectors, LISA may suffer an em barrassment of riches. This is natural since the frequency range of the space based interferometer (down to 10~ 4 Hz) is a good match to the timescale of many astronomical systems (hours). There are in fact, many classes of galactic binary systems that will radiate gravitationally in the LISA band and which should lead to detectable signals. Typical such systems are i) bi nary white dwarf systems, ii) binaries comprising an accreting white dwarf
35
and a Helium donor star, iii) low-mass X-ray binaries. In fact, it is antici pated that there are more than 108 galactic binary systems in the relevant frequency range. This could lead to a massive confusion problem 5 , where the signal from individual systems may not be easy to distinguish. One can classify the anticipated gravitational-wave sources by the na ture of the waves. This is useful since it helps us appreciate that different kinds of signals require different signal analysis strategies for their detec tion. Chirps.— As a binary system radiates gravitational waves and loses energy the two constituents spiral closer together. As the separation de creases the gravitational-wave amplitude increases, leading to a character istic "chirp" signal. The second generation of ground based interferometers should be able to detect many extragalactic binaries per year. Bursts.— Many scenarios lead to burst-like gravitational waves. A typ ical example would be black-hole oscillations excited during binary merger. Burst signals are difficult to search for in a noisy data stream, but there may be coincidences with electromagnetic observations. Potentially very exciting would be an association with gamma-ray bursts 6 , for which a ro tating black hole surrounded by an accretion torus is a plausible central engine. Periodic.— Systems where the gravitational-wave backreaction leads to a slow evolution (compared to the observation time) may radiate persistent waves with a virtually constant frequency. This would be the gravitationalwave analogue of the radio pulsars. The associated waves are expected to be weak, but they may nevertheless prove detectable from within the Galaxy if one can integrate the signal for several weeks to months. Stochastic.— A stochastic (non-thermal) background of gravitational waves is expected to have been generated following the Big Bang. These wavesb were created when the Universe was younger than ~ 1 0 - 2 4 s. One may also have to deal with stochastic gravitational-wave signals when the sources are too abundant for us to distinguish them as individuals 5 . This is the expected situation for LISA, where the population of galactic binaries may to some extent limit gravitational-wave observations below 1 0 - 3 Hz.
Despite the fact that an observation of a cosmological background may be the most fundamental physics observation that gravitational-wave detection may facilitate, I will not review this topic in this article. The interested reader is referred to Ref. 4.
36
3.1. Compact
binary inspiral
and
merger
In contrast to the case in Newtonian gravity the two-body problem remains unsolved in General Relativity. Given the lack of suitable exact solutions to the Einstein field equations significant effort has gone into developing various approximations and numerical approaches to the problem. Within the post-Newtonian approximation 2 ' 7 to General Relativity the leading order radiation effects are described by the so-called quadrupole formula. It states that the gravitational-wave strain follows from the second time derivative of the source's quadrupole moment Qjk = f pXjXkdV ->• Q ~ MR2
(11)
in such a way that ^
2GcPQjk M2 = ^ - ^ - ^ ~ 7 ] j
(12)
In the simple estimates we have assumed that we are considering an equal mass binary with separation R. The emission of gravitational waves leads to a shrinking of the orbit of the system. This takes place on a timescale R ichirp ~ - ^ 3
. . (13)
Assuming that this timescale is shorter than the observation time, we can estimate, using (8), the effective amplitude of the binary signal as he « Vftcurp h~ — \jj)
(14)
This result, illustrated in Fig. 1, shows that even though the actual signal gets stronger its detectability decreases as the orbit shrinks. Hence, it is important to have detectors that are sensitive at low frequencies where the binary system spends a lot of time. A neutron star binary which is observed from (say) 10 Hz to coalescence will radiate something like 104 cycles in the process. This means that one can enhance the detectability by roughly a factor of 100 if one can track the signal through the entire evolution. This motivates the development of high order post-Newtonian approximations to the waveforms (especially the phase) as well as detailed signal analysis algorithms. One can readily show that any binary system which is observable from the ground will coalesce within one year. Statistics based on the pulsar population indicates that these events would happen less than once in every
37
105 yrs in our Galaxy. This means that we would need to detect events from a volume of space containing at least 106 galaxies in order to see a few mergers per year. In other words, we need to be able to see coalescing binaries at a distance of at least a few hundred Mpc. Since a black hole system is more massive than one comprising neutron stars it will lead to a stronger gravitational-wave signal. With a second generation interferometer one can expect to see neutrons stars spiraling into I O M Q black holes at a distance of 650 Mpc, while a binary with two IOMQ black holes should be observable out to a redshift of 0.4 (see Ref. 8 for a more detailed discussion). This means that black-hole binaries may be the most promising source for the first generation of interferometers. However, it is important to realize that the event rates for black-hole merger are essentially unknown. Different estimates lead to varying rates, but it is possible that black-hole binaries are (at least) as abundant as neutron star ones. In the case of LISA, one would expect to be able to observe mergers of supermassive black holes with very high signal-to-noise ratios (several 1000s), cf. Fig. 1. This means that one may be able to see such events no matter where in the Universe they occur.
graviational-wave frequency (log
f/lHz)
Figure 1. Comparing the typical gravitational-wave strength for various binary systems to the sensitivity curves for the new generation of interferometric detectors.
More detailed calculations show that in the case of unequal masses the leading order signal depends only on the so-called "chirp-mass", i.e. the combination /t 3 / 5 M 2 / 5 , where /J, is the reduced mass and M the total mass. If one observes the shrinking time of the orbit as well as the gravitational-
38
wave amplitude then one can infer the chirp mass and the distance to the source. In principle, this means that coalescing binaries are "standard candles" which may be used to infer the Hubble constant and other cosmological parameters. By extracting higher order post-Newtonian terms one can hope to infer the individual masses, the spins and maybe also put constraints on the graviton mass. Another closely related problem concerns the infall of smaller bodies into black holes. In the case of an extreme mass ratio one can model one of the bodies as a "point particle" and study the problem within black hole perturbation theory. That is, one would use Eq. (1) with the metric of a black hole as background. One can show that (in vacuum) the correspond ing linear perturbations are governed by a wave equation, albeit with a non-trivial effective potential that accounts for the backscattering of waves due to the spacetime curvature. There is strong motivation for studying this problem since LISA may be able to detect many such events. The detailed waveforms carry information about the nature of the spacetime geometry in the vicinity of the black hole, and could therefore plausibly be used to test eg. the black-hole uniqueness theorems. To model these systems is, however, far from trivial (see Ref. 9 for a recent review). In particular since the orbits may be highly eccentric. The main challenge concerns the calculation of the effects of radiation reaction on the point particle. The inspiral phase will be followed by a burst of radiation generated when the two objects merge. This burst should depend on the masses and spins of the two objects. In order to model this signal one need to account for the full nonlinearities of Einstein's theory. This means that one must resort to large scale simulations, and the challenge of modeling black-hole mergers has long been the main driving force behind the development of numerical relativity 10 . As computers are becoming more powerful, and our understanding of the intricate issues involved improves, this problem is beginning to seem tractable.
3.2. Black-hole
oscillations
When a black hole is perturbed by an external agent, or when it settles down after formation following supernova core collapse or binary merger, it oscil lates. These oscillations, known as the "quasinormal modes" of the black hole, have characteristic frequencies that depend only on the parameters of the black hole 11 . This is very exciting since it may enable direct identifica-
39 tion of black holes, and the inference of their parameters (the mass M and the rotation rate, here represented by the Kerr parameter 0 < a < M). Fully nonlinear calculations within numerical relativity have shown that the quasinormal mode ringing dominates the radiation from most dynam ical processes that involve a black hole. No matter how you kick a black hole, it's response will be dominated by the quasinormal modes. This is fortuitous since the mode-problem can be approached at the level of lin ear perturbations. One finds that (non-rotating) black holes oscillate with a frequency / w 12(M Q /M) kHz, while the associated e-folding time is r ?s 0.05(M/M Q ) ms. The quasinormal modes of a black hole are clearly very shortlived. Hence, the associated gravitational-wave signals will be extremely short bursts in the detector, which may make them very hard to distinguish from various noise sources. Not surprisingly, rotation has a significant effect on the quasinormal modes. As the black hole spins up, the modes that are co-rotating as a —> 0 will be speeded up by inertial frame dragging. In addition, they become much longer lived. The available numerical results for the leading co-rotating quadrupole mode are well approximated by /«32
l-^(l-«/M)3/
1 0
^
] kHz ,
a / M ) 3 / 1 0
\MQJ
(15)
and 1.9 x 10 - 2 (1 - a/M)*/10
1
-^
( 1
-
ms .
(16)
From this we can see that the mode becomes undamped in the limit a = M. That the modes of a black hole rotating close to the extreme limit may be very long lived would, provided that these modes will actually be excited in a realistic process (which is not at all clear) — improve the chances of detection. 3.3. Supernova
core
collapse
Intuitively one might expect gravitational collapse to lead to a very strong gravitational-wave signal. However, the outcome depends entirely on the asymmetry of the collapse process and at the present time the available numerical evidence suggests that the level of radiation may be rather low. Typical results suggest that an energy equivalent to 8 x 10~ 8 M Q c 2 (or less!) may be radiated 12 . The signal from a collapse that leads to the formation of a black hole is likely to be dominated by the slowest damped quadrupole
40
quasinormal mode of a black hole. Let us assume that a (slowly rotating) I O M Q black hole is formed and that a typical timescale for the event is of the order of a millisecond. Then we can use Eq. (7) to estimate that the gravitational-wave amplitude may be of the order of hc ~ 10~ 22 for a source in the Virgo cluster. This estimate (which accords reasonably well with full numerical simulations) suggests that such sources are unlikely to be seen beyond the local group of galaxies. This would make observable events very rare. It is expected that three to four supernovae will go off every century in a typical galaxy. This means that we would be very lucky to see one in our Galaxy given only a decade or so of observation. However, even if we only detect a single unique event, we may gain significant insights into su pernova physics. Gravitational waves carry unique information away from a collapsing system. While the optical signal emerges hours, and the neu trino burst several seconds, after the collapse, the gravitational waves are generated during the collapse itself. This means that the gravitational-wave signal carries a clean signature of the collapse dynamics. This information may be impossible to extract in any other way. It may be that the current estimates of the radiated energy are too pessimistic. In fact, the large proper velocities of many neutron stars lend some observational support for the presence of asymmetries in the super nova mechanism. It is easy to show that the release of about 10~ 6 MQ(? of energy could explain a velocity of 1000 km/s. Thus it is not unreason able to expect that future studies of the fully three-dimensional collapse problem may indicate an enhanced radiated energy and maybe even pre dict clearly detectable waves. To solve this problem is, of course, a far from trivial task. Nevertheless, there has been interesting recent progress in this direction 12>13. Numerical relativity is rapidly maturing and the problem may soon be manageable.
4. Neutron stars as gravitational-wave sources Neutron stars are tremendously complicated objects, models of which need to account for supranuclear physics, general relativity, superflu idity/superconductivity, strong magnetic fields, exotic particle physics etcetera 14 . Through the detection of gravitational waves from neutron stars we may be able to probe the very extremes of physics. Hence, it is imper ative that we understand the various possible scenarios in detail. This is, however, a serious challenge for theoretical physics and we may actually require observations to help narrow down the theoretical possibilities.
41
4.1. Lumpy
stars
As soon as a newly born neutron star cools below roughly 10 10 K (within a few minutes) its outer layers will begin to crystallize, forming what is known as the neutron star crust. Even though this crust is not very rigid — in fact, it is rather jelly-like — it can sustain shear stresses. This means that one can envisage a neutron star with a "mountain" which radiates gravitational waves at twice the stars spin-frequency. This would lead to a continuous low-amplitude gravitational-wave signal, which would require integration over long stretches of data for detection. The challenge of detecting such signals from unknown sources is immense. Fortunately, there are many candidate objects for which we know both the frequency and the position in the sky. Given this information one can devise a search algorithm similar to that used by radio astronomers to study pulsars 15 . We know very little about the crustal asymmetries of astrophysical neu tron stars. Given this, the observed spin-down rates of, for example, the Crab and Vela pulsars are often quoted as absolute upper limits. However, this is likely to paint a seriously over-optimistic picture 0 . It seems clear from the observed braking indices of these pulsars that they spin down mainly because of electromagnetic radiation emission. Of course, a small gravitational-wave component may well be hidden somewhere in the data. Current models are certainly not detailed enough to rule this out. Parameterizing the star's deformation in terms of an ellipticity e we can use standard results for a slightly deformed rotating solid body to estimate the strength of the associated gravitational waves. We find
where we have used the fact that the gravitational-wave frequency / is twice the rotation frequency (and assumed a canonical M = 1.4MQ and R = 10 km neutron star). This signal would obviously be too weak to be detected directly, but the effective amplitude increases (roughly) as the square-root of the number of detected cycles. Accounting for this and assuming an observation time of a year, we arrive at „r
,« e / 1 0 0 H z \ 5 / 2 /
r
\ /
he
\
c In order to explain the spindown rate of typical young pulsar one would need e ~ 10 — four orders of magnitude larger than detailed calculations suggest is possible.
3
42
Comparing this result to the noise-curves in Fig. 2 we see that a deformation corresponding to e ~ 1 0 - 6 may lead to a detectable signal. The key question is how large asymmetries the neutron star crust can sustain. An upper limit is provided by the breaking strain of the crust material. Based on results for terrestrial materials Ruderman has estimated that the breaking strain will lie in the range 1 0 - 4 < Ubreak < 1 0 - 2 . Given this, a "reasonable theory" 16 would suggest that e
< 5 x 10- 7 ( ^ f )
(19)
From this we see that neutron star can only sustain "mountains" a tiny fraction of a centimeter high. Yet these deformations may be large enough that the associated gravitational waves can be detected. At the very least, a search for these signals would provide constraints on our theoretical models. In particular, a search for gravitational waves from fast spinning pulsars could set interesting upper limits on the ellipticity, cf. Eq. (18).
id , y i . . i 100 1000 gravitational-wave frequency (Hz)
i 1
i i_ 2 3 gravitational-wave frequency (log10 f /1Hz)
Figure 2. Estimated signal strengths for gravitational waves from nascent (left panel) and mature (right panel) neutron stars.
4.2. Wobbling
stars
Since free precession is the most general motion of a solid body it would not be surprising if spinning neutron stars were found to be precessing. For an axisymmetric solid rigid body the precessional motion consists of the symmetry axis moving along a cone of half-angle 6 — the "wobble angle"
43
— around the (fixed) angular momentum vector. Even though neutron stars are not rigid bodies (far from it!) this general picture holds also for fluids contained inside an elastic shell (cf. geophysics models for the Chandler wobble of the Earth). One key additional feature is related to the elasticity of the crust. During one free precession period, the shape of the crust will change. Since the induced strain is limited by the crustal breaking strain one can derive an upper limit on the wobble angle which can be used to set an upper limit on the achievable gravitational-wave amplitude. This exercise suggests that the detection of gravitational waves from neutron stars undergoing free precession due to crustal deformations is unlikely, cf. Figs. 2-3 in Ref. 17. There have recently been two interesting twists to the free precession story. The first was provided by the observational evidence that PSR B1828-11 may be precessing with a period of about 1000 days. This data provides the strongest available case for neutron star free precession18. Fur thermore, it seems to indicate that the superfluid that penetrates the in ner crust is virtually unpinned. The observations are inconcistent with significant superfluid pinning since this would necessarily lead to a much shorter precession period. The second recent result is neat theoretical idea: Cutler 19 has discussed the possibility that neutron stars may have sizeable (internal) toroidal magnetic fields. This is interesting because a toroidal magnetic field will lead to a prolate deformation of the star, and such de formations tend to increase the wobble angle until the star has become an orthogonal rotator. Such a system would be an ideal emitter of gravita tional waves. One can estimate that the toroidal field must be 19
B
ll2 I
f
^ 3 - 4 X 1 ° l30¥HzJ
G
^
in order for the (prolate) magnetic deformation to overcome the (oblate) one due to crustal stresses. It is not implausible that such strong fields exist in mature neutron stars. 4.3.
Pulsating
(unstable)
stars
Neutron stars have a rich spectrum of pulsation modes. These oscillations may be excited to a relevant level in a newly born neutron star. Gravita tional waves from a pulsating neutron star would provide an excellent probe of the stars properties, and could lead to quite accurate "measurements" of the stars mass and radius 20 . This would obviously help constrain the
44
supranuclear equation of state, which is currently known only up to factors of two or so. The most promising scenarios leading to large amplitude oscillations and relevant gravitational-wave signals involve either dynamical or sec ular instabilities. As an unstable pulsation mode grows it may reach a sufficiently large amplitude that the emerging gravitational waves can be detected. Neutron star instabilities are often discussed in terms of the ratio 0 = T/|W|, where T is the rotational kinetic energy and W the gravitational potential energy of the star. Prom classical studies of the Maclaurin spheroids we know that stars become secularly unstable (e.g. due to gravitational-wave emission) at 0S ft* 0.14 and dynamically unstable at 0d » 0.27. The dynamical bar-mode instability.— Several interesting scenarios may lead to a compact star becoming dynamically unstable. For example, since 0 ~ 1/R one might expect a collapsing star to suffer a triaxial instability at some point during its evolution. In order to study the nonlinear bar-mode evolution one must resort to large scale numerical simulations. Such work, carried out over the last two decades (see Ref. 21 for a detailed review), shows that the nature of the barmode instability depends on the magnitude of 0 compared to the critical value. For large values, 0 » 0a, the initial exponential growth of the unstable mode (on the dynamical timescale) is followed by the formation of spiral arms. Gravitational torques on the spiral arms lead to the shedding of a mass and angular momentum. Through this process the unstable mode saturates and the star reaches a dynamically stable state. In this scenario gravitational waves are emitted in a relatively short burst 2 2 . For some time it was thought that this was the generic behaviour, but recent work indicates that when 0 is only slightly larger than 0a a long-lived ellipsoidal structure may be formed. If this is the case, the bar-mode may decay slowly (on the viscosity/gravitational-wave timescale) until the star reaches the point where it is seculary stable. This could lead to a relatively long lasting gravitational-wave signal. It is straightforward to estimate the strength of the gravitational waves emitted by a sizeable bar-mode. Let us assume that the mode saturates at an amplitude e (representing the ellipticity induced by the unstable mode). Typical values may lie in the range e « 0.2 - 0.4. From Eq. (17) we readily
45
get (again for a canonical neutron star)
This estimate (which is illustrated in Fig. 2) compares reasonably well with the more detailed results available in the literature (see, for example, Ta ble 7 in Ref. 22). A signal with this strength may be detectable for sources in local galaxy group. Furthermore, the detectability of the signal can be significantly improved if the instability leads to the formation of a persis tent bar-like structure. Should a long-lived bar form and last for hundreds of rotation periods, one can easily gain a factor of ten in the signal-to-noise ratio. Gravitational-wave driven instabilities.-— Rotating perfect fluid stars are generically unstable due to the emission of gravitational waves (see Ref. 21 for a detailed review). The criterion for a pulsation mode to be unstable is that the mode pattern speed is retrograde in the rotating frame but prograde according to an inertial observer. When this is the case the star "thinks it is losing negative energy" through the emitted gravitational waves and as a result the unstable mode grows. The modes that are thought to lead to the strongest instabilities are the acoustic f-modes and the inertial r-modes. Until about four years ago, the main focus was on the f-modes. This is natural since they lead to considerable density variations, and hence significant gravitational wave emission. Detailed calculations show that only f-modes with m < 5 are expected to grow fast enough to lead to an astrophysically relevant instability. On the other hand, the low order modes only become unstable at extremely high rotation rates (and the quadrupole mode may not be unstable at all). It was somewhat of a surprise when it was realized that the Coriolis restored r-modes, which lead to comparatively small variations in the stellar density, may lead to a significant instability. While the f-modes only become unstable beyond a critical rotation rate (which turns out to be close to the break up speed), the r-modes are always unstable in a rotating perfect fluid star. For the most important r-mode one can show that the growth time is of the order of a few tens of seconds for a star rotating near the mass-shedding limit. The growth-time estimates indicate that the unstable r-modes may be of significance for rapidly spinning neutron stars (see Ref. 21 for a detailed discussion). However, in order to assess the true relevance of these insta bilities we must also consider possible damping effects. In particular, an
46
unstable mode must grow fast enough that it is not completely damped out by viscosity. To assess the strength of viscous damping one typically con siders the effects of bulk and shear viscosity. At relatively low temperatures (below a few times 109 K) the main viscous dissipation mechanism in a fluid star arises from momentum transport due to particle scattering, modeled as a macroscopic shear viscosity. At high temperatures (above a few times 109 K) bulk viscosity is the dominant dissipation mechanism. Bulk viscosity arises as the mode oscillation drives the fluid away from beta equilibrium. It corresponds to an estimate of the extent to which energy is dissipated from the fluid motion as weak interactions try to re-establish equilibrium. The mode energy lost through bulk viscosity is carried away by neutrinos. Esti mates of these damping mechanisms show that a gravitational-wave driven instability will only be active in a certain temperature range. Around 109 K there is a temperature window where the growth time due to gravitational radiation is short enough to overcome the viscous damping and drive the mode unstable. For simple neutron star model, the r-modes are unstable at rotation periods shorter than 25 ms. The fact that this estimate is close to the initial spin period of the Crab pulsar inferred from observational data, P RJ 19 ms, led to the suggestion that the r-mode instability may play a significant role in the spin-evolution of nascent neutron stars. This possibility caused some excitement and spawned a multitude of studies into the r-modes and the instability mechanism. Much of this work is reviewed in Ref. 21. The detailed studies illustrate that this is an incredibly difficult problem, and that we need to understand many extremes of physics before we can draw any reliable conclusions regarding the relevance of the unstable r-modes. At the time of writing, the two most important/interesting issues concern: Exotic bulk viscosity: The presence of exotic particles in the core of a neutron star may lead to significantly stronger viscous damping than assumed in the "standard" instability analysis. Of particular relevance may be the presence of hyperons. Recent results show that the dissipation due to hyperons is, indeed, overwhelming23. The bulk viscosity of strange matter is also many orders of magnitude stronger than its neutron star counterpart. This leads to the instability window shifting towards lower temperatures in a strange star. Nonlinear saturation mechanisms: To model the gravitational waves associated with a secular instability we need to understand how an unstable mode evolves and what the "backreaction" on the bulk of the star is. This is a very difficult problem. Intuitively, one would expect the growth of the
47
mode to be halted at some finite amplitude. It seems plausible that the excess angular momentum will be radiated away as the mode saturates, and that the star will spin down as a consequence. Most models to date have been phenomenological. .The key question concerns the nonlinear saturation amplitude, typically parametrized in terms of a dimensionless parameter a such that 6v oc ailR. In the first studies, it was assumed that a could reach values of order unity. This would lead to a newly born neutron star spinning down significantly on a timescale of a few weeks-months. The resultant gravitational-wave signal would be detectable by LIGOII for sources in the Virgo cluster 2 4 , cf. Fig. 2. The mode-saturation has recently been investigated in two different ways. The first studies were based on fully nonlinear hydrodynamics. The results indicated that the r-modes would saturate at an amplitude large compared to unity 25 ' 26 . The problem has also been addressed within second-order perturbation theory 27 . The obtained results indicate that a strong res onant coupling to short wavelength inertial modes would lead to r-mode saturation at an amplitude at least two orders of magnitude smaller (for rotation rates near the Kepler limit) than that used in the early studies of the r-modes. This means that the effect of the instability is likely be less dramatic than originally assumed.
4.4. Accreting
stars
An interesting class of potential gravitational-wave sources are the lowmass X-ray binaries (LMXB). In the last few years, observations by the Rossi X-ray Timing Explorer satellite have provided indications that the accreting neutron stars in these systems may be clustered in a relatively narrow range of spin frequencies (vs), between perhaps 250 Hz and 590 Hz. These systems are thought to be the progenitors of the millisecond pulsars. Intuitively one might expect them to exhibit a wide range of spin rates. The evidence that this may not be the case requires an explanation. The possibility that the spin-up of these systems may be stalled because of gravitational radiation was first suggested by Bildsten 28 . In his model accretion from the companion star triggers nuclear burning in the deep crust of the neutron star. This leads to temperature/density gradients in the crust and asymmetries which generate gravitational radiation at twice the stars spin frequency. The deformation required to balance the accretion
48
torque can be estimated to be
Two alternative mechanisms have been proposed. The first of these is based on the notion that the unstable r-modes provide the gravitationalradiation torque. This idea would be viable as long as the r-mode instability can operate in a cold mature neutron star. Recent evidence (see Ref. 21 for a detailed discussion) suggests that this will be the case, and that the presence of hyperons or deconfined quarks in the core of the star may lead to the accreting system reaching a quasiequilibrium where the gravitational waves balance the accretion torque while the shear viscosity heating due to the mode motion balances the cooling of the star. The final possibility depends on internal toroidal magnetic fields lead ing to a prolate deformation of the star (see discussion above). In order to generate the required deformation the toroidal field component must be in the range 19 2 x 10 12 — 2 x 10 14 G. This would be perhaps five orders of mag nitude stronger than the expected poloidal field. Yet it is not inconceivable that such strong internal fields exist. Since all three mechanisms seem viable (at least given the present evi dence) the LMXBs must be considered a key target for observations. Par ticularly interesting would be if it were to be possible not only to detect the waves, but also to distinguish the radiation mechanism. If any of the proposed models is correct Sco X-l, the strongest X-ray source in the sky, should be easily detectable with second generation interferometers (like LIGO II), cf. Fig. 2. Detecting these sources is obviously not straightfor ward given the fact that the accretion rate is variable (likely linked to the observed X-ray variability). This could mean that it may only be meaning ful to integrate the signal coherently for 20 days or so 8 .
5. Final remarks Even though we are still waiting for the first indisputable direct detection of gravitational waves this is an area of study permeated by optimism and great hope for the next decade(s). This article was an attempt to illustrate the current excitement by describing (some of) the truly exotic physics that we can hope to probe via gravitational-wave observations.
49
References 1. B.F. Schutz, Gravitational Radiation, in the Encyclopedia of Astronomy and Astrophysics (IOP Publishing, Bristol 2000), preprint gr-qc/0003069 2. T. Damour, pp. 128-198 in 300 Years of Gravitation, Ed: S.W. Hawking and W. Israel (Cambridge Univ. press 1987) 3. P.R. Saulson, Class. Quantum Grav. 17 2441 (2000) 4. B.F. Schutz Gravitational-wave astronomy preprint gr-qc/9911034 5. E.S. Phinney, A Practical Theorem on Gravitational Wave Backgrounds preprint astro-ph/0108028 6. T. Piran Gamma-Ray Bursts - a Primer For Relativists in the proceedings of the 16th International conference of General Relativity and Gravitation, preprint gr-qc/0205045 7. L. Blanchet Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Relativity 5 3, (2002), Online article http://www.livingreviews.org/Articles/Volume5/2002-3blaiichet/ 8. C. Cutler and K.S. Thome An Overview of Gravitational-Wave Sources in the proceedings of the 16th International conference of General Relativity and Gravitation, preprint gr-qc/0204090 9. S. Hughes Listening to the Universe with Gravitational-Wave Astronomy preprint astro-ph/0210481 10. L. Lehner, Numerical Relativity: Status and Prospects in the proceedings of the 16th International conference of General Relativity and Gravitation, preprint gr-qc/0202055 11. K.D. Kokkotas and B.G. Schmidt Quasi-Normal Modes of Stars and Black Holes Living Rev. Relativity 2 2, (1999), Online article http://www.livingreviews.org/Articles/Volume2/1999-2kokkotas/ 12. H. Dimmelmeier, J.A. Font and E. Mueller, Astron. Astrophys. 388 917 (2002) 13. M. Shibata and S.L. Shapiro, Ap. J. 572 L39 (2002) 14. N.K. Glendenning, Compact stars (Springer 1997) 15. P.R. Brady and T. Creighton, Phys. Rev. D 61 082001 (2000) 16. G. Ushomirsky, C. Cutler and L. Bildsten, MNRAS 319 902 (2000) 17. D.I. Jones and N. Andersson, MNRAS 331 203 (2001) 18. I.H. Stairs, A.G. Lyne and S.L. Shemar, Nature 406 484 (2000) 19. C. Cutler, Phys. rev. D. 66 084025 (2002) 20. N. Andersson and K.D. Kokkotas, MNRAS 299 1059 (1998) 21. N. Andersson, Gravitational waves from instabilities in relativistic stars preprint astro-ph/0211057 22. J.L. Houser, MNRAS 299 1069 (1998) 23. L. Lindblom and B.J. Owen, Phys. Rev. D 65 063006 (2002) 24. B.J. Owen et al, Phys. Rev. D 58 084020 (1998) 25. N. Stergioulas and J.A. Font, Phys. Rev. Lett. 86 1148 (2001) 26. L. Lindblom, J.E. Tohline and M. Vallisneri, Phys. Rev. Lett. 86 1152 (2001) 27. P. Arras et al Saturation of the r-mode instability preprint astro-ph/0202345 28. L. Bildsten, Ap. J. Lett. 501 89 (1998)
COSMOLOGICAL PARAMETERS: FASHION AND FACTS
A. B L A N C H A R D Laboratoire d'astrophysique de I'OMP, GNRS, UPS 14, Av. E.Belin, 31 400 Toulouse, FRANCE E-mail:
[email protected]
We are at a specific period of modern cosmology, during which the large increase of the amount of data relevant to cosmology, as well as their increasing accu racy, leads to the idea that the determination of cosmological parameters has been achieved with a rather good precision, may be of the order of 10%. There is a large consensus around the so-called concordance model. Indeed this model does fit an impressive set of independent data, the most impressive been: CMB C; curve, most current matter density estimations, Hubble constant estimation from HST, apparent acceleration of the Universe, good matching of the power spectrum of matter fluctuations. However, the necessary introduction of a non zero cosmo logical constant is an extraordinary new mystery for physics, or more exactly the come back of one of the ghost of modern physics since its introduction by Einstein. Here, I would like to emphasize that some results are established beyond reason able doubt, like the (nearly) flatness of the universe and the existence of a dark non-baryonic component of the Universe. But also that the evidence for a cosmo logical constant may not be as strong as needed to be considered as established beyond doubt. In this respect, I will argue that an Einstein-De Sitter universe might still be a viable option. Some observations do not fit the concordance pic ture, but they are generally considered as not to be taken into account. I discuss several of the claimed observational evidences supporting the concordance model, and will focus more specifically on the observational properties of clusters which offer powerful constraints on various quantities of cosmological interest. They are particularly interesting in constraining the cosmological density parameter, nicely complementing the CMB result and the supernova probe. While early estimations were based on the of the MjL ratio, i.e. a local indirect measure of the mean density which needs an extrapolation over several orders of magnitude, new tests have been proposed during the last ten years which are global in nature. Here, I will briefly discuss three of them: 1) the evolution of the abundance of clusters with redshift 2) the baryon fraction measured in local clusters 3) apparent evolu tion of the baryon fraction with redshift. I will show that these three independent tests lead to high matter density for the Universe in the range 0.6 — 1.. I therefore conclude that the dominance of vacuum to the various density contributions to the Universe is presently an interesting and fascinating possibility, but it is still premature to consider it as an established scientific fact.
50
51
1. Introduction: the contents of the Universe Cosmology is a rather young field still undergoing a very fast evolution. Twenty years ago the nature of the microwave background was still a matter of debate, although it was generally believed that the origin was mainly from the Big Bang, the exact shape of the spectrum was still uncertain. The measurement of the spectrum by COBE, which was concomitant to the Gush et al. measurement (1990), showing that the spectrum was a nearly perfect blackbody has been a fundamental result in modern cosmology, by establishing in a definitive way one of the most critical prediction of the standard hot Big Bang picture. The determination of cosmological parameters is a central question in modern cosmology and it has become more central after the next fundamental result established by COBE: the first robust detection of the CMB fluctuations, nearly thirty years after the prediction of their presence (Sachs and Wolf, 1967). This detection has opened a new area with the perspective of reaching high "precision cosmology". However, it is also important to mention the fact that the Inflation paradigm (Guth, 1981) has represented an enormous attraction for theorist towards the field of cosmology, opening the perspective of properly testing high energy physics from cosmological data, while such a physics will probably remain largely unaccessible from laboratory experiment. Even if the data from the CMB fluctuations were not taken at face values as a proof of inflation the need for new physics appear very strongly (it is interesting to mention that the origin of the asymmetry between matter and antimatter was a fundamental problem which solution involves physics of the very early universe). Moreover, the presence for non-baryonic dark matter can be now con sidered as a well established fact of modern physics. This was far from being obvious twenty years ago. By present days the abundances of light elements is well constrained by observations, consistent with a restricted range of baryon abundance (O'Meara et al., 2001): ^baryons
= 0 . 0 2 f t - 2 ± 0.002
(1)
where h is the Hubble constant in unit of 100 km/s/Mpc. The above baryon abundance is in full agreement with what can be inferred from CMB (Le Dour et al., 2000; Benoit et al., 2002b). There are differences in matter density estimations, but nearly all of them lead to a cosmological density parameter in the range [0.2-1.], and therefore those estimates imply the presence of a non-baryonic component of the density of the universe. An other implication is that most baryons are dark: the amount of baryons
52
seen in the Universe is mainly in form of stars: fistars = 0.003 - 0.010
(2)
much less than predicted by primordial nucleosynthesis. This picture, the presence of two dark components in the Universe, has gained considerable strength in the last twenty years, first of all because the above numbers have gained in robustness. However, it is now believed that a third dark constituent has been discovered: the dark energy.
2. Observations a n d cosmological p a r a m e t e r s 2.1. What the CMB does actually
tell
us?
2.1.1. The curvature of space The detection of fluctuations on small angular scale, mainly by the Saska toon experiment (Netterfield et al, 1995) more than 7 years ago provided a first convincing piece of evidence for a nearly flat universe (Linewaever et al., 1997; Hancock et al., 1998; Lineweaver and Barbosa, 1998), or more precisely evidence against open models which were currently favored at that time. This conclusion is now firmly established thanks to high precision re cent measurements including those of Boomerang, Maxima and DASI (de Bernardis et al;, 2000; Hanany et al., 2000; Halverson et al., 2002): open models are now entirely ruled out: fit > 0.92 at 99% C.L., it should be noticed that upper limit on fit are less stringent, fi4 < 1.5 at 99% C.L., unless one add some prior, for instance on the Hubble constant. The most recent measurements of CMB anisotropies, including those obtained after this conference by Archeops (Benoit et al., 2002a), provide a remarkable success of the theory: the detailed shape of the angular power spectrum of the fluctuations, the theoretical predictions of the C\ curve, is in excellent agreement with the observational data. This success gives confidence in the robustness of conclusions drawn from such analyzes, while alternative theories, like cosmological defects (Durrer et al, 2002) are al most entirely ruled out as a possible primary source of the fluctuations in the C.M.B. This gives strong support for theories of structures formation based the gravitational growth of initial passive fluctuations, the gravita tional instability scenario, a picture sketched nearly seventy years ago by G. Lemaitre (1933). At the same time this implies that conclusions on cosmological parameters from CMB have to be considered as robust: the
53
1.4 1.2 1.0
cf 0.8 0.6
0.4 40
60 Ha
80
100
Figure 1. On this picture the likelihood contours from the CMB constraints are given: dashed lines, when projected provide the 68%, 95%, 99% confidence intervals, while shade area correspond to the contours on two parameters. The likelihood is maximized on the other parameters. This diagram illustrates several aspects of constraints that can be obtained from CMB: flatness of the universe follows from the fact that fit > 0.92 at 99% C.L., but Ho is very poorly constrained. Indeed CMB allows to severely tighten the model parameters space, but can leave us with indetermination on specific individual parameter because of degeneracies. See Douspis et al., (2000) for further details. spectacular conclusion that the universe is nearly flat space a is a major sci entific result of modern science which is certainly robust and is very likely to remain as one of the greatest advance of modern Cosmology.
2.1.2. A strong test of General Relativity Contrarily to a common conception, General Relativity (GR) is weakly tested on cosmological scales: the expansion of the Universe can be de scribed in a Newtonian approach, while departure from the linear Hubble diagram are weak, and therefore does not provide strong test of GR. Aca
It is sometimes believed that a space cannot be "nearly" flat, because mathematically space is flat or not. This is not true in Cosmology where there is a natural scale which is c/ffo- Stating that the Universe is nearly flat means that its curvature radius Rc is much larger than this scale.
54
tually the observed Hubble diagram is used to fit the amplitude of the cosmological constant, i.e. ones assumes (a non-standard version of!) GR and fits one of the parameter, therefore this does not constitute a test of the theory. However, the Ci curve of CMB fluctuations provides an interesting test of GR on such scales: the angular distance to the CMB accordingly to RG is such that: ■Dang ~ 300^0
_
*lss)
(3)
(t0 being the present age of the universe, and ii ss the age of the universe at the last scattering surface (lss) from where the C — I curve is produced). This means that the angular distance to the CMB is of the same order than the one to the Virgo cluster! Therefore the Cj curve can be obtained only within a theory where photons trajectories are essentially those predicted byGR. 2.2. I s the Universe
accelerating
?
It is often mentioned that the present day data on the CMB excluded a model without a cosmological constant. Given the present-day quality of the data, and the anticipated accuracy one can hope from satellite experi ments, this is a crucial issue. Actually, what's happen is that an Einstein-de Sitter is at the boarder of the 3 — a contour in likelihood analysis. But this is not sufficient to claim that the model is excluded at 3 —CT!Actually, a model without a cosmological constant provides a very acceptable fit to the data in term of a goodness of fit. Therefore, CMB data do not request a non-zero cosmological constant. The possible detection of a cosmological constant from distant supernovae has brought the essential piece of evidence largely comforting the socalled concordance model: the apparent luminosity of distant supernovae now appears fainter, i.e. at larger distance, than expected in any deceler ating universe (Riess et al, 1998; Perlmutter et al., 1999) and can therefore be explained only within an accelerating universe (under the assumption of standard candle). Indeed a CDM model in a flat universe dominated by a cosmological constant is in impressive agreement with most of existing data: such a model is consistent with the HST measurement of the Hubble constant, the age of the Universe, the power spectrum and the amplitude of matter fluctuations as measured by clusters abundance and weak lensing on large scale, as well as most current measurements of the mass content on small scales obtained by various technics. The concordance model offers
55 n„, = 1.0s
n„, = 1.0 A Q,h' H, n q
A aji' H, n Q 0,„ a nt
=0.0 = 0.02 = 43.72' = 0.87 = 22.4
n„ = o.o a
nt
= 0.0 = 0.0 CO -
!-*V* 1000
1500
T
= = = = = = = = =
0.7 0.023 57.82 0.99 ■ 19.4 . 0.0 0.2 0.0 0.0 -
1^ . L A . . . , 1000
1500
Figure 2. An example of an acceptable model to the CMB data without inclusion of a cosmological constant (left) The Hubble constant has been taken to a low value of 44 km/s/Mpc. An exemple of optimal model is shown, corresponding to the concordance model. Both provide an acceptable fit in term of goodness of fit. Courtesy of M. Douspis.
therefore a remarkable success for the CDM theory, but at the expense of the introduction of a non-zero cosmological constant. 2.3. Some reasons for
caution
Despite the above impressive set of agreements cited above, one should keep an open mind. The question of the age of the Universe is not an issue: models fitting adequately the C; curve leads to similar ages, consistent with existing constraint. For instance, the model drawn on figure 2 (left side) has an age of 15 Gyr, well consistent with age estimates (actually a model with t0 ~ 10 Gyr should probably not be securely rejected on this basis). Identically the amplitude of matter fluctuations on small scales is some times claimed to be inconsistent with a high density universe, while there is actually a degeneracy between this amplitude and the matter density parameter Clm. Very often, authors implicitly refer to the standard CDM scenario (ftm = 1, h ~ 0.5, n = 1). Actually this simplest CDM model is known to be ruled out from several different arguments, but there exists also different way one can imagine the spectrum to be modified in order to match the data (an example is a possible contribution of hot dark matter of the order of 20%).
56
A high density universe is actually inconsistent (because of the age prob lem) with value of the Hubble constant as high as those found by the HST. However, the HST measurement of the Hubble constant has been questioned (Arp, 2002). In order to illustrate the argument I show the figure given by Arp, which is claimed to represent the Hubble diagram from the HST data. Clearly, a firm conclusion on the Hubble constant from this data seems difficult and actually Arp claims that data can favor H0 ~ 55 km/s/Mpc. An other doubt on the Hubble constant comes from the Sunyaev-Zeldovich measurements: in a recent review Carlstrom et al. (2002) found that the best value slightly depends on the cosmology, but that in an Einstein-de Sitter model one finds an average values of H0 ~ 55 km/s/Mpc, furthermore given that such determination suffers from possi ble clumping of the gas (Mathiesen et al., 1999; see below), the actual value could be 25% less! Let us now examine observational direct evidences for or against a non zero cosmological constant. Distant SNIa are observed to be fainter than ex pected (in a non-accelerating universe) given their redshift, indicating very directly that the universe is accelerating should they be standard candles. The signal is of the order of 0.3 magnitude (compared to an Einstein-de Sitter universe). It is important to realize that several astrophysical effects of the same order are already existing, and that their actual amplitude might be difficult to properly evaluated. Rowan-Robinson (2002) argued for instance that the dust correction might have been underestimated in high redshift SNIa, while such a correction is of the same order of the sig nal. Identically, the K-correction that has to be applied to high redshift supernovae is large (in the range 0.5-1. mag) and is estimated from zero redshift spectral templates; one can therefore worry whether some shift in the zero-point would not remain from the actual spectrum, with an ampli tude larger than the assumed uncertainty (2%). Identically, the progenitor population at redshift 0.5 is likely to be physically different from the pro genitors of local SNIa (age, mass, metalicity). Consequence on luminosity are largely unknown. Finally it is worth noticing that the first 7 distant SNIa which were analyzed conducted to conclude to the rejection of a value of A as large as 0.7: A < 0.51 (95%) (Perlmutter et al., 1997). Several arguments have been used in the past or recently to set upper limit on a dominant contribution of A (Maoz et al., 1993; Kochanek, 1996; Boughn et al., 2002). There is
57
therefore a number of arguments for caution:
Figure 3. Hubble diagram from HST cepheids according to Arp (2002). Clearly the derivation of a value from this data set is uncertain. But a value of 55 km/s/Mpc seems as least as adequate as the HST finding (72 km/s/Mpc).
1) SNIa measurements provide the single direct evidence for a cosmological constant, 2) most measurements of fim are local in nature (mostly inferred from clus ters), 3) some upper limits have been published on A which do not agrre with recent measurements , 4) a non-zero cosmological constant is an extraordinary new result in physics and therefore deserves extraordinary piece of evidence. Before the existence of the cosmological constant can be considered as scientifically established, it is probably necessary to reinforce evidence for the convergence model by obtaining further direct evidence for a cosmologi cal constant. Because there exist degeneracies in parameters determination with the CMB, even the Planck experiment will not allow to break these degeneracies. It is therefore necessary to use tests which provide comple-
58
mentary information. The data provided by the distant SNIa satisfies well this requirement. As it is difficult to think of a new test measuring directly the presence of a cosmological constant, the best approach is probably still to try to have reliable estimates of the matter density from global tech nics. In this respect, clusters are probably the most powerful tool, as they provide several major roads to measure the density matter of the Universe and which exploration is still in its infancy. Here I will concentrate on this perspective.
3. The mean density of the Universe from Clusters The classical way to use clusters to constraint the average matter density in to try to obtain a direct measurement of the local density. This is the principle of M/L test. Because the matter content of the universe is essentially in a dark form, we do not have direct measurement of the mass content even at the local level. This is the reason why in practice we rely on a two-steps procedure: first the average luminosity density of the universe is estimated from galaxy samples, this quantity is now relatively well known thanks to the large redshift surveys like the SLOAN or the 2dF (although difference of the order of 50% might still exist); the second ingredient is the value of the M/L ratio obtained from data on clusters (total luminosity and mass estimations). There might be a factor of two of uncertainty in this quantity. For instance Roussel et al. (2000) found that the average M/L could be as large as 750/i when the mass-temperature relation for clusters is normalized from numerical simulations of Bryan and Norman (1998), while values twice smaller are currently obtained by other technics. The basic principle for estimating the average density of the Universe is then to write: pm = pix
M/L
However, it should be realized that the volume occupied by clusters is a tiny fraction of the total volume of universe, of the order of 1 0 - 5 . The application of the M/L relies therefore on an extrapolation over 105 in volume! 4. N e w global tests In order to have a reliable estimation of the mass density of the universe, it is vital to have the possibility to use global tests rather than local ones.
59
Figure 4. These plots illustrate the power of the cosmological test of the evolution of the abundance of X-ray clusters: the T D F (temperature distribution function) has been normalized to present day abundance (blue - dark grey - lines). The abundance of local clusters is given by the blue (dark grey) symbols (Blanchard et al., 2000). Present abundance allows one to set the normalization and the slope of the spectrum of primordial fluctuations on clusters scale (which is H m dependant). The evolution with redshift is much faster in a high matter density universe (left panel, fim = 0.89) than in a low density universe (right panel, Qm = 0.3): z = 0.33 (yellow - light) the difference is already of the order of 3 or larger. It is relatively insensitive to the cosmological constant. We also give our estimate of the local T D F (blue symbols) derived by Blanchard et al. (2000), as well as our estimate of the T D F at z — 0.33 (yellow symbols- light grey). Also are given for comparison data (Henry 2000) and models predictions at z = 0.38 (red dark grey - symbols and lines). On the left panel, the best model is obtained by fitting simultaneously local clusters and clusters at z = 0.33 leading to a best value of f l m of 0.89 (flat universe). The right panel illustrates the fact that an flat low density universe Qm = 0.3 which fits well local data does not fit the high redshift data properly at all.
4.1. The evolution
of the abundance
of
clusters
The abundance of clusters at high redshift has been used as a cosmological constraint more than ten years ago (Peebles et al., 1989; Evrard, 1989). Ten years ago, Oukbir and Blanchard (1992) emphasized that the evolution of the abundance of clusters with redshift was rather different in low and high density universe, offering a possible new cosmological test. The interest of
60
this test is that it is global, not local, and therefore allows to actually con straint directly flm. It is relatively insensitive to the cosmological constant. In principle, this test is relatively easy to apply, because the abundance at redshift ~ 1. is more than an order of magnitude less in a critical universe, while it is almost constant in a low density universe. Therefore the mea surement of the temperature distribution function (TDF) even at z ~ 0.5 should provide a robust answer. In recent years, this test has received considerable attention (Borgani et al, 1999; Eke et al., 1998; Henry, 1997; Henry, 2000; Viana and Liddle, 1999, among others). The first practical application was by Donahue (1996) who emphasized that the properties of MS0451, with a temperature of around 10 keV at a redshift of 0.55, was already a serious piece of evidence in favor of a low density universe. This argument has been comforted by the discovery of a high temperature cluster at redshift z ~ 0.8, MS1054, which has a measured temperature of ~ 12 keV (Donahue et al, 2000). In the mean time, however, the redshift distribution of EMSS clusters was found to be well fitted by a high density universe under the assumption of a non evolving luminosity-temperature relation (Oukbir and Blanchard, 1997; Reichert et al., 1999), as seems to follow from the properties of distant X-ray clusters (Mushotsky, R.F. and Sharf, 1997; Sadat et al., 1998). Application of this test is the purpose of the XMM Q program during the guaranty time phase (Bartlett et al., 2001). In principle, this test can also be applied by using other mass estimates, like velocity dispersion (Carlberg et al, 1997), Sunyaev-Zeldovich (Barbosa et al, 1996), or weak lensing. However, mass estimations based on X-ray temperatures is up to now the only method which can be applied at low and high redshift with relatively low systematic uncertainty. For instance, if velocity dispersions at high redshift (~ 0.5) are overestimated by 30%, the difference between low and high density universe is canceled. Weak lensing and SZ surveys of clusters to allow this test remain to be done.
4.2. The local temperature
distribution
function
In order to estimate the amount of evolution in the number of clusters, one obviously needs a reliable estimate of the number of clusters at z ~ 0. This already is not so easy and is a serious limitation. The estimation of the local temperature distribution function of X-ray clusters can be achieved from a sample of X-ray selected clusters for which the selection function is known, and for which temperatures are available. Until recently, the
61
standard reference sample was the Henry and Arnaud sample (1991), based on 25 clusters selected in the 2. - 10. keV band. The ROSAT satellite has since provided better quality samples of X-ray clusters, like the RASS and the BCS sample, containing several hundred of clusters. Temperature information is still lacking for most of clusters in these samples and therefore such clusters samples do not allow yet to improve estimations of the TDF in practice. We have therefore constructed a sample of X-ray clusters, by selecting all X-ray clusters with a flux above 2.210 - 1 1 erg/s/cm 2 with \b\ > 20. Most of the clusters come from the Abell XBACS sample, to which some non-Abell clusters were added. The completeness was estimated by comparison with the RASS and the BCS and found to be of the order of 85%. This sample comprises 50 clusters, which makes it the largest one available for measuring the TDF at the time it was published. The inferred TDF is in very good agreement with the TDF derived from the BCS luminosity function or from more recent comprehensive survey (Reiprich and Bohringer, 2002) (with ~ 65 clusters). The abundance of clusters is higher than derived from the Henry and Arnaud sample as given by Eke et al. (1998) for instance. It is in good agreement with Markevitch (1998) for clusters with T > 4 keV, but is slightly higher for clusters with T ~ 3 keV. The power spectrum of fluctuations can be normalized from the abundance of clusters, leading to a» = ac — 0.6 (using PS formula) for Clm = 1 and to ac = 0.7 for Clm = 0.35 corresponding to as = 0.96 for a n = —1.5 power spectrum index (contrary to a common mistake the cluster abundance does not provide an unique normalization for as in low density models, but on a scale ~ - 3 ■^flrn8h~1Mpc), consistent with recent estimates based on optical analysis of galaxy clusters (Girardi et al., 1998) and weak lensing measurements (Van Waerbeke et al., 2002).
4.3. Application
to the determination
of Qm
The abundance of X-ray clusters at z = 0.33 can be determined from Henry sample (1997) containing 9 clusters. Despite the limited number of clusters and the limited range of redshift for which the above cosmological test can be applied, interesting answer can already be obtained, demonstrating the power of this test. Comparison of the local TDF and the high redshift TDF clearly show that there is a significant evolution in the abundance of X-ray clusters (see figure 1), such an evolution is unambiguously detected in our analysis. This evolution is consistent with the recent study of Donahue et al. (2000). We have performed a likelihood analysis to estimate the mean
62
density of the universe from the detected evolution between z = 0.05 and z = 0.33. The likelihood function is written in term of all the parameters entering in the problem: the power spectrum index and the amplitude of the fluctuations. The best parameters are estimated as those which maximize the likelihood function. The results show that for the open and flat cases, one obtains high values for the preferred Clm with a rather low error bar : flm = O.92I022 (open case)
(4)
flm = 0.86±g;l| (flat case)
(5)
(Blanchard et al., 2000). Interestingly, the best fitting model also repro duces the abundance of clusters (with T ~ 6 keV) at z — 0.55 as found by Donahue and Voit (2000).
1.0 0.8 0.6 0*4 oje 0.0
0.0
0.5
1.0
\A
2.0
% Figure 5. Likelihoods from the measured abundance of EMSS clusters in the redshift range (0.3,0.4) based on the Henry's sample (1997). The dashed line is for aflat universe while the continuous line is for an open cosmology.
63
4.4.
Systematic
uncertainties
in the determination
of
flm
The above values differ sensitively from several recent analyzes on the same test and using the same high redshift sample. It is therefore important to identify the possible source of systematic uncertainty that may explain these differences. The test is based on the evolution of the mass function (Blanchard and Bartlett, 1998). The mass function has to be related to the primordial fluctuations. The Press and Schechter formalism is gener ally used for this, and this is what used in deriving the above numbers. However, this may be slightly uncertain. Using the more recent form pro posed by Governato et al. (1999) we found a value for Ctm slightly higher (a different mass function was used in Figure 1). A second problem lies in the mass temperature relation which is necessary to go from the mass function to the temperature distribution function. The mass can be esti mated either from the hydrostatic equation or from numerical simulations. In general hydrostatic equation leads to mass smaller than those found in numerical simulations (Roussel et al., 2000; Markevitch, 1998; Reiprich and Bohringer, 2002; Seljak, 2002). Using the two most extreme masstemperature relations inferred from numerical simulations, we found a 10% difference. We concluded that such uncertainties are not critical. An other serious issue is the local sample used: using HA sample we found a value smaller by 40%. Identically, if we postulated that the high redshift abundance has been underestimated by a factor of two, Clm is re duced by 40%. The determination of the selection function of EMSS is therefore critical. An evolution in the morphology of clusters with redshift would result in a dramatic change in the inferred abundance (Adami et al., 2001). This is the most serious possible uncertainty in this analysis. How ever, the growing evidences for the scaling of observed properties of distant clusters (Neumann and Arnaud, 2001), rather disfavor such possibility. 4.5. An other global test : the baryon fraction
in local
clusters
This is a very interesting test proposed by White et al. (1993) which in principle offer a rather direct way to measure Clm. It relies on one side on the fact that one should be able to measure the total mass of clusters, as well as their baryon content and on the other side that the primordial abun dance of baryons can be well constrained from the predictions of primordial nucleosynthesis and the observed abundances of light elements. Further more, the CMB is providing interesting constraints on the baryon density
64
of the universe, that are essentially consistent with values inferred from nucleosynthesis (Eq. 1). X-ray observations of clusters allow to measure their gas mass which represents the dominant component of their (visible; baryonic content (the stellar component represents around 1% of the total mass). In this way one can measure the baryon fraction fa and infer Qm:
where 7 represents a correction factor between the actual baryon fraction and the naive value fi&&n/^m; typically, 7 ~ 0.9. This method has been used quite often (Evrard, 1997; Roussel et al., 2000). There are some differences between measurements, mainly due to the mass estimators used. One key point is that the baryon fraction has to be estimated in the outer part of clusters as close as possible to the virial radius. However, the outer profile of the X-ray gas has been shown by Vikhlinin et al. (1999) not to follow the classical f3 profile, usually assumed, but being actually steeper; consequently derived gas masses are somewhat lower than from usual analysis. Recently, several consequences of this work were derived on the baryon fraction (Sadat and Blanchard, 2000): • the scaled baryon fraction flattens in the outer part of clusters. • the global shape of the baryon fraction from the inner part to the outer part follows rather closely the shape found in numerical sim ulations from the Santa Barbara cluster project (Frenk et al, 1999). • when mass estimates are taken from numerical simulations the baryon fraction, corrected from the -rather uncertain- clumping factor (Mathiesen et al, 1999) could be as low as 10% (h = 0.5). The consequence of this is that a value of Qm as high as 0.8 can be ac ceptable. Large systematic uncertainties are still possible, and value twice lower can certainly not be rejected on the basis of this argument, but sim ilarly a value Qm ~ 1 can not be securely rejected. 4.6. The baryon fraction
in high redshift
clusters
A reasonable assumption is that the baryon fraction in clusters should re main more or less constant with redshift, as there is no motivation for intro ducing a variation with time of this quantity. When one infers the baryon fraction from X-ray observations of clusters at cosmological distances, the background cosmology is coming in the inferred value, through angular and luminosity distances. Therefore for a given observed cluster, the inferred
65
0.10
O
0.01 0.01
0.10 R/R,
1.00
Figure 6. From the observed X-ray surface brightness of the distant cluster R.XJ1120 (Arnaud et al., 2002) the gas fraction density profile (red filled circles) is compared to the results from the local clusters derived by Roussel et al. (2000) (blue open triangles) and those found in the outer regions by Sadat and Blanchard (2000) (red rhombuses). The profile shape is very close to those of local clusters. The amplitude is right for an H m = 1. model, while a lambda model (open red triangles) is in strong disagreement with the data.
gas fraction would vary accordingly to the cosmology. This opens a way to constraint the cosmology, if one assumes that the apparent baryon fraction has to be constant (Sasaki, 1996; Pen, 1997), or equivalently that the emissivity profiles of clusters has to be identical when scaling laws are taken into account (Neumann and Arnaud, 2001). Application of this test probably needs a large statistical sample, but a preliminary application can be done on a distant cluster observed by XMM: RXJ1120. This distant cluster is a perfect candidate for the application of this test: the X-ray emission has been detected up to a distance close to the virial radius (Arnaud et al., 2001), the cluster is a ~ 6 keV cluster, with a relaxed configuration. The gas profile can be derived up to a radius of the order of the virial radius without extrapolation. The inferred radial gas profile possesses two remarkable properties: i) the shape of the gas profile in this distant cluster is in very good agreement with the shape of the profile inferred from local clusters by Sadat and Blanchard (2000), giving an interesting further piece of evidence in favor of this shape ii) the
66
amplitude matches the amplitude of the local sample only for a high matter density universe, while an universe dominated by a cosmological constant is strongly disfavored.
5. Conclusion In this paper I have presented a personal point of view on the observa tional determination of cosmological parameters and especially on question of the possible non-zero value of the cosmological constant. Although, the concordance model provides a nice agreement with several observational data sets, I have argued that i) the only direct case foran accelerating uni verse, implying the domination of the vacuum density over the other type of dark matter already assumed to be present in the Universe (baryonic dark matter, non-baryonic dark matter), is coming from the distant SNIa and is not sufficient to be considered as robustly established, ii) some evi dences against the concordance model are systematically rejected, because they are judged as insufficiently robust. The global picture drawn by the concordance model might be right after all! But I still consider that the case for a cosmological constant is oversold. It would be crucial in order to strength the case to have independent evidence either direct or indirect. A possible way for this would be to achieve a reliable measurement of the matter density of the Universe, which in conjunction with the CMB evi dence for flatness, would allow an estimate of the cosmological constant. I have argued that clusters are in several ways the best tool to achieve such a measurement. Again contrary to a common prejudice I have illustrated that there are different values obtained by such methods, some correspond ing to high matter density consistent with an Einstein-de Sitter model. Summarizing results on clusters, I have shown an up-to-date local tem perature distribution function obtained from a flux limited ROSAT sample comprising fifty clusters. When compared to Henry's sample at z — 0.33, obtained from the EMSS, this sample clearly indicates that the TDF is evolving. This evolution is consistent with the evolution detected up to redshift z = 0.55 by Donahue et al. (1999). This indicates converging evi dences for a high density universe, with a value of Clm consistent with what Sadat et al. (1998) inferred previously from the full EMSS sample taking into account the observed evolution in the Lx—Tx relation (which was found moderately positive and consistent with no evolution). From such analyzes, low density universes with Clm < 0.35 are excluded at the two-sigma level.
67
This conflicts with some of the previous analyzes on the same high redshift sample. Actually, lower values obtained from statistical analysis of X-ray samples were primarily affected by the biases introduced by the local refer ence sample, which lead to a lower local abundance and a flatter spectrum for primordial fluctuations (Henry, 1997, 2000; Eke et al., 1998; Donahue and Voit, 1999). Our result is consistent with the conclusion of Viana and Liddle (1999), Reichert et al. (1999) and Sadat et al (1988). The possible existence of high temperature clusters at high redshift, MS0451 (10 keV) and MS1054 (12 keV), cannot however be made consistent with this picture of a high density universe, unless their temperatures are overestimated by a large factor or the primordial fluctuations are not gaussian. The baryon fraction in clusters is an other global test of flm, provided that a reliable value for Qb is obtained. However, it seems that the mean baryon fraction could have been overestimated in previous analysis, possibly being closer to 10% rather than to 15%-25%. This is again consistent with a high density universe. Finally, we have seen in one case that the apparent evolution of the baryon fraction in clusters could also be consistent with a high density universe. In conclusion, I pretend that the determination of cosmological parame ters and especially the evidence for a non-zero cosmological constant is still an open question which needs to be comforted and that the exclusion of an Einstein de Sitter model is over-emphasized. References 1. Adami, C. et al. XXIth Moriond Astrophysics Meeting. http://wwwdapnia.cea.fr/Conferences/Morion_astro_2001/ (2001). 2. Arp, H. ApJ, 571, 615 (2002). 3. Arnaud, M. et al. astro-ph/0204306, A&A, 390, 27 (2002). 4. Barbosa D., Bartlett J.G., Blanchard A. __ Oukbir, J., A&A, 314, 13 (1996) 5. Bartlett, J. et al. astro-ph/0106098, XXIth Moriond Astrophysics Meeting. http://www-dapnia.cea.fr/Conferences/Morion_astro-2001/ (2001) 6. Boughn, S. P., Crittenden, R. G., Koehrsen, G. P. ApJ, 580, 672 (2002) 7. Benoit et al., astro-ph/0210305, A&A in press (2002a) 8. Benoit et al., astro-ph/0210306, A&A in press (2002b) 9. Blanchard, A., Sadat, R., Bartlett, J. & Le Dour, M. astro-ph/9908037, A&A, 362, 807 (2000) 10. Bryan, G.L. k, Norman, M.L. ApJ, 495, 80 (1998) 11. Carlberg, R.G., Morris, S.L., Yee, H.K.C. __ Ellingson, E., ApJ, 479, L19 (1997) 12. Carlstrom, J. E.; Holder, G. P., Reese, E. D. ARA&A, 40, 463 (2002)
68 13. 14. 15. 16.
de Bernardis, P. et al., Nature, 404, 955, (2000) Donahue, M. ApJ, 468, 79 (1996) Donahue, M. k Voit, G. M. astro-ph/9907333, ApJ, 523, L 137 (2000) Donahue, M., Voit, G. M., Scharf, C. A., Gioia, I., Mullis, C. P., Hughes, J. P. & Stocke, J. T. astro-ph/9906295, ApJ, 527, 525 (2000) 17. Douspis, M., Blanchard, A., Sadat, R., Bartlett, J.G., Le Dour, M. A&A, 379, 1 (2001) 18. Durrer, R., Kuntz, M., Melchiorri, A. Phys. Rept, 364, 1 (2002) 19. Eke, V.R., Cole, S., Frenk, C.S. k Henry, P.J. MNRAS, 298, 1145 (1998) 20. Evrard, A.E. ApJ, 341, L71 (1989) 21. Evrard, A.E. MNRAS, 292, 289 (1997) 22. Evrard, A.E, Metzler, C.A., k Navarro, J.P., ApJ, 469, 494 (1996) 23. Frenk, C.S. et al., ApJ, 525, 554 (1999) 24. Girardi, M., Borgani S., Giuricin, G., Mardirossian, F. k Mezzetti, M. ApJ, 506, 45 (1998) 25. Governato, F., Babul, A., Quinn, T., Tozzi, P., Baugh, C. M., Katz, N. k Lake, G. MNRAS, 307, 949 (1999) 26. Gush, H.P., Halpern, M., Wishnow, E. Phys. Rev. Lett, 65, 537 (1990) 27. Guth, A. Phys. Rev. D, 23, 347 (1981) 28. Halverson, N.W. et al., ApJ, 568, 38 (2002) 29. Hanany, S. et al., ApJ, 545, L5 (2000) 30. Hancock, S., Rocha, G., Lasenby, A. N., Gutierrez, C. M. MNRAS, 294, LI (1998) 31. Henry, J.P. k Arnaud, K.A. ApJ, 372, 410 (1991) 32. Henry, J.P. ApJ, 489, LI (1997) 33. Henry, J.P. ApJ, 534, 565 (2000) 34. Kochanek, C.S. ApJ, 466, 638 (1996) 35. Le Dour, M., Douspis, M., Bartlett, J.G., Blanchard, A., A&A 364, 369 (2000) 36. Lemaitre, G. Ann. Soc. Sci. Bruxelles A 5 3 , 51P (1933) 37. Lineweaver, C , Barbosa, D., Blanchard, A. k Bartlett, J. A&A, 322, 365 (1997). 38. Lineweaver, C. k Barbosa, D. ApJ, 496, 624 (1998) 39. Maoz et al., ApJ, 409, 28 (1993) 40. Markevitch, M. ApJ, 503, 77 (1998) 41. Mathiesen, B., Evrard, A.E. k Mohr, J.J. ApJ, 520, L21 (1999) 42. O'Meara, J. M. et al., ApJ, 552, 718 (2001) 43. Mushotsky, R.F. k Sharf, C.A. ApJ, 482, L13 (1997) 44. Netterfield, C. B. et al., ApJ, 445, L69 (1995) 45. Neumann, D. k Arnaud, M. A&A, 373, L33 (2001) 46. Oukbir, J. k Blanchard A. A&A, 262, L21 (1992) 47. Oukbir, J. k Blanchard A. ,40.4, 317, 10 (1997) 48. Peebles, P. J. E., Daly, R. A. k Juszkiewicz, R. ApJ, 347, 563 (1989) 49. Pen, U., New astron., 2, 309 (1997) 50. Perlmutter, S. et al., ApJ, 483, 565 (1997) 51. Perlmutter, S. et al., ApJ, 517, 565 (1999)
69 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.
Reiprich, T. H. & Bohringer, H. ApJ, 567, 716 (2002) Reiss, D. et al., AJ, 116, 1009 (1998) Reichart, D.E. et al., ApJ, 518, 521 (1999) Roussel, H., Sadat, R. & Blanchard, A. A&A, 361, 429 (2000) Rowan-Robinson, M. astro-ph/0201034 (2002) Sachs, R. K.& Wolfe, A. M., ApJ, 147, 73 (1967) Sadat, R., Blanchard, A. & Oukbir, J. A&A, 329, 21 (1998) Sadat, R. k Blanchard, A. A&A, 371, 19 (2001) Sasaki, S., PASJ, 48, L119 (1996) Seljak, U., astro-ph/0111362, MNRAS, 334, 797 (2002) Van Waerbeke, L. et a l , astro-ph/0202503, A&A, in press (2002) Viana, P.T.R. & Liddle, A.R. MNRAS, 303, 535 (1999) Vikhlinin, A., Forman, W. k. Jones, C. ApJ, 525, 47 (1999) White S. D. M., Navaxro J. F. & Evrard A. E., Nature, 366, 429 (1993)
This page is intentionally left blank
PART 2
CONTRIBUTIONS
This page is intentionally left blank
Astroparticle Physics Beyond the Standard Model
This page is intentionally left blank
P H Y S I C A L IMPLICATIONS OF Q U I N T E S S E N T I A L B R A N E COSMOLOGIES
KERSTIN E. KUNZE Fakultdt fur Mathematik und Physik Universitdt Freiburg Hermann Herder Strafie 3 D-79104 Freiburg, Germany E-mail:
[email protected] M I G U E L A. V A Z Q U E Z - M O Z O Theory Division, CERN CH-1211 Geneva 23 Switzerland E-mail: Miguel.
[email protected]
A braneworld model of quintessence is considered where late time acceleration is due to the speeding of a brane in a static five-dimensional bulk space-time. T h e physical and cosmological implications of the model are studied.
1. I n t r o d u c t i o n The experimental realization that the Universe is undergoing a phase of accelerated expansion 1 has triggered a lot of activity oriented to accom modate this fact within the theoretical framework of our understanding of the Universe. Taken at face value, this acceleration seems to indicate the presence of a non-vanishing, though small, cosmological constant indicating that we are living in a patch of de Sitter space-time. From a string-theoretic point of view, live in de Sitter space is not easy. Applying the "strong" holo graphic principle one is led to the conclusion that the number of states in the (visible) universe is finite. In Ref. 2 it was argued that an ergodic interpretation of this fact seems to imply that "intelligible" universes, like ours seems to be, are extremely rare. This revives once more, at an even greater scale, the problem of the fine tuning of the initial condition of the Universe, something that it was thought to have been dealt with by the inflationary paradigm.
75
76
Of course, a non-vanishing cosmological constant is not the only way to explain why the Universe seems to be expanding at an accelerating rate. An interesting alternative is quintessence, where the vacuum energy dominance has its origin in the the existence of a classical scalar field rolling down a potential. Interestingly, the riddles posed by the authors of 2 are avoided in quintessence models as far as the asymptotic value of this potential is equal to zero. Speculative theoretical cosmology has been dominated in recent year by a new proposal according to which the observed universe is some kind of three-dimensional topological defect, a brane, embedded in a (4 + 1)dimensional bulk space-time. The main characteristic of these models is the fact that some physical mechanism ensures the confinement of matter to the brane, whereas gravity is free to propagate into the bulk. Originally, the model was intended to explain, among other facts, the weakness of gravity as compared to gauge interactions, a version of the old problem of explaining the energy hierarchy between the electroweak scale and the Planck scale. Although it is not really clear yet to what extend the scenario solves the hierarchy problem or just reformulate it, this alternative to the standard dimensional reduction indeed deserves further study in order to test it again cosmological data.
2. Quintessential brane cosmology Recently, a brane model of quintessence has been proposed where the late time acceleration of the universe is produced by the rolling of the brane world in a static bulk space time in the presence of a scalar field with a Liouville potential 3 . The action governing the dynamics in this model is
Ifc-I(00»-A*rt* d4x^
^K± _ A(^) - e46^brane (e2b V )
brane
(1)
As shown in 3 , there are solutions to the equations of motion derived from this action that describes a flat Friedmann-Robertson-Walker (FRW) brane moving in a static bulk space-time, which reflects in the four-dimensional space-time as a cosmological evolution. For the family of solutions consid ered, the metric in the bulk takes the form ds25B = l r t ( f c 2 - 3 ) d r 2 + r 2 (~dt2 + Sydx'cbt)
(2)
77
whereas the dilaton is logarithmic in the fifth coordinate,
(r) = klogr. The constant £ is related to the amplitude of the Liouville potential in the bulk by A = i ( f c 2 - 1 2 ) £ 2 . The braneworld (our universe) moves in the geometry (2) following the trajectory t = t(r), r = R(T) and x,y,z = constant, where r is the proper time. This means that the induced geometry in four-dimensions is a FRW metric, ds20 = —dr2 + R{T)2S^dx1 dx3. The dynamics of the scale factor -R(T) is determined by the junction conditions on the brane location. In the following, we will assume for simplicity that the bulk space-time has Z2-reflection symmetry around the position of the braneworld. Let us further assume that the matter on the brane is represented by a perfect fluid with equation of state p = wp. In this case, writing down the Israel junction conditions as well as the corresponding matching conditions for the bulk scalar one arrives at the Friedmann-like equation (see 3 for the details) 2
= *1GN(R)PQ(R)
+ ^-GN(R)p(R)
+ ^p(R)2
(3)
We recognize the structure of the typical braneworld Friedmann equation, with a term proportional to the energy density squared coupled directly to the five-dimensional Planck mass. The effective four-dimensional Newton constant GN(R) is given by GN(R)
= GNfiR-%
(4)
where GJV,O is the Newton constant at present given as a combination of the brane tension, the five-dimensional Planck scale and £. On the other hand, the energy density p(R) scales as
p(R) ~ R-3^<<+1\
WeS
= w + (w - i J kb
(5)
with w the barotropic index of the brane matter and k and b the constant appearing in the action (1). We find that, because of the conformal coupling between the bulk scalar field and the brane matter, the scaling of the density with the scale factor is governed by an "effective" barotropic index. Only in the absence of such a coupling (b = 0) or for a scale-invariant fluid (w = 5) the anomalous index coincides with the "bare" one a . a
A s explained in 3 , there is also a multiplicative rescaling of the energy density appearing in Eq. (3) with respect to the one calculated from the matter Lagrangian -Cbrane = — Pbare in Eq. (1). The factor differs from one by terms of order k2.
78
Finally we come to the interesting term, as far as late time acceleration of the universe is concerned. The "quintessence" energy density in Eq. (3) is given by PQ(R) = PQfiR-^
(6)
with the present time value pQto being expressed in terms of the fivedimensional Planck scale, the brane tension and the amplitude of the bulk potential. For "small" k we see that the barotropic index associated with ,2
i
the "quintessence" term, WQ = ^ — 1 will be in the region — 1 < WQ < — 5 and therefore will act as a veritable quintessence component in four dimen sions. A detailed study 3 shows indeed that this is the case. If k2 < 12 and PQ,O > 0 a radiation or matter dominated brane will inflate at late times, after going through a phase of standard decelerated expansion governed by the matter energy density. Since k also governs the time-variation of the Newton constant, a bound on this parameter can be obtained by demanding that this variation in time of GJV does not spoil big-bang nucleosynthesis. Using the results of Ref. 4 one finds that k2 < 0.03ft"1
(7)
At the same time pQto is estimated by the CMB+SNIa results indicating that Qdark — 0.7. These two results, put together, imply that the contri bution of the quintessence component to the energy density of the universe at the time of nucleosynthesis has to be of the order ^ IO- 3 5
^Q(TBBN) 5
(8)
much inside the bounds given in Ref. . Thus, for sufficiently flat potentials (i.e. k small enough) we have constructed a quintessence braneworld model that can accommodate big bang nucleosynthesis. It is important to stress that, regarding the dimensionful parameters in the action (1), cosmological observations can only put bounds on certain combinations of these parameters which are related to four-dimensional (brane) quantities, like pQto and Gjv.o- A direct determination of £ or K5, for example, would require direct measurement of "bulk" physics. By inspection of the effective Friedmann equation (3) wc conclude that the only window to bulk effects, available at this level, is the p2 term. As it was already mentioned the proposed model of brane quintessence has two distinct features: first, the existence of a time-varying Newton constant whose scaling with R is governed by the slope of the potential.
79
Secondly, the existence of an "anomalous" effective barotropic index due to the coupling of the bulk scalar field to the matter on the brane. If the time variation of the Newton constant depends only on k, the difference between the bare and effective barotropic indices for the brane'matter Actually, using the bound for k one can write weS-w<0.1b
(9)
i.e. a bound on b would require a direct measurement of the effective barotropic index to compare it with the "microscopic" one. An alternative way to put bounds on b is to look for time variation of the fundamental constants due to the coupling of the Lagrangian on the brane to the bulk scalar field. The Yang-Mills action coupled to massless fermions is classically scale invariant, so the corresponding terms in the £brane w m ^e insensitive to the coupling to the scalar field. The consequence is that the fundamental coupling constant a, astrong and the weak-mixing angle &w will be time-independent. This classical conformal invariance is only broken by scalar (Higgs) fields, fermion and gauge boson mass terms and/or Yukawa couplings. Therefore, the masses of the brane fields will acquire a time-dependence through a scaling with the scale factor of the form m{R) = moRhk which implies (-)
=bkH0
(10)
VWo where Ho is the Hubble parameter at the present time. Notice that the Fermi constant actually does vary with time, since its variation is induced by that of the mass of the W-boson. Since the variation of the mass is universal for all particles, mass ratios will be time-independent. 3. Concluding remarks We have shown how a phenomenologically viable model of quintessence can be implemented in the context of the braneworld scenario by considering a three-dimensional brane moving in a static bulk space-time with a Liouville scalar field. This "mirage" cosmological evolution induced on the braneworld is governed by an effective Friedmann equation of the type found in the brane cosmological scenarios, i.e. it contains a term proportional to the energy density squared that modifies the standard cosmological evo lution at early times. In our model, in addition, the nontrivial potential for the bulk scalar field induces an additional term in the Friedmann equa tion that, when the potential is flat enough, effectively acts a quintessence
80
component, producing an accelerated expansion at late times. Notice that since in our model the scale factor R(T) also parametrizes the position of the brane along the extra dimension this acceleration in the expansion corresponds to a kinematical acceleration of the brane in the bulk. It is important to stress that here we have taken a "phenomenological" attitude, in the sense that we are not trying to provide a first principle derivation of our model on string/M-theoretic grounds. Therefore, we re gard the constants appearing in the Lagrangian (1) as parameters to be fixed or bounded from observations. It is however interesting that the kind of exponential potential considered here for the scalar field is quite generic in supergravity. It would be very interesting to further study some of the implications of these models in the context of particle physics. Regarding the now fashionable variation of the fundamental constants, as explained above, the conformal coupling between the scalar field and the Lagrangian on the brane seem to preclude any time variation of the coupling constants of the standard model. Only masses and, indirectly, Fermi's constant, are subject to time variation governed by the constant b in the Lagrangian. This fact could be used in principle to put bounds on this parameter in the same way that the time variation of Newton constant provides a bound on k, the slope of the potential. Notice, however, that the quintessential behavior of the model presented here does not seem to depend strongly on the particular coupling consid ered between the scalar field and the matter on the brane. More general couplings between <j> and the standard model Lagrangian, like the ones considered for example in Ref 6 , might result in an effective variation of fundamental constans while preserving the late time acceleration of the four-dimensional universe. This, and other issues, will be addressed else where. Acknowledgments M.A.V.-M. thanks the organizers of Astro 2002, and in particular Robertus Potting, for their kind invitation to present this work and their hospitality in Faro. References 1. S. Perlmutter et al., Astrophys. J. 517 (1999) 565; A. G. Riess et al, Astron. J. 116 (1998) 1009.
81
2. L. Dyson, M. Kleban and L. Susskind, J. High Energy Phys. 0210 (2002) Oil. 3. K.E. Kunze and M.A. Vazquez-Mozo, Phys. Rev. D 6 5 (2002) 044002. 4. E.W. Kolb, M.J. Perry and T.P. Walker, Phys. Rev. D 3 3 (1986) 869 F.S. Ascetta, L.M. Krauss and P. Romanelli, Phys. Lett/248 (1990) 146. 5. R. Bean, S.H. Hansen and A. Melchiorri, Phys. Rev. D 6 4 (2001) 103508. 6. P. Brax, C. van de Bruck, A.C. Davis and C.S. Rhodes, Varying constants in brane world scenarios, hep-ph/0210057.
D A R K RADIATION A N D LOCALIZATION OF GRAVITY ON T H E B R A N E
R U I N E V E S A N D C E N A L O VAZ Area Departamental de Fisica/CENTRA, FCT, Universidade Campus de Gambelas, 8000-117 Faro, Portugal E-mail: [email protected], [email protected]
do
Algarve
We discuss the dynamics of a spherically symmetric dark radiation vaccum in the Randall-Sundrum brane world scenario. Under certain natural assumptions we show that the Einstein equations on the brane form a closed system. For a de Sitter brane we determine exact dynamical and inhomogeneous solutions which depend on the brane cosmological constant, on the dark radiation tidal charge and on its initial configuration. We define the conditions leading to singular or globally regular solutions. We also analyse the localization of gravity near the brane and show that a phase transition to a regime where gravity propagates away from the brane may occur at short distances during the collapse of positive dark energy density.
1. Introduction In the search for extra spatial dimensions the Randall and Sundrum (RS) brane world scenario is particularly interesting for its simplicity and depth 1 . In this model the Universe is a 3-brane boundary of a noncompact Z 2 symmetric 5-dimensional anti-de Sitter space. The matter fields live only on the brane but gravity inhabits the whole bulk and is localized near the brane by the warp of the infinite fifth dimension. Since its discovery many studies have been done within the RS scenario (see Ref. 2 for a recent review and notation). For a brane bound observer 3,4 s ' the interaction between the brane and the bulk introduces correction terms to the 4-dimensional Einstein equations, namely, a local high energy embedding term generated by the matter energy-momentum tensor and a non-local term induced by the bulk Weyl tensor. Such equations have an intrincate non-linear dynamics. For example, the exterior vaccum of collapsing matter on the brane is now filled with gravitational modes origi nated by the bulk Weyl curvature and can no longer be regarded as a static space 6 ' 7 .
82
83
Previous research on the RS scenario has been focused on static or ho mogeneous dynamical solutions. In this proceedings we report some new results on the dynamics of a spherically symmetric RS brane world vaccum. For a de Sitter brane we present exact dynamical and inhomogeneous so lutions, define the conditions to characterize them as singular or globally regular and discuss the localization of gravity to the vicinity of the brane (see Ref. 8 for more details). 2. Brane Vaccum Field Equations In the Gauss-Codazzi formulation of the RS model 3 ' 4 ' 5 , the Einstein vac cum field equations on the brane are given by G^v = — A#M„ — E^,
(1)
where A is the brane cosmological constant and the tensor S^u is the limit on the brane of the projected 5-dimensional Weyl tensor. It is a symmetric and traceless tensor constrained by the following conservation equations V„5£ = 0.
(2)
The projected Weyl tensor £M„ can be written in the following general form 5 K
I 4
U [ U^Uv + - h ^ ) + Vftv + Q^Uu +
QvUti
(3)
where uM such that u^u^ = - 1 is the 4-velocity field and /iM„ = (?M„ + wMu„ is the tensor projecting orthogonaly to uM. The forms U, V^v and Q p represent different aspects of the effects induced on the brane by the 5dimensional gravitational field. Thus, U is an energy density, Vp,v a stress tensor and QM an energy flux. Since the 5-dimensional metric is not known, in general £^v is not com pletely determined on the brane 3 ' 4 and so the effective 4-dimensional theory is not closed. To close it we need simplifying assumptions about the effects of the gravitational field on the brane. For instance we may consider a static and spherically symmetric brane vaccum with QM = 0, V^v ^ 0 and i / ^ 0 . This leads to the Reissner-Nordstrom black hole solution on the brane 9 . It is also possible to close the system of Einstein equations when consid ering a dynamical and spherically symmetric brane vaccum with QM = 0, U ^ O , and PM„ ^ 0. The general, spherically symmetric metric in comoving coordinates (t, r, 6, <j>) is given by ds2 = gllvdx>idxv = -eadt2
+ A2dr2 + R2dVl2,
(4)
84
where dtt2 = d62 + sm26d<j)2, a - a(t,r), A = A(t,r), R = R(t,r) and R is the physical spacetime radius. If the traceless stress tensor 7 V is isotropic then it will have the general form ? V = V [rfirv - -h^v
,
(5)
where V = V(t, r) and rM is the unit radial vector, given in the above metric by !•„ = (0,4,0,0). Then £1= ( - J diag (p, -pr, -Pr, ~PT) ,
(6)
where the energy density and pressures are, respectively, p — U, pr = (1/3) {U + IV) and pT = (1/3) (U - V). Consequently, the conservation Eq. (2) read 10
2j(p+pr)
= -2p-4-(p
+ pT),
JDl
a'(p+Pr)
= -2Pr'+4—(pT-pr),
(7)
where the dot and the prime denote, respectively, derivatives with respect to t and r. A synchronous solution is permitted with the equation of state p + pr = 0, equivalent to V = -111 where U has the dark radiation form
The constant Q is the dark radiation tidal charge. Hence, we get GV = ~ A 3 M " + -pi ( U M U ^
_ 2r r
n v + hnv),
(9)
an exactly solvable closed system for the unknown functions A(t, r) and R(t, r) which depends on the free parameters A and Q. Indeed, its solutions are of the LeMaitre-Tolman-Bondi type ds2 = -dt2 + ^—-dr2
+ R2dfl2,
(10)
where R satisfies (11) ^ = i * 2 ~ R^2 + fThe function / = f(r) > - 1 is interpreted as the energy inside a shell labelled by r in the dark radiation vaccum and is fixed by its initial config uration.
85
3. Localization of Gravity near the Brane As is clear in Eq. (9) the dark radiation dynamics depends on A and Q. It is important to point out that these parameters have a direct effect on the localization of gravity in the vicinity of the brane. Indeed, the tidal acceleration away from the brane 5 is given by 9 - lim kABGDnAuBncuD
= ^ A + %,
(12)
where UA is the extension off the brane of the 4-velocity field satisfying UAUA = 0 and UAUA = — 1. The gravitational field is only bound to the brane if the tidal acceleration points towards the brane. It must then be negative implying that ~AR4 < - %
(13)
As a consequence, gravity is only localized for all R if A < Ac with Ac = rc4A2/12 and Q < 0 or A = Ac and Q < 0. For A < Ac and Q > 0 the gravitational field will just remain localized if R > Rc where i? 4 = 3Q/(A C — A). On the other hand for A > Ac and Q < 0 localization is limitted to the epochs R < Rc. If A > Ac and Q > 0 then gravity is always free to propagate far away into the bulk. According to recent supernovae measurements (see e.g. Ref. n ) A ~ 10- 84 GeV 2 . On the other hand M p > 108GeV and M p ~ 1019GeV imply A > 108GeV4 12 because 6/t2 = AK 4 . Since Ac = K 2 A / 2 then Ac is bound from below, Ac > 10 _29 GeV 2 . Hence, observations demand A to be positive and smaller than the critical value A c , 0 < A < Ac. Note that is means an anti-de Sitter bulk, A < 0. The same conclusion is true if M p is in the TeV range because Ac increases when M p decreases. Since current observations do not yet constrain the sign of Q 13 we conclude that for 0 < A < Ac only for Q < 0 gravity is bound to the brane for all R. If Q > 0 then for R < Rc the tidal acceleration is positive and gravity is no longer localized near the brane. 4. Inhomogeneous Dynamics for a de Sitter Brane Assume from now on that 0 < A < A c . Non-static solutions correspond to / ^ 0. An example is
R
+
2A
= A//? cosh ±2\
-t + cosh~l '
3
I V ?
(14)
86
where /? = (3/A)[3/ 2 /(4A) + Q] and + or - correspond respectively to ex pansion or collapse. If Q > 0 then / > — 1 but for Q < 0 the energy function / must satisfy in addition | / | > 2y/—QA/3. Since R is a non-factorizable function of t and r these solutions define new exact and inhomogeneous cosmologies for the spherically symmetric dark radiation de Sitter brane. 5. Singularities and Regular Bounces The dark radiation dynamics defined by Eq. (11) may produce shell focusing singularities at R — 0 or regular bouncing points at some R ^ 0. To see this consider R2R2 = V(R,r)
= ^R4 + fR2-Q.
(15)
If for all R > 0 the potential V is positive then a shell focusing singularity forms at R — Rs = 0. Alternatively, if there is an epoch R = R* ^ 0 for which V — 0 then a regular rebounce point appears at R = i?*. For the dark radiation vaccum at most two regular rebounce epochs can be found. Since A > 0 there is always a phase of continuous expansion to infinity with ever increasing rate. Depending on f(r) other phases may exist. To ilustrate take j3 > 0 and compare the settings Q < 0, / > —1, | / | > 2y / -<2A/3 and Q>0,/>-l. 0.1
0.08
0.06
V 0.04 0.02
0.2
0.4
R2
0.6
0.8
1
Figure 1. Plot of V for /3 > 0 and Q < 0. Non-zero values of / belong to the interval — 1 < / < —2^/—QA/3 and correspond to shells of constant r. If for Q < 0 we have / > 2^/-QA/3 then V > 0 for all R > 0. There are no rebounce points and the dark radiation shells may either expand continuously or collapse to a singularity at Rs = 0. However for — 1 < / < —2yJ— QA/3 (see Fig. 1) we find two rebounce epochs at R = i?*± with
87
Rl± = - 3 / / ( 2 A ) ± V/3- Since 1/(0, r) = -Q > 0 a singularity also forms at i? s = 0. Between the two rebounce points there is a forbidden zone where V is negative. The phase space of allowed dynamics is thus divided in two disconnected regions separated by the forbidden interval' i?*_ < R < R*+. For 0 < R < R*~ the dark radiation shells may expand to a maximum radius R = i?*-, rebounce and then fall to the singularity. If R > i?* + then there is a collapsing phase to the minimum radius R = R*+ followed by reversal and subsequent accelerated continuous expansion. The singularity at Rs = 0 does not form and so the solutions are globally regular. Since Q < 0 gravity is bound to the brane for all the values of R. 10
8 6
V 4 2
0.5
1
1.5
2
2.5
3
3.5
4
R2 Figure 2. Plot of V for (3 > 0 and Q > 0. Non-zero values of / belong to the interval / > — 1 and correspond to shells of constant r. The shaded region indicates where gravity is not localized near the brane. If Q > 0 (see Fig. 2) then we find globally regular solutions with a single rebounce epoch at R = R* where R% = - 3 / / ( 2 A ) + ^/0. This is the minimum possible radius for a collapsing dark radiation shell. It then reverses its motion and expands forever. The phase space of allowed dynamics denned by V and R is limitted to the region R> R*. Below i?» we find a forbidden region where V is negative. In particular, V(0,r) = —Q < 0 implying that the singularity at Rs = 0 does not form and so the solutions are globally regular. Note that if gravity is to be bound to the the brane for R > R* then i?» > Rc. If not then we find a phase transition epoch R = Rc such that for R < Rc the gravitational field is no longer localized near the brane. 6. Conclusions In this work we have reported some new results on the dynamics of a RS brane world dark radiation vaccum. Using an effective 4-dimensional
88
approach we have shown that some simplifying but natural assumptions lead to a closed and solvable system of Einstein field equations on the brane. We have presented a set of exact dynamical and inhomogeneous solutions for A > 0 showing they further depend on the dark radiation tidal charge Q and on the energy function f(r). We have also described the conditions under which a singularity or a regular rebounce point develop inside the dark radiation vaccum and discussed the localization of gravity near the brane. In particular, we have shown that a phase transition to a regime where gravity is not bound to the brane may occur at short distances during the collapse of positive dark energy density on a realistic de Sitter brane. Left for future research is for example an analysis of the dark radiation vaccum dynamics from a 5-dimensional perspective. Acknowledgements We thank the Fundacao para a Ciencia e a Tecnologia (FCT) for fi nancial support under the contracts POCTI/SFRH/BPD/7182/2001 and POCTI/32694/FIS/2000. We also would like to thank Louis Witten and T.P. Singh for kind and helpful comments. References 1. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); Phys. Rev. Lett. 83, 4690 (1999). 2. R. Maartens, Geometry and Dynamics of the Brane World, arXiv:grqc/0101059. 3. T. Shiromizu, K. I. Maeda and M. Sasaki, Phys. Rev. D62, 024012 (2000). 4. M. Sasaki, T. Shiromizu and K. I. Maeda, Phys. Rev. D62, 024008 (2000). 5. R. Maartens, Phys. Rev. D62, 084023 (2000). 6. M. Bruni, C. Germani and R. Maartens, Phys. Rev. Lett. 87, 231302 (2001). 7. M. Govender and N. Dadhich, Phys. Lett. B538, 233 (2002). 8. R. Neves and C. Vaz, Dark Radiation Dynamics on the Brane, arXiv:hepth/0207173, to be published in Phys. Rev. D. 9. N. Dadhich, R. Maartens, P. Papadopoulos and V. Rezania, Phys. Lett. B487, 1 (2000). 10. T. P. Singh, Class. Quant. Grav. 16, 3307 (1999). 11. N. A. Bahcall, J. P. Ostriker, S. Perlmutter and P. J. Steinhardt, Science 284, 1481 (1999). 12. D. Langlois, R. Maartens, M. Sasaki and D. Wands, Phys. Rev. D63, 084009 (2001). 13. K. Ichiki, M. Yahiro, T. Kajino, M. Orito and G. J. Mathews, Phys.Rev. D66, 043521 (2002).
A S T R O P H Y S I C A L TESTS OF LORENTZ S Y M M E T R Y I N ELECTRODYNAMICS
MATTHEW MEWES Physics Department, Indiana University, Bloomington, IN 47405, E-mail: [email protected]
U.S.A.
In this talk presented at the Fourth International Workshop on New Worlds in Astroparticle Physics, I discuss recent constraints on Lorentz violation in electro dynamics. The observed absence of birefringence of light that has propagated over cosmological distances bounds some coefficients for Lorentz violation to 2 x 10 - 3 2 .
1. Introduction The exact character of physics beyond the standard model is an open ques tion. The standard model is commonly believed to be the low-energy limit of Planck-scale physics which unifies all known forces. Due to the energy scales involved, an experimental search for this new physics would seem pointless given the current technology. However, some high-energy theories may lead to violations in symmetries which hold exactly in the standard model. 1 ' 2 In particular, spontaneous symmetry breaking in the fundamental theory might result in apparent violations in the Lorentz and CPT symme tries. Furthermore, Lorentz and CPT violations can be tested to extremely high precision using today's technology.3 A general Lorentz-violating extension to the standard model has been constructed. 3 ' 4 It consists of the minimal standard model plus small Lorentz- and CPT-violating terms. The standard-model extension has pro vided a theoretical framework for many searches for Lorentz and CPT vio lations. To date, experiments involving hadrons, 5 ' 6 protons and neutrons, 7 electrons, 8 ' 9 photons, 10 ' 11 and muons 12 have been performed. In practice, one often works with a particular limiting theory extracted from the standard-model extension. For example, the photon sector of the standard-model extension yields a Lorentz-violating modified electro dynamics. The theory predicts several unconventional features that lead to sensitive tests of Lorentz symmetry. For example, in the presence of
89
90 certain forms of Lorentz violation, light propagating through the vacuum will experience birefringence. The absence of birefringence in light emitted from distant sources leads to tight bounds on some of the coefficients for Lorentz violation. 10 ' 11 In this work, I review some of these bounds. This research was done in collaboration with Alan Kostelecky. A detailed discussion can be found in the literature. 11 2. Lorentz-Violating Electrodynamics The modified electrodynamics maintains the usual gauge invariance and is covariant under observer Lorentz transformations. It includes both CPTeven and -odd terms. The CPT-odd terms have been the subject of nu merous experimental and theoretical investigations. 4 ' 10 ' 13,14 For example, some of these terms have been bounded to extremely high precision using polarization measurements of distant radio galaxies. 10 In contrast, until re cently, the CPT-even terms have received little attention. Here, I review a recent study of these terms. 11 The CPT-even lagrangian for the modified electrodynamics is 4 £ = -\F^F"V
- \{kF)KXliVFKXF^
,
(1)
where F^v is the field strength, FM„ = d^Av — duA^. The first term is the usual Maxwell lagrangian. The second is an unconventional Lorentzviolating term. The coefficient for Lorentz violation, {kp)^^, is real and comprised of 19 independent components. The absence of observed Lorentz violation implies {kp)^^ is small. The equations of motion for this la a grangian are daF^ + (kp)^a/}JdaFf}'Y = 0. These constitute modified source-free inhomogeneous Maxwell equations. The homogeneous Maxwell equations remain unchanged. A particularly useful decomposition of the 19 independent components can be made. 11 The lagrangian in terms of this decomposition is C = |[(1 + ktv)E2 - (1 - ktr)B2} + \E ■ (ke+ + ke_) ■ E -\B
■ (ke+ - « e _) -B + E- (k0+ +
K0_)
•B ,
(2)
where E and B are the usual electric and magnetic fields. The 3 x 3 matrices Ke+, Ke_, k0+ and K 0 _ are real and traceless. The matrix k0+ is antisym metric, while the remaining three are symmetric. The real coefficient ktr corresponds to the only rotationally invariant component of (/CF)/KI/37From the form of Eq. (2), we see that the component ktr can be thought of as a shift in the effective permittivity e and effective permeability \x by
91
(e — 1) = —(A*"1 — 1) = ktI. The result of this shift is a shift in the speed of light. Normally, this may be viewed as a distortion of the metric. In fact, this result generalizes to the nine independent coefficients in « t r , ke- and k0+. To leading order, these may be viewed as a distortion of the spacetime metric of the form rfv —> ri^" + h1*", where ¥lv is small, real and symmetric. Small distortions of this type are unphysical, since they can be elimi nated through coordinate transformations and field redefinitions. However, each sector of the full standard-model extension contains similar terms. Eliminating these terms from one sector will alter the other sectors. There fore, the effects of such terms can not be removed completely from the theory. As a consequence, in experiments where the properties of light are compared to the properties of other sectors, these terms are relevant. How ever, in experiments where only the properties of light are relevant, the nine coefficients in ktT, ke- and k0+ are not expected to appear. The tests discussed here rely on measurements of birefringence. This involves com paring the properties of light with different polarizations. Therefore, these tests compare light with light and are only sensitive to the ten independent components of Ke+ and k0-. Constraints on birefringence have been expressed in terms of a tendimensional vector ka containing the ten independent components of ke+ and Ho-.11 The relationship between ke+, «„_ and ka is given by k
(ke+y
f-(k3+k4) =A;5 V k6
(2k2 -k9 9 («„_)''* = -k -2k1
V ks
k5 k6\ k3 k7 , k7k*J ka k10
\ .
(3)
kw 2{kl - k 2 ) J
Bounds on birefringence appear as bounds on \ka\ = Vkaka, the magnitude of the vector ka. 3. Birefringence In order to understand the effects of Lorentz violation on the propagation of light, we begin by considering plane-wave solutions. Adopting the ansatz F^u{x) — Ftlve~WaX'" and solving the modified Maxwell equations yields the dispersion relation p°± = (l +
p±a)\p\.
(4)
92 In a frame where the phase velocity is along the z-axis, the electric field takes the form E± o c ( s i n f , ± l - c o s f , 0 ) + O ( f c f ) .
(5)
To leading order, the quantities p, a sin £ and a cos £ are linear combinations of {kp)K\]xv and depend on v, the direction of propagation. A prediction of these solutions is the birefringence of light in the vacuum. Birefringence is commonly found in conventional electrodynamics in the presence of anisotropic media. In the present context, the general vacuum solution is a linear combination of the E+ and £?_. For nonzero a, these solutions obey different dispersion relations. As a result, they propagate at slightly different velocities. At leading order, the difference in the velocities is given by Aw = v+ — V- = 2(7 .
(6)
For light propagating over astrophysical distances, this tiny difference may become apparent. As can be seen from the above solutions, birefringence depends on the linear combination a sin £ and a cos £. As expected, these only contain the ten independent coefficients which appear in ke+ and re0_. Expressions for a sin £ and a cos £ in terms of these ten independent coefficients and the direction of propagation can be found in the literature. 11 Next I discuss two observable effects of birefringence. The first effect is the spread of unpolarized pulses of light. The second is the change in the polarization angles of polarized light.
3.1. Pulse-Dispersion
Constraints
The narrow pulses of radiation from distant sources such as pulsars and gamma-ray bursts are well suited for searches for birefringence. In most cases, the pulses are relatively unpolarized. Therefore, the components E± associated with each mode will be comparable. The difference in velocity will induce a difference in the observed arrival time of the two modes given by At ~ AvL, where L is the distance to the source. Sources which produce radiation with rapidly changing time structure may be used to search for this difference in arrival time. For example, the sources mentioned above produce pulses of radiation. The pulse can be regarded as the superposition of two independent pulses associated with each mode. As they propagate, the difference in velocity will cause the two
93
pulses to separate. A signal for Lorentz violation would then be a mea surement of two sequential pulses of similar time structure. The two pulses would be linearly polarized at mutually orthogonal polarization angles. The above signal for birefringence has not yet been observed. However, existing pulse-width measurements place constraints on Lorentz violation. To see this, suppose a source produces a pulse with a characteristic time width w$. As the pulse propagates, the two modes spread apart and the width of the pulse will increase. The observed width can be estimated as w0 ~ ws + At. Therefore, observations of w0 place conservative bounds on At ~ AvL ~ 2aL. The resulting bound on a constrains the tendimensional parameter space of ke+ and /c 0 _. Since a single source con strains only one degree of freedom, at least ten sources located at different positions on the sky are required to fully constrain the ten coefficients. Using published pulse-width measurements for a small sample of fifteen pulsars and gamma-ray bursts, we found bounds on a for fifteen different propagation directions v. Combining these bounds constrained the tendimensional parameter space. At the 90% confidence level, we obtained a bound of \ka\ < 3 x 10~ 16 on the coefficients for Lorentz violation. 11 3.2. Polarimetry
Constraints
The difference in the velocities of the two modes results in changes in the polarization of polarized light. Decomposing a general electric field into its birefringent components, we write E(x) = (E+e~ip+t + E_e~ip-t)etp''s. Each component propagates with a different phase velocity. Consequently, the relative phase between modes changes as the light propagates. The shift in relative phase is given by A = (p0+-p°_)t~4iraL/\,
(7)
where L is the distance to the source and A is the wavelength of the light. This phase change results in a change in the polarization. The L/X dependence suggests the effect is larger for more distant sources and shorter wavelengths. Recent spectropolarimetry of distant galaxies at wavelengths ranging from infrared to ultraviolet has made it possible to achieve values of L/X greater than 10 31 . Given that measured polarization parameters are typically of order 1, we find an experimental sensitivity of 10~ 31 or better to components of ( & F ) K A ^ In general, plane waves are elliptically polarized. The polarization el lipse can be parameterized with angles tp, which characterizes the orienta tion of the ellipse, and x — ±arctan " ' * ] " !"ds' w m c n describes the shape
94 of the ellipse and helicity of the wave. The phase change, Acf>, results in a change in both tp and x- However, measurements of \ are not commonly found in the literature. Focusing our attention on if), we seek an expression for Sip = tjj — ip0, the difference between ip at two wavelengths, A and AoWe find11 5ib = - tan" 1
sin
£ c o s C o + cos|sinCocos((ty - — (po)
where we have defined S(j> = iira(L/X — L/\Q), £ = £ — 1i\)§ a n d
95
2.
3. 4.
5.
6.
7.
8.
9. 10. 11. 12. 13. 14.
Phys. Rev. D 63, 046007 (2001); V.A. Kostelecky, M. Perry, and R. Potting, Phys. Rev. Lett. 84, 4541 (2000); M.S. Berger and V.A. Kostelecky, Phys. Rev. D 65, 091701(R) (2002). S.M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001); Z. Guralnik et al., Phys. Lett. B 517, 450 (2001); A. Anisimov et al., hep-ph/0106356; C.E. Carlson et al., Phys. Lett. B 518, 201 (2001). For overviews of various ideas, see, for example, V.A. Kostelecky, ed., CPT and Lorentz Symmetry II, World Scientific, Singapore, 2002. V.A. Kostelecky and R. Potting, Phys. Rev. D 51, 3923 (1995); D. Colladay and V.A. Kostelecky, Phys. Rev. D 55, 6760 (1997); 58, 116002 (1998); V.A. Kostelecky and R. Lehnert, Phys. Rev. D 63, 065008 (2001). KTeV Collaboration, H. Nguyen, in Ref. [3]; OPAL Collaboration, R. Ackerstaff et al., Z. Phys. C 76, 401 (1997); DELPHI Collaboration, M. Feindt et al., preprint DELPHI 97-98 CONF 80 (1997); BELLE Collaboration, K. Abe et al., Phys. Rev. Lett. 86, 3228 (2001); FOCUS Collaboration, J.M. Link et al., hep-ex/0208034. D. Colladay and V.A. Kostelecky, Phys. Lett. B 344, 259 (1995); Phys. Rev. D 52, 6224 (1995); Phys. Lett. B 511, 209 (2001); V.A. Kostelecky and R. Van Kooten, Phys. Rev. D 54, 5585 (1996); V.A. Kostelecky, Phys. Rev. Lett. 80, 1818 (1998); Phys. Rev. D 6 1 , 016002 (2000); 64, 076001 (2001); N. Isgur et al., Phys. Lett. B 515, 333 (2001). L.R. Hunter et al., in V.A. Kostelecky, ed., CPT and Lorentz Symmetry, World Scientific, Singapore, 1999; D. Bear et al., Phys. Rev. Lett. 85, 5038 (2000); D.F. Phillips et al, Phys. Rev. D 63, 111101 (2001); M.A. Humphrey et al., Phys. Rev. A 62, 063405 (2000); V.A. Kostelecky and C D . Lane, Phys. Rev. D 60, 116010 (1999); J. Math. Phys. 40, 6245 (1999); R. Bluhm et al., Phys. Rev. Lett. 88, 090801 (2002). H. Dehmelt et al., Phys. Rev. Lett. 83, 4694 (1999); R. Mittleman et al., Phys. Rev. Lett. 83, 2116 (1999); G. Gabrielse et al, Phys. Rev. Lett. 82, 3198 (1999); R. Bluhm et al, Phys. Rev. Lett. 82, 2254 (1999); Phys. Rev. Lett. 79, 1432 (1997); Phys. Rev. D 57, 3932 (1998). B. Heckel in Ref. [3]; R. Bluhm and V.A. Kostelecky, Phys. Rev. Lett. 84, 1381 (2000). S. Carroll, G. Field, and R. Jackiw, Phys. Rev. D 4 1 , 1231 (1990). V.A. Kostelecky and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); Phys. Rev. D 66, 056005 (2002). V.W. Hughes et al., Phys. Rev. Lett. 87, 111804 (2001); R. Bluhm et al, Phys. Rev. Lett. 84, 1098 (2000). R. Jackiw and V.A. Kostelecky, Phys. Rev. Lett. 82, 3572 (1999). V.A. Kostelecky et al., Phys. Rev. D 65, 056006 (2002).
A P P A R E N T LORENTZ VIOLATION T H R O U G H SPACETIME-VARYING COUPLINGS
RALF LEHNERT Physics Department, Universidade do Algarve 8000 Faro, Portugal E-mail: [email protected] In this talk, we explore the relation between smoothly varying couplings and Lorentz violation. Within the context of a supergravity model, we present an ex plicit mechanism that causes the effective fine-structure constant and the effective electromagnetic 6 angle to acquire related spacetime dependences. We argue that this leads to potentially observable Lorentz violation and discuss some implications for the standard-model extension.
1. Introduction Originally proposed by Dirac, 1 the idea of spacetime-dependent couplings has remained the subject of a variety of experimental and theoretical in vestigations. As a result of current claims of observational evidence for a time variation of the electromagnetic coupling2 and the recent realiza tion that such effects are a natural consequence of many unified theories, 3 this idea has regained considerable interest. 4 Although there has been sub stantial theoretical progress in the subject, a realistic underlying theory allowing concrete predictions is presently still lacking. For the identifica tion of high-precision tests, it is therefore important to determine generic physical effects caused by spacetime-dependent couplings. In models with varying couplings, the usual spacetime symmetries de scribed by the Poincare group can be effected. For example, translational invariance is normally lost in such models. In this talk, we argue that vary ing couplings can also lead to violations of the remaining spacetime sym metries associated with the Lorentz group, a point not widely appreciated. Intuitively, this can be understood when the effective vacuum is interpreted as a spacetime-varying medium, in which, for example, isotropy can be lost so that certain rotations, which are contained in the Lorentz group, may no longer be associated with a symmetry transformation.
96
97
A partial motivation for this study is provided by the extreme sensi tivity of experimental Lorentz tests and by recent progress in the under standing of a general Lorentz- and CPT-violating extension of the stan dard model, 5 a framework that includes all possible coordinate-invariant Lorentz- and CPT-breaking interactions. It describes the low-energy limit of possible Lorentz and CPT violation at a more fundamental level, such as strings, 6 nontrivial spacetime topology,7 and realistic noncommutative field theories. 8 The standard-model extension has provided the basis for numer ous experimental investigations with hadrons, 9 ' 10 protons and neutrons, 11 electrons, 12 ' 13 photons, 14 ' 15 and muons 16 placing constraints on possible violations of Lorentz and CPT symmetry. We remark also that in this context, the inverse line of reasoning has already been discussed: Certain constant parameters in the Lorentz- and CPT-breaking standard-model ex tension are equivalent to spacetime-dependent masses.5 As part of our analysis, we construct a classical cosmological solution in the framework of the JV = 4 supergravity in four spacetime dimensions demonstrating how the fine-structure constant a and the electromagnetic 9 angle can acquire related spacetime dependences. Although this model is known to be unrealistic in detail, it is a limit of the N = 1 supergravity in 11 dimensions, which is directly related to M-theory. It could therefore yield valuable insight into generic features of a promising candidate fundamental theory. Moreover, a smoothly varying 9 angle can be associated with a Lorentz-breaking Chern-Simons-type interaction. Our explicit mechanism for a varying 9 in the context of a consistent supergravity model therefore sheds some light on the usual theoretical difficulties associated with this term and how they may be avoided. 2. Cosmology When only one graviphoton, F^, is excited and Planck units are adopted, the bosonic lagrangian for the N = 4 supergravity in four dimensions is 17 £ = y/g(-±R-
\MFIXVF>"' - \NF^F»V
+
dpAd'lA + 4B2
where g^ represents the metric, F^v = e^l'paFpa/2,
dftBditB\ ) ' (1)
and
B(A* + B* + 1) A(A* + B*-l) [) (1 + A2 + 5 2 ) 2 - 4A2 ' (1 + A2 + B*)*-4A2' The conventional complex scalar denoted by Z, which contains an axion and a dilaton, 17 is related to A and B via a canonical transformation, 18 such that B can be identified with the string-theory dilaton.
98 Within this model, one can determine a simple classical solution. To this end, we set FM„ to zero for the moment and assume a flat (k = 0), ho mogeneous and isotropic Universe. In comoving coordinates, the associated metric has the usual Friedmann-Robertson-Walker (FRW) line element ds2 =dt2-a2(t)(dx2
+ dy2 + dz2),
(3)
where a(t) denotes the cosmological scale factor. The above assumptions imply also that A and B can depend on t only. For a more realistic situation, the known matter content of the Universe needs to be modeled. An often employed approach is to include the energy-momentum tensor of dust, Ty.v. If uM is the unit timelike vector orthogonal to the spatial surfaces and p(t) is the energy density of the dust, it follows that T^ = pu^u^, as usual. In the present context, this type of matter arises from the fermionic sector of our supergravity model. Since the scalars A and B do not couple to the fermion kinetic terms, 17 TM„ is conserved separately. In this model, the evolution of the scale factor is determined by 18 a(t) = ( | c n ( t + y / 4ci/3c„) 2 — ci) 1 / 3 . Here, the integration constants c n and c\ control the amount of energy stored in the dust and the scalar fields, respectively. Another integration constant has been chosen such that the initial singularity a(t) — 0 occurs at t = 0. The dependences of A and B on a parameter time defined by r = \/3/4 arcoth(i/3c n /4ci t +1) are given by 18 A = ±A tanh (
c 3 ) + A0,
B = A sech (
c3 ) ,
(4)
where A, C3, and AQ are integration constants. It can be verified that both A and B tend to constant values as t —> 00. It follows that in our supergravity cosmology the values of the axion A and the dilaton B are fixed despite the absence of a dilaton potential. This essentially results from the conservation of energy. 3. Spacetime-varying couplings We proceed by considering localized excitations of F^v in the scalar back ground given by (4). Since experimental investigations are often performed in spacetime regions small on a cosmological scale, it is appropriate to work in local inertial frames. In the presence of a nontrivial #-angle, the conventional electrodynamics lagrangian in inertial coordinates can be taken as
Cem = - ^ V "
- 1 ^ V"*.
(5)
99
where e is the electromagnetic coupling. Comparison with our supergravity model shows that e 2 = 1/M and 6 = 4TT2N. Note that M and N are functions of the axion-dilaton background (4), so that e and 8 acquire related spacetime dependences in an arbitrary local inerti-al frame. The spacetime dependence of both a = e2/47r and 8 can be relatively complicated and can vary qualitatively with the choice of model parame ters. Figure (1) depicts relative variations of a for 1/TO = 0 in comoving coordinates. The fractional look-back time to the big bang is defined by 1 - t/tn, where tn denotes the present age of the Universe. Each broken line is associated with a set of nontrivial choices for A, y / 3c„/4ci tn, and AQ. Parameter sets consistent with the Oklo constraints 19 are marked with an asterisk. Note the qualitative differences in the various plots, the non linear features, and the sign change for a in the two cases with positive AQ. Also shown in Fig. (1) are the recent experimental results 2 obtained from measurements of high-redshift spectra over periods of approximately 0.6i„ to 0.8i„ assuming H0 — 65 km/s/Mpc and (f)m,f)A) = (0.3,0.7).
l
'
1
1
1
1
'
1
! : 1 ;'
1
0.5
l
i
o Z> -0.5
-5. <
1
.•'
i
1
-
|
_
i
-*.
! .
A -
.-.^
-l -1.5 \
1
0
'
1
0.2
0.4 0.6 fractional look-back time
V3c„/4e, /„
A 2TT/137
•)
27T/137 4
(1 + 5.245 x l 0 " ) 2 n / l 3 7
Figure 1.
1
2xl0
-
0.8
1
. A i!
5
V 1 -(2TT/137)2
5
- V'-(2T/137)2
3 * 10 3 »10
i
1
3
V 1 -(27T/137) 2 + 9 . 3 x 1 0 " "
Sample relative variations of a versus fractional look-back time 1 — t/t,
100
4. Lorentz violation In the presence of charged matter described by a 4-current j " , the equations of motion for ,FM„ become
l a , i ^ - | ( V ) i ^ + ^(d»8)F^
= f.
(6)
Note that in the limit of spacetime-independent e and #, the usual inhomogeneous Maxwell equations are recovered. In the present context, however, the last two terms on the left-hand side of Eq. (6) lead to apparent Lorentz violation despite being coordinate invariant. This fact becomes particular transparent on small cosmological scales, where d^M and d^N are approxi mately constant, and hence, select a preferred direction in the local inertial frame. As a consequence, particle Lorentz covariance5 is violated. It is important to note that this is not a feature of the particular coordinate system chosen. Once <9MM, for example, is nonzero in one local inertial frame associated with a small spacetime region, it is nonzero in all local inertial frames associated with that region. By contrast, such Lorentz-violating effects are absent in conventional FRW cosmologies that fail to generate spacetime-dependent scalars coupled to nongravitational fields. Although global Lorentz symmetry is usually broken, local Lorentz-symmetric inertial frames always exLc. It is also im portant to note that the above mechanism for generating Lorentz-breaking effects is not a unique feature of our supergravity model. Equation (6) shows that any similarly implemented smooth spacetime dependence of e, 8, or both on cosmological scales can lead to such effects. This suggests that this type of apparent Lorentz breaking could be a common feature of models incorporating spacetime-varying couplings. An equivalent form of the electrodynamics lagrangian (5) in a local inertial frame can be obtained via an integration by parts: £
- = ~^F^lV
+ ^(d,0)AvF^.
(7)
The the second term on right-hand side of Eq. (7) gives a Chern-Simonstype contribution to the action. Such a term is contained in the standardmodel extension, and one can identify (A:/IF)M — e29M^/87r2. The presence of a nonzero (k^F)^ in (7) shows explicitly Lorentz and CPT violation at the lagrangian level. The situation in which e and (k^F)^ are constant has been discussed extensively in the literature. 14 ' 5 ' 20 In this limit, lagrangian (7) becomes translationally invariant. However, the associated conserved energy fails to be positive definite, which usually leads to instabilities in
101
the theory. The question arises how this problem is avoided in the present context of a positive-definite supergravity model. 21 Although in most models a Chern-Simons-type term is assumed to arise in an underlying framework, its treatment at low energies usually involves a constant nondynamical (kAF)n- In the present context, however, (AIAF)/* is associated with the dynamical degrees of freedom A and B. Excitations of FM„ therefore lead to perturbations 8A and SB in the axion-dilaton back ground (4). As a result, the energy-momentum tensor ( T 5 ) ^ of the back ground receives an additional contribution, (T b )'" / -» (Thy + 5{ThYu. It can be demonstrated 18 that this contribution does indeed compensate the negative-energy ones associated with a nonzero {kAp)n5. Summary Our analysis suggests that couplings varying on cosmological scales can be obtained as simple solutions of theories beyond the standard model and that such couplings may generically lead to local particle Lorentz violation. As an illustration, we have constructed a classical cosmological solution within the N = 4 supergravity in four dimensions that exhibits spacetime-varying electromagnetic couplings a and 9 and establishes the resulting Lorentz and CPT breaking. In this model, a Chern-Simons-type term is generated but the usual associated stability difficulties are circumvented. References 1. P.A.M. Dirac, Nature (London) 139, 323 (1937). 2. J.K. Webb et al., Phys. Rev. Lett. 87, 091301 (2001). 3. See, e.g., E. Cremmer and J. Scherk, Nucl. Phys. B 118, 61 (1977); A. Chodos and S. Detweiler, Phys. Rev. D 21, 2167 (1980); W.J. Marciano, Phys. Rev. Lett. 52, 489 (1984); T. Damour and A.M. Polyakov, Nucl. Phys. B 423, 532 (1994). 4. For a review, see, e.g., J.-P. Uzan, hep-ph/0205340. 5. V.A. Kostelecky and R. Potting, Phys. Rev. D 51, 3923 (1995); D. Colladay and V.A. Kostelecky, Phys. Rev. D 55, 6760 (1997); 58, 116002 (1998). 6. V.A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989); 40, 1886 (1989); Phys. Rev. Lett. 63, 224 (1989); 66, 1811 (1991); V.A. Kostelecky and R. Potting, Nucl. Phys. B 359, 545 (1991); Phys. Lett. B 381, 89 (1996); Phys. Rev. D 63, 046007 (2001); V.A. Kostelecky, M. Perry, and R. Potting, Phys. Rev. Lett. 84, 4541 (2000); M.S. Berger and V.A. Kostelecky, Phys. Rev. D 65, 091701(R) (2002). 7. F.R. Klinkhamer, Nucl. Phys. B 578, 277 (2000); 8. S.M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001); Z. Guralnik et al.,
102
9.
10.
11.
12.
13.
14. 15. 16. 17. 18. 19. 20. 21.
22.
Phys. Lett. B 517, 450 (2001); A. Anisimov et al, hep-ph/0106356; C.E. Carlson et al, Phys. Lett. B 518, 201 (2001). KTeV Collaboration, H. Nguyen, in V.A. Kostelecky, ed., CPT and Lorentz Symmetry II, World Scientific, Singapore, 2002; OPAL Collaboration, R. Ackerstaff et al, Z. Phys. C 76, 401 (1997); DELPHI Collaboration, M. Feindt et al, preprint DELPHI 97-98 CONF 80 (1997); BELLE Collabora tion, K. Abe et al, Phys. Rev. Lett. 86, 3228 (2001); FOCUS Collaboration, J.M. Link et al, hep-ex/0208034. D. Colladay and V.A. Kostelecky, Phys. Lett. B 344, 259 (1995); Phys. Rev. D 52, 6224 (1995); Phys. Lett. B 511, 209 (2001); V.A. Kostelecky and R. Van Kooten, Phys. Rev. D 54, 5585 (1996); V.A. Kostelecky, Phys. Rev. Lett. 80, 1818 (1998); Phys. Rev. D 61, 016002 (2000); 64, 076001 (2001); N. Isgur et al, Phys. Lett. B 515, 333 (2001). L.R. Hunter et al, in V.A. Kostelecky, ed., CPT and Lorentz Symmetry, World Scientific, Singapore, 1999; D. Bear et al, Phys. Rev. Lett. 85, 5038 (2000); D.F. Phillips et al, Phys. Rev. D 63, 111101 (2001); M.A. Humphrey et al, Phys. Rev. A 62, 063405 (2000); V.A. Kostelecky and C D . Lane, Phys. Rev. D 60, 116010 (1999); J. Math. Phys. 40, 6245 (1999); R. Bluhm et al, Phys. Rev. Lett. 88, 090801 (2002). H. Dehmelt et al, Phys. Rev. Lett. 83, 4694 (1999); R. Mittleman et al, Phys. Rev. Lett. 83, 2116 (1999); G. Gabrielse et al, Phys. Rev. Lett. 82, 3198 (1999); R. Bluhm et al, Phys. Rev. Lett. 82, 2254 (1999); Phys. Rev. Lett. 79, 1432 (1997); Phys. Rev. D 57, 3932 (1998). B. Heckel, in V.A. Kostelecky, ed., CPT and Lorentz Symmetry II, World Scientific, Singapore, 2002; R. Bluhm and V.A. Kostelecky, Phys. Rev. Lett. 84, 1381 (2000). S. Carroll, G. Field, and R. Jackiw, Phys. Rev. D 4 1 , 1231 (1990). V.A. Kostelecky and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); Phys. Rev. D 66, 056005 (2002). V.W. Hughes et al, Phys. Rev. Lett. 87, 111804 (2001); R. Bluhm et al, Phys. Rev. Lett. 84, 1098 (2000). E. Cremmer and B. Julia, Nucl. Phys. B 159, 141 (1979). V.A. Kostelecky, R. Lehnert, and M.J. Perry, in preparation. T. Damour and F. Dyson, Nucl. Phys. B 480, 37 (1996); Y. Fujii et al, Nucl. Phys. B 573, 377 (2000); K. Olive et al, hep-ph/0205269. R. Jackiw and V.A. Kostelecky, Phys. Rev. Lett. 82, 3572 (1999); C. Adam and F.R. Klinkhamer, Nucl. Phys. B 607, 247 (2001); and references therein. A constant timelike {k^p)^ also violates microscopic causality. 20,22 Our supergravity model may avoid this, but a complete analysis lies outside the scope of this talk. V.A. Kostelecky and R. Lehnert, Phys. Rev. D 63, 065008 (2001); R. Lehnert, in V.A. Kostelecky, ed., CPT and Lorentz Symmetry II, World Scientific, Singapore, 2002.
T H E C E R N A X I O N SOLAR TELESCOPE
M.D. HASINOFF"'6? S. ANDRIAMONJEc, E. ARIKd, D. AUTIERO", F. AVIGNONE6, K. BARTH", E. BINGOL", H. BRAUNINGER-^, R. BRODZINSKI5, J. CARMONA'1, E. CHESI0'6, S. CEBRIAN'1, S. CETIN0* J. COLLARS R. CRESWICK6, T. DAFNP, R. DE OLIVEIRA", S. DEDOUSSIS-7, A. DELBARTC, L. DI LELLA°, C. ELEFTHERIADISJ', G. FANOURAKISfc, H. FARACH6, H. FISCHER', F. FORMENTI0, T. GERALIS*, I. GIOMATARISc, S. GNINENKOm, N. GOLOUBEV™, R. HARTMANN^.D. HOFFMANN", I.G. IRASTORZA'1, J. JACOBY", D. KANG', K. KONIGSMANN', R. KOTTHAUS0, M. KRCMARP, M. KUSTER-f, B. LAKICP, A. LIOLIOS-?', A. LJUBICICP, G. LUTZ°, G. LUZON'1, H. MILEY9, A. MORALES'1, J. MORALES'1, M. MUTTERER™, A. NIKOLAIDISJ', A. ORTIZ'S T. PAPAEVANGELOU^', A. PLACCIa, G. RAFFELT0, H. RIEGE°, M. SARSA7*, I. SAVVIDISj, R. SCHOPPER™, I. SEMERTZIDIS™, C. SPANO', J. VILLAR^, B. VULLIERMEa, L. WALCKIERS0, K. ZACHARIADOU*, K. ZIOUTASaj' a European Organization for Nuclear Research (CERN), Geneve, Switzerland Dept. of Physics & Astronomy, University of British Columbia, Vancouver, Canada C DAPNIA, Centre d'Etudes de Saclay (CEA-Saclay), Gif-Sur-Yvette, France Department of Physics, Bogazici University, Istambul, Turkey e Department of Physics and Astronomy, U. South Carolina , Columbia, Sc, USA * Max-Planck-Institut fur Extraterrestrische Physik, MPG, Garching, Germany 9 Pacific Northwest National Laboratory, Richland, Wa, USA Instituto de Fisica Nuclear y Altas Energias, Universidad de Zaragoza, Zaragoza, Spain % Enrico Fermi Institute, University of Chicago, Chicago, II, USA 3 Aristotle University of Thessaloniki, Thessaloniki, Greece National Center for Scientific Research "Demokritos" (NRCPS), Athens, Greece Albert-Ludwigs-Universitat Freiburg, Freiburg, Germany m Institute for Nuclear Research (INR), Russian Academy of Sciences, Moscow, Russia n Institut fur Kernphysik, Technische Universitat Darmstadt, Darmstadt, Germany ° Max-Planck-Institut fur Physik, Munich, Germany p Ruder Boskovic Institute, Zagreb, Croatia
The CAST experiment at CERN is using a decommissioned LHC prototype magnet to search for solar axions through their Primakoff conversion into x-ray photons. The magnet (B == 9.0 Tesla, L = 10 m) can track the sun each day for a total exposure time of ~180 minutes (sunrise + sunset). We expect to reach a sensitivity in axion-photon coupling, ga-yy < 5 x 1 0 - 1 1 G e V - 1 for ma < 10~ 2 eV after ~ 1 year's running time. By filling the beam tube with 4 He or 3 He gas we should be able to extend the sensitive axion mass region into the eV mass range.
* attending speaker, e-mail: [email protected]
103
104
1. Introduction The Standard Model of Electroweak Physics has had remarkable success explaining all the existing experimental data. Its extension to strong inter actions, Quantum Chromodynamics, has also proven remarkably success ful. However, QCD does have one loose end-its non-Abelian nature intro duces T-, P- and CP-violating effects, and, in its basic form, it predicts a substantial CP-violating electric dipole moment (edm) for the neutron in rather sharp disagreement with experiment (dtheory ~ 10 _ 1 5 e • cm and dexp ~ 10 _ 2 5 e-cm). The simplest and most elegant solution to this problem is the one proposed by Peccei and Quinn 1 . They showed that a mimimal extension of the Higgs sector endows the SM with a global U(l) symmetry, the Peccei-Quinn(PQ) symmetry, which is broken at some new scale, fpQ. Subsequently Weinberg2 and Wilzeck3 pointed out that, since a continuous symmetry has been broken, there must be an associated Goldstone boson (the AXION). Although the axion starts out as a massless Goldstone bo son, it eventually acquires an effective mass through intermediate states coupled to its axial colour anomaly. Axions are attractive candidates for Cold Dark Matter since they could have been produced in the early stages of the Universe. Higher energy axions might also contribute to Hot Dark Matter and help to produce a flat Universe. Axions could also be produced in the core of stars by means of the Primakoff conversion of the blackbody photons in the fluctuating electromagnetic fields of the hot dense plasma. The solar axion flux from our own Sun can easily be estimated 4 using the standard solar model and the conservative assumption of a "hadronic" axion with very small leptonic couplings. The resulting axion flux has a broad spectrum which peaks at about 4 keV as shown in Figure 1. These solar axions can then be converted back into real photons in the presence of an intense magnetic field here in the laboratory. This is the basic principle of the CERN Axion Solar Telescope (CAST) experiment 5 described in the next section. Although the motivation given above for our experiment has been centred on the axion because of its special theoretical significance, our telescope will search for any type of low mass pseudoscalar or scalar particles which couple to photons and such a discovery would have profound implications in Particle Physics. A combination of astrophysical and nuclear physics constraints, plus the requirement that the relic axion abundance does not overdose the Universe, restricts the allowed range of viable axion masses to the region
105 8 7 6
y 5 's s" 4 O
f 3 E 2 1 °0
2
4
6
8
10
Energy (keV) Figure 1. Differential solar axion flux at the Earth assuming Primakoff conversion of the blackbody photon as the only way of solar axion production (hadronic axions).
10- 6 < ma < 20 eV 6 . This mass range and the allowed range of
= -ga770oE ■ B
(1)
where a is the axion field and the coupling strength is a
(E
2(4 + z ) V
9
!
(
.
106
The quantity, E/N is the PQ symmetry anomaly which is the ratio of the electromagnetic and colour anomalies, a model-dependent ratio of small integers. Two popular models are the GUT axion 13 (E/N = 8/3) and the KSVZ 14 model (E/N = 0). The second term in the parenthesis is the chiral symmetry breaking correction which is a function of z = mu/md, the up and down quark masses. The probability that an axion passing through a transverse magnetic field B over a length L will convert to a photon is given by Pai = 2.4 x 10- 1 7 {gail x 1010 GeV-1)2
|M| 2 (JL)f
(^-^
(3)
where the matrix element |M| = 2(1 - cos qL)/(qL)2 accounts for the coherence of the process and q is the momentum exchanged between the axion and the magnet. Since ma ^ 0, the axion and photon waves become out-of-phase after a certain length of travel in the magnet. For the keV energies for solar axions this coherence is preserved up to ma values ~ 10~ 2 eV over a length of 10 m. For larger axion masses \M\ becomes < 1 and we lose sensitivity so we plan to fill the beam pipe inside the magnet with 4 He or 3 He gas to give an effective mass to the photon through the index of refraction of the gas. Then the photon mass becomes equal to the plasma frequency of the gas, m 7 = wp = ^4irnero where ne is the spatial density of the electrons in the plasma and ro is the classical electron radius. When ma ~ m 7 coherence is once again restored. By changing the pressure of the gas inside the beampipe m 7 can be varied and so the sensitivity of the experiment can be extended to higher ma values as shown in Figure 3. The diagonal band indicates the range allowed by the various model-dependent E/N values. Given the improvement of x 10 in our B-L value our sensitivity should be ~ 5 — 10x better than the Tokyo axion helioscope result 11 , 12 .
3. The CAST Experiment The decommissioned LHC super-conducting prototype magnet (B = 9.0 Tesla, L = 10 m) has twin-apertures with an effective cross sectional area = 2 x 14 cm 2 . It is mounted on a girder capable of moving ±8° vertically and ±40° horizontally; hence we can track the sun for about 90 minutes at sunrise and also 90 minutes at sunset. The vertical limitation is determined by the cryogenic system which supports the cold mass and the horizontal limitation is a result of the size of the hall. Because of this horizontal limitation we have chosen to mount detectors at both ends of the magnet
107
Figure 2. Schematic view of the CAST experimental setup. The 10 m long LHC pro totype magnet can move ± 8 ° vertically and ±40° horizontally.
in order to double our observing time. A schematic view of the CAST experimental setup is shown in Figure 2. The solar axion flux can be computed for a given axion-photon cou pling, 5a 77 ; hence the expected limit on gayl in terms of the experimental parameters (assuming total coherence) becomes: **" *
1A
X 10
" 9 *I*{BL)WAV*
GeV X
~
(4)
where b is the background of the x-ray detector (in counts/day), t is the time of alignment with the sun (in days), BL (in Tesla meters) and A is the area of the magnet beampipe (in cm 2 ). All these quantities are already fixed for CAST except for the background, b, which we have tried to make as low as possible by using very low radioactivity materials for the construction of the detectors. Fortunately the dependence of <7a77 on the background is only to the 1/8 power. We have developed three different types of detectors to detect the lowenergy x-rays: a small plexiglass time projection chamber (TPC), a position sensitive CCD detector, and a small MicroMegas detector(MM) - also con structed from plexiglass. The TPC has 48 anode wires and 96 cathode wires placed perpendicular to each other at 3 mm wire spacing. Each wire is readout by a 10 MHz flash ADC so that very accurate positions can be obtained. In this way we can easily separate the spatially localized x-ray events from the long tracks produced by cosmic-rays. The TPC has an active volume = 30 x 15 x 10 cm 3 and is operated with a 95:5 Ar:Ethane
108 -7
ox -10
-11 ■ 4 - 3
5
-
1
0
1
logfm a / eV\ Figure 3. Expected exclusion limit of CAST (solid line) taking into account both data taking phases (vacuum and gas). Also shown are the limits obtained by the SOLAX 1 5 and COSME 1 6 experiments using solid state detectors (thin line), and the Tokyo helioscope 1 1 , 1 2 (dashed line) which uses the same principle as CAST. The region favoured by theoretical axion models is indicated by the light grey band. The dotted line represents the theoretical red giant bound 1 7 and the dark grey region on the right is excluded by the absence of an axion-decay quasi-monochromatic photon line from galactic clusters 1 8 .
gas mixture at a drift field of 700 V/cm. The energy resolution for an 55 Fe source is about 12% (a) and the noise threshold is about 300 eV. Using various software cuts we have obtained a background counting rate ~ 10~ 5 counts/keV/cm 2 /s and we expect this to improve by another factor of 5-10 once the shielding of copper/cadmium/lead/polyethelyne has been installed around the detector. This is roughly equal to the background level estimated in the original proposal. The TPC detector covers both of the magnet exit windows and will observe the sunset axions. At the other end of the magnet, the sunrise axions will be detected by a CCD + and a MicroMegas detector. In order to improve our signal/noise ratio we have obtained a focussing mirror system which can focus the x-ray beam from the 43 mm beampipe diameter down to a 1 mm spot, thereby greatly improving the signal/noise ratio. A position sensitive CCD detector or a second MicroMegas detector will be mounted behind this focussing telescope.
109
4. Present Status and Future Plans The magnet was cooled down to 1.8K in August and it reached full current (13,330 amps) on Sept 9. The first solar data was collected by the TPC detector on Sept 19. The installation of the focussing telescope and the MM detector will take place during October and we expect to collect data with all 3 detectors in a full Engineering run in November. The background shielding for the TPC detector will be installed during the CERN Winter shutdown and we should then be able to run continuously from March un til Dec'03. We expect to install the thin Be windows necessary for the implementation of the low pressure gas measurements during the Dec'03 shutdown and we will then run during 2004 with 4 He gas and 2005 with 3 He gas in the beam tubes in order to extend the axion mass range into the eV region. This will allow CAST to challenge the various theoretical model predictions (see Figure 3). The possibility of adding a high energy 7-ray calorimeter behind the MM detector on the sunrise side of the mag net in order to search for higher mass pseudoscalar particles is also being discussed. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38,1440 (1977). S. Weinberg, Phys. Rev. Lett. 40,223 (1978). F. Wilczek, Phys. Rev. Lett. 40,279 (1978). K. van Bibber, P.M. Mclntyre, D.E. Morris and G.G. Raffelt, Phys. Rev. D39,2089 (1989). K. Zioutas et al, Nucl. Instrum. Meth. A425,482 (1999). E.W. Kolb and M.S. Turner, "The Early Universe", Addison Wesley Publishing (1990). H. Schlattl, A. Weiss and G. Raffelt, Astropart. Phys. 10,353 (1999). G. Raffelt, Phys. Rep. 198,1 (1990). T. Moroi and H. Murayama,Phys. Lett. B440,69 (1998). D. M. Lazarus et al, Phys. Rev. Lett. 69,2333 (1992). S. Moriyama et al, Phys. Lett. B434,147 (1998). Y. Inoue et al, Phys. Lett. B536,18 (2002). M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 1 0 4 , 199 (1981). J.E. Kim, Phys. Rev. Lett. 43,103 (1979), M. A. Shifman, A.I. Vainschtein and V.I. Zakharov, Nucl. Phys. B166.493 (1980). F.T. Avignone et al. [SOLAX Collaboration], Phys. Rev. Lett. 81,5068 (1998). A. Morales et al. [COSME Collaboration], Astropart. Phys. 16,325 (2002). G. Raffelt, Phys. Rev. D33,897 (1986). M.T. Ressell, Phys. Rev. D44.3001 (1991).
This page is intentionally left blank
Matter Under Extreme Conditions
This page is intentionally left blank
PROPERTIES OF D E N S E A N D COLD QCD
H. J. DE VEGA LPTHE, Universite Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), Tour 16, ler. Stage, 4, Place Jussieu, 75252 Paris, Cedex 05, France
We study equilibrium and non-equilibrium aspects of the normal state of cold and dense QCD and QED. The exchange of dynamically screened magnetic gluons (photons) leads to infrared singularities in the fermion propagator for ex citations near the Fermi surface and the breakdown of the Fermi liquid de scription. We implement a resummation of these infrared divergences via the Euclidean renormalization group to obtain the spectral density, dispersion re lation, widths and wave function renormalization for single quasiparticles near the Fermi surface. We find that all feature scaling with anomalous dimensions: 1
1
2\
wp(k) oc Ifc-fciH 1 " 2 * ; F(fe) oc \k - kF\i=5X ; Zp{k) oc \k - kF\i-*>> with \ = ^ r for QED , f £ % ^ for QCD with Nc colors and NF flavors. The discontinuity of the quasiparticle distribution at the Fermi surface vanishes. For k » kF we find nkakF = s-^ - T j l f c * / 4 A ) + 0(fc - kF)2 with M the dynamical screening scale of magnetic gluons (photons). The dynamical renormal ization group is implemented to study non-equilibrium relaxation. The ampli tude of single quasiparticle states with momentum near the Fermi surface falls off as |*jfess*F(*)l « |**ss*; F (
There is a substantial theoretical and experimental effort to map the phase diagram of QCD as a function of temperature (T) and chemical potential (/z). Understanding the region of the phase diagram for T < 300 - 400 Mev and fi < 0.6 Gev can provide insights into the QCD phase transition in the Early Universe, about 10 fxs after the Big Bang as well
113
114
as the equation of state of hot and dense QCD. Matter at low temperature < 10 Mev and up to nuclear matter density po ~ 0.16 f m - 3 ; fi w 300 Mev is amenable of study by low energy nuclear systems such as multifragmentation phenomena in nuclei. Cold-dense nuclear matter for densities larger than a few times nuclear matter density cannot be studied with terrestrial accelerators and is the realm of astrophysical compact objects, such as neu tron stars 3 . The fascinanting possibility of detecting a phase transition in quark matter in neutron star X-ray binaries was raised recently 4 , where the signal would be a pronounced peak in the frequency distribution of X-ray neutron stars due to a long spin-up stage and the cooling history as revealed by the (soft) X-ray spectra. Thus while QCD at high tempera ture and relatively small chemical potential can be experimentally studied with ultrarelativistic heavy ion collisions, astrophysical observation of the properties of neutron stars can provide observable signatures from cold and dense QCD if quark matter is the correct description of the core of spinning neutron stars. There is the tantalizing possibility that at the core of neu tron stars where the density is up to 5 times that of nuclear matter, there could be a component of cold, degenerate quark matter with temperatures T < 1 Mev and chemical potentials /i ~ 300 - 500 Mev 3 . Goals: The common framework to study degenerate correlated fermion systems is that of Fermi liquid theory. The emergence of superconductivity (diquark condensation in the case of QCD) is associated with the instability of the normal Fermi liquid towards an attractive pairing interaction. In the case of a weakly interacting Fermi system the starting point is the free Fermi gas and pairing results in the opening of a gap in the single particle spectrum at the Fermi surface. Fermi liquid theory is argued to describe the low energy effective field theory of nuclear matter and is therefore an important tool to study nuclear matter and its impact in astrophysical compact objects. Studying Fermi liquid aspects of the normal phase is an important part of the program towards understanding cold-dense QCD since understanding the properties of the normal phase is the first step towards a complete assessment of the pairing instabilities and properties of the superconducting state. Furthermore, if the pairing interaction opens a gap at the Fermi surface of some quarks, such as the two or three color superconducting phases (2SC or 3SC) 7 , the remaining gapless quarks will be described by the concomitant Fermi liquid. Our goal 1 was to provide a comprehensive study of non-Fermi liquid aspects in the normal phase of cold-dense QCD. We study both equilib-
115
rium and non-equilibrium aspects of the non-Fermi liquid behavior. In particular we focus on i) dispersion relation and damping rates of quasiparticles near the Fermi surface, these reveal anomalous dimensions resulting from the breakdown of Fermi liquid theory and ii) the relaxation of these quasiparticles: again we find anomalous relaxation with origin in the same infrared divergences responsible for the breakdown of Fermi liquid theory. Our main motivations for initiating this study and long term goals are man ifold: a) to obtain a further assessment of non-Fermi liquid corrections to the superconducting gaps, critical temperature and spectrum of excitations in the superconducting phase, b) a study of the potential implications of non-Fermi liquid corrections to the neutrino emissivity and the cooling rate of neutron stars with degenerate quark-matter cores, c) a more complete and detailed understanding of the properties of dense QCD in a regime which is not yet amenable to lattice simulations, d) a study of transport phenomena in the non-Fermi liquid, which is relevant to cooling and ther modynamics of neutron stars. In 1 we begin this program by studying in detail the breakdown of the Fermi liquid description and by providing a comprehensive analysis of the spectrum of single quasiparticle excitations in the normal phase along with their relaxation properties. Strategy: We study both QED and QCD at zero temperature but large (baryon) density so that a perturbative analysis is reliable. In this regime the hard-dense-loop (HDL) approximation, which is the finite density equiv alent of the hard-thermal-loop program for finite temperature, is reliable and describes the main aspects of static and dynamical screening of gluons (and photons). The leading order in the HDL approximation is the same in the abelian (QED) and the non-abelian (QCD) theories and screening of gauge fields is completely determined by the one loop quark polarization at finite (and large) density. In this approximation, static longitudinal glu ons (instantaneous Coulomb interaction) are screened by a Debye screening mass mo oc g/j, while transverse gluons are only dynamically screened via Landau damping. To this order in the HDL approximation the polariza tion tensor for gluons in QCD is similar to that for photons in QED save for trivial color and flavor factors. The main difference between QCD and QED in this approximation is that while one gluon exchange leads to an attractive (pairing) interaction in the antitriplet particle-particle channel and therefore to diquark condensation, there is no such attractive channel in QED i.e, particles above and holes below the Fermi surface bound by their mutual (screened) Coulomb attraction. These are the counterpart of exciton bound states in condensed matter. Thus to this order in the HDL
116
approximation, the Fermi liquid aspects of the normal state of cold-dense QCD are similar to those of QED. As described in detail 1 , a Fermi liquid description has an associated 'order parameter', this is the jump discontinuity of the Fermi distribution function (of the interacting) system at the Fermi momentum. This discon tinuity is given by the residue (wave function renormalization) of the quasiparticle pole for quasiparticles with the Fermi momentum. The breakdown of the Fermi liquid picture is associated with the vanishing of this order parameter, i.e, the Fermi distribution function is continuous at the Fermi momentum. We begin by obtaining the quark propagators to leading order in the HDL approximation, corresponding to one bare gluon exchange in the quark self-energy and show explicitly that to this order there is a sharp discontinuity at the Fermi surface determined by the wave function renor malization of the quasiparticle pole. However this picture does not survive screening corrections to the gluon propagator, whereas longitudinal gluon exchange leads to an infrared finite contribution, the exchange of magnetic gluons which are only dynamically screened by Landau damping introduce logarithmic divergences in the quark propagator for quasiparticles near the Fermi surface. For excitations with Fermi momentum we argue that these infrared divergences are similar to those of a critical theory at the upper critical di mensionality. Thus we provide a resummation of the quark propagator for particles with the Fermi momentum using the Euclidean renormalization group which reveals the emergence of anomalous dimensions in the spectral density. Non-equilibrium aspects are studied by implementing a dynam ical renormalization group 6 which provides a resummation of the quark propagator directly in real time. The dynamical renormalization group re veals power law relaxation with anomalous dimension for quasiparticles with Fermi momentum. Summary of the results 1 : The exchange of dynamically screened magnetic gluons leads to infrared divergences in the single particle propa gator for particle excitations near the Fermi surface. We implemented 1 a resummation of the perturbative expansion via the Euclidean renormaliza tion group. We find that the particle component of the quark propagator for excitations near the Fermi surface is a scaling function of the two vari ables u> = u] — /i ; k = k — /i with anomalous exponents that depend on the gauge coupling. For k ^ 0 the spectral density near the Fermi surface has
117
the form (
u\\
-2A
sinJTrA]
(Q\tt\
2A
-^COS[TTA])
+ (fcsin[7rA]V
where k = = k — /J, and Q == LJ — JJ,. T h e q u a s i p a r t i c l e d i s p e r s i o n r e l a t i o n follows, i
up(k) = sign(fc) \k\M~2X cos[7rA]
1_2
*
(1)
We note that the group velocity of the quasiparticles near the Fermi surface , r, 2X
vg(k)
= [M~
COS[TTA]] '
2A
(1 - 2A)
(2)
vanishes as k -> kF. We interpret this novel phenomenon in terms of a collective backflow that surrounds the quasiparticle. Near the position of the resonance, i.e, for u> « (bp the spectral density can be approximated by a Breit-Wigner form and the residue of the quasi particle pole and the quasiparticle width vanishing near the Fermi surface as 2A
Zp[k] =
M
2A
ex \k-kF\*=sx
1-2A
,
T(k) = Zp[k] \k\ sm[n\] oc |*-Jfe F |i=ax(3)
The effective coupling is given by #- forQED 2_
f%^2FT~
for
QCD
(4)
witn N
c colors and NF flavors.
The residue of the quasiparticle pole vanishes as k -> kF, leading to the following form of the (quasi) particle distribution function near the Fermi surface nkRikF
=
sin[7rA]
k
2TTA
TTM(1 - 4A)
+0(k)2
M=s<£
f^
^QED for QCD with NF flavors.
revealing the vanishing of the discontinuity of the Fermi distribution func tion at the Fermi surface and therefore the vanishing of the Fermi-liquid order parameter. Implementing a real-time, i.e, a dynamical version of the renormalization group to study the non-equilibrium relaxation of single quasiparticles
118
near the Fermi surface. We find that the amplitude of the wave function of single quasiparticle states near the Fermi surface fall off as ■
Ilk**,(*)l « |V***F(*o)| e- r ^- f °>
^ A
(5)
Thus quasiparticles with the Fermi momentum have vanishing group veloc ity and relax with an anomalous power law. We obtained the expression for the quasiparticle distribution function in terms of the quark spectral density and obtain the equation of motion for quarks which will be used to study non-equilibrium aspects 1 . We study the equilibrium aspects of single quasiparticles. We begin by studying the quark propagator to lowest order in the HDL approximation, i.e, with the self energy given by the exchange of hard (bare) gluons and make contact with the Fermi liquid description to this order. Soft (q < g/j.) gluons require HDL resummation, and the propagator for particles near the Fermi surface is computed by including HDL (screening) corrections to the exchanged gluon. The resulting infrared divergences are recognized to be similar to those of a critical theory at its upper critical dimension and resummed using the Euclidean renormalization group. The renormalization group improved spectral density features scaling behavior that lead to the single quasiparticle dispersion relation and lifetime that scales with anomalous dimensions. We show that the jump discontinuity of the single particle distribution function vanishes at the Fermi surface as a consequence of the vanishing of the single quasiparticle residue at the Fermi momentum. After that we explore non-equilibrium aspects: the relaxation of single quasiparticle excitations near the Fermi surface. Implementing a real-time version of the renormalization group reveals that single quasiparticle states with Fermi momentum relax with a power law with anomalous dimension. Finally, we summarize the connection between QED and the normal state of QCD to the order studied and address the important issue of vertex corrections. We also discuss the striking resemblance of the spectral density and relaxation to that obtained in a Luttinger liquids and some related conjectures on non-Fermi liquid aspects of high Tc superconductivity, we elaborate on the potential impact of the results and discuss their range of validity. The existence of quark matter phases in the cores of neutron stars and pulsars could have distinct observational consequences. The equation of state of quark matter may lead to pronounced delays in the spin-up history of neutron stars in low mass X-ray binaries and may explain recently ob-
119
served anomalous frequency distributions 4 ' 5 , a deconfinement transition can lead to observational consequences in the rotational properties of pulsars 4 , superconducting quark matter can influence the strength and distribution of magnetic fields of pulsars and could influence the cooling history of young neutron stars and protoneutron stars. The properties of neutron stars and pulsars are studied by a satellite program that includes the Einstein Observatory, ROSAT, AXAF, RXTE and more recently CHANDRA and XMM that measure the (soft) X-ray emission from neutron stars, thus studying their cooling history as well as rotational properties (spin-up and spin-down). A few seconds after the collapse of a Type II supernovae the newly born neutron star cools via the emission of neutrinos and antineutrinos and within few minutes the temperature falls to about l O 9 ^ (~ 0.1 Mev) cool ing proceeds via neutrino emission for another 1 0 5 - 6 years before photon emission becomes the most important cooling mechanism. The most efficient neutrino emission mechanisms are the direct URCA processes f\ +1 -> fa + vi ; / 2 -» / i + / + ui with fi being either baryons or quarks and / either electrons or muons with vi ; vi their respective neutrinos and antineutrinos. In the region between the crust and the core, where nuclear matter is the dominant component this process occurs with baryons and corresponds to the beta decay of the neutron and electron capture in a proton. If the core is composed of quark matter the corresponding processes are direct quark Urea, d —> u + e~ + Pe ; u + e~ -> d + ve. The cooling rate is determined by the equation dU _ dT ., , , -^ = a = -(L, + L 7 ) (6) with Cv the specific heat (at constant volume and baryon number) and L„^ are the neutrino and photon luminosities, any other potential sources of energy loss such as magnetic field decay, differential rotation etc. had been neglected in the above equation. There are two important quantities that determine the cooling rate: the specific heat Cv and the neutrino emissivity. Both are completely deter mined by the physics near the Fermi surface of the degenerate quarks and leptons (the only important ones are electrons). Our goal2 was to study in detail the non-Fermi liquid corrections to the specific heat to establish if these result in an enhancement or suppression of the cooling rate via neutrino emission through direct quark Urea pro cesses, postponing the study of non-Fermi liquid effects upon the neutrino emissivity to a forthcoming article.
120
Our strategy 2 to obtain the specific heat (at constant volume and baryon density) was to first obtain the internal energy and take its temperature derivative. The internal energy in turn is the expectation value of the full Hamiltonian H = Hq + Hg + Hq-g, the sum of the quark, gluon and interaction parts. The regime of interest for neutron star phenomena is T « 0.1 - 1 Mev ; fiq « 0.3 - 0.5 Gev, i.e, T « fi. Furthermore the scale that enters in the resummed quark propagator 1 is M « g^/2-K at energy scales ~ 1 Gev the strong coupling constant as ss 0.6 hence M ~ \iq implying the hierarchy T C M ~ \i. The contribution from massless gluons or Goldstone bosons remaining from the superconducting transition, to the internal energy is Ug(T) oc T4. Therefore the gluon contribution to the internal energy is subleading by a factor T2/fi2 as compared to the quark contribution which, as argued above is SU ~ fi2T2. Hence we will neglect the contribution from Hg to the internal energy. Temperature affects only states a distance » T from the Fermi surface, hence only quasiparticle states are important, and antiparticle states are exponentially suppressed. Therefore the contribution to the internal energy, specific heat, and (quasi) particle density for states near the Fermi surface are given by U(T,ri
= 2Jdqoq0r,(qo)N(q0)
,
Cv = ^
,
N - y =2
jdq0r,(q0)N(q0)
where we have introduced the single quasiparticle density of states v(Qo)=J
-7^p-(qo,k)
(7)
We find by straightforward integration and approximating the above form of r)(q0) for z
.
1 //i\2
M
1 — —-r — sin ■K A + u, a cos n A M 2 / 7T
o M \2aa . „ , M z —^ sin 27rA —— cos 2-nX + 0(z3) TTZ
7T
(8)
2
leading to the following form of the temperature dependence of the chemical potential ^(T)=M2(0)
-2 A
l
-^(m)(mY
/ / r r . \ 3 /
+
m
\ -4 A
° 1(M) im)'
where we have introduced the functions F(X) = r ( 3 - 2 A) C(2 - 2 A) (1 - 2 2 A _ 1 )
(9)
121
K(X) = 8 cos(TrA) r(2 - 2 A) <(2 - 2 A) (1 - 22X~1) =
4cos
^ A F(X) 1—A
in terms of the Gamma and Riemann zeta functions. The non-analiticity of the temperature derivative of the chemical poten tial reflects the divergence of the frequency derivative of the single quasiparticle density of states near the Fermi surface. The leading order (in coupling) contribution to the specific heat takes the form2 M
m
m,+mdT III
dqo <7o v(Qo)
/ ■M
D(q0)
/.I
I
I
(10)
with D(x) After a rescaling % = xT, and neglecting terms perturbative in the coupling A we find2
Ov —
M2(o)r
i + A[F(2A_1)cos27rA
-«GSJ)
(mY
+0
+ 8i , (A)2 cos^x]
[(w)) {m)
( ^ )
2
( ^ )
-4 A
(ii)
with the functions F(X) ; K(A) given by eq. (9)-(10) above. The non-Fermi liquid contributions are explicit in the anomalous power laws with the temperature, which are a result of the divergent frequency derivative of the single quasiparticle density of states near the Fermi surface. We emphasize that for T C M the non-fermi liquid corrections to the specific heat are much larger than those of the massless gluons or any other Goldstone bosons to the specific heat, which is of order T3. To illustrate the magnitude of the non-Fermi liquid corrections to the specific heat we show in fig. 1 the function G(z) — 3CV//J,2(0)T vs. z = T/fi(0) for three colors and two flavors of massless quarks. The solid line corresponds to /z(0) = 0.5 Gev and the dashed line to ^(0) = 1 Gev, for these values of the strong coupling constant, the value of the effective coupling at these scales are A = 0.152 for fi = 0.5 Gev and A = 0.046 for fi = 1 Gev therefore the reliability of the perturbative calculation for the values of the chemical potentials expected at the core of neutron stars is at best questionable. This issue notwithstanding this qualitative estimate points out that non-Fermi liquid corrections to the specific heat result in a change of a few percent.
122 1.00
a 0.95
0.90
0.85
0.80
0.00
0.02
0.04
0.06
0.08
0.10
z Figure 1. G{z) = JfyT vs. z = ^ dashed line: fi(0) = 1 Gev.
for Nc = 3; Nf = 2. Solid line: /i(0) = 0.5 Gev,
References 1. D. Boyanovsky and H. J. de Vega, Phys. Rev. D63, 034016 (2001). 2. D. Boyanovsky and H. J. de Vega, Phys. Rev. D 6 3 , 114028 (2001). 3. N. K. Glendenning, Compact Stars, Nuclear Physics, Particle Physics and General Relativity , New York, Springer, 1997. F. Weber, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics, Bristol, IOP, 1999; J. Phys.G: Nucl. Part. Phys. 25, R195 (1999); Acta Phys.Polon. B 3 0 (1999) 3149. 4. N. K. Glendenning and F. Weber, Ap. J. 559, L119 (2001). 5. D. Blaschke et al. Physics of Neutron Star Interiors, Springer LNP 578, 285 (2001), E. Chubarian et al, A&A. 357, 968 (2000). 6. D. Boyanovsky, H.J. de Vega, R. Holman, and M. Simionato, Phys. Rev. D 60, 065003 (1999); D. Boyanovsky, H.J. de Vega, and S.-Y. Wang, Phys. Rev. D 6 1 , 065006 (2000); S.-Y. Wang, D. Boyanovsky, H. J. de Vega and D.-S. Lee, Phys. Rev. D62, 105026 (2000). 7. For reviews, see: F. Wilczek, hep-ph/0003183, T. Schafer, nucl-th/9911017, T. Schafer and E. Shuryak, Phases of qcd at high baryon density. M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422, 247 (1998); Nucl. Phys. B537, 443 (1999). D. H. Rischke, Phys.Rev. D62, 054017 (2000); ibid034007.
STABILITY OF QUARK MATTER A N D Q U A R K STARS*
M. F I O L H A I S Department
of Physics and Center for Computational 3004-516 Coimbra, Portugal E-mail: tmanuelQteor.fis.uc.pt
Physics,
M. M A L H E I R O Instituto
de Fisica, Universidade Federal 24210-340 Niteroi, Brazil E-mail: [email protected]
Fluminense,
A. R. T A U R I N E S Instituto
de Fisica, Universidade Federal do Rio Grande CP 15051, 91501-970 Porto Alegre, Brazil E-mail: [email protected]
do Sul
We use the confining chromodielectric model to study quark matter. In the ver sion of the model with a quartic potential for the confining field we obtain two solutions for the mean-field equations. The resulting equations of state for strange quark matter are degenerate at large densities with a small energy barrier between them. As the density decreases the barrier gets higher lowering the probability for transitions between the two solutions. The solution that is lower in energy saturates around the nuclear matter density and corresponds to a chiral broken phase with massive quarks. The other solution, saturates at around five times the nuclear matter saturation density and corresponds to a chiral symmetric phase with massless quarks. Using the Tolman-Oppenheimer-Volkoff equations we study the structure of compact objects emerging from the equations of state. It turns out that the metastable equations yield pure quark stars with roughly a solar mass, and radii below 10 km. These objects, however, are not absolutely stable.
1. Introduction According to the so-called strange matter hypothesis, formulated by Witten 1 , strange quark matter could be the ground state of strongly in*This work was supported by FCT ( P O C T I / F E D E R program), Portugal and by CNPq/ICCTI through the Brazilian-Portuguese scientific exchange program.
123
124
teracting matter. The energy per baryon number of such matter would be lower in energy than 56 Fe. The strange matter hypothesis means that a nu cleus would be a metastable state and could convert into a drop of strange quark matter (a strangelet). Because of energy limitation of present day accelerators, they cannot provide direct experimental information on quark matter. However, re cently, a number of different analysis of observational data suggest the possibility for the existence of compact objects with very small radii. Such compact objects cannot be easily accommodated in conventional models of baryon matter since their smallness requires a much softer equation of state (EOS) than those provided by the models. A rather obvious alterna tive is to consider that those small compact objects are made out of quarks. For such quark stars to exist, their density at the surface should be well above the nuclear matter saturation density. Taken Witten hypothesis for granted, quark stars would be an ultrastable bulk of matter. From the theoretical point of view, this kind of issues should be ulti mately addressed by quantum chromodynamics (QCD) because it is the theory for the strong interactions. It is well known, however, that it is a prohibitively complicated theory at the hadronic energy scales, due to its non-linearities. Alternative approaches are therefore in place. Essentially they consist on phenomenological descriptions of baryons and quark mat ter. The approaches are based on effective models which incorporate as much as possible the properties of QCD. A number of phenomenological models for the strong interactions at intermediate energies have been developed since QCD was proposed. The MIT bag was the first relativistic model to be considered as an effective theory of the nucleon. The model was object of numerous refinements, in particular the inclusion of chiral mesons to endow it with an important property of QCD in the light quark sector, namely chiral symmetry. One must say that the bag model, as well as various versions of soliton models, provided a reasonable description of the nucleon properties and also of its excitations. These effective models, originally designed for the nucleon, involving quarks as fundamental dynamical fields, have also been successfully used to describe infinite quark matter. The resulting equations of state have been applied to investigate the structure of compact s t a r s 2 - 6 . The chromodielectric model (CDM) 7 is one of such models, providing a reasonable phenomenology for the nucleon8. On the other hand it also allows us to obtain EOS's for dense quark matter. In the baryon number one
125
sector of the model, it yields soliton solutions representing single baryons with three quarks dynamically confined by a scalar field, Xi whose quanta can be assigned to 0 + + glueballs. When it is applied to quark matter in two or three flavors9 the resulting EOS turns out to be relatively soft at large densities. It is therefore tempting to try to describe quark stars in the framework of this model. With a quadratic potential for the % field, the model was applied 10 to describe the inner part of hybrid stars. The resulting masses lie in the range 1 — 2M Q , with the radii of the order of 10 km or higher, and a hadron crust of about 2 km. In this work we consider an extension of the model used in Ref. 10, taking quartic instead of quadratic potentials. In addition to these structures, the quartic model predicts another type of compact objects made out of quarks only, smaller and denser than neutron stars. Interestingly enough, these stars are metastable, and therefore they may decay into hybrid stars. However, it turns out that for such processes to take place, a reasonably high energy barrier has to be transposed, and therefore they may live for long periods. Prom the observational point of view, the recent discovery of X-ray sources, by the Hubble and Chandra telescopes, increased the plausibility that these sources might be strange quark stars 11 . In particular, the com pact objects RXJ1856.5-3754 and 3C58, with apparently small radii, do not show evidence of spectral lines or edge features 12 , reinforcing the conjecture for the existence of stars made out of strange matter. The phenomenology of these objects seems to be compatible with the small and dense quark stars reported in this work, but one should be aware that the data for those compact objects might not be accurate enough and eventually they are neutron stars. This contribution is organized as follows. In section 2 we present the chromodielectric model, underlying its most important properties. In sec tion 3 we describe quark matter in the mean field approximation and, fi nally, in section 4 we show the results obtained in our study of compact objects using the EOS predicted by the model.
2. The Chromodielectric model The CDM contains quark and chiral meson degrees of freedom, in addition to a scalar-isoscalar chiral singlet chromodielectric field. The coupling of this field to fermions leads to quark confinement and this is an important
126
feature of the model. We write the CDM Lagrangian in the form8 *-* ~ ^q ~r -^a-,7r T ^q — meson T t^x >
\ '
,
(2)
where Cq =
ty7"0^
A,,* = | ^ ^ " * + 5 ^ • 0"# - W(#, a) ,
and W(7?,IT) = (A/4)((j2 + 7r2 - / 2 ) 2 is the Mexican hat potential. In the u, d sector the quark-meson interaction is described by £ , - m e s o n = ~ ^ ( o - + IT ■ n-y5)tp . A
(3)
The current quark masses are zero but, due to the spontaneous chiral sym metry breaking enforced by the Mexican hat potential, the quarks acquire a dynamical mass. The last term in (1) contains the kinetic and the potential piece for the X-field: £x = kd»xd»x-U(x).
(4)
The vector current is conserved for the Lagrangian (2), which is chiral SU(2)xSU(2) symmetric. The vacuum expectation values of the chiral mesons are (0|7f|0) = 0 and (0|cr0jO) = /„.. We work in the chiral limit, i.e. we take m^ = 0 for the pion mass. For the other parameters in the Mexican hat potential, which are kept fixed in our calculations, we take 8 / w = 0.093 GeV for the pion decay constant, and ma = 1.2 GeV for the sigma mass [in the Mexican hat, A = m 2 / ( 2 / 2 ) ] . Besides the parameters in the potentials there is one more parameter in the model, namely the coupling constant, g. The potential term for the \ n e ld is 8r 4 u(x) = \M2X2 1 + M l2r - 2n\
6r 4N * , f, ? \ X2 1- 2 (5) 7 ) 7M V 7 / (7M)2 where M is the \ mass. The parameterization used in (5) allows for a physically meaningful interpretation of the parameters 7 and 77: U{\) has a global minimum at \ — 0 a n d a local one at x = iM, and U(-jM) = r]4M4. The height of the local minimum, B = (t]M)4, is interpreted as a "bag pressure" and this is used to fix the parameters in U(x)- Assuming the wide range 0.150 < Bl>A < 0.250 GeV, one has 0.08 < 77 < 0.15, using M = 1.7 GeV. We note that 7 is not a free parameter since the quartic term of U(x) must be positive and the cubic term negative, which implies 7 2 > 6?74. In the soliton sector of the model, best nucleon properties
-4T+
127 are obtained for G = y/gM ~ 0.2 GeV (only G matters for the nucleon properties) and we keep that G in our quark matter calculations. In order to study strange quark matter, we add to the interaction Lagrangian (3) the term 13 ^s — meson ~
tysVs
>
\P)
X accounting for the coupling between the strange quark and the \
ne
ld.
3. Quark matter in the mean field approximation In the mean-field approximation, the pion field vanishes in homogeneous infinite matter. Therefore, the energy per unit volume of homogeneous u,d quark matter interacting with sigma and chi is E
N =
f "
„ / 4
«P\jP2
J2^J0
+ MtdP
+ U(x) + W(cT)
(7)
where N = 12 [2(spin) x 2(isospin) x 3(color)] is the degeneracy factor and kp is the Fermi momentum related to the quark density, p, through
N ,,
..
p=—2k*F
_f6*V1/3
,
or
kF={—)
,
(8)
and Mq is the quark dynamical mass given by M
9
= ^ .
(9)
A
Since the vacuum expectation value of chi is zero [\ = 0 is the global minimum of the potential U(x)} it is clear that a vacuum in the NambuJona-Lasinio sense cannot be defined in the present model. This was already anticipated in (7), where no negative energy particles have been considered. From Eq. (7) one readily obtains variational equations for a and \They are the following gap equations: 27T 2 MV-P(x) _
W
r[*"
2
p dp ?*>
1 Jo ,/p^TW*
2^\Ul-o*)X2 Ng'
kF P2dp = /f* _^L ===
JO
do) [
(11)
y/?TM*
where we have introduced the function % ) = H
^ 2 ^ 72
2
i ^ ) lM
l - ^ 2 ) z ^MV 7 ) (l )
(12)
128
For quadratic potentials (7 —> 00) this function is D = 1. In order to study strange quark matter in beta equilibrium, an electron gas must also be considered. The mean-field energy per unit volume for strange quark matter in the CDM (plus electrons) is now given by
£
= aY.Jo' j^\Ik2+mf^^2+al
( ^ ^ 2 + ms(x)2
+ 2 J'" ^ 0 V^T^f + U(X) + 0i(a2 - tl)\
(13)
where the first two terms refer to quarks and the third one to the electrons, all described by plane waves. The degeneracy factor is now a = 6 (for spin and color). The last term is the Mexican hat potential (with 7? = 0 and / w = 93 MeV). The fcj in (13) are the Fermi momenta of quarks and electrons. The quark masses in (13) are 13 : mUid = gu,d
(14)
[here, pi = afc3/(67r)2 stand for each flavor density] and IJ-d = P-u + Me , Pd = Ps (15) where fi = \Jm2 + k2F is the chemical potential for each particle. These conditions should supplement the gap equations, and altogether we have a system of six algebraic equations to solve at each baryon density P= ^{pu + Pd + Ps) ■
(16)
The solution of the system of equations are the meson fields, a and \: and the Fermi momenta, ku, kd, ks and ke. For the same set of model parameters we found two stable solutions, hereafter denoted by I and II (for details see Ref. 14). For both solutions a is always close to /„.. In solution I, the x n e ld is a slowly increasing function of the density, remaining always smaller than ~ 0.05 GeV. For such a small x, the quartic potential and the quadratic
129
potential are indistinguishable, thus, in practice, solution I corresponds to the one obtained and used by Drago et al. 10 in the framework of the quadratic potential. Due to the smallness of the \ field, quark masses are large and the system is in a chiral broken phase. The solution II exhibits a large confining field, x ~ 1^ (local minimum of £/), independent of the density. The resulting quark masses are similar for the three flavors and very close to zero (chiral restored phase). Therefore, the chemical potentials in solution II are dominated by the Fermi momentum contribution, pu ~ Hd ~ ps and pe ~ 0, i.e. in solution II there are almost no electrons. Besides solutions I and II, there is an additional unstable solution corresponding to X ~ 7 M / 2 [local maximum of U(x)]The energy per baryon number as a function of the baryon density (EOS) is readily evaluated for each solution. For each solution we obtained the corresponding energy per baryon number as a function of the baryon density (EOS) (see Fig. 1). EOS-I is not sensitive to 7 and 77 (since x is small), just depends on G, and it is rather similar to the one used in Ref. 10. The saturation density occurs at a low density, slightly higher than the nuclear matter equilibrium density, po- Its shape, at intermediate densities, is similar to hadronic EOS's (see Ref. 9 for the two flavors sector). The EOS-II is also insensitive to 7, but does depend on 77 [in fact, the dependence is on (77M)4, as we have already discussed]: the energy per baryon number increases with 77 and so does the saturation density. Depending on 77 the minimal energy per baryon number of solution II can be either below or above solution I. For 77 ~ 0.12 the saturation occurs at p ~ 5po and the energy per baryon number is some 230 MeV higher than for solution I at its saturation density. The two stable solutions are almost degenerated at high densities in the narrow range 0.1 < 77 < 0.12. In Fig. 1 the dotted lines refer to the EOS for the unstable solution of the gap equations (supplemented by electric neutrality and beta equilibrium conditions), for which the x is a t the local maximum of the potential (5). In order to undergo a transition from I to II, the system has to go through the energy barrier represented by the dotted EOS. The barrier gets higher at small densities and, for 77 = 0.12 (third panel in Fig.l), at p ~ 5p0, e/p ~ 2.7 GeV for the unstable solution. Therefore a transition from one regime to the other is not likely to occur and both minima in the EOS I and II are stable. In a 3D plot of the energy per baryon number versus (p, x) the stable solutions correspond to two distinct "valleys", and the unstable solution mentioned above corresponds to the top of the barrier between the two
130
11=0.10 2.01.51.0. II 0 5.
Figure 1. Energy x) and II (dashed M = 1.7 GeV, g = solution with \ ~
per baryon number versus density for solutions I (solid line, small line, x ~ lM) and various parameters r\ (other model parameters: 0.023 GeV and 7 = 0.2). The dotted line corresponds to the unstable "fM/2.
Figure 2.
3D plot of the energy per baryon number versus (p, x)-
valleys (see Fig 2). Regarding energetics, both phases are almost degenerated at high densi ties and have similar shapes in the Pxe plane even at intermediate densities
131
(or energy densities) in the narrow range 0.1 < 77 < 0.12. In that region of 77, one solution is not clearly lower in energy than the other. However, we should point out that they correspond to two different x values and for the system to undergo a transition from the chiral restored to the chiral broken phase it has to go through a high potential energy barrier. 4. Quark stars In order to investigate the structure of stars, for both EOS, we solved the Tolman-Oppenheimer-Volkoff (TOV) equations: d P _ [e(r) + P{r)][M{r) + 4?rr 3 P(r)] ~6r ~~ 2M(rj\ ' r2 A
dM dr
Awr2e(r)
(17)
where P(r) = p 2 f- ( - ) is the pressure and (18) M(r) = / 4irr2 e(r)dr Jo is the mass contained is a sphere of radius r. Since EOS-I is identical to the one using a quadratic potential, it leads to stars that have the same phenomenology as the hybrid stars obtained by Drago et al. 10 : R ~ 10 — 12 km, a hadron crust and a mass M ~ 1 — 2M Q . At low densities, hadronization occurs and an hadronic equation of state should be used, replacing EOS-I. The EOS-II saturates at a high density and, in addition, the system is not likely to undergo a transition to solution I, so that one should not perform any connection to the hadronic sector: the EOS-II alone generates a new family of strange quark stars. In Fig. 3 it is shown the mass-radius relation for different values of 77. These quark stars are smaller and denser in comparison with those resulting from EOS-I. For 77 ~ 0.115 (and M = 1.7 GeV, yielding B1/4 ~ 0.195 GeV) one obtains a maximum radius R ~ 6 km and a corresponding mass M ~ 0.9M Q . According to our calculation, such star has a central density of lOpo (po is the nuclear matter density) and a central energy density e ~ 3xl0 1 5 g/cm 3 . At the edge, the density drops to 5p0 and e ~ 1.35xl0 15 g/cm 3 . The ratio e/p remains approximately constant inside the star. From Fig. 3 one concludes that the mass-radius relation for these strange small stars mainly depends on the height of the local minimum of the \ potential. These stars, as already discussed, are not absolutely stable and therefore they may decay into hybrid stars.
132
n =o.io
il=O.I2 i.o-
=0.15
n
0.5 -
"." |
0
/
" T -
1
2
1
1
.
4
1
6
.
1
8
.
1
10
R/km Figure 3. Mass versus radius for the pure quark stars (solution II) in the CDM model.
References 1. E. Witten, Phys. Rev. D 30, 272 (1984) 2. N. K. Glendenning, Compact Stars - Nuclear Physics, Particle Physics, and General Relativity (Springer, New York, 1997) 3. F. Weber, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics, (IOP, UK, 1999) 4. H. Heiselberg, M. Hjorth-Jensen, Phys. Rept. 328, 237 (2000) 5. M. Hanauske, L. M. Satarov, I. N. Mishustin, H. Stocker, W. Greiner, Phys. Rev. D 64, 043005 (2001) 6. A. R. Taurines, C. A. Z. Vasconcellos, M. Malheiro, M. Chiapparini, Phys. Rev. C 6 3 , 065801 (2001) 7. H. B. Nielsen, A. Patkos, Nucl. Phys. B 195, 137 (1982); H. J. Pirner, Prog. Part. Nucl. Phys., 29, 33 (1992) 8. T. Neuber, M. Fiolhais, K. Goeke, J. N. Urbano, Nucl. Phys. A 560, 909 (1993); A. Drago, M. Fiolhais, U. Tambini, Nucl. Phys. A 609, 488 (1996) 9. A. Drago, M. Fiolhais, U. Tambini, Nucl. Phys. A 588, 801 (1995) W. Broniowski, M. Cibej, M. Kutschera, M. Rosina, Phys. Rev. D 41, 285 (1990) 10. A. Drago, U. Tambini, M. Hjorth-Jensen, Phys. Lett. 5 380, 13 (1996); Prog. Part. Nucl. Phys. 36, 407 (1996); A. Drago, A. Lavagno Phys. Lett. B 511, 229 (2001) 11. I. Bombaci, A. V. Thampan, B. Datta, Astrophys. J. 541, L71 (2000); X. D. Li, I. Bombaci, M. Dey, J. Dey, E. P. J. van den Heuvel, Phys. Rev. Lett. 83, 3776 (1999); M. Dey, I. Bombaci, J. Dey, S. Ray, B. C. Samanta, Phys. Lett. B 438, 123 (1998) 12. J. A. Pons, F. M. Walter, J. M. Lattimer, M. Prakash, R. Neuhauser, P. An, Astrophys. J. 564, 981 (2002); J. J. Drake et al., astro-ph/0204159 13. J. A. McGovern, M. Birse, Nucl. Phys. A 506, 367 (1990); Nucl. Phys. 506, 392 (1990) 14. M. Malheiro, E.O. Azevedo, L.G. Nuss, M. Fiolhais and A.R. Taurines, AIP Conference Proceedings, 631 (1), 658 (2002)
H E A V Y Q U A R K S OR C O M P A C T I F I E D E X T R A D I M E N S I O N S I N T H E C O R E OF H Y B R I D STARS
G.G. BARNAFOLDI, P. LEVAI* B. LUKACS KFKI Research Institute for particle and Nuclear Physics P.O.B. 49, Budapest, H-1525, Hungary
Neutron stars with extremely high central energy density are natural laboratories to investigate the appearance and the properties of compactified extra dimensions with small compactification radius, if they exist. Using the same formalism, these exotic hybrid stars can be described as neutron stars with quark core, where the high energy density allows the presence of heavy quarks (c, b, t). We compare the two scenarios for hybrid stars and display their characteristic features.
1. Introduction Neutron stars are natural laboratories to investigate the overlap of strong, electro-weak and gravitational interaction. Many theoretically determined properties of these astrophysical objects were tested by the observed prop erties of pulsars, and we have detailed calculations about these stars 1 ' 2 ' 3 ' 4 . However, if new perspectives appear in the description and understand ing of the gravitational interaction or in the unification of the above in teractions, then revisiting of the models becomes necessary. Such a reinvestigation was triggered by the refreshed attention on compactified ex tra dimensions 5 . Extra dimensions inside neutron stars were investigated earlier 6 , but the Kaluza-Klein (K-K) excitation modes were not considered in the equation of state (EoS). These modes are important constituents of the recent gravitation theories. Introducing the K-K modes into the EoS of fermion stars at their central core, new features and properties emerged 7 . Here we display a few of our ideas about these extra dimensions and their possible connection to particle physics. We summarize our numerical results on neutron stars with different interiors (heavy quarks vs. extra dimensions) and discuss the visibility of extra dimensions in these objects. •Talk was presented in the workshop by P. L<§vai. E-mail: [email protected]
133
134
2. The Fifth Dimension and the Eotvos Experiment The introduction of the 5 t h dimension into the real World has a long history. One interesting attempt is related to the effort of Fishbach et al. 8 in finding the "Fifth Force" in the Eotvos Experiment. Originally the Eotvos Experiment 9 has proved the Equivalence Principle (the proportionality of inertia and gravitating mass) with high precision. Deviation appeared in the 9 t h digit, only. This deviation was connected to the "Fifth Force", which may be coupled to the hypercharge Y = B + S and rather weak. Considering infinite range, the gw coupling constant of this interaction is in the order of fl^/e2 ~ 10~ 38 — 10~ 41 . Such a weak force is able to explain simultaneously the CP violation of hyperweak interaction, which has a terrestrial background in this interpretation 8 . This fifth force can be a weak force disturbing the gravity measurement, or the World is at least five-dimensional, mimicking the existence of an extra force. Earlier papers 10 ' 11 investigated five-dimensional geodesic motions as suming that metric has a Killing symmetry in the extra direction, x5, which is space-like. Since we do not observe a macroscopical 5 t h dimension, then x5 must be compactified on a microscopic scale. If we cannot observe u5, then we measure a false ul. This u% satisfies an equation of motion. In lowest order the leading "force" term mimics Coulomb force. The specific "charge" in this force starts as Krur, where u% is the true 5-velocity and Kl is the Killing vector working in the extra direction. One can obtain a constrain 10 : q^/GrriQ < 16n, where q is the charge, G is the gravitational constant, mo is the rest mass. Charge q may have sign, following the sign of u5, so 5 t h dimension offers a possibility of geometrizing vectorial forces not stronger than gravity. There is a chance that the fifth-dimensional motions of par ticles is connected to the quantum number hypercharge 8 or strangeness 11 . Quantization puts a serious constraint on five-dimensional motion. If there is an independence on x5, then the particle is freely moving in x5. However, being that direction compactified leads to an uncertainty in the position with the size of 2TTRC, where Rc is the compactification radius. An angular momentum-like quantum rule appears on the "charge" connected to the 5 t h dimensional motion, thus the smallest possible charge is Q= n
-^RV-
(1)
Because of the extra motion into the fifth dimension, an extra mass appears in 4D descriptions. Considering Rc ~ 10~ 12 — 10~ 13 cm, together with the extra interaction in the range of q, this extra "mass" is m ~ 100 MeV.
135
3. Strange Compact Stars and the Cyg X-3 We do know that pulsars are very compact objects with mass M ~ 1 M& in solar mass unit and radius R ~ 10 km. Neutron matter in its own grav ity can produce such configurations, thus the simplest explanation of the pulsars is connected to neutron stars produced in gravitational collapse 1 ' 2 . The observed maximal mass for pulsars is ~ 1.5 M Q . Early theoreti cal calculations 1 with non-interacting one component neutron matter have yielded to maximum mass w 0.67 M Q . The introduction of nuclear inter action among neutrons 2 can increase this mass by a factor of 2, however details may be still crucial, because our knowledge is very much limited about the properties of interactions at 10 times of normal nuclear densities. This leads to the conclusion, that some heavy pulsar may not be a simple neutron star, but a more complicated object with an exotic core 12 . Further astrophysical observations strengthen this expectation. In 1987 two disjoint neutrino bursts were measured from the supernova (SN) 1987A, separated by several hours. Various SN models predict one neutrino burst when/if the neutron star is formed, but never two. One could understand the double burst with the formation of two possible compact star configurations: hyperon star containing strange hadrons heavier than neutron or quark star containing a deconfined quark matter core. Between 1981 and 1991 strong muon showers were detected with a 4.8 hours periodicity 13 . The direction of these showers is that of the Cyg X-3 object, which is a close binary object: one component is a normal mas sive star with 4 M©, the other is a compact star with orbital period of 4.8 hours 14 and the muons have shown the same periodicity 15 . Their distance from Earth is « 40,000 ly. Cyg X-3 is a very intensive source in a wide spectral range and surely one important phenomena is the impact of stellar wind on the surface of the compact component generating particle pack ages, which will hit the Earth later. These packages must consist of neutral particles traversing the 40,000 light years, otherwise galactic magnetic fields would have smeared away the 4.8 hour period. However, photons or neu trinos would generate far too few muons in the terrestrial atmosphere 15 . So dilambdas 16 or (so far undiscovered) strange nuggets were suggested to be the messengers, which can be stable 16,17 and Cyg X-3 B was identified to be a hyperon (or a strange quark) star generating these strange particle packages in its surface. Theoretical neutron star calculations have found such a strange quark star configuration18, which was stable. Such an object could be the source of these strange messengers.
136
4.
T h e Tolman-Oppenheimer-Volkov E q u a t i o n
The formation of neutron stars is based on the existence of a hydrostatic equilibrium between strong and gravitational interaction. In 4 dimensions a static spherically symmetric fluid configuration has to satisfy 3 nontrivial components of the Einstein equation. Two of them give pure quadratures, the third constrain leads to the Tolman-Oppenheimer-Volkov equation 1 ' 2 ' 19 : dp dr
=
\p{r) + e{r)][GM(r) + 4TT G r3 p(r)} r[r - 2 GM(r)] '
(
'
where p(r), e(r) are the radial distribution of the pressure and the energy density, and M(r) is the mass of the neutron star within radius r. Before solving eq.(2) we have to specify the EoS of the interior matter in the form: p = p(e). Since the configuration is static, we can take cold matter in equilibrium and considering the n density of some conserved particle number (e.g. baryon number for neutron): e = e(n). Applying thermodynamic identities the pressure can be extracted: p = n de/dn — e. In eq.(2) the central energy density, ecent, can become the initial condi tion. Starting from a homogen central core with radius RQ ~ 1 cm, one can integrate the TOV equation until the surface. Because the static interior solution needs to match the exterior vacuum condition on this surface, the p = 0 condition will fix the radius at R ~ 10 km. Finally, the equilibrium configurations depend on a single parameter, ecentFor stability criteria see Ref.1, where this question is investigated in details. Practically: stability can change only at extremum points of the curve M(ecent), where either the smallest real eigenfrequency of oscillation becomes imaginary through 0, or vice versa. The actual change depends on the sign of di?/de c e n t at that point (see Chap. 6 and 7 in Ref.1 ). In case of quark stars a first order phase transition can appear inside the star. We obtain a critical surface at the critical pressure, p c r , where two different e values can be found, depending on the EoSs. Since p is continuous, then the sectionwise solutions of the TOV equation can be determined. However, the solution of the TOV equation jumps from ex to e 2 . This jump is absent in the case of second order phase transition. Introducing extra dimensions, in the general case the 5-dimensional Ein stein equations cannot be reduced into a single, TOV-type first order differ ential equation, but a coupled differential equation system is generalized 6 ' 7 . Assuming a radially independent compactification circumference the TOV equation reappears with a 5-dimensional interpretation. The details of this correspondence is discussed in the Appendix.
137
5. Neutron Star in 4 Dimensions Our reference object is an "ideal" neutron star, which consists of pure neutron matter. We use free fermion gas EoS with multiplicity djv = 2. We neglect the "normal matter" constituent, which is a very thin and negligible layer at the surface of neutron stars. The radius is obtained in the range of R ~ 5 — 15 km for /J,N > 1 GeV, reproducing earlier results from Ref.1. Real neutron star calculations are more complicated (see Refs. 2 ' 18 ), however we want to demonstrate some features and the above simple model is just appropriate for this task. On Figure 1 the mass and the radius of the neutron star are displayed as the function of central energy density. The first upgoing part of M(ecent) is stable until the star (*), the next is unstable, and all downslope parts of M(eceni) are unstable. In parallel, we constructed the M(R) curve (see right hand side), where the spiraling behaviour can be clearly identified1. i
IT,
2
S
T
\ ! =/ 1
Quark core
Si
0.75 0.5
IT,
1.75
r
1.5
:
1.25
-
Neutron star
■f
T,
f I Quark core
0.25 ■
T,
1
\
Interacting neutron matter
:
0.5 Neutron star IT, Qj
^
_lTj_^,
^ —
0.25
W
Non-interacting neutron matter
Quark core 10' Iff" , ^(GeV/fm1)
R(km)
Figure 1. The mass and the radius of neutron stars containing pure neutron matter or light quark core. The M(R) function is displayed on the right hand side for both cases. Dotted line between full dot and open dot indicates a discontinuity before the appearance of quark core (Q). The T\, T2, T3 display turning points for quark star.
Introducing a quark core with light (u and d) quarks, a first order phase transition occurs inside the neutron star. Considering an interacting EoS for the neutron matter 20 and a free fermion gas EoS for the quark matter with a bag constant B = 0.25 GeV/fm 3 , the quark core appears (Q) at £cent = 1-7 GeV/fm 3 . No stable configuration was found between turning points Ti and T3, e.g. light quark star is unstable around T 2 . The region between Q and 7\ is promising, but a more sophisticated model is needed.
138
6. Heavy Flavours in Hybrid Stars One can observe particles heavier than the neutrons, carrying conserved quantum numbers, namely the S strangeness, C charmness, B bottomness (and maybe T topness). Weak interaction is allowed to create these flavours inside the core of a neutron star at extremely high energy density. Let us construct an "ideal heavy hadron star" consisting of neutrons and other heavier hadrons with these heavy flavours. For simplicity we consider the neutral As(1115), Ac(2452) and A B ( 5 G 2 4 ) (we stop at central chemical potential fj,cent — 25 GeV, thus we miss the A T ) . The multiplicities are di = 2 and free fermion EoS is used in the TOV equation as previously. Figure 2 displays our results for heavy hadron star, which are very close to the pure neutron star case, e.g. the stability and the spiraling feature remain the same. Just before the first peak stable configuration appears with As core and N mantles ("hyperon star" 2 ) . In the first maximum (*), far before the appearance of Ac, the stability ceases and never returns.
Figure 2. The mass and the radius of heavy hadron stars with and without heavy quark core. The M{R) function is displayed in the right hand side for both cases.
Introducing a quark core with light (u,d) and heavy (s,c,b) quarks, a first order phase transition have to be considered, again. Since the s quark is relatively light, then some modifications appear, but the investigated fea tures remain similar. The quark core appears (Q) at ecent = 1-3 GeV/fm 3 , when u, d, s quarks are already present. No stable configuration was found between turning points Ti and T 3 , but close to Q stable states ("strange quark stars" 18 ) may appear. However, no charm, bottom or top quark stars are expected to be stable at high central energy densities 21 .
139
7. Hybrid Stars in 5 Dimensions Now let us consider a neutron star with excitations in the 5 t h dimension, which are compactified. In this case neutrons are moving into the direction of x5. In the special case of x5-independence this motion appears as an extra degrees of freedom with larger masses:
(m$)2 = (i/Re)2 + m%.
(3)
Here the integer i means the excitation level, Rc is the compactification radius of the 5 t h dimension and m ^ ' = mAr(940). The EoS is generated as the superposition of the different K-K modes 7 , summing up the free fermion EoS at masses from eq.(3). If we choose Rc = 0.33 fm, then the first exci tation has the mass identical with that of the A particle, nvN' = A(1115). The TOV equation remains valid to find the appropriate equilibrium states. Figure 3 displays the (non-interacting) heavy hadron matter case (full line) and the 5-dimensional cases with two different compactincation radii, i? c = 0.33 fm (dashed line) and Rc = 0.66 fm (doted line). The filled and open triangles indicate the K-K modes. Jo.8 ^0.7
0.5
Hyperon star Extradim. core (0.33) Extradim. core (0.66)
LE, *N*
0.6 -
0.4 L
=
0.5
/ FA ! /
0.4
__—•&•■
Fz
F.-^sJL***^. F ^-A -A-AA-
0.3
5
0.2
zr^
-
Re-0.66 fm
0.35 0.3 0.25
0.1
F3
0.45
k
c
~")
A - . . A — A - ^ R ^ O . 3 3 fm
0.2 R(km)
:As\ %E,
Fi
...v^****^
F„ -A~4-»'
10' W £„*(GeV/fm 3 )
R(km)
Figure 3. The mass and the radius of heavy hadron stars in 4 dimensions and neutron stars including extradimensional K-K modes into the core.
In the case of Rc = 0.33 fm the first K-K mode (Ei) appears in a stable configuration indicated until the star (*). Choosing Rc = 0.66 fm, we have Fx and F2 modes before the end of stability (the star (*)) on the dotted line. In the case of Rc > 0.25 fm, one or more extradimensional modes can fit into the core before loosing the stability of the hybrid star.
140
8. More Extra Dimensions The introduction of the 5 t h dimension is a minimal extra-dimensional model. One may assume an x6, or more extra dimensions, roughly on the scale of x5. (The idea of six-dimensional microphysics goes back to 35 years 27 .) The opening of the 6 t h dimension leads to similar results as dis played in Figure 3, however now more combinations of extra-dimensional excitations may appear in the stable region. In Ref.7 even two stable regions and two maximums on M(R) appeared, indicating the complexity of the higher dimensional results. It is interesting to mention, that the existence of such a second stable peak can explain the double neutrino burst of SN 1987A. However, detailed calculations are needed to verify this explanation. 9. Conclusions We have demonstrated in a simple model for compact star that neutron stars with hyperon or extradimensional core are very similar objects. The TOV equation leads to similar structure with well-defined stability region, where the lowest K-K modes may appear in the extradimensional case. The main reason is connected to the size of the compactification dimension: assuming Rc = 0.33 fm one obtains the mass of A0 for the first K-K mode, which dominates the inner structure of the hybrid star in the stable region. We saw in Section 2 that motion in the fifth dimension results in an apparent new quantum number, similar in structure to strangeness, so a neutron moving into the 5 t h dimension may be seen as a As particle. (The resulting apparent "violation of equivalence" will appear roughly in the order of Ref.8 as shown in Ref. 25 .) Now the dilambda explanation of muon bursts triggered by Cyg X-3 for short periods is not utterly hopeless, although rather cataclysmic events are needed to get strange matter to the surface. The introduction of more extra dimensions leads to the appearance of further stability regions and the existence of more than one stable hybrid star configuration. Such a result is supported by the double neutrino shower arrived from SN 1987A. The investigation of neutron stars with light and heavy quark core leads to different characteristics and to higher star masses. It remained open the question of stability of these objects, more sophisticated models 28 ' 29 ' 30 are needed for interacting neutrons and quarks. Interesting question is if any extra-dimensional setup is able to mimic the features of the heavy quark stars. The connection between strange quark and extra-dimensional propagation of light quark deserves further studies.
141
Appendix A: 5D Static Equilibrium and the TOV Equation Let us consider a spherical equilibrium situation in 5 dimensions. The solu tion of the Einstein equation must be stationary and spherically symmetric. Since we do not have information about the dependence on the microscopic, compact dimension, and cannot observe either, let us assume x 5 indepen dence. Thus we have an U(l) ® SO(3) ® U(l) Killing symmetry, 5 Killing vectors altogether, with 4-dimensional transitivity; SO(3) is transitive in 2 dimensions. Using permitted coordinate transformations, on the analogy of the derivation of Ref. 22 we arrive at the general form of the line element: ds2 = g00dt2 + ffndr2 - r 2 dfi 2 + s 55 d:r 5 d:r 5 + 2g0Sdtdx5 2
.
(4)
Here dfi is the usual spherical elementary surface. The still unknown components 26. The more general stationary class, with some interplay or drag between dt and d i 5 would deserve further attention in a subsequent paper. Now we are confronted with a further choice. We are restricted to static fluids, and by definition, the 3-dimensional stress tensor of a static fluid is isotropic (Pascal law). However, we are in 5 dimensions, so the space-like section is of 4 dimensions. What is the proper generalisation of a fluid now? The problem is discussed but by no means obligatorily solved in Ref. 2 3 . For a macroscopical 5 t h dimension full equipartition, (thus 4 dimensional isotropy) would have good arguments. However, in our case the 5 t h dimen sion is microscopic, when usual interactions in fluids (e.g. van der Waals forces) could not establish full spatial isotropy. Until a convincing ansatz is found, we decouple the 55 component of the Einstein equation. The re maining equations have always two sets of solution: either dg^/dr = 0 or not. In the first case the equations can be converted into the TOV equa tion, and T 55 can be calculated afterwards. In the second case a system of coupled differential equations appears and the compactification radius has radial dependence 7 .
142 Acknowledgments P.L. t h a n k s for t h e warm hospitality of A. Krasnitz a n d t h e Workshop organizers, they have created a wonderful meeting a t Faro. T h i s work was supported by t h e O T K A T032796 a n d T034269. References 1. B.K. Harrison, K.S. Thome, M. Wakano, J.A. Wheeler, Gravitation theory and gravitational collapse, University of Chicago Press, 1965. 2. N.K. Glendenning, Compact Stars, Springer, 1997; and references therein. 3. F. Weber, Astrophysical Laboratories for Nuclear and Particle Physics, IOP Publishing, Bristol, 1999. 4. D. Blaschke, N.K. Glendenning, A. Sedrakian (Eds.), Physics of neutron star interiors, Springer, Heidelberg, 2001. 5. L. Randall, R. Sundrum, Phys. Rev. Lett. 8 3 , 3370 (1999); ibid. 4690 (1999). 6. A.R. Liddle, et al. Class. Quantum Grav. 7, 1009 (1990). 7. N. Kan, K. Shiraishi, Phys. Rev. D 6 6 , 105014 (2002). 8. E. Fishbach et. al. Phys. Rev. Lett. 56, 3 (1986); Ada Phys. Hungarica 69, 335 (1991). 9. R. von Eotvos, D. Pekar, E. Fekete, Annalen der Physik 68, 11 (1922). 10. B. Lukacs, T. Pacher, Phys. Lett. A 1 1 3 , 200 (1985). 11. B. Lukacs, in Relativity Today, Proc. of the Sixth Hungarian Relativity Work shop, Akademia, Budapest, Eds. C.A. Hoenselaers, Z. Perjes, p. 161. 12. See N.K. Glendenning, this Proceedings. 13. M.A. Thomas et al. Phys. Lett. B269, 220 (1991). 14. M. van der Klis, J.M. Bonnet-Bidaud, A. & A. 95, L5 (1981). 15. M.L. Marshak et al., Phys. Rev. Lett. 54, 2079 (1985); ibid. 55, 1965 (1985). 16. G. Baym et al. Phys. Lett. B160, 181 (1985). 17. S. Fleck et al. Phys. Lett. B220, 616 (1989). 18. Ch. Kettner, F. Weber, M.K. Weigel, N.K. Glendenning, Phys. Rev. D 5 1 , 1440 (1995). 19. J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55, 374 (1939). 20. G.G. Barnafoldi, P. Levai, B. Lukacs, in preparation. 21. M. Prisznyak, B. Lukacs, P. Levai, (KFKI-1994-24/A, astro-ph/9412052). 22. S.W. Hawking, G.F. Ellis, The large scale structure of space-time, Cambridge University Press, 1973. 23. D. Sahdev, Phys. Rev. D 3 0 , 2495 (1984); ibid D 3 9 , 3155 (1989). 24. G.G. Barnafoldi, Fermion stars with extra dimensions (in Hungarian), M.S. Thesis, 2001. Budapest 25. B. Lukacs, J. Ladik, Phys. Ess. 6, 39 (1993). 26. Z. Perjes, The properties of stacionary gravitation fields (in Hungarian) Ma gyar Fizikai Folyoirat, IV. kotet 3. fuzet, Budapest, 1976. 27. H.F. Ahner, J.L. Anderson, Phys. Rev. D l , 488 (1970). 28. D. Blaschke, H. Grigorian, D.N. Voskresensky, A. & A. 368, 561 (2001). 29. D. Blaschke, S. Fredriksson, M. Oztas, (astro-ph/0111587). 30. D. Blaschke, et al. (hep-ph/0301087).
RATIOS OF A N I T B A R Y O N / B A R Y O N YIELDS IN HEAVY ION COLLISIONS
YU.M. SHABELSKI Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300 Russia E-mail: shabelskQthd.pnpi.spb.ru
We discuss model predictions for antibaryon/baryon production ratios in high en ergy heavy ion collisions. The role of string junction mechanism of baryon number transfer seems to be very important here and we consider some quantitative results.
The first RHIC experimental data collected in 1 show that the values of antibaryon/baryon production ratios in midrapidity region of Au-Au central collisions at ^/SNN = 130 GeV are rather small in comparison with most theoretical predictions. Really, the values of the order of 0.6 for p/p and 0.73 ±0.03 for A/A were measured whereas the standard Quark-Gluon String Model (QGSM) 2>3'4>5 predicts in both cases the values more than 0.9 and in the String Fusion Model 1 the predicted values are about 0.8 and about 0.87, respectively. The main part of B and B should be produced at high energies as B — B pairs. It means that the additional source of the baryons in the midrapidity region is needed. The realistic source of the transfer of baryons charge over long rapidity distances can be realized via the string junction (S J) diffusion where it can combain three sea quarks into a secondary baryon (but not antibaryon), see 6 . In such a picture where the transfere of baryon charge is connected with SJ exchange there exist three different possibilities to produce a secondary baryon which are shown in Fig. 1 7 . This additional transfer of baryon number to the midrapidity region is governs by contribution 7 Glu = Gpud = aN^[v0e(l
- zf~a"
+ vqz3/2(l
- z) + vqqz2} ,
(1)
where the items proportional to vqq, vq and VQ correspond to the contribu tions of diagrams Fig. la, lb and lc, respectively. The most important in
143
144
the midrapidity region at high energies is the diagram Fig. lc which obeys the baryon number transfer to rather large rapidity region. In agreement with the experimental data the parameter e in Eq. (1) is rather small. However different data are in some disagreement with each other, see the detailed analyses in 7 . Say, XF -distributions of secondary protons and antiprotons produced at 100 and 175 GeV/c and at y/s — 17.3 GeV are in better agreement with e = 0.05, whereas ISR data and the p/p asymmetry at HERA energy are described better with e = 0.2. Some part of this disagreement can be connected with different energies. In Fig. lc, as a minimum, two additional mesons M should be produced in one of the strings, as it is shown in this Fig., that can give the additional smallness 8 at not very high energy due to decrease the available phase space. Another source of disagreement of the data at low and high energies can come from the fact that the suppression of baryon number transfer to large rapidity distance 5y should be proportional to e-(aSj-l)-Ay
?
(2)
and the effective value of agj depends on the energy due to Regge cut contribution 9 . In the case of -K~p ->■ ClX, QX reactions the contribution (1) leads to the contradiction with the simplest version of additive quark model 1 0 because experimentally (see 7 ) the yields of fl in the central region is permanently larger than the yields of fl. Let us note that it is rather dangerous to use large value of e as well as the value of asj close to unity in Eq. (1). The reason is that the string junction mechanism can not transfer more baryon charge than we have in an incident state. The hadron content of sea-quark baryons can be written "as (u + d + As)3 = Ap + An + 12A(A + S) + ... ,
(3)
where A is the suppression factor for strange quark production. So the integral multiplicity of the protons produced via SJ mechanism from one incident baryon can not be larger than Wp = 4/((8 + 12A) RJ 0.3 — 0.4 (for A = 0.2 - 0.4) : /•CO
/ e(asj-VAydAy < Wp (4) Jo The HERA data on p/p asymmetry can be described 7 with the param eters which are near to the presented boundary, namely asj = 0.5, e = 0.2. e
145
However, if we use the large part of the initial baryon charge for its diffu sion to the mid-rapidity region, the multiplicity of secondary baryons in the fragmentation region should significantly decrease. It will result in addi tional mechanism of Feynman scaling violation in the fragmentation region in comparison with the estimations which were claimed in 12 . In the case of hadron-nucleus collisions the yields of secondaries can be calculated in QGSM by similar way 3 ' 1 2 and also with including SJ contributions.
p
B
c c c c c a
b
e
Figure 1. Three different possibilities of secondary baryon production in pp collisions: SJ together with two valence and one sea quarks (a), together with one valence and two sea quarks (b), together with three sea quarks (c).
In the case of heavy ion (A-B) collisions the multiple scattering theory allows one to account for the contribution af all Glauber-type diagrams only using some Monte Carlo method where the integrals with dimention of about 2 • A ■ B should be calculated in coordinate space 13>14. The analitical calculations allow one to account only some classes of diagrams 15,16,17 Q n e 0 £ approaches here is so called rigid target approximation where it is assumed that for the forward hemisphere we can neglect by the binding of projectile nucleons (i.e. consider them as a beam of free nucleons and every of them can interact with target nucleus). The last one is considered as a dense medium. And vice versa, for the backward hemisphere we consider the target nucleons as a beam of free nucleons and every of them can interact with dense medium of projectile nucleus. All details and needed formulae can be found in 5 . The resulting expression for
146
secondary h production in A — B collisions reads as 1 prod
AB-+hX 1 NB -+hX = e(y){NA)- prod dy dy T NB NA^hX 6(-V)(NB) prod dy T NA
+ (5)
where {NA) and {NB) are the average numbers of interacting nucleons in nuclei A and B. They depend on the A — B impact parameter, A/B ratio, etc. 18 . The calculated ratios oip/p and A/A production in Au — Au collisions at RHIC, predicted by Eq.(5) with accounting for the percolation effects 19 are presented in Table 1 20 . One can see that small, e = 0.05, SJ contribution can not explain the data. Comparatively large contribution (e = 0.2), close to the upper limit (4) gives the ratios more close to their experimental values. The more accurate accounting of inelastic shadowing/percolations (multipomeron interactions) 19 ' 21 can lead to better agreement with the data.
p/p A/A
QGSM e = 0.05 e = 0.2 0.83 0.67 0.64 0.83
Exper. 0.6 0.73 ± 0.03
In conclusion we note that B/B asymmetry in midrapidity region at RHIC energies is rather large. It can be explained by large SJ contribution that is in reasonable agreement with HERA data.
Acknowledgements This work was supported by grants NATO PSTCLG 977275 and RFBR 9802-17629. I am very grateful to G.Arakelyan, A.Capella, J.Dias de Deus, A.Kaidalov, C.Pajares and R.Ugoccioni for collaboration and useful discus sions.
147
References 1. N.Armesto, C.Pajares and D. Sousa; hep-ph/0104269. 2. A.B.Kaidalov and K.A.Ter-Martirosyan. Yad.Fiz. 39 (1984) 1545; 40 (1984) 211. 3. A.B.Kaidalov, Yu.M.Shabelski and K.A.Ter-Martirosyan. Yad.Fiz. 43 (1986) 1282. 4. Yu.M.Shabelski. Z.Phys. C57 (1993) 409. 5. J.Dias de Deus and Yu.M.Shabelski; hep-ph/0107136. 6. D.Kharzeev. Phys.Lett. B378 (1996) 238. B.Kopeliovich and B.Povh. Phys.Lett. B446 (1999) 321. 7. G.Arakelyan, A.Capella, A.B.Kaidalov and Yu.M.Shabelski. hep/ph/0103337. 8. K.A.Ter-Martirosyan and Yu.M.Shabelski. Yad.Fiz. 25 (1977) 403; 570. 9. A.Schaale and Yu.M.Shabelski. Yad.Fiz. 46 (1987) 594. 10. V.V.Anisovich, M.N.Kobrinski, J.Nyiri and Yu.M.Shabelski. Sov.Phys.Usp. 27 (1984) 901. 11. A.Capella, C.-A.Salgado. Phys.Rev. C60 (1999) 054906. 12. Yu.M.Shabelski. Z.Phys. C38 (1988) 569. 13. Yu.M.Shabelski. Yad.Fiz. 51 (1990) 1396. 14. D.Krpic and Yu.M.Shabelski. Yad.Fiz. 52 (1990) 766; Z.Phys. C48 (1990) 483. 15. M.A.Braun. Yad.Fiz. 45 (1987) 1525.3 16. K.G.Boreskov and A.B.Kaidalov. Yad.Fiz. 48 (1988) 575. 17. V.M.Braun and Yu.M.Shabelski. Int J. of Mod.Phys. A3 (1988) 2417. 18. C.Pajares and Yu.M.Shabelski. Phys.Atom.Nucl. 63 (2000) 908. 19. J.Dias de Deus, Yu.M.Shabelski and R.Ugoccioni. hep-ph/0108253 20. Yu.M.Shabelski. hep-ph/0205136. 21. A.Capella, A.B.Kaidalov and J.Tran Thanh Van. Heavy Ion Physics 9 (1999) 169.
This page is intentionally left blank
Cosmic Rays
This page is intentionally left blank
EUSO: BASIC PARAMETERS
M. C. ESPIRITO SANTO LIP, Av. Elias Garcia, 14-1, 1000 Lisboa, Portugal E-mail: [email protected]
Extreme energy cosmic rays are one of the most challenging topics of theoretical and experimental Astroparticle Physics today. The EUSO experiment will collect a high statistics of such events by looking downwards to the dark Earth atmosphere and detecting the light produced by the interaction of high energy cosmic particles with the atmosphere. EUSO will measure the energy and direction of high energy cosmic rays, and give information on their composition. The importance of the EUSO results will be driven both by the collected statistics and by the precision of the performed measurements. Parameters that influence these measurements in a critical way are discussed below, as well as ongoing efforts to measure them.
1.
Introduction
The main goal of EUSO - Extreme Universe Space Observatory is to detect Extreme Energy Cosmic Rays (EECR) and neutrinos, that may be indicative of unknown particle production and acceleration mechanisms in the Universe [1]. EUSO will look downwards to the dark Earth atmosphere from the International Space Station (ISS) and image the Ultraviolet (UV) fluorescence light traces produced by charged secondary particles along the Extensive Air Shower (EAS) initiated by cosmic relativistic particles and developing in the Earth atmosphere. The Cherenkov Figure 1: The EUSO operating principle.
151
light accompanying the shower reflected/diffused back by
an(j
152
the Earth surface or the clouds will also be detected by the EUSO telescope. The EUSO operating principle is depicted in Figure 1. The EUSO instrument [2] is schematically shown in Figure 2. It consists of a wide-angle optical system (Fresnel lenses, about 2.5 m diameter) concentrating the UV light onto a large focal surface made up of thousands of multipixel photomultipliers. On-board electronics will handle instrument control, triggering and data taking, selection and transmission. EUSO is foreseen to be installed as an external payload on Figure 2: The EUSO instrument.
the Columbus module of the ISS in 2009. EUSO is currently approved by ESA for a Phase A study, to be completed by the Summer of 2003. The EUSO design criteria are based on an orbital altitude of about 380 Km, a field of view of ±30° around the zenith, a pixel area on ground of 0.8x0.8 Km2 and an energy threshold of about 3xl0 19 GeV. About 1000 events per year for three years are expected. EUSO will overcome the problem of the very low flux of EECR by looking downwards to the Earth atmosphere, with an observed area on ground of over 100000 Km2 , and a target mass of the order of 2.1012 Tons of air. The EUSO observation wavelength range is 300-400 nm (near-UV). EUSO will measure the energy of EECR with a precision of about 20%, and their arrival direction within l°-2°. Information on their composition can be obtained namely by the measurement of the depth of the shower maximum. Which are, then, the parameters that influence the these measurements in a critical way and what can we do about them? The different parameters and their relevance are discussed in section 2. Ongoing experimental studies are discussed in section 3. 2.
Basic Parameters
The Earth atmosphere is the ideal large scale detector needed for EECR detection. Interacting with the atmosphere, they give raise to EAS, as a result of a complex relativistic cascade process. EAS are accompanied by the isotropic emission of UV fluorescence light. A collimated beam of Cherenkov light
153
accompanies the cascade. The atmosphere is, thus, the medium where the EUSO signal is produced and propagates. Furthermore, any UV background in the Atmosphere will be seen by EUSO. Thus, all aspects relevant for UV light production and propagation in the Earth atmosphere will affect the EUSO measurement. The atmospheric characterization, as well as the study of such aspects, are crucial for EUSO. The main parameters relevant for the EUSO measurement fall, thus, into three categories: the ones related to light production, light propagation and background. 2.1. Light 2.1.1.
Production
Air Fluorescence
Yield (AFY)
The development of an EAS in the atmophere is accompanied by the emission of UV fluorescence light, which is produced by the de-excitation of Nitrogen molecules excited by the electrons of the shower moving in the air. The fluorescence yield spectrum extends over a large range of wavelengths, but has a number of important peaks in the region of the near-UV (300-400 nm), as shown 1U " in Figure 3. The AFY is d affected by the pressure, ~Z temperature and compo| 10 sition of the atmosphere, * and the dependence on t> these factors may be S. io0
"2 I *
different for the different spectral lines. This is illustrated in
1(r, 320
330
340
350
360
370
380
390
Wavelengths in nm
Figure 3: Fluorescence lines in air.
400
TTimiro
A
wVis>rf»
tVio
rlgure
4,
wnere
me
AFY dependence on the altitude is shown. In EUSO, the shower energy will be estimated from the integral of the fluorescence light detected. The imaging of the fluorescence trace over time will give us the shower direction and information on the shower longitudinal profile. In fact, the AFY is used by several experiments to estimate EECR energy, and uncertainties on its knowledge can be an important contribution to their energy uncertainty. An accurate knowledge of the conversion from fluorescence yield to ionization energy loss by EAS particles is thus crucial for this technique. AFY studies have been carried on for decades and several different measurements exist [3]. However, some opened questions remain, in particular in what
154
Figure 4: AFY dependence on the altitude. concerns the behaviour of individual spectral lines and differences between the various experimental results. 2.1.2.
Cherenkov light
The development of an EAS is accompanied by the emission of a very large amount of Cherenkov photons by the ultrarelativistic electrons of the shower. This light is emmited in a narrow cone (about 1°-1.5°) around the direction of the particle, and it is thus beamed forward along the direction of the shower axis. The number of photons (mainly UV) and the cone aperture depend on the refractive index at each altitude. In EUSO, the Cherenkov light diffusively reflected back from ground can be detected. It will provide and alternative measurement of the shower energy and give the point of Atmosrjhere impact of the shower front on ground, allowing an absolute measurement of the depth of shower maximum. Figure 5 illustrates in a schematic way how the combined detection „. , TT , , , , . _. _. of the fluorescence trace and Figure 5: Hadron shower and Neutrino Shower as imaged by the EUSO focal surface. Cherenkov peak of a shower
155 in EUSO can provide information on the nature of the primary particle, through the determination of the depth of shower maximum. For this purpose, the knowledge of the albedo and altitude of the reflecting surface are crucial. In summary, Cherenkov detection and interpretation in EUSO crucially depend on the knowledge of the reflecting surface altitude and albedo. 2.2. Light Propagation The Earth atmosphere is not transparent for UV light, as different scattering and absortion processes exist [4]. The scattering in air molecules (Rayleigh scattering) is an important effect. It is, nevertheless, rather stable, and its dependence with 1/A.4 is well known. This is the dominant effect in the upper atmosphere. Mie scattering involves particles of size comparable to the light wavelength (dust particles, water droplets, ...). Contrary to Rayleigh scattering, Mie scaterring is higly variable with time and position, and thus harder to know with precision. It is relevant mostly in the first few kilometers above ground. Thus, in the case of EUSO, Mie scattering affects may not be vary ;1976 U.S Standard Model relevant for the fluorescent trace detection, but crucial for the Cherenkov peak interpretation. The effect of Rayleigh and Mie / / // A=337 nm A=391 nm - /* / // scattering is shown in Figure 6 for Total two different wavelengths, using if fl J is the US standard atmospheric Rayleigh model. Mie '// Abosrtion by the Ozone layer '• is expected to affect only the lower 10 15 20 25 30 35 40 edge of the EUSO wavelengths altitude of the starting point band (below 330 nm). Figure 6: UV light scattering in the atmosphere. Clouds are a crucial element in the description of light propagation in the Earth atmophere On average, at any moment half of the surface of the Earth is covered by clouds, which can have very different characteristics (altitude, optical thickness, ...). The knowledge of the cloud coverage is crucial for EUSO, as clouds define the lower boundary of our natural detector and will affect both the fluorescence and Cherenkov light detection. As illustrated in Figure 7 (based on preliminary simulation studies performed for EUSO) the presence of a thick cloud at a significant altitude (2 Km) will considerably affect i . . . .
156
the flurescence light detection. In particular, dense clouds with top altitudes not allowing EAS observation beyond the shower maximum will in practice disable the EUSO fluorescence observation capability, while they may be a good reflecting surface for Cherenkov. Optically thin clouds may originate more complex Cherenkov reflection patterns. Cloud height determination is essencial for the interpretation of Cherenkov signal in EUSO. Last but not least, the knowledge of the cloud coverage and characteristics on the EUSO field of view is crucial for the determination of the EUSO effective aperture and makes it a rather complex.
Figure 7: Preliminary simulation of the shower profile observed by EUSO with clear sky (left) and in the presence of a thick cloud (right).
2.3. Background The UV light signals arriving to the EUSO telescope on the ISS are expected to be quite faint (as a reference, AFY is of the order of 4 photons per meter of propagation length). The knowledge of the UV background in the atmosphere throughout all the ISS path is crucial for EUSO. Different background sources can be considered. Sky objects have a clear effect on the EUSO duty cycle definition. Both the Sun and the full Moon will prevent operation. While operation in different Moon phases is under study, diffuse light from the Stars is expected to have a negligible effect. Transient phenomena in the atmosphere (storms, meteors, auroras) are much slower events than EECR and can thus be separated. Besides, they are interesting by themselves and will be studied by EUSO. Human light sources (namely city lights) have an important UV
157
component and might create very bright spots in the EUSO focal surface (factor of about 200 on the background over a few hundred pixels, according to preliminary studies). A particularly relevant background source for EUSO is the Ionosphere Nightglow, which is fluorescence light emission caused by photochemical processes or low energy cosmic ray interaction in the upper atmosphere (50-110 Km). The Nightglow brightness can be higly variable with time and latitude. As an example, measurements from Cosmos45 [5] give an estimation of 500 to 1500 photons/ns for EUSO (about 0.3 photons per pixel and per integration time window of 1 milisecond). The understanding of the background for EUSO is important for detector design and trigger algorithm optimization. 3.
Measuring the basic parameters
Several experiments and projects have been identified, in the context of EUSO, as supporting activities aiming at a better knowledge of the different parameters and problems listed above. They are essentially three-fold, and address for the moment the detailed knowledge of: air fluorescence yield, UV background in the atmosphere, Cherenkov light propagation and diffusion. They will be briefly discussed below. 3.1. Air fluorescence Yield As discussed above, the precise understanding of AFY is crucial for all the experiment using the fluorescence technique to determine the energy of EECR. Open questions concern the discrepancies between existing measurements, as well as the pressure and particle energy dependence of individual spectral lines. At present, there are several ongoing experimental efforts in the USA, Japan and Europe. A recent discussion and overview of the ongoing projects took place at the First International Workshop on AFY (FIWAFY) in October 2002, in Utah [6]. 3.2. The BABY programme The BABY programme is a series of balloon flights aiming at the study of the UV background in the atmosphere. Three BABY flights took place up to now, in clear, moonless nights, from the Milo-Trapani launch base in Sicily, in the summers of 1998, 2001 and 2002. The BABY apparatus [7] is a set of photomultipliers looking down to Earth and measuring the light in the 300-400 nm range. In the most recent flights, different narrow band UV filters were also
158
used. The spectrum measure over land and sea by BABY98 (no filter) is shown in Figure 8. A brightness of about 400 photons/m2.sr.ns over sea was measured. The recent BABY2002 flight was a trans-Mediterranean flight from Sicily to Spain and its data is currently being analysed. Flights at different latitudes are foreseen for the near future. BABY data profile. Milo-Trapani July 30 1998 Ifcai - CNR, Palermo, Italy 10000
1000
E
100
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
time (sec.)
Figure 8: BABY98 light spectrum over land and sea. 3.3. The ULTRA experiment The ULTRA (Uv Light Transmission and Reflection in the Atmosphere) experiment has been designed to provide quantitative mformation on the reflection/diffusion signal produced by the EAS impacting on the Earth surface [8]. The ULTRA apparatus is schematically shown in Figure 9.
Figure 9: Schematic view of the ULTRA apparatus.
159
A scintillator array using conventional sampling technique will trigger on EAS impacting on ground, while an UV light detector (300-400 nm wavelength interval) placed uphill and operating in coincidence with the array will detect the Cherenkov light from the EAS reflected on ground. The synchronization of the system will be performed using a GPS based system. While triggering on the EAS, the scintillator array will also measure its properties, namely the shower size (proportional to the total number of Cherenkov photons) and the shower direction (for which a fast and precise timing is a crucial element). The UV light detector will be similar to the one used in the BABY flights, but with a much wider filed of view. The atmospheric transmission properties will also be studied, using the UV detector and a laser emitter, which will also be used for calibration purposes. The reflected Cherenkov light will be measured in different locations, allowing the determination of the albedo of different types of surfaces (water, snow, desert, forest,...). Early tests of ULTRA have been performed in the Alps in 2002 (using three scintilator units and an UV light detector). Physics runs are foreseen for 2003. 3.4. Atmospheric sounding As discussed above, the knowledge of the atmospheric parameters is crucial for EUSO. The transmission properties in the light path are crucial for an accurate energy measurement, and the knowledge of the cloud coverage in the field of view matters for the acceptance determination. Online atmospheric monitoring is foreseen for EUSO. As a baseline, a LIDAR system is under study. A LIDAR can detect the intensity, time delay and frequency shifts of the echoes backscattered as a laser beam propagates through the atmosphere. Scanning the EECR path would allow to obtain the atmospheric transmission profiles, thick clouds top heights, extinction profile of thin clouds and aerosol layer observation. Information from dedicate satellites databases will also be used. Other atmospheric sounding strategies, like the use of infrared images of the field of view for cloud coverage and height determination, are also under study. 4.
Conclusions
The existence of EECR constitutes one of the great puzzles of astroparticle physics today. Experimentally, the present situation is characterized by the rather low statistics of observed events. Although the existence of such events seems confirmed, there are differences in the fluxes measured by today's largest experiments [1]. The solution seems to be to perform higher statistics, reliable measurement. This will be done in the near future with Auger, which has also the
160
advantage of allowing the intercalibration of the two energy determination techniques (AFY and sampling ground arrays). EUSO will allow to go one step further. For that, a good knowledge and control of atmospheric parameters affecting light production, light propagation, and background levels is crucial. Several projects aiming at a better knowledge of the different relevant aspects are ongoing. Acknowledgments I wish to thank all my EUSO colleagues for the help and material provided. In particular, I thank C. Berat and E. Plagol for the results of the simulations of the atmosphere effect and J. Adams for the useful discussions and information on Nightglow brightness measurements. I thank O. Catalano, D. Lebrun, and B. Tome for the material on BABY and ULTRA. I thank also M. C. Maccarone, M. Pimenta and M.Teshima. References 1.
2.
3.
4. 5. 6. 7.
8.
See Talk by L. Scarsi in these Proceedings; Many recent reviews exist, e.g. V. Berezinsky, astrojph/0107306 (2001); G. Sigl, astro-ph/0210049 (2002); F. Stecker, astro-ph/0208507 (2002); X. Bertou et al, astro-ph/0001516 (2000). L. Scarsi et al., "EUSO - Extreme Universe Space Observatory: Doing Astronomy by looking downward the Earth atmosphere", Proceedings of ICRC 2001 and references therein. A.N. Bunner, PhD Thesis, Cornell University, Ithaca, N.Y. (1964); F.Kakimoto et al., MM, A372, 527-533 (1996); Sakaki, Aspen winter conference Ultra high energy particles from space, January 2002. D.G. Andrews, "An introduction to atmospheric physics", Cambridge University Press (2000). A.I. Levedinsky et al., "Space Research", p.77-88, Nauka, Moskow (1965). http://www.physics.utah.edu/~fiwaf/done/index.html O.Catalano et al., NIM A480/2-3, 547-554, (2001); S.Giarusso et al, "Nocturnal atmosperic UV background measurements in the 300-400 nm wavelengths band with BABY 2001: A transmediterranean balloon borne experiment", Proceedings of ICRC 2001. O.Catalano et al., "ULTRA Technical Report" (2002).
T H E R A D I O T E C H N I Q U E 40 Y E A R S LATER: W H E R E DO W E S T A N D ?
ENRIQUE ZAS Departamento de Fisica de Particulas, Universidad de Santiago de Compostela, E-15706 Santiago, Spain. E-mail: [email protected] In this article I stress the merits of the radio technique for detecting ultra high energy showers based on the knowledge acquired through the study of the radio pulses generated by the excess charge development. Experimental initiatives in the radio technique are also addressed.
1. Introduction It was about 40 years ago that the Armenian physicist Askar'yan proposed the detection of very high energy particles through the coherent radio emis sion from the excess electrons in high energy showers 1. If the wavelength of the radiation is larger than the shower dimensions, the particles emit co herently, and, as the energy rises, the relative contribution of radio emission increases with respect to other wavelengths (say optical Cerenkov). This fact together with the technical simplicity of detectors are often argued in favor of this technique for detecting very high energy showers. Askar'yan suggested detecting showers in dense medium, as those that could be in duced by deeply penetrating particles underground or by cosmic rays on the Moon surface. This is natural because in a dense medium showers have much smaller physical dimensions and emission is coherent up to higher frequencies. The existence of Ultra High Energy Cosmic Rays (UHECR) at energies in excess of 10 20 seems an established fact which is difficult to understand 2 . Unfortunately the spectrum is known with little precision, the UHECR data are very scarce and we have little knowledge about their composition. It is nevertheless quite well accepted that, whatever their nature, they must come associated with neutrino fluxes of similar energies. Moreover it has been argued that the ratio of the fluxes of these UHE neutrinos to the
161
162
UHE hadrons, believed to be the main component of the UHECR, gives a good handle to distinguishing between two classes of models to explain their origin. One class of models rely on quark and gluon fragmentation into protons and the other class on stochastic acceleration of hadrons at a variety of astrophysical sources. Neutrino fluxes in the first group are induced by fragmentation and decay of charged pions. The neutrino fluxes expected are about an order of magnitude above the proton flux. In the second group pions are produced when accelerated hadrons interact with the surrounding matter, radiation or with the cosmic microwave background. In those cases neutrino fluxes relative to hadrons are expected to be smaller provided the target is thin. It is thus hardly surprising that there have been quite a number of efforts to understand the emission patterns in detail, to test them at ac celerator experiments and to exploit the advantages of coherence to detect ultra high energy showers induced by particles of whatever nature. The technique seems very appropriate for detecting neutrinos that traverse the atmosphere without interacting and can produce high energy showers in large volumes of natural media such as ice or sand. In this article I discuss the Aska'yan effect, I then remark important advantages of detecting coher ent radio-pulses making use of what has been learned by pulse simulations in dense media, and I finally address some experimental initiatives in this field, including the experiment at SLAC confirming the Askar'yan effect3, remarking the peculiarities of the different media being considered.
2. Askar'yan effect: The excess charge Electromagnetic showers are induced by electrons or photons and they con tain mostly large numbers of electrons and photons that are created in successive bremsstrahlung and pair production processes. Showers can be produced by hadrons and in that case electromagnetic sub-showers develop mainly because of neutral pion decays. As a shower develops in a medium there is a significant excess of electrons. This is because when the electrons and photons of the electromagnetic showers or sub-showers get to energies below the critical energy (Ec ~ 80 MeV in ice or air), they start having sig nificant probabilities to interact with the medium matter electrons. Matter electrons are accelerated into the shower by Compton interactions of the shower photons and by Bhabha (Moller) interactions of the shower positrons (electrons). Also shower positrons can be annihilated by matter electrons, thus removing positive charge from the shower. All of these processes con-
163
tribute to a charge imbalance which becomes most important in the regions of the shower where the particles have lower energy, namely after shower maximum and at large distances from the shower axis. Askar'yan was the first to note this effect and for that reason it has been addressed as the Askar'yan effect. Advances in theoretical work in the past decade have also been signifi cant particularly after the development of a specific Monte Carlo program for the calculation of the excess charge development of electromagnetic showers in ice and its contribution to the distant radio signals 4 . These simulations have shown that the excess charge is close to 20% of the to tal number of electrons and positrons, that is about twice the magnitude first calculated by Askar'yan from theoretical arguments 4 . Concerning the Cerenkov radio-emission associated with this excess, it became clear that the crucial magnitude was not the excess charge which varied with shower depth and distance to shower axis but the difference between the projected track-lengths a contributed by the electrons and by the positrons. This quantity is much more stable and represents 21% of the total track-length of all the particles. It is usually referred to as the excess projected tracklength and it is shown on Fig. 1. The excess electrons emit coherent Cerenkov radiation in the radio band, up to the few GHz region in a medium with density comparable to water or higher. This mechanism has been now tested at Stanford Linear Accelerator (SLAC) by dumping photon bunches onto sand to simulate a high energy shower and measuring the radio emission with antennas 3 . The results of this experiment confirm both the generation of excess negative charge in the shower and also the main properties of coherent radiation as predicted by Askar'yan and later calculated by detailed simulation. This confirmation has given an enormous thrust to many initiatives trying to exploit the technique in very different ways.
3. Advantages of the radio technique Most of the advantages of the radio technique are due to the fact that radio-waves can be searched for in a frequency range in which the emission induced by all the individual shower particles is coherent. The coherent character of this multi-particle emission process is responsible for a number of very interesting properties that make radio stand out in comparison a
Track-lengths are projected onto the shower axis
164
10000
n
■—■—■ i ■ ■ ■■!
Total Tracklength % 5000 Total Projected Tracklength ^2000 g1 1000 o a u
\
Excess Tracklength ***»,. ^**» %
1 PeV
500
E->
100 TeV
X
10 TeV
200
1 TeV 100
0.1
0.5 1.0 5.0 10.0 , 50.0 Kinetic Energy Cutoff (MeV)
Figure 1. Simulation results for the track-length and Excess Projected Tack-length (see text) as a function of threshold energy in the shower simulation.
with other detection techniques. I will try to illustrate these through the experience reached in the process of simulating radio-emission from high energy showers 4 ' 5 . When a particle of charge z moves through a medium of refractive index n with velocity \v\ = /?c > c/n Cerenkov light is emitted at the Cerenkov angle 6c, verifying cos6c = (/?n) - 1 , with a power spectrum given by the well known Frank-Tamm result 6 : cfW dudl
\Ait2h
2 a zv
1
/?22n:2 A-,
(1)
with v the frequency, c the speed of light, dl = cfidt a small element of particle track length, and a the fine structure constant. This is the standard approximation used for most Cerenkov applications for wavelengths orders of magnitude smaller than the tracks. When the radiation is induced by a set of particles such as a shower, and the wavelength of the radiation considered is larger than the size of the region containing the particles, the emission is coherent so that the emission of positrons would cancel that of the electrons were it not for the Askar'yan effect. The electric field radiated becomes proportional to the
165
excess projected track-length, which has been shown to be a stable 21% of the total shower track-length. As the total track-length induced by a photon or an electron is known to scale very well with the shower energy, the radiated power must scale with the square of the primary particle energy as pointed out by Askar'yan. But this is not the end of the story because the mere fact that the emis sion from all particles is coherent implies that the radiation pattern contains much more information on the distribution of the excess charge through the shower, both its longitudinal development and its lateral spread. The rela tion between the charge distribution and the radiated pattern is far from simple and involves both the angular distribution of the signal and its fre quency spectrum. The angular distribution of the pulse has many features that closely resemble the pattern produced by a diffraction slit. It can be shown that if the shower is simplified to a one dimensional model in which all the particles travel along the shower axis at the speed of light, the time Fourier transform of the electric field amplitude in the Fraunhofer approx imation, i.e. E(u>,x) at a given point x situated at a large distance R from the shower and making an angle 9 to the shower axis, simply becomes the Fourier transform of the longitudinal charge distribution Q(z): ikR
r
£(w,x) = - ^ - 5 - iu sin^ % - n x / dz' Q(z') eipz'. (2) 2neoCz R J Here /j,r is the relative permeability of the medium, eo is the permittivity of the vacuum and u the angular frequency. We have introduced for con venience the parameter p(0, u) = (1 — n cos 6) u/c in Eq. 2. The angular pattern around the Cerenkov direction is the analog of the classical diffrac tion pattern of an aperture function. The appearance of p insures that the main diffraction peak is along the Cerenkov direction. Its width relates to the length over which the shower develops and to the frequency. For a given shower higher frequencies and/or longer showers imply narrower diffraction peaks. It is not too difficult to see that the frequency spectrum must even tually become suppressed as the frequency rises. At the Cerenkov angle this suppression is mostly due to the lateral spread of the shower and the corresponding frequency fixes the transition between coherent emission of all particles to strong interference effects. At angles which are different from the Cerenkov angle the same behavior is observed but with a lower transition frequency, which is due to the combined effect of the longitudi nal development and the lateral spread. The frequency spectrum of the
166
signal in a given direction thus carries combined information on both the longitudinal development and lateral spread of the showers. In principle a thorough study of the electric field surrounding the shower would allow precise evaluation of the excess charge distribution. Last but not least the one-dimensional model also illustrates the stabil ity of electric field measurements. It is easy to see that along the Cerenkov direction the parameter p = 0, so that the Fourier integral shown in Eq. 2 simply becomes the integral of the charge distribution. This integral is sim ply the excess projected track-length in this approximation. Track-lengths scale extremely well with energy as can be seen in Fig. 1, mostly because of energy conservation. Moreover, although it is well known that the num ber of particles in a shower fluctuates, and that these fluctuations can be quite large typically of order 20%, the fluctuations in track-length are much smaller. As a result radio measurements at the Cerenkov angle are a natural way to measure shower energy. Fluctuations in track-length as obtained in simulations are well below a 1% limit, but the systematic uncertainties are on the other hand un known, and certainly expected to be much higher than this uncertainty. Understanding of systematics in the track-length calculations are thus of utmost importance. In this respect work has been done to quantify the uncertainty in the absolute value of the simulated track-length results by checking them using a different shower simulator, in this case GEANT. Al though early results showed that there were significant discrepancies in the absolute value of the track-length between the two simulation packages 7 , it has only been recently established that the agreement between them is actually quite remarkable when the threshold energy cut-offs are taken into account in equivalent ways 8 .
4. Experimental progress The radio emission by showers was first tested by looking for coincidences with extensive air shower arrays in the late 1960's and 1970's 9 and radio pulses were routinely observed in coincidence with other air shower detec tors. Efforts to substitute particle detectors by arrays of antennas were later almost completely abandoned because of difficulties in the interpreta tion of the measured signals. In the 1980's the radio technique in antarctic ice was re-discussed as an alternative to high energy neutrino detection 10,11 . A competitive experiment must have an effective volume above the km 3 and calls for an array of antennas possibly embedded in the ice. As a
167
result in the 1990's the detailed simulation program was developed for ice 4 and an experimental initiative to search for radio pulses from high en ergy neutrinos interacting in the Antarctic ice sheet has been going on since 1996, the RICE experiment 12 . Finally the regolith of the Moon surface has also been considered for radio detection. Two attempts have been made to search for pulses produced under the Moon surface using radio-telescopes, the Parkes telescope in Australia 13 and the GLUE experiment with two tracking telescopes in Goldstone, US 14 . Although no signals have been observed in any of these experiments, both the RICE and the GLUE ex periments have provided competitive bounds to neutrino fluxes, the GLUE experiment extending up to 10 23 eV. A good comparative review of most initiatives can be found in 15 . In the year 2000 radio-pulses induced by 3 GeV photon bunches on a sand target were measured for the first time with a microwave horn that was placed at different orientations with respect to the target 3 . The bunches were selected to have an overall energy comparable to that of interest up to the 1019 eV range. The experiment measured the strength of the pulse and the electric field in the signal was shown to scale with the primary energy up to 10 19 eV, thus confirming the coherent character. The polarization and the frequency spectrum of the pulse were shown to have the right features although a precise comparison with theory was complicated by Fresnel and near field effects that were difficult to establish. The measurement can be considered as the establishment of the Askar'yan effect which had never been observed b The experiment also tested the coherent behavior of the signal scaling as the primary energy squared up to equivalent energies of order 10 19 eV. Although this was ex pected, it is clear that the scaling must break down at a sufficiently high energy, otherwise the pulse would be more energetic than the shower itself. This breakdown of coherence would most likely come from charge sepa ration of the shower because of the electric field due to the accumulated charge. A strict upper limit of about 2 10 25 eV can be easily obtained for ice equating the pulse energy estimate (Eq. 22 in Ref. 4 ) to half of the shower energy. It is apparent that the effect must become significant at a lower energy and that it is also sensitive to the medium. The pulse energy measurements at SLAC agree with theoretical calculations and therefore confirm that such an effect is not to be expected at energies below 1019 eV. b
It is believed that the radio pulses that have been observed in association to extensive air showers are not due to this effect but to other mechanisms.
168
There was little doubt about the excess charge mechanism because it is a combination of Maxwell's laws and quantum electrodynamics in an energy region which is rather well studied and understood. However the measurement has attracted a lot of attention, and efforts worldwide to search for radio-pulses from high energy particles are increasing and are likely to proliferate in the near future. These experiments are going to search for pulses emitted by showers in different media and depending on it their properties are going to be very different. Simple scaling of knowledge acquired from simulations in ice suggests that the maximum frequencies for full coherence are in the one MHz range for air, where attenuation of waves is thousands of kilometers and thus opens the possibility to cover very large areas with few detectors. It is likely that these pulses are very complicated because measurements are usually made in the 10-100 MHz band and in this region there is bound to be some degree of coherence but many incoherent effects as well. The observation distance is typically comparable to the shower dimensions and near field and Presnel effects are likely to complicate the coherent signal. Lastly several mechanisms have been suggested as competing for the pulse generation, from transverse currents in the showers, to dipole radiation and synchrotron emission. The complexity of the problem calls for a careful experiment in coincidence with an air shower detector. An array of antennas is being deployed in the Kascade project in Karslruhe 16 and there are plans to look for these showers from space using a weather satellite 17 . Although the signals are expected to be dominated by cosmic ray showers it is possible that satellite measurements search for neutrino showers from space. In a denser media such as natural ice (high purity salt), full coherence is kept to frequencies in the 1 GHz (3 GHz) range and typical attenuation distances are of order 1 km (300 m). A new initiative has been recently approved to search for neutrino pulses in Antarctica from a balloon exper iment, this is the ANITA project 1 8 which would naturally search for high energy showers in very large volumes of ice. Other projects 19 are aiming to establish the viability of using different natural salt domes for establishing an array of antennas in a volume of order 100 km 3 , in a similar way to RICE. Work is in progress to simulate radio-pulses in salt domes. A similar situation happens to dry sand which is expected to have large attenuation lengths and can also be found in large quantities. The showers have similar dimensions to those in salt and the maximum frequency for full coherence is thus quite similar. Up to now sand has only been used to test the radio properties of showers at Argonne 2 0 and SLAC 3 .
169
For the Moon regolith the situation is similar with respect to the max imum frequency for full coherence which is close to 4 GHz. Although the attenuation is of order 10 m, extremely large effective areas for cosmic rays and volumes for neutrinos can be accessible using appropriate radiotelescopes. It is interesting to remark that the GLUE experiment has ob tained a not yet understood excess signal from the Moon direction when a low quality trigger is used 14 . If this signal was due to cosmic rays hit ting the Moon surface, the acceptance would be of order 3 105 km 3 sr 15 corresponding to a threshold energy in the 3 10 19 eV range, which is quite consistent with calculations 2 1 . The search for UHECR with radio tele scopes thus deserves further efforts. 5. S u m m a r y I have reviewed some progress in understanding and using the radio tech nique since it was proposed. The radio technique has three big advantages because the emission from all the particles can be coherent, namely the signal scales with the square of the shower energy, it carries information on the spatial distribution of the excess charge, and it is fairly insensitive to fluctuations, at least in the cases in which the pulses are generated by the Askar'yan effect. Since this effect and the technique have been shown to work at experiment, and there are good reasons to think it can be used for both ultra high energy cosmic ray and neutrinos searches, we are likely to see very much activity in radio in the near future. Hopefully some of these efforts will also give answers to some of the most intriguing physics questions about the highest energy tail of the cosmic particles that have been observed. Acknowledgments I thank J. Alvarez-Muiiiz for carefully reading the manuscript, J. AlvarezMuniz, F. Halzen, E. Marques, T. Stanev and R.A. Vazquez for many discussions and collaboration in the work discussed here and D. Besson, P. Gorham, D. Saltzberg and D. Seckel for discussions and facilitating easy access to information on their work. This work was partly supported by the Xunta de Galicia (PGIDIT02 PXIC 20611PN), by MCYT (FPA 200201161), by the European Science Foundation (ESF Scientific Network N86), and by FEDER funds. The work of R.A. V. is supported by the "Ramon y Cajal" program. We thank the 'Centro de Supercomputacion de Galicia" (CESGA) for computer resources.
170 References 1. G.A. Askar'yan, Zh. Eksp. Teor. Fiz 41, 616 (1961) [Soviet Physics J E T P 14 441, (1962)]; 48, 988 (1965) [21, 658 (1965)]. 2. For a review see for instance: M. Nagano, A. A. Watson, Rev. Mod. Phys. 72, 689(2000) and for more recent articles, M. Takeda, et aJ. (AGASA Coll.) e-Print Archive: astro-ph/0209422 and T. Abu-Zayyad et al. (HiRes Coll.), e-Print Archive: astro-ph/0208301, subm to Astropart. Phys. (2002). 3. D. Saltzberg, P. W. Gorham, et al. Phys. Rev. Lett. 86, 2802 (2001). 4. F. Halzen, E. Zas, and T. Stanev, Phys. Lett. B 2 5 7 432 (1991) and Phys. Rev. D 4 5 , 362 (1992). 5. J. AIvarez-Mufiiz, R.A. Vazquez and E. Zas, Phys. Rev. D 62, 063001 (2000) and D 61, 023001 (2000); J. Alvarez-Muniz and E. Zas, Phys. Lett. B 4 3 4 , 396 (1998) and B 4 1 1 , 218 (1997). 6. Frank, I. and Tamm, I., Dokl Akad. Nauk SSSR 14 109 (1937). 7. S. Razzaque, et al. Phys. Rev. D 6 5 , 103002, (2002). 8. J. Alvarez-Muniz, E. Marques, R.A. Vazquez, and E. Zas e-Print Archive: astro-ph/0206043, (submitted to PRD). 9. T. Weekes, Proc. of the "First International Workshop on Radio Detection of High Energy Particles (RADHEP-2000), UCLA, November 2000; Ed. D. Saltberg and P. Gorham, American Institute of Physics, New York (2001), A I P Conf Proc N 579 pp 3, and references therein. 10. M.A. Markov, I.M. Zheleznykh, Nucl. Instr. and Methods Phys. Res. A 2 4 8 , 242 (1986). 11. Ralston, J.P. and McKay, D.M., Proc. Astrophysics in Antarctica Conference, ed. Mullan, D.J., Pomerantz, M.A. and Stanev, T., (American Institute of Physics, New York, 1989) Vol. 198, p. 241 12. I. Kravchenko, et al, e-Print Archive: astro-ph/0206371 and astroph/0112372, submitted to Astropart. Phys. (2002). 13. T.H. Hankins, R.D. Eckers, and J.D. O'Sullivan, MNRAS 283 (1996) 1027; and Proc. of RADHEP-2000 UCLA, Nov 2000; Ed. D. Saltberg and P. Gorham, A I P Conf Proc N 579 pp 168. 14. P. W. Gorham, et al., Proc. of 26th ICRC, Aug 1999, Salt Lake City, HE 6.3.15, Vol. 2 pp. 479 (1999); Proc. of RADHEP-2000; Ed. D. Saltberg and P.W. Gorham, New York (2001), A I P Conf Proc N 579 pp 177. 15. P. Gorham in Ultra High Energy Particles from Space Aspen Feb. 2002. (See http://astro.uchicago.edu/ olinto/aspen/program) 16. LOFAR: H. Falcke, P. Gorham, e-Print Archive: astro-ph/0207226. As tropart. Phys. (in press). 17. FORTE satellite, see "http://nis-www.lanl.gov/nis-projects/forte/". 18. See "http://www.ps.uci.edu/ barwick/anitaprop.pd". 19. P. Gorham, et al. Nucl. Instrum. Meth. A490 (2002) 476. 20. P.W. Gorham, D. Saltzberg, et al. Phys. Rev. E62, 8590 (2000). 21. J. Alvarez-Muniz and E. Zas; Proc. of RADHEP-2000, UCLA, Nov,. 2000; Ed. D. Saltberg and P. Gorham, A I P Conf Proc N 579 pp 128, astroph/0102173.
GPS SYNCHRONIZATION IN COSMIC RAY EXPERIMENTS P. ASSIS LIP Av. Elias Garcia, 14-1, 1000-149 Lisboa, Portugal E-mail: [email protected]
A wireless system to synchronize ground array cosmic ray detectors is presented. The system developed is composed by a GPS receiver and a PCI board with a time-tagging system. Tests performed on the GPS receiver accuracy are also presented.
1.
Introduction
The synchronization of Ground Based Cosmic Ray Detectors is a recurring problem. Traditional acquisition systems tend to drive signal cables from each station of an array of detectors to a central acquisition system. In the context of ULTRA - a support experiment for the EUSO mission - it was investigated the possibility of using a wireless synchronization system based on the GPS system. EUSO [1] is a space mission devoted to the study of Extreme Energy Cosmic Rays (EECR). The interaction of EECR with the atmosphere gives birth to Extensive Air Showers (EAS) which produce UV light by fluorescence of the Nitrogen present in the atmosphere and by Cherenkov effect. EUSO detection method is based on the UV light emitted in the interaction between EECR and the atmosphere. The knowledge of the production, propagation, diffusion and reflection of this light assumes a crucial role [2] and this study is the main intent of the ULTRA [3] experiment. ULTRA plans to collect the UV light generated by an EAS that is reflected on several surfaces (land, water, snow, etc.). ULTRA is composed by a UV telescope to collect the UV light (UVScope) and a conventional ground array of scintilator detectors (ETscope) responsible for the detection of the EAS. In the ETScope, an EAS is detected if, in a given time window, there is signal in more than one station. The estimation of the direction of the primary Cosmic Ray is based on the measurement of the time differences between the arrival of the shower front at the different stations, the distance between them, and the assumption that the EAS propagates at the speed of light (see fig. 1).
171
172
Figure 1: Estimation of arrival direction of an EAS The ULTRA requirement of 8° [3] on the error of the direction estimation is thus translated on a precision of 10ns on the measurement of the time differences, assuming that the stations are 20m apart from each other. 2.
Time-Tagging System
The Time-Tagging system under development is composed by several units, one per station, and must time-tag each event in each station. The timedifferences between the several stations' events are computed offline thus allowing to trigger on EAS and to reconstruct the arrival direction. Each unit includes a low-cost, commercial GPS receiver (GPSboard) to grosse time-tag and custom electronics (TMS - Time Measuring Subsystem) to perform the fine time-tagging. The GPS board generates a pulse that marks the UTC seconds (PPS). The identification of the second (year, month, day...) is provided through a RS-232 protocol. The TMS measures the time between the PPS and the local event trigger. TMS also measures the time between consecutive PPS pulses for calibration purposes. The time of an event in a station will be the time of the previous PPS summed with the time measured by TMS. The time-tagging error of an event in a local station is therefore the convolution of the PPS error with the TMS error. 2.1. GPS Board (GPSb) The advent of GPS receivers that can synchronize to UTC with a precision better than 50ns has introduced GPS synchronization in several areas such as telecommunications. In GPS synchronization it is assumed that both the positions of the satellites and of the receiver are known. GPS satellites act as beacons emitting a synchronization signal at each UTC second. The GPS receiver acquires the signal and, since the distance between the receiver and the satellite is known the
173
propagation delay of the signal can computed3 and corrected for, enabling the GPS receiver to output a signal (PPS) synchronized to the UTC seconds. The errors present in GPS synchronization are mainly due to the inaccurate knowledge of the propagation of the synchronization signal from the satellites to the receivers, namely the propagation in the atmosphere under several atmospheric conditions. Nevertheless the important parameter for ULTRA is the relative accuracy of the PPS pulses. The stations of the ETScope are very near each other and it can be assumed that the signal from a satellite has the same propagation characteristics for the different stations. Thus most of the errors can be viewed as systematic errors that cancel when taking time differences from different stations. The search, based on the accuracy reported, for a low cost, commercially available GPS receiver led to the choice of Motorola UT+ ONCORE which was also previously chosen by the Auger Collaboration.
Figure 2: Photograph of two Motorola UT+ GPS receiver modules The UT+ is an eight channel GPS receiver optimized for timing applications. It has a serial protocol for data transfer such as position, visible satellites and time (year, month, day, ...). UTC seconds are marked by the PPS signal and are identified through the serial interface. The PPS pulse is a TTL signal and is outputted with an offset, different for each PPS and each receiver. The offset value is made available, in each second, through the serial interface and varies roughly from -50 to 50ns. It is therefore imperious to take into account this correction in order to achieve maximum accuracy. Motorola states an absolute accuracy, of the PPS signal to UTC second, of 40ns for the UT+ model [4]. The differential accuracy can be estimated to be better than 8ns from the report on the accuracy of the Auger Observatory time tagging system [5]. Assuming, for simplicity, that the signal propagates at the velocity of light - c
174
Tests of the GPS receivers were performed in this work to estimate the differential error inherent to the GPS. The experimental setup consists, basically, of two co-located GPS receivers, a commercial time measuring device from ACAM (ATMD) and a PC. The pulses, from the GPS receivers were fed into the ATMD start and stopb and, in each second, the time difference between them was measured. The PC controls and reads the ATMD and the GPS receivers. r
JT
#1
SPS
l"L Start ATMD Stop
GPS#:
Jl
X
TL
PC
RS232
We want to measure At
_J1
JJL it„.
At,
Figure 3: Measurement Scheme Figure 4 shows the time differences, as measured by the ATMD and the correction to be applied to these data. The data and the correction to be subtracted have very similar structure. D i l Between corrections and data measured 110 o 4
13]
Stop-Start Correction TDC-20O
1D0
W
3D
t \
\
L
* t
-i
* i
i
*
\ i
"W «>
* * * * * * i
hti
*
/
t^°* S 3
:
** 'Mi*
, *
i *
i
*! t * ,
* * *
* j * ** i *
SO Run#(s)
Figure 4: Raw Data obtained and correction to be subtracted
b
The GPS receiver, whose signal was fed into the stop input of ATMD, was set to delay the pulse by 200 ns to ensure a positive time difference between stop and start. This offset was later removed and is not shown.
175
Figure 5 shows the time differences with the correction applied versus the number of the measurement. In figure 5 the distribution of this data is presented. The distribution is Gaussian shaped with a standard deviation of ~2ns which we consider to be the accuracy of the GPS board and is well below the requirement of 10ns. Time measured after correction
| •
mPti0m..Ml~tnriiumi0t»ii'"
dataCorrected|
n i i ""*" " 'II
Run#(s)
Figure 5: Data after correction Histogram of i t after corr. & trim
AT(ns)
Figure 6: Data after correction Distribution It is needed to emphasize that the errors of the GPS will only cancel, in differential mode, if the conditions for the different receivers are similar. Namely it is necessary to ensure that the set of satellites used to compute time are the same for all the receivers. This can be done by pre-programming the set of satellites to be used at each time. 2.2. Time Measuring Subsystem (TMS) The TMS is responsible for the measurement of the time difference between the synchronization pulse and an event. This subsystem must comply with the accuracy required and it has to have a dynamical range of 1 s, since the event can occur at any part of the second.
176
The system developed consists of a clock (50 MHz oscillator and counter) combined with a TDC. The use of a clock allow the Is dynamic range while the use of a TDC allows the precision necessary without having a huge frequency oscillator (e.g. to have a precision of Ins precision the clock oscillator would have to have a frequency of 1GHz). The oscillator period is calibrated with the signal from the GPS. The hardware implementation of TMS results from a collaboration with the Lisbon Cosmic Ray Telescope and was implemented as a part of a PCI board LIP-PAD board 3.
LIP-PAD Board
The TMS was implemented as an autonomous part of the LIP-PAD board. This board was developed with the aim of being able to perform a full data acquisition of a scintilator ground array detector. The board can receive the PMT signals, digitize them and digitally trigger on events.
Figure 7: Photograph of the LIP-PAD board
The components of the board are: a TDC-part of the TMS six analog acquisition channels, each having a shaper, an amplifier and an eight bit FLASH ADC that runs at 100 MHz. A programmable logic device in which it is implemented the control logic (PCI protocol, status and signal control), the logic associated with the TMS and also the logic related with the analog acquisition.
177
GP1 RS232 port Altera PLD
Shaper
&
H
ADC PCI Interface
PLD programming
interface Osc I00MH2
Figure 8: Block diagram of the LIP-PAD Board
The analog signals are shaped and amplified and then digitized by the ADC. The ADC samples the signal voltage with a frequency of 100MHz and writes the values to a buffer memory. The digital trigger unit can implement various trigger conditions (e.g. simple threshold, muon, double pulse, shower). The board is read and controlled through the PCI protocol, Preliminary tests were already performed and indicate that the TMS has a precision better than the specification required. 4.
Conclusions
A system for wireless synchronization for ground based scintilator array detector was developed. The requirement was that the system had an accuracy better than 10ns. The system is based on a low cost commercial GPS receiver and on custom electronics implemented as part of a PCI board. The preliminary tests indicate that the system is well below the specification required. References 1. 2. 3. 4.
See Talk by L. Scarsi in these proceedings See Talk by M. C. Espirito Santo in these proceedings O. Catalano et al, "ULTRA Technical Report" (2002) Oncore users guide, Motorola, available at http://www.motorola.com/ies/GPS 5. F. Meyer, F. Vernotte, Time Tagging Board Tests at Besanqon Observatory, GAP-2001-050 (2001). Can be found at http://www.auger.org/amin/GAP_Notes/GAP2001/gap_2001_050.pdf
RESULTS F R O M T H E A M S 0 1 1998 S H U T T L E F L I G H T
MANFRED STEUER MIT
and the AMS
Collaboration
The Alpha Magnetic Spectrometer (AMS) was flown in June 1998 on the space shuttle Discovery during flight STS-91 in a 51.7° orbit at altitudes between 320 and 390 km. The major detector elements were a permanent magnet with an analyzing power B * L2 of 0.14 Tm2, a six layer, double sided silicon tracker, time of flight hodoscopes, an aerogel threshold Cerenkov counter and anti-coincidence counters. A total of 2.86 x 10 6 helium nuclei were observed in the rigidity range 1 to 140 GeV. No antihelium nuclei were detected at any rigidity. The upper limit on the flux ratio of antihelium to helium was determined as 1.1 X 1 0 - 6 . Below the geomagnetic cutoff a second helium spectrum is observed and more than 90% of the helium was identified as 3He. The primary proton spectrum in the kinetic energy range 0.2 to 200 GeV was measured at an altitude of 380 km and is parameterized by a power law above 10 GeV. Below the geomagnetic cutoff a substantial secondary spectrum was observed. It is concentrated at equatorial latitudes with a flux of around 70 per (m 2 x s X sr). The lepton spectra in the kinetic energy ranges 0.2 to 40 GeV for electrons and 0.2 to 3 GeV for positrons were measured at altitudes near 380 km. Two distinct spectra were observed, a higher energy spectrum and a substantial secondary spectrum with positrons much more abundant than electrons. Tracing leptons from the second spectra shows that most of these travel for an extended period of time in the geomagnetic field and t h a t the positrons and electrons originate from two complementary geographic regions. The 10 day test flight of the AMS detector has shown its viability for an extended period of several years of data taking at the International Space Station (ISS). Currently the detector is undergoing several upgrades, the most prominent one being the replacement of the permanent by a superconducting magnet, thus greatly extending the sensitive region of the experiment. The addition of transition radiation and enhanced Cerenkov (RICH) detection capabilities as well as an electromagnetic calorimeter allows to diversify the physics program. The new AMS02 detector will commence a three year physics data taking period with its next shuttle flight and installation on the ISS foreseen for October 2005.
1. Introduction AMS is an international collaboration now spanning 3 Continents, 16 Coun tries, 46 Institutions. It is based on an agreement between NASA and DOE. It is a CERN Recognized Experiment. Results are published in Physics Let-
178
179
Figure 1. AMS (near the tail) as seen from MIR during STS-91 docking on June 4, 1998.
ters B and Physics Reports 1>2>3'4>5'6J which can be consulted for details. The apparatus, as flown on STS-91, is described in detail in a NIM paper 7 . The purpose of the AMS experiment is to study the antimatter composition of the primary cosmic rays as well as looking for extraterrestrial matter, for missing matter and dark matter. The existence (or absence) of antimatter nuclei in space is closely connected to the theories of elementary particle physics, like CP-violation, baryon number nonconservation, Grand Unified Theory (GUT), etc. Balloon-based cosmic ray searches for antinuclei at altitudes up to 40 km have been carried out for more than 20 years; all such searches have been negative. The Alpha Magnetic Spectrometer (AMS) is scheduled for a particle physics program on the International Space Station. This program will search for antinuclei using an accurate, large acceptance magnetic spec trometer. In addition searches for supersymmetric dark matter, origin and transport mechanisms of cosmic rays will be carried out. The AMS flight on the space shuttle Discovery (STS-91) in June 1998 was primarily a test flight to verify the detector's performance under actual space flight condi tions.
180
2. D a t a taking After the shuttle had attained orbit, data collection commenced on 3 June 1998 and continued over the next nine days for a total of 184 hours, re sulting in 100 million triggers recorded, half during the MIR docking pe riod. Around 12% of physics data were sent via slow links to various NASA/USAF ground stations and were delivered in almost real time. Dur ing data taking the shuttle altitude varied from 320 to 390 km and the latitude ranged between ± 51.7°. Before the rendezvous with the MIR space station the attitude of the shuttle was maintained as to keep the zaxis of AMS pointed within 45 ° of the zenith. While docked, the attitude was constrained by MIR requirements and varied substantially. After undocking the pointing was maintained with a precision of 1° at 0°, 20° and then 40 ° of the zenith. Shortly before descent the shuttle turned over and the pointing was towards the nadir. Events were triggered by the coincidence of signals in all four scintillator (TOF) planes consistent with the passage of a charged particle through the active tracker volume (see Figure 2). Triggers with a coincident signal from the anti-counters (ACC) were vetoed. The detector performance as well as temperature and magnetic field were monitored continuously.
3. Event reconstruction - The sign of the particle charge Z was derived from the deflection in the rigidity fit and the particle direction. - The particle mass was derived from \Z\R and /?. - The particle rigidity, R = pc/\Z\e(GV) , was obtained from the measurement of the deflection of the trajectory measured by the tracker in the magnetic field. Hits in at least four tracker planes were required and the fitting was performed with two different al gorithms, the results of which were required to agree. - The particle velocity, /J, and direction, z = ± 1 , was obtained from the TOF, where z = — 1 signifies a downward going particle in the AMS coordinate system. - The magnitude of the particle charge Z was obtained from the measurements of energy losses in the TOF counters and tracker planes (corrected for /?).
181
Figure 2. a,b: Views of AMS as flown on STS-91.
4. Search for antihelium in cosmic rays The results of our search are summarized in Figure 3. As seen, we obtain a total of 2.86 x 106 He events up to a rigidity of 140 GV. We found no He event at any rigidity. Since no He nuclei were observed, we can only establish an upper limit
182
on their flux. If the incident He spectrum is assumed to have the same shape as the He spectrum over the range 1 < R < 140 GV, then one obtains at the 95 % C.L. a limit of
WHe NKe
< 1.1 x 10~ 6 .
AMS 10
10"
STS - 91
He
He
0 events
2.86x10 events
0)
fio3 >
102 10
1 fe -150
-100
-50
0
50
SignxRigidity
00
150 GV
Figure 3. Measured rigidity times the charge sign for selected \Z\ = 2 events.
5. Protons in near earth orbit Data were collected in the cosmic ray proton spectrum region from kinetic energies of 0.1 to 200 GeV, taking advantage of the large acceptance, the accurate momentum resolution, the precise trajectory reconstruction and the good particle identification capabilities of AMS. The high statistics (~ 107) available allow to determine the variation of the spectrum with shuttle position both above and below the geomagnetic
183
cutoff. Because the incident particle direction and momentum were accu rately measured in AMS, it is possible to investigate the origin of protons below cutoff by tracking them in the earth's magnetic field. The spectrum above cutoff is referred to as the "primary" spectrum and below cutoff as the "second" spectrum. 5.1. Properties
of the Primary
Spectrum
Since the observed primary proton spectrum was verified to be isotropic, all data collection periods were combined to obtain a total of 5.6 x 106 identified primary protons. The primary proton spectrum can be parameterized by a power law in rigidity, $o x R _ T - Fitting the measured spectrum over the rigidity range 10 < R < 200GV, i.e., well above cutoff, yields: 7 = 2.78 ± 0.009 (fit) ± 0.019 (sys), GeV 2 7 8 $ 0 = 17.1 ± 0.15 v(fit) ± 1.3 w(sys) ± 1.5 w(7) —— . / — 2 n ' ' m secsrMeV The systematic uncertainty in 7 was estimated from the uncertainty in the acceptance (0.006), the dependence of the resolution function on the particle direction and track length within one sigma (0.015), variation of the tracker bending coordinate resolution by ± 4 microns (0.005) and variation of the selection criteria (0.010). The third uncertainty quoted for $0 reflects the systematic uncertainty in 7. 5.2. Properties
of the Second
Spectrum
A substantial second spectrum of downward going protons is clearly visible for all but the highest geomagnetic latitudes. The upward and downward going protons of the second spectrum have the following unique properties: (i) At geomagnetic equatorial latitudes, @M < 0.2, this spectrum ex tends from the lowest measured energy, 0.1 GeV, to ~6 GeV with a flux ~ 7 0 m _ 2 s e c _ 1 s r _ 1 . (ii) The second spectrum has a distinct structure near the geomagnetic equator: a change in geomagnetic latitude from 0 to 0.3 causes the proton flux to drop by a factor of 2 to 3 depending on the energy. (iii) Over the much wider interval 0.3 < @M < 0.8, the flux is nearly constant. (iv) In the range 0 < © M < 0.8, detailed comparison in different lat itude bands (Figure 4) indicates that the upward and downward fluxes are nearly identical, agreeing within 1%.
184
0<6M<0.3 M
-J.
«
•••# _•• • - \ •
•
■
I »
■
*
■
10
•
•
) Z
1
*
1 1
"
■
-1
■
t,t1 l
■
t •"
: 5 ™ •# •
i
|
a)
o.9<eM
•
»«»
v ■ '•■
10
Downward upward
A
'■
■
•. ,
a
r\a ,n n-> O.b<6M<0.7
° Downward • Upward
-1
«
>
.
.
10 10
1
10
10 10
\
2
2 - 1
1
10
10
Kinetic Energy (GeV)
Figure 4. Comparison of upward and downward second proton spectrum at different geomagnetic latitudes. As seen, below cutoff, the upward and downward fluxes agree in the range 0 < 0 M < 0.8.
To understand the origin of the second spectrum, we traced back 105 pro tons from their measured incident angle, location and momentum, through the geomagnetic field for 10 sec flight time or until they impinged on the top of the atmosphere at an altitude of 40 km, which was taken to be the point of origin. All second spectrum protons were found to originate in the atmosphere, except for few percent of the total detected near the South Atlantic Anomaly (SAA). The trajectory tracing shows that about 30 % of the detected protons flew for less than 0.3 sec before detection. The origin of these "short-lived" protons is distributed uniformly around the globe. In contrast, the remaining 70 % of protons with flight times greater than 0.3 sec, classified as "long-lived", originate from a geographically restricted zone. Figure 5 shows the strongly localized distribution of the point of origin of these long-lived protons in geomagnetic coordinates. Though data is presented only for protons detected at ©M < 0.3, these general features hold true up to 0 M ~ 0.7. 6. Leptons in near earth orbit Data were collected to study the spectra of electrons and positrons in cosmic rays over the respective kinetic energy ranges of 0.2 to 40 GeV and 0.2 to 3 GeV, the latter range being limited by the proton background. The large acceptance of AMS and high statistics (~ 105) enable us to study the variation of the spectra with position and angle both above and below the geomagnetic cutoff. The origin of particles below cutoff can be obtained by
185
Long Lived p Detected at ®M < 0.3 rad $ CD
CD T3 CD
T3
80 60 40 20 0
-4—»
03
-20
—'
-40
"O)
-60
O
-80
-150
-100
-50
0
50
100
150
Origin Longitude (degrees) Figure 5. Origin of long-lived protons ( 0 M < 0.3,p < ZGeV/c coordinates.
) in geomagnetic
tracking them in the geomagnetic field. For this study the acceptance was restricted to events with an incident angle within 25° of the positive z-axis of AMS and data from four periods are included. In the first period the z-axis was pointing within 1° of the zenith. Events from this period are referred to as "downward" going. In the second period the z-axis pointing was within 1° of the nadir. Data from this period are referred to as "upward" going. The effect of the geomagnetic cutoff and the decrease in this cutoff with increasing 0 M is particularly visible in the downward electron spectra. The spectra above and below cutoff differ. To understand this difference the trajectory of electrons and positrons were traced back from their measured incident angle, location and momentum, through the geomagnetic field. This was continued until the trajectory was traced to outside the Earth's magnetosphere or until it crossed the top of the atmosphere at an altitude of 40 km. The spectra from particles which were traced to originate far away from Earth are classified as "primary" and those from particles which originate in the atmosphere as "second" spectra. In practice, particles below the geomagnetic cutoff are from the second spectra, however this classification provides a cleaner separation in the transition region. Similar to the procedure for second spectra protons, leptons with flight times < 0.2sec are defined as "short-lived", the remaining as "long-lived". For 0 M < 0.3, most (75% of e + , 65% of e~) leptons are long-lived.
186
6.1. Origin of long lived
leptons
Figure 6 shows for long-lived second spectra leptons (< 3GeV) the geo graphical origin of (a) electrons and (b) positrons. The lines indicate the geomagnetic flield contours at 380km. One can see two (A and B) strongly localized distributions of the point of origin for long-lived leptons. Tracking shows, that regions of origin for positrons coincide with regions of sink for electrons and vice versa. Long-lived second spectra positrons have the same points of origin as long-lived second spectra protons.
80 60 40 20 0 -20 -40
1
-80
£ _l
80 60 40 20 0 -20 -40 -60 -80 -150
-100
-50
0
50
100
150
Longitude Figure 6. The geographical origin of long-lived second spectra leptons (< 3GeV).
187
6.2. Lepton charge
ratio
An interesting feature of the observed second lepton spectra is the predom inance of positrons over electrons. The energy dependence of the e + / e ~ ratio for 0° attitude and 0 M < 0.3 is shown in Figure 7. As seen, short lived and long-lived leptons behave differently. For short-lived leptons the ratio does not depend on the particle energy in the range 0.2 to 3 GeV but for long-lived leptons the ratio does depend on the lepton energy, reaching a maximum value of ~ 5. Their origin lies within the earth's atmosphere and, as the analysis shows, nuclear interactions together with the earth magnetic field are responsible for this otherwise unexpected ratio.
b) Long Lived
a) Short Lived
CD CD
Ek (GeV) Figure 7. The e+/e~ ratio of second spectra leptons (< 3GeV, 0 M < 0.3) as a function of energy.
7. Helium in near earth orbit The high statistics (~ 106) allow measurements over a range of geomagnetic latitudes. Below the geomagnetic cutoff the origin is studied by tracking the particles in the earth's magnetic field. Figure 8 shows the effect of the geomagnetic cutoff which decreases in R with increasing ©M7.1. Above the geomagnetic
cutoff
A fit to the primary He spectrum shows similarity to the primary proton spectrum. This points to the same source for both particles. In the rigidity
IU :
..^
„s^a,ss„
£3
\J
&
X10 e
>
O
®
s *«
a
*S
-■ y « s
0.4 < e^| < 0,8 0.8 <
3
1 -
S
o
m
\3
X
m
-1
cv
. -^
<"\
V < -V
y
IS
«
w
.
$°$
10
-
E
&
A
*»■—''
*
lht\ : I| ;
310
EL 10
s
J,
¥
-3
$
r,i 1
10
t
10
Rigidity (GV) Helium flux spectra for zenith pointing data, separated according to 0 M 35 30 25
2 20 C CD
|3 1 5 10 5 0
■
■ ■
[
1
'
■
'
■■ ' I ' . ■ . ' . ■.
■ I ' . ■ . ■. .■ » ■ I ' ■
2 Z P 3He
ZV4 *He
■
I
■^■^
■
5
Mass (GeV/c2) Figure 9. Helium second spectrum reconstructed mass for 0 M < 0.6.
189
range 20 < R < 200 GV a power law in rigidity $ = $o>
T
yields
7 = 2.740 ± 0.010 (stat) ± 0.016 (sys), GV 2 ' 7 4 $o = 2.52 ± 0.09 (stat) ± 0.13 (sys) ± 0.14 (7) —2 — . m secsrMV 7.2. Below the geomagnetic
cutoff
For 0 M < 0.8 a second spectrum is observed. From the lowest mea sured rigidity 0.8GV until 3GV the integrated flux is (6.3 ± 0.9) x 1 0 3 m - 2 s e c - 1 s r - 1 . More than 90% of the helium was reconstructed as 3 He at the 90% C.L. This second flux has been traced to originate from the same locations within the atmosphere as the corresponding second proton and positron fluxes, with the long lived component (flight time above 0.3 sec) originating from the same two restricted geographic regions A and B. 8. Conclusion The short technical test flight of the AMS detector has shown its viability for an extended period of several years of data taking on the International Space Station (ISS). In addition to the presented results on protons and He versus He in primary cosmic rays, on near earth spectra of protons, He, e + and e~, we obtained antiproton and deuteron spectra too. An extensive upgrade program is underway in order to enhance the sensitivity, background rejection power and Rgidity range, as well as to improve the identification of nuclei greater than He. AMS02 represents major improvements over AMS01 using a supercon ducting magnet, a transition radiation detector, a ring-imagaing Cerenkov counter, and an electromagnetic calorimeter. Dedication This paper is dedicated to the crew of Space Shuttle Columbia, to be re membered for their excellent scientific work during their 16 day STS-107 mission ending in the tragic accident on February 1, 2003, while returning to earth. Acknowledgment Many thanks to my colleagues in the AMS data acquisition and analysis groups.
190
References 1. J. Alcaraz et al (the AMS collaboration), Search for Antihelium in Cosmic Rays, Phys. Lett. B 4 6 1 , 387 (1999). 2. J. Alcaraz et al (the AMS collaboration), Protons in Near Earth Orbit, Phys. Lett. B 4 7 2 , 215 (2000). 3. J. Alcaraz et al (the AMS collaboration), Leptons in Near Earth Orbit, Phys. Lett. B484, 10 (2000). 4. J. Alcaraz et al (the AMS collaboration), Cosmic Protons, Phys. Lett. B 4 9 0 , 27 (2000). 5. J. Alcaraz et al (the AMS collaboration), Helium in Near Earth Orbit, Phys. Lett. B 4 9 4 , 193 (2000). 6. M. Aguilax et al (the AMS collaboration), The Alpha Magnetic Spectrometer (AMS) on the International Space Station, Part I, Results from the test flight on the Space Shuttle, Phys. Rep. 366, 331 (2002). 7. G.M. Viertel, M. Capell, The ALPHA Magnetic Spectrometer, Nuclear In struments and Methods A419, 295 (1998).
ELECTRIC CHARGE RECONSTRUCTION W I T H RICH DETECTOR OF THE AMS EXPERIMENT
L U I S A A R R U D A , F . B A R A O , J . B O R G E S , F. C A R M O , P . G O N C A L V E S , A. K E A T I N G *
M.PIMENTA
LIP/IST Av.
Elias Garcia,
1000-149
14, 1°
Lisboa,
e-mail:
andar
Portugal
[email protected]
I.PEREZ FCUL,
Campo
Grande,
1749-Lisboa
The Alpha Magnetic Spectrometer ( A M S ) to be installed on the International Space Station (ISS), in 2005, will be equipped with a proximity focusing Ring Imaging Cerenkov detector. This detector, will be equipped with an aerogel radiator, a lat eral conical mirror and a detection plane made of around 700 photomultipliers and light-guides, enabling measurements of particle electric charge and velocity. When a charged particle traverses a dielectric medium witli speed greater than the velocity of light in that medium, emission of Cerenkov photons occurs, their number being proportional t o the particle's charge squared. Therefore, different charged nuclei can be discriminated with basis on their Cerenkov signals. In order t o determine the number of emitted photons, attenuation factors such as scattering and absorption in the radiator, the mirror reflectivity, the geometrical acceptance, the light guide effi ciency and photomultiplier's ( P M T ) quantum efficiency have to be evaluated. The method presented, has its foundations on the estimation of the overall event efficiency, which takes all the attenuation factors into account. Some of these factors have a wide range of variation and an event-by-event based charged reconstruction method is required.
1. The AMS02 detector AMS 1 (Alpha Magnetic Spectrometer) is a precision spectrometer designed to search for cosmic antimatter, dark matter and to study the relative abundance of elements and isotopic composition of the primary cosmic rays. It will be 'currently at ESTEC/ESA Netherlands
191
192 installed in the International Space Station (ISS), in 2005, where it will operate for a period of three years. T h e spectrometer will be capable of measuring the rigidity (R = the charge (Z),
the velocity (/?) and the energy (E)
a geometrical acceptance of ~ 0.5 m2.sr. of the A M S spectrometer.
pc/\Z\e),
of cosmic rays within
Figure 1 shows a schematic view
On t o p , a Transition Radiation Detector ( T R D ) ,
will discriminate between leptons and hadrons. It will be followed by the first of the four T i m e - o f - F l i g h t ( T O F ) system scintillator planes.
T h e T O F will
provide a fast trigger, charge and velocity measurements for charged particles, as well as, information on their direction of incidence. T h e tracking system will be surrounded by V e t o Counters and embedded in magnetic field of about 0.9 Tesla produced by a superconducting magnet. It will consist on a Silicon Tracker, constituted of 8 double sided silicon planes, providing both charge and momentum measurements w i t h a resolution Ap/p GeV/c/nucleon.
of at most 2 % up t o 100
T h e Ring Imaging Cerenkov Detector (RICH), described in
the next section, will be located right after the last T O F plane and before the Electromagnetic Calorimeter (ECAL) which will enable e/p separation and will measure the energy of the photons detected.
Figure 1.
A whole view of the A M S Spectrometer.
1 . 1 . T h e R I C H detector A charged particle crossing a dielectric material of refractive index n, w i t h a velocity /3, greater than the speed of light in t h a t medium emits photons. The aperture angle of the emitted photons w i t h respect t o the radiating particle is
193 known as the Cerenkov angle, 9C, and it is given by (see 5 ) :
cos8c = —— p n
(1)
The RICH is a proximity focusing Cerenkov radiation detector. It will be composed by a low refractive index radiator (aerogel 1.030, 3 cm thick), a lateral conical mirror of high reflectivity increasing the reconstruction efficiency and a detection matrix with 680 photomultiplier and light guides. The active pixel size of the PMTs is planned to be of 8.5 mm with a spectral response ranging from 300 to 650 nm with a maximum at A = 420nm. There will be a large non-active area at the center of the detection area due to the insertion of the ECAL. For a more detailed description of the RICH detector see reference 3 . Figure 2 shows a view of the RICH with the corresponding dimensions.
Figure 2.
The RICH detector.
The RICH detector of AMS was designed to measure the velocity of charged particles with a resolution A/3//? of 0.1%, to extend the electric charge sepa ration capability, to provide more information on the albedo rejection and to contribute in e/p separation. Its acceptance is of ~ 0.35 m2.sr, that is, around 80% of the AMS acceptance.
2. Photon Pattern Tracing The Cerenkov photons emitted by the charged particles crossing the RICH radiator are either refracted or fully reflected at the boundaries, depending on their incident angle (9i). The photons leaving the radiator and reaching the photomultiplier plane can come either directly or after being reflected by the
194 conical mirror. These photons will form a hit pattern in the detection plane, with a geometrical acceptance depending on the particle's impact point (P) on the radiator, on its direction (9,(j)) and on the C*erenkov angle (6C). This is schematically represented in Figure 3. 8,$
RBMIR
Figure 3. Representation of the photon's path length through the RICH detector. The left photon is reflected and the right one reaches directly the basis of the detector.
Complex detected photon patterns can occur due to mirror reflected pho tons. This can be seen in the two examples in Figure 4, where the detected patterns expected for two different radiator materials, aerogel (n=1.030) and Sodium Fluoride - NaF (n=1.334), are shown.
-60
-40
-20
0
20
40
60
-60
-40
-20
0
20
40
60
Figure 4. Simulated patterns for a n=1.030 aerogel (left) and NaF, n=1.334, (right). The reconstructed photon pattern (full line) includes both reflected and non-reflected branches. The inner and outer circular lines correspond respectively to the upper and lower boundaries of the conical mirror. The square is the electromagnetic calorimeter boundary.
195 3. Charge reconstruction T h e Cerenkov photons produced in the radiator are uniformly emitted along the particle path inside the dielectric medium, L, and their number per unit of energy depends on the particle's charge, Z,
and velocity, (3, and on the
refractive index, n, according t o the expression:
di\L dE
Z2L
\
1
1
Z2Li
[32n2/
(2)
So t o reconstruct the charge the following procedure is required: Cerenkov angle reconstruction. method see reference
4
For a complete description of the
in these proceedings.
Estimation of the particle path, L, which relies on the information of the particle direction provided by the Tracker. Counting the number of photoelectrons. The number of photoelectrons related t o the Cerenkov ring has to be counted w i t h i n a fiducial area, in order to exclude the uncorrelated background noise. Therefore, photons which are scattered in the radiator are excluded. Figure 5 shows the integrated number of hits as function of a distance to the ring. A distance of 15 m m t o the ring was defined as the limit for photoelectron counting. •^7000 *fc 6000 5000 4000
-
,<*»"?"*""-—' cut
- * -*
3000 2000
-
1000 0
'■_,,!_, 1
1
:
,.,,,_..,
,.,,:.,
, 1 , . L 1 ..I...i_,.. : . ..,.<
35
40
45
d (hit-pattern) (mm) Figure 5. Distribution of the integrated number of hits as function of the hit distance (right) to the reconstructed Cerenkov pattern (left).
• Evaluation of the photon detection efficiency. T h e number of radiated photons ( i V 7 ) which will be detected ( n p . e ) is reduced due t o the interactions w i t h the radiator (erac{), geometrical acceptance {sgeo), light guide (siig) and photomultiplier efficiency (eprrlt) g)
N7 er
id £geo &lg
£pmt
(3)
196 3.1. Photon Detection Efficiency 3.1.1. Radiator effects The main interactions suffered by the Cerenkov photons inside the aerogel radiator are Rayleigh scattering and absorption, for NaF radiator the first one doesn't exist. The absorption rate is 2 orders of magnitude below the scattering rate so it can be neglected in a first approach 6 . The radiator factor depends on the distance, d, crossed by the photons inside the radiator. It is calculated by integrating the probability of a photon not to interact in the radiator, p1 = e-d(z,ip)/Lint^ a | o n g t h e rac jjator thickness and along the photon azimuthal angle (ip). For Rayleigh scattering, the interaction length depends on the wavelength of the photons, according to expression:
Lint = \A/C
(4)
where C is the aerogel clarity. Therefore, the radiator efficiency can be evaluated through the following ex pression: rH d
2 rnrarad rfmax e -T—75:— / dz / dip AipHrad Jo J
"Lint
(5)
where Hrad is the radiator thickness. Figure 6 presents the evaluated radiator efficiency, erad, (shadow) for an aerogel radiator (1.030), 3 cm thick compared with the simulation results (thick marks). The simulation results were obtained for a Carbon sample of events.
1
Figure 6. Radiator efficiency. Evaluated radiator efficiency, £ra
197 3.1.2. Geometrical
acceptance
T h e geometrical acceptance, sgeo, takes into account t h e photons lost through the radiator walls or totally reflected on media transitions, photons absorbed by the mirror and photons falling into the non-active detection area. It is evaluated calculating the portion of the visible photon pattern in units of photon azimuthal angle, egeo
= Aip/2ir.
Figure 7 shows the geometrical acceptance
calculated for an aerogel radiator of 1.030 and for events within the A M S fiducial volume. T h e extreme variation of egeo f r o m event t o event is evident and shows clearly t h a t such a factor is fundamental t o be evaluated.
§ 500
>
400
300
200
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1
0.9
e geo
Figure 7. Photon geometrical acceptance for events in the RICH detector for an aerogel 1.030 radiator, 3cm thick.
3.1.3. Light guide
efficiency
Photons can be reflected when reaching the light guides' surface or be trans mitted between adjacent light guide divisions. T h e light guide efficiency factor €ig depends on t h e incidence angle of the photons on the t o p of it ( # 7 ) . It is calculated t a k i n g into account the probability of a given photon t o get into the photomultiplier cathode once it entered t h e light guide, and integrating it along the reconstructed photon pattern: £
ig=
tig{Q1(Q,Oc,
(6)
■J
Figure 8 shows the distribution of t h e photon incident angles Q1 on the t o p of light guide for an aerogel radiator (1.030) and variation of the light guide
198 efficiency eig with the same photon incident angle (left). Distribution of the light guide efficiencies for events within AMS acceptance (right).
Figure 8. Distribution of the photon incident angles on the t o p of light guide for an aerogel radiator (1.030) and variation of the light guide efficiency w i t h the photon incident angle (left). Distribution of the light guide efficiencies for events within A M S acceptance (right).
3.1.4. PMT quantum efficiency The PMT quantum efficiency is defined as the ratio between the number of photons reaching the PMT photocathode and the number of photoelectrons produced. It will be assumed as a constante value (~15%) for each event since the photon wavelenght spectrum is not affected by the reducing factors. It will result from a convolution of photons radiated energy spectrum with quantum efficiency curve of the photomultiplier. 3.1.5. Total efficiency The overall event efficiency can be written as: I
l-Hrad
"paths
d(«,¥>)
(7) where Hrad is the radiator thickness, 91 is the polar angle of the radiated photon, npaths is the number of visible branches constituting the reconstructed pattern (i.e. reflected and direct branches), and p,; is the reflectivity for the ith path. Figure 9 (left) presents the overall efficiency Stot f ° r a sample of events passing within the AMS acceptance, before being applied the multiplicative correction of PMTs efficiency.
199 A multiplicative correction factor can be evaluated from simulation by c o m paring the number of photoelectrons w i t h the number of photons expected after applying all the previous corrections. In fact not only the effect of P M T s quan t u m efficiency but also the presence of a plastic foil after radiator and the dead spaces between light guides can be extracted (Figure 9, right), from the slope of the line ajusted to the population of points.
Figure 9. Distribution of the overall event efficiency etot before applying P M T efficiency for a sample of events passing through A M S (left). Comparison between the evaluated efficiency and the value obtained from simulation (right).
3 . 2 . Counting t h e photoelectrons' n u m b e r To count the number of photoelectons in each pattern, direct branches and reflected branches can be used. However only those w i t h acceptance greater than 2 0 % and t h a t are not closer t o the particle's impact point than 5 cm are selected. Consequently the number of radiated photons t h a t suffer reflection could be obtained by the number of photoelectrons in the reflected branch corrected by the corresponding efficiency factor, and the direct by the number of photoelectrons of a direct portion divided by the corresponding efficiency factor.
4.
Results
T h e charge is then calculated according t o expression (2), where the normali sation constant can be evaluated f r o m a calibrated beam of charged particles. In the case of the present results it was obtained from 10000 simulated he lium nuclei events.
Figure 10 shows the reconstructed charge for elements
ranging from Helium t o Nitrogen. It was obtained using a 3 cm t h i c k , 1.030 refractive index aerogel radiator, for events w i t h geometrical acceptance greater than 6 0 % . T h e charge resolution ranges f r o m £\ZjZ
~ 15% for Helium t o
200 A Z / Z ~ 5.5% for Nitrogen. The charge determination is stable with de ge ometrical acceptance of the path considered which can be seen from the plot at right. ... 500
§
*
450
■
400 350
He
B
r
300 250 200 150
-
n= 1.030 rad=3cm
1 i Be C J
N
1 \J
100 50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
visible acceptance Figure 10. Charge reconstruction with simulated data from Helium to Nitrogen in the RICH detector. Gaussian fits are superimposed t o the distributions. A t right the stability of charge reconstruction with the geometrical acceptance. Stability is obtained using pattern with a Geometrical Acceptance greater than 0.2.
A RICH prototype, Figure 11 (left), was built and was tested with cos mic muons events. The algorithm provides a good reconstruction for unitary charges as can be seen from Figure 11 (right).
Figure 11. Squematic view of the RICH prototype (left). Charge reconstruction using cosmic events (right).
5. Conclusions AMS is a spectrometer designed for anti-matter, dark matter searches and for measuring relative abundances of nuclei and isotopes. The instrument will be
201 equipped w i t h a proximity focusing RICH detector based on an aerogel ra diator, enabling velocity measurements w i t h a resolution of about 0 . 1 % and extending the charge measurements up t o the Iron element.
Charge recon
struction is made in an event-by-event basis. It is based both on the velocity reconstruction procedure, which provides a reconstructed photon pattern, and on a semi-analytical calculation of the overall efficiency t o detect the radiated Cerenkov photons belonging to the reconstructed photon ring: radiator inter actions, geometrical acceptance, light guide efficiency, P M T efficiency.
The
algorithm was successfully applied t o simulated data samples for flight config uration. Evaluation of the algorithm on real data is also expected w i t h data obtained in the test beam w i t h RICH prototype at C E R N , in October 2002.
References 1. S. P. Ahlen et al., Nucl. Instrum. Methods A 350,34 (1994). V. M. Balebanov et al., AMS proposal to DOE, approved April 1995. 2. J.AIcaraz et al., Phys. Gett. B 4 6 1 , 387 (1999). J.AIcaraz et al., Phys. Lett. B 490, 27 (2000). J.AIcaraz et al., Phys. Lett. B 494, 193 (2000). J.AIcaraz et al., Phys. Lett. B 484, 10 (2000). M.Aguilar et al., Phys. Reports 3 6 6 / 6 , 331 (2002). 3. M.Buenerd. Proceedings of the Fourth Workshop on Rich Detectors (RICH02) June 5-10, 2002, Pylos, Greece. 4. J.Borges These proceedings. 5. T.Ypsilantis and J.Seguinot, Nucl. Instrum. Methods A 343, 30 (1994). 6. M.Brassed, Handbook of Optics, McGraw Hill 1995.
VELOCITY R E C O N S T R U C T I O N W I T H T H E RICH D E T E C T O R OF T H E A M S E X P E R I M E N T
L. A R R U D A , F . B A R A O , J O A O R. B O R G E S , M. P I M E N T A LIP, Av Elias Garcia, 14 - 1° andar, 1000-149 Lisboa, Portugal I. P E R E Z FCUL,
Campo
Grande,
1749-Lisboa
AMS will be the first complete spectrometer in space to study charged cosmic-rays. This spectrometer will comprehend a proximity focusing Ring Imaging Cherenkov (RICH) detector. Among other roles for this subdetector, a very precise measure of the cosmic particles velocity will be required. A possible velocity reconstruction algorithm and results of its aplication are presented.
1. The A M S experiment and its R I C H detector AMS is a spectrometer to be instaled in the International Space Station by 2005 W. This spectrometer will be able to measure the rigidity, the charge, the velocity and the energy of charged cosmic rays. These measurements are acomplished by the different elements of the spectrometer as indicated in figure 1. The main goal of this experiment is the search of heavy antimatter (Z ^ 2) in the cosmic rays. An improvement on the measurements of the relative abundances of elements and isotopes composing the cosmic rays spectrum is also intended. An example of such a measure is the abundances ratio 10Be/9Be. Because of the large halflife time of wBe (1.5 xlO 6 years) this ratio provides us information about the confinement time of cosmic rays in the Galaxy, being thus sensitive to different cosmic rays propagation models. A proximity focusing RICH detector has been included in the AMS aparatus to contribute to particle identification. Its purpose is to measure the (charged) cosmic rays velocity with a relative accuracy of 0.1% for singly charged particles (and thus better for higher charges). Charge measurement is also intended to be done for nuclei up to Z ~ 25 (see ^). This detector will also contribute to do e + jp and e~~ /p separation and to reject albedo
202
203 • particle bending Superconducting magnet • particle direction of inci dence Time-of-Flight and RICH . Ridgidity (p/Z) Silicon Tracker • Velocity (/3) Time-of-Flight and RICH • Charge magnitude |Z| Tracker, T O F and RICH • e/p separation TRD and calorimeter Figure 1.
the AMS spectrometer.
ECAL
• photons ECAL calorimeter
particles like He coming from downwards that could mimic He events. The RICH detector can be splitted in 3 main composing parts (see Fig. 2): (i) a low refractive index (n=1.03) radiator made of Silica Aerogel (ii) an expansion volume of vacuum (with an height of about 46 cm) surrounded by a conical mirror, in order to increase the acceptance of the detector (iii) a matrix of 680 photomultiplier tubes (PMT) to detect the Cherenkov light. In order to overcome dead areas between adjacent PMT 's, an array of light guides covers the PMT readout matrix. This matrix has an non active central squared area without PMTS's, due to the existence of the electromagnetic calorimeter (ECAL) placed bellow. Aerogel is a porous material with microscopic air pockets. This structure is responsible for the Rayleigh scattering of the light that crosses it. The loss of directionality for the Cherenkov photons constitutes a background source for this detector . The production light yield is ~ 50 photons/cm for charges Z = 1 and ft ~ 1. Du to the light guides and the PMT efficiencies (and in some extent to the loss of acceptance due to the ECAL hole), that light yield is decreased to a mean value of around 4-5 detected photons. The light guide unit is a pyramidal polyedron composed of 16 plexiglass pieces glued to each other because of the 16 channels per PMT. The Light guide top pixel size, which determines the resolution on the position detection for the photons hits, is 8.5mm x 8.5mm 2 .
204
Figure 2.
side-view of the RICH detector.
2. The Reconstruction M e t h o d The velocity measurement in a RICH detector is based on the Cherenkov relation that relates the particle velocity (/?) with the radiation emission angle 6C and the refractive index of the radiator (n) : cos0 c = — (1) np The reconstruction of the Cherenkov angle presented here relies on a pattern fitting technique. As depicted by the schematic draw of Figure 3, the idea is to find the pattern that best fits the collection of detected photons in the PMT readout matrix. This predicted pattern is computed as a function of
Figure 3. Schematic view of an incoming particle generating hits in the P M T readout matrix that provide the input to the pattern fit.
the cone aperture angle, i.e. the Cherenkov angle (6C) to be reconstructed. This is the only free parameter during the fit. A procedure to generate Cherenkov patterns has thus been developed for the setup of the AMS RICH
205
detector. In this procedure, a geometrical tracing of the possible Cherenkov photons trajectories, is performed up to their hitting point on the PMT matrix (for details see ^). Each one of these trajectories is described by two parameters: the azimutal photon angle (if) and the Cherenkov angle (6C) (see Fig.3). The pattern tracing procedure takes into account the refraction between different media crossed and the reflection in the conical mirror. The needed input for this tracing is the particle trajectory that will correspond to the axis of the cone described by our photons trajectories. This information is provided by the AMS tracker. 2 . 1 . A likelihood
fit
With this tool, one is able to compute distances between data hits and some predicted pattern. Thus the best 8C can be obtained based on the Minimum Squares method by the minimization of the following function: N
X2(0c) = £ > 2 ( 0 c ) , i=l
where ri(9c) is the distance (or residual) between hit # i and pattern with 6C aperture a . In fact a cut-off should be introduced to deal with the background hits, ending up with the minimization of the following merit function: N / ] ri2(6c)H(rj), where H is the usual step function i=i
„, . I 1 if x < cut-off value H(x) = < „ [ 0 otherwise
The disadvantage of such an approach is the intrisic discontinuous charac ter of the merit function. To overcome this feature, the next step was to look for a method with the capacity to deal more smoothly with the back ground. This was achieved by considering a maximum likelihood method in which there is no need to reject the background hits because these are naturally weighted out by a given probability density function (p.d.f.) V. The reconstructed 6C is now given by the maximum of: N
Wc) = Y[V{n(
(2)
i=i a
D u e to the refraction radiator-air, our pattern curve is non-analytical. For this reason, the distances r\ (i = 1,N) have to be computed by numerical minimization of a distance function on the variable ip for each hit and each 0C. In the following we stress this by writing the dependence r;(# c ) = rj(
206
The choice of the p.d.f. is made on the light of the distribution of residuals (see Fig. 4) computed with respect to the simulated Cherenkov pattern. In
Figure 4.
residuals to Cherenkov pattern (Flight Setup).
this distribution, two different populations can be isolated. A main com ponent corresponding to the signal (i.e. the ring hits) having a Gaussian shape, and a second with an aproximate flat shape corresponding to the background (hits corresponding to photons that did suffer Rayleigh scat tering or simply ocasional PMT noise). Then the likelihood p.d.f. can be expressed as : V{r) = Vsignalir)
with
+ Vnoise
Vsignai(r) = (1 - b) r— exp V27r<7
and
Vn0iSe = -p
(3)
1 /r\2l
-
(;)'
(4) (5)
where b represents the mean background ratio ( i.e. the ratio of the noisy hits to the total number of hits ) and R represents the matrix dimension. The typical values for the parameters a and b are about 0.5 cm and 10% respectively (and -^ ~ 1 0 _ 3 c m _ 1 ) .
207
2.2. Clusters
and photon
emission
point
An important issue is the ocurrence of clusters in the PMT matrix. Most of it are due to the passage of the particle across the light guides, which generates a huge production of photons in a very small area of the matrix (left fig 5), because of the very high refractive index of Plexiglass (~ 1.5). These clusters have to be removed to prevent incorrect reconstructions of 6C. Therefore, the hits in a 4cm radius around the particle passage point on the PMT matrix (this is an extrapolated point of the particle track) are not considered for the fit, otherwise one would obtain biased reconstructions like the one in the event Display of right Fig. 5.
-60
-40
-20
0
20
40
60
Figure 5. in the event display of the right we show the effect of considering the particle cluster hits in the pattern fit, obtaining a reconstructed Cherenkov pattern completely biased (smaller and full pattern) instead of the expected pattern (larger dotted one ).
Being a proximity focusing detector, the RICH of AMS does not produce perfect such rings as the ones of the optical focusing RICH detectors found in accelerators experiments. As photons are emitted along the particle path through the radiator, they produce a slightly "unfocused" or blurred ring in the matrix instead of a sharp one. Nevertheless, for a matter of simplicity, the reconstruction assumes a single vertex for the photons emission. As it can be seen in Fig. 6, this optimal point is not the midle point of the particle track inside the radiator, but rather a point bellow. This is a consequence of the Rayleigh scattering that gives rise to a transmittance law that we can write as t(x) oc exp(—Cx/X4), where x is the length traveled by the photon across the radiator and C characterizes the radiator clarity or transmitance. According to this, the photons produced earlier in the
208
particle track across the radiator are the most scattered and this is why the majority of the photons that arrive at the ring comes from the lower part of the radiator. The greater transparence of aerogel with index n=1.03 compared with aerogel n=1.05 becomes evident by comparing the optimal vertices for both in Fig. 6. : Systematic A0C (mrad)
; Systematlc A6C (mrad) 3
: Agilteo
Agll.< 50
:
» .i
* '
L
*
60
a
o.
; •
a 0
.
.
0.70
'
: ■
>
,
0,2
0,4
0.6
O.S
z
Vertex
I
0
»
0.2
0.4
/HRAD
0.6
O.S
z
Vertex
1
/HRAD
Figure 6. Systematic error on 8™° (i.e. 6"° ~ #S' m ) as a function of the z coordinate (normalized to radiator thickness) used for the photons-tracing emssion point in the reconstruction.
3. Results 3.1. Flight Setup
Geometry
(simulation
results)
The results presented in this subsection have been collected in the frame work of a simulation of the setup described in section 1 based on GEANT3. Some of the parameters used here differ from the more final established values presented in the detector description : radiator thickness: 2 cm (instead of 3 cm) clarity: 0.0042 /j.m4/cm b(for a refractive index n = 1.03) expansion volume height: 46.8 cm (instead of 45.8 cm) The achievable limits for the resolution on 6rcec depend on different sources such as the chromaticity of the radiator c (=^ 80cchro) and on geometrical b this nominal value announced by the manufacturer seems not to be reached according to measurements made inside the RICH colaboration that point a value closer to 0.009 fim4 /cm c as for any optical medium, aerogel is characterized by a chromatic dispersion,i.e. the refractive index depends on the wavelenght of photons by some relation n = n(\). This leads to an uncertainty 8n on the used mean value (n).
209 factors related to the detector dimensions, pixel uncertainty and the direc tion of the photons (=$- 59%e°). We have the following expression for the single hit resolution on 9C : 59c = 59sceo e Sd?ro
with
2„
$r
h where 5r is the uncertainty on the Cherenkov ring radius and h is the expansion volume height. As summarized in Table 1 the dominant source is the pixel dimension. Table 1. contribuitons of different error sources to final single-hit 9C resolution. pixel size
(~ 8.5mm)
=> 59c ~ 5 mrad
radiator thick. ( t ~ 2 — 3cm)
=> 59c < 5 mrad
chromaticity
=> 59c < 5 mrad
So, a single hit resolution on 9C of the order of 5mrad is then expected. The total resolution scales down with the number of hits according to : Mc(total)
=
^sins'ehit) V Numb hits
As the histogram of Fig. 7 shows, the achieved resolution (3 mrad) is in good agreement with previous apriori estimations given the fact that the mean number of hits for the selected reconstructions is ~ 4. Figure 8 shows
c*rc.„,„,
.
r! r.
™ 1.1.61 0.1777
//
protons (20 GeV)
f i I t
/ j
-
l
i
\\ \\ \1
+ \
f
_-7
\
V
9rec (Degree)
p (GeV/c/nucleon)
Figure 7. left: distribution of reconstructed 9C for 20000 triggers by protons of 20 GeV/c in momentum. The triggers were simulated in all AMS acceptance, right : reconstructed 9C versus momentum per nucleon
210
that despite the very different resolutions -4- obtained for protons and he lium nuclei (0.07 % and 0.04 % respectively for /? ~ 1), due to the different number of hits (there is a factor of 2.8 between heliums and protons), one can estimate the single-hit resolution independently, obtaining the asymp totic value of 0.15 % . The reason why the single hit resolution on'/? gets
p (GcV/c/nudeon)
p (G
Figure 8. relative resolution on /3 as a function of the momentum and the charge of the particles (total at left and single-hit at right).
worse with the increase of the particle velocity in fig. 8 is the following scalling relation with 8e: - j - (single hit) = t a n 6C 56<.
3.2. Prototype test-runs
Setup Geometry data)
(preliminary
results
from
The following results correspond to a very preliminary analysis made on data collected by a RICH prototype assembled at ISN (Grenoble-France) which setup is the one depicted by Fig. 9. The setup of this prototype re produces one small part of the final flight setup (it has a detection matrix of 96 PMT's instead of 680). The main difference is the absence of the conical mirror and the pixelsize (~ 8mm) slightly smaller than the final one. The collected data consisted of cosmic muons triggered by 2 scintilators placed above and bellow the tracking system composed by 3 wire chambers. On the run here analysed ~ 190K events were taken in a 3 days run. In order to grant a better track defininition of the mother particle, a 4th point has been
211
Figure 9.
Prototype Setup used in the early cosmic muons runs at Grenoble .
used in adition to the 3 hits given by the wire chambers. This 4th point is provided by the particle cluster generated in the light guides as previously described. In fact this procedure turned out to be a real event selection, in view of a occasional malfunctioning of the wire chambers. The selection procedure was then to only accept events for which the extrapolation of the particle track to the detection matrix gives a point close enough to the particle cluster, this last being defined by a strong enough PMT signal to reveal the particle passage trough the light guides. On left figure (10) we give an idea of what strong enough means. The hits close to the track ex trapolated point (named "particle hits" on left figure (10)) clearly present a higher PMT signals than the hits far apart from it (named "particle hits" on the same figure). The superposition part of the 2 signal populations is
212
due to the ocasional malfunctioning of the tracking system that from timeto-time gave a completely wrong track. The separation cut was established in 5 photoelectrons (p.e.). The right figure (10) shows what close enough means. The required matching between the 2 refered points was one sigma of the plotted Gaussian. This severe cut is justified by the preliminary character of this analysis. In Fig. 11 we compare some results obtained from this data run to the corresponding simulation results. At left, we can see the number of hits on the reconstructed Cherenkov rings. The general agreement shows that in a first aproximation all RICH components (from the radiator up to the PMTS) are working as expected. At right, we com pare the reconstructed incidence angle of the photons on light guides. The transmission light guide efficiency depends strongly on this angle (see ^). This comparison turns out to be an indirect test to this eficiency and as it can be seen, this test is well fulfilled. As a final result we present the reconstructed muons spectrum in Fig 12 .
4. Conclusions The RICH detector of AMS was designed to provide the spectrometer with an high precision velocity measurement (8/3/0 < 0.1%); this in order to contribute to: - the measurement of He/He - e/p separation - isotopic mass separation of heavier particles like 3He/4He,9
Be/10Be
In addition, RICH allows a redundant charge measurement t 2 '. A prototype is already working and has been tested with cosmic muons runs. In the meanwhile (last October), it has also been submitted to an ion beam test at CERN. A velocity reconstruction based on a pattern fit likelihood approach was developed and here presented. This algorithm was succesfully applied to both simulated and real data.
213
Figure 10. Selection event procedure is based on the matching between a strong signal on the P M T matrix and the track information.
Figure 11. Comparison between reconstruction of data and simulation. Dots stand for data, bars for simulation.
214
lO
lO
l 0.975
0.98
0.985
0.99
0.995
1
1J0O5 J3
Figure 12. cosmic muons reconstructed spectrum. Dots correspond to the reconstructed data. Full histogram corresponds to expected velocity spectrum, simulated according to (6). Light hollow histogram corresponds to the reconstruction of the simulated spectrum. The muons momentum spectrum at sea level and low latitude is given by: —
= 3.09 • 10-3p-0«54-0.3406 I„(p)
( g e e
^ [3])
dp [cm-2s-1sr-1(GeV-1)}
(6)
References 1. S.P.Ahlen et al., Nucl. Instrum. Methods A 350,34 (1994) V. M. Balebanov et al., AMSpropossal to DOE, aproved April 1995. 2. L.Arruda et al, Electric Charge reconstruction with RICH detector in AMS (in this proceedings) 3. Peter K. F. Grieder, Cosmic Rays at earth ,2001 Elsevier, page 359
Neutrino Physics and Astrophysics
This page is intentionally left blank
RESULTS F R O M T H E S U D B U R Y N E U T R I N O OBSERVATORY
J. MANEIRA FOR THE SNO COLLABORATION Department of Physics, Queen's Kingston, Ontario K7L 3N6,
University, Canada
The Sudbury Neutrino Observatory is a 1 kton D2O Cerenkov detector, sensitive to 8 B solar neutrinos. The energy, radius, and direction with respect to the sun of each neutrino event are used to determine the rates of the charged current (CC) and neutral current (NC) reactions of neutrinos on deuterium and neutrinoelectron scattering (ES). Comparison of the CC, NC and ES rates from the analysis of the pure-D20 phase of the experiment provides a measurement of the non-!/ e solar flux component of 4>y.T = 3Alt%\H(sta.t.)±l\H (syst.) x 10 6 c m _ 2 s _ 1 , 5.3NC = S . O S ^ Q ' ^ s t a t . ) ^ ' ^ ! (syst.) x 10 6 c m - 2 s - 1 , consistent with solar models. Measurements of the day-night v flux asymmetry are also reported. A global analysis of the data from SNO and all the other solar neutrino experiments in terms of matter-enhanced oscillations strongly favors the Large Mixing Angle (LMA) solution.
1. Introduction The discrepancy between measured 1 ' 2 ' 3,4 ' 5 ' 6 and predicted 7 ' 8 solar neutrino fluxes, that lasts for over thirty years, is known as the solar neutrino prob lem. A possible solution is that the f e 's produced in the Sun core undergo a flavor change on their way to the Earth, and an elegant mechanism for this is supplied by matter-enhanced oscillations 9 ' 10 . The Sudbury Neutrino Observatory was specifically designed to test the flavor change hypothesis by separately measuring the flux of electron and all flavor neutrinos, and these results were recently published 12 , as well as the measurement of the day-night asymmetry of the energy spectrum 13 . A previous publication 11 compared the rate of the CC reaction to the rate of the ES reaction as measured in the SuperKamiokande experiment and SNO.
217
218
Figure 1. Cross-sectional view of the SNO detector.
2. The S N O experiment 2.1. The
detector
14
SNO is an underground heavy water Cerenkov detector located in the INCO, Ltd. Creighton mine near Sudbury, Ontario, Canada, at a depth of about 2 km (6010 m of water equivalent). The active target of the detector, shown in Fig. 1, consists of 1000 tonnes of heavy water(D20), and is contained in a 12 m diameter transparent acrylic vessel. The Cerenkov photons produced in the D2O are detected by 9456 8 inch photomultiplier tubes (PMTs) mounted on a 17.8 m diameter stainless steel geodesic sphere and coupled to light concentrators that raise the effective coverage to 55 %. Ultra-pure light water surrounds the acrylic vessel and the PMT support structure, in order to shield the heavy water from radioactivity both in the PMTs and in the surrounding rock.
2.2. Neutrino
interactions
in SNO
8
SNO detects B solar neutrinos through the processes: ve+d->p + p + e~ vx+d-*p + n + vx vx + e~ ->■ vx + e-
(CC), (NC), (ES).
219
The charged current (CC) and neutral current (NC) reactions are made possible by the use of heavy water and are exclusive to SNO, while the elastic scattering reaction (ES) was used by other experiments. The charged current (CC) reaction is sensitive exclusively to iVs in the energy range of solar neutrinos. The energies of the electron and the neutrino are strongly correlated, so the CC reaction also provides a good measurement of the 8 B energy spectrum. In addition, also the directions of the electron and neutrino are correlated, according to (1 — 0.340cos(#)). The neutral current reaction (NC) is equally sensitive to all active neu trino flavors (x = e, fi, r ) , so it provides a model-independent measurement of the total flux of active solar neutrinos. The neutron is detected indirectly by deuteron capture followed by the emission of a 6.25 MeV 7 ray. The 7 ray Compton scatters an electron that can then be detected. The Compton effect cross section is sharply peaked at this energy so the electron energy is strongly correlated to the 7 ray energy. This reaction has a low energy threshold for the incident neutrino - 2.2 MeV - but its energy and direction information are lost. The elastic scattering (ES) reaction is sensitive to all electron flavors as well, but with reduced sensitivity to i/M and vT. The direction of the recoil electron is strongly correlated to the direction of the Sun, but the cross section is about 10 times smaller than that of the CC reaction.
3. Calibration The detector is equipped with a flexible system capable of positioning sources in two perpendicular vertical planes inside the acrylic vessel, as well as in vertical axes in the light water volume between the PMTs and the acrylic vessel. Five different sources are used to calibrate the detector response: a light diffusing sphere connected to a tunable N2 laser, a trig gered 16 N source 15 emitting 6.13 MeV 7 rays, a 8 Li source 16 , a 3H(p, j)4He ('pT') source 17 providing 19.8 MeV 7's and a 252 Cf spontaneous fission neu tron source. The optical calibration of SNO consists in determining the attenuation lengths of D2O and H2O, as well as the effective angular response of the PMTs for different wavelengths, by comparing the occupancy distributions with the optical source in several positions. These are the input parameters for an optical model of the detector, used to calculate the most likely energy of a neutrino event from its number of hit PMTs, position and direction. The optical source, however, does not provide an absolute calibration of
220
Figure 2. The energy response of SNO to different sources compared to Monte Carlo simulations.
the energy scale. This is obtained by comparing data from the 16 N source with Monte Carlo simulations using the optical parameters previously de termined. Fig. 2 shows the measured energy response for three sources: 16 N 7, 8 Li /?, and pT(19.8 MeV 7), compared to Monte Carlo calculations for each of the sources and for CC solar neutrino interactions. Our overall systematic uncertainty for energy scale was 1.21%, and the uncertainty in the determination of energy resolution was 4.5%. The vertex resolution is determined to be 16 cm at 6.25 MeV by com paring the reconstructed position to the known position of the 16 N and 8 Li sources. The angular resolution is determined to be 26.7° at the same energies, by comparing the reconstructed direction to the vector from the source to the vertex, for 16 N events that reconstruct far from the source. The neutron capture efficiency was measured with the 252 Cf source to be 29.90±1.10%, and when the fiducial volume and energy threshold cuts are applied, the overall detection efficiency is 14.38±0.53%. 4. Backgrounds 4.1.
Instrumental
The highest rate background in SNO is the instrumental background, due primarily to PMT light emission ("flasher" events), static discharges in the neck of the acrylic vessel or electronic pickup. A set of cuts was devel oped to reject these backgrounds while minimizing the impact on neutrino events. The low level cuts are based on time and charge patterns of the
221
1W>
ZW)
3M
«W
Days Since November 2,1999
Figure 3. Thorium(a) and Uranium(b) equivalent equilibrium concentrations in D20measured by ex situ and in situ techniques. The time-integrated averages on the right-hand side include an additional sampling systematic uncertainty for the ex situ measurement.
hit PMTs and their application causes a data loss of 0.34^002%) evaluated by source calibration data. The following step in data processing is the reconstruction of the most likely position and direction of the event from the time and position information of the hit PMTs. The high level set of cuts is based essentially on reconstruction fit quality figures-of-merit and other parameters that indicate how close the time and angular distributions of the event are to those expected for Cerenkov light. The data loss from these cuts is 1.43±g;|?% for CC, 1.46±g;|?% for ES, and 2.28±g;||% for NC.
4.2. Radioactive
Backgrounds
The radioactivity levels in D 2 0 and H 2 0 were measured by regular radioassays of the U and Th chain daughters and the detector response to their decays was studied by means of acrylic encapsulated sources. The background events from radioactivity are present essentially only at low energies, below 6 MeV, and are due mainly to the decays of 208 T1 in the Thorium chain and 214 Bi in the Uranium chain. These isotopes emit 7 rays with energies above 2.2 MeV, enough to cause photodisintegration of the deuteron. In addition, a fraction of the 7 and /? — 7 Cerenkov events is above the analysis energy threshold of 5.0 MeV. The ex situ techniques that were used to measure the levels of U and Th in water were based on the extraction of Ra ions onto MnO x or hy-
222
drous Ti oxide ion exchange media, followed by membrane degassing and the counting of Rn daughters decays. The in situ measurements consist on the analysis of the low energy (4 - 4.5 MeV) Cerenkov signal region. Statistical separation of the Bi and Tl in situ signal is possible by analyz ing the hit PMT pattern isotropy. The measured levels for the whole data set are shown in Fig. 3. From the average values for D 2 0 , we expect 44t 8 photodisintegration neutrons, and 20+g3 Cerenkov events. Cosmic rays and atmospheric neutrinos can also produce photodisintegration neutrons. In order to reduce this background, a dead-time of 250 ms was imposed after every event in which the total number of hit PMTs was larger than 60. For a list of all contributions to the background, see Table I in reference 12 . 5. D a t a Analysis The results presented here are based on the analysis of the full pureD 2 0 data set, collected between Nov. 2, 1999 and May 28, 2001. This corresponds to 306.4 days of live time, 128.5 during the day and 177.9 dur ing the night. The application of the cuts reduces the data to 2928 events, shown in Fig. 4. Fig. 4(a) shows the kinetic energy(Te-f-f) spectrum of the selected events for the analysis threshold of Teff> 5 MeV and fiducial volume selection of R < 550 cm, where R is the reconstructed event radius. The data are resolved into contributions from CC, ES, and NC events above threshold using probability density functions (pdfs) in kinetic energy, cosine of the angle with respect to the Sun, and radial position. These functions are derived from Monte Carlo calculations generated assuming no flavor transformation and the standard 8 B spectral shape 18 . Background event pdfs are included in the analysis with fixed amplitudes determined by the background calibration. The extended maximum likelihood method used in the signal decomposition yields I967.7t%'l CC events, 263.6J]25g ES events, and 576.5l4g'g NC events, where only statistical uncertainties are given. Systematic uncertainties on fluxes derived by repeating the signal decomposition with perturbed pdfs (constrained by calibration data) are shown in Table II in reference 12 . Normalized to the integrated rates above the kinetic energy threshold of Teff> 5 MeV, the flux of 8 B neutrinos measured with each reaction in SNO, assuming electron neutrino cross sections and the standard spectrum shape 18 is (all fluxes are presented in units of 106 cm~ 2 s _ 1 ):
4c
= i.76±g:gS(Btat.)lS:S (<*«*■)
223
Figure 4. (a)Kinetic energy for R < 550 cm. Also shown are the Monte Carlo pre dictions for CC, ES and NC + bkgd neutron events scaled to the fit results, and the calculated spectrum of Cerenkov background (Bkgd) events. The dashed lines represent the summed components, and the bands show ±15 MeV. (b)Flux of 8 B solar neutrinos of y, or T flavor vs the flux of electron neutrinos. The diagonal bands show the total SSM predicted 8 B flux (dashed) and that measured with the CC, NC and ES reactions in SNO (solid).
4f
= 2.39±°;2S(stat.)±£i2 (syst-)
41°
= 5.09tS;**(stat.)iS:?3 (syst.).
The CC and ES results reported here are consistent with the earlier SNO results 11 for Teff>6.75 MeV. The NC flux is in good agreement with solar model predictions 7 , while the excess of the NC flux over the CC and ES fluxes implies neutrino flavor transformation. A simple change of variables resolves the data directly into electron (cj>e) and non-electron (0MT) components, 4>e = 1.76±°;°5(stat.)+°;099 (syst.) 0„ T = 3.4li°;4455(stat.)+°;4485 (syst.). This change of variables allows a direct test of the null hypothesis of no flavor transformation (0MT = 0) without requiring calculation of the CC, ES, and NC signal correlations. Combining the statistical and systematic uncertainties in quadrature, 0 Mr is 3.4li°'64) which is 5.3
224
1
£20
1
10 :
1
I
o
1
I
I
1
-10
- Calibrations -20 . and Checks i
°-
' i
a
9
i
gg Sa
i
a
1 i
Neutrino Flu* Asymmetries. i
| 8
;
10 11 12 13 20 Kinetic energy (MeV)
§0.2
So.i
.-ff|-H-f+-f.
I o l-o.i o "-0.2
5
6
7
8
9
10 11 12 13 20 Kinetic energy (MeV)
Figure 5. (a) Various event classes used to determine systematic differences between day and night measurements. Also shown are measured asymmetries on the CC flux, and on the electron neutrino flux derived from the CC, ES, and NC rates when the total neutrino flux is constrained to have zero asymmetry. (b)Energy spectra and residuals for day and night data.
CC, ES, and NC rates. The error ellipses represent the 68%, 95%, and 99% joint probability contours for (j>e and far. 6. Day-Night analysis The day-night analysis of the SNO data consists on the standard analysis of the separate day and night data sets, but with a different treatment of systematic uncertainties. Tests were performed to probe possible daynight variations of detector response that could falsify the measurement of asymmetries in the neutrino signals. These consist of an east/west division of the neutrino data based on the Sun's position: a 5 Hz pulser, to ver ify livetime accounting, muon-induced neutrons and the solitary point of high background radioactivity, or "hot spot", on the upper hemisphere of the SNO acrylic vessel. The rate asymmetries for each test are shown in Fig. 5(a). For details on the systematic uncertainties, see 13 . The measured night and day fluxes fa- and fa. were used to form the asymmetry ratio for each reaction: A = 2(fa — fa)/(fa + fa). The asymmetries for the extracted signals are (in %) Ace — +14.0 ± 6.3J^'jj, AES = -17.4 ± 19.5±i;|, ANC = -20.4 ± 16.9J£.B- Fig. 5(b) shows the energy spectra for day and night and night - day residuals. All signals and backgrounds contribute. A change of variables from fac fas and fa~c to electron neutrino flux, <j)e and total flux, fat, allows us to impose
225
- 50% CL - 95% CL ■ 99% CL - 99.73% CL
1
0
,
1
log(taire)
0 , 1 log(tarre)
Figure 6. Allowed regions for MSW oscillation parameters using only SNO data (a-left) and using a global analysis with all the experiments(b-right).
the condition Atot = 0, as predicted for matter oscillations between active flavors only. In this case, we obtain, Ae = 7.0 ± 4.9J;];;3).
7. Neutrino oscillation analysis The SNO data was fit to two active flavor MSW oscillation models, in or der to place limits on neutrino flavor mixing parameters. The model for the predicted CC energy spectrum was obtained by combining the neutrino spectrum 18 with the survival probability, cross section 19 and SNO's re sponse functions. The free parameters of the fit are Am 2 ,the squared mass difference between the neutrino mass eigenstates, the mixing angle 0, and the total 8 B flux. The flux of higher energy neutrinos from the solar hep reaction is fixed 7 at 9.3 x 103 c m - 2 s _ 1 . MSW exclusion contours were cal culated in and tan28 for Ax 2 (c./.) = 4.61 (90%), 5.99 (95%), 9.21 (99%), and 11.83 (99.73%). Figure 6(a) shows the allowed mixing parameter regions using the mea sured CC energy spectrum (day and night separated), the NC and ES fluxes with no additional experimental or solar model constraints. The measured rates from the Cl l and Ga 4,3 ' e experiments, the measured day and night spectra from the SuperK experiment 5 and the solar model predicted fluxes for pp, pep, 7Be and CNO 7 neutrinos were also included in the calcula tion of the contours shown in Figure 6(b). The best fit points are shown in Table 1 and indicate that the global analysis strongly favors the LMA solution. Using the total SNO CC energy spectrum instead of separate day and night spectra yields nearly identical results.
226 Table 1. Best fit points in the MSW plane for global MSW analysis using all solar neutrino data. <J>B is the best-fit 8 B flux for each point, and has units of 10 6 c m - 2 s _ 1 . A m 2 has units of eV 2 . Ae is the predicted asymmetry for each point. Region LMA LOW
*L:„/dof 57.0/72 67.7/72
4>B 5.86 4.95
A(%) 6.4 5.9
Am2 5.0 x 10~ 5 1.3 x l O " 7
tan20 0.34 0.55
c.l.(%) 99.5
8. Summary The SNO measurement of CC, NC and ES interactions of 8 B neutrinos on D2O has provided at the same time strong evidence for neutrino flavor transformation and confirmations of the solar model flux predictions. The measured fluxes and the CC day and night energy spectra are consistent with matter oscillation scenarios and the LMA solution is strongly favored in a global analysis including information from all other experiments. References 1. B.T. Cleveland et al, Astrophys. J. 496, 505 (1998). 2. K.S. Hirata et al, Phys. Rev. Lett. 65, 1297 (1990); K.S. Hirata et al, Phys. Rev. D 44, 2241 (1991), 45 2170E (1992); Y. Fukuda et al, Phys. Rev. Lett. 77, 1683 (1996). 3. J.N. Abdurashitov et al, Phys. Rev. C 60, 055801, (1999). 4. W. Hampel et al, Phys. Lett. B 447, 127 (1999). 5. S. Fukuda et al, Phys. Rev. Lett. 86, 5651 (2001). 6. M. Altmann et al, Phys. Lett. B 490, 16 (2000). 7. J.N. Bahcall, M. H. Pinsonneault, and S. Basu, Astrophys. J. 555, 990 (2001). 8. A.S. Brun, S. Turck-Chieze, and J.P. Zahn, Astrophys. J. 525, 1032 (1999); S. Turck-Chieze et al, Ap. J. Lett., v. 555 July 1, 2001. 9. L. Wolfenstein, Phys. Rev. D17, 2369-2374 (1978) 10. S.P. Mikheyev, A.Y. Smirnov, Sov. Jour. Nucl. Phys. 42, 913-917 (1985) 11. Q.R. Ahmad et al, Phys. Rev. Lett. 87, 071301 (2001) 12. Q.R. Ahmad et al, Phys. Rev. Lett. 89, 011301 (2002) 13. Q.R. Ahmad et al, Phys. Rev. Lett. 89, 011302 (2002) 14. The SNO Collaboration, Nucl. Instr. and Meth. A449, 172 (2000). 15. M.R. Dragowsky et al, Nucl. Instr. and Meth. A481, 284-296, (2002). 16. N.J. Tagg et al, Nucl. Instr. and Meth. A489, 178-188, (2002). 17. A.W.P. Poon et al, Nucl. Instr. and Meth. A452, 115, (2000). 18. C.E. Ortiz et al, Phys. Rev. Lett. 85, 2909 (2000). 19. S. Nakamura et al (2002), nucl-th/0201062 20. Details of SNO's response functions are available from the SNO web site http://sno.phy.queensu.ca 21. Z. Maki, N. Nakagawa, S. Sakata, Prog. Theor. Phys 28, 870 (1962) 22. V. Gribov and B. Pontecorvo, Phys. Lett. B28, 493 (1969)
STATUS REPORT O N BOREXINO*
ALAIN D E B E L L E F O N f PCC College de France, 11 Place Marcelin Berthelot, F 75231 Paris Cedex05 E-mail: [email protected]
Borexino aims to detect low energy 7 Be solar neutrinos with a good energy res olution, via the elastic neutrino electron scattering ve + e - —► ve + e~. The experiment is installed at LNGS and now is in its pre-filling phase.
1. Introduction The solar neutrino puzzle consists in the deficit observed in the different experiments (chlorine, gallium or water targets) compared to the predic tions of the standard (or even non standard) solar models (see for example ref.1 for a review of the experimental results). The most attractive inter pretation today is the oscillation between ve and/or v^ , i/T which has been confirmed with the SNO results 2 . Even though solar neutrinos are no more a problem details on the solu tions are still needed. There are four possibilities, the "vacuum" solution (small probability) and three solutions via the MSW effect: the small mix ing angle one (SMA), the large-mixing angle one (LMA) and the "low" one (LOW); a global and recent analysis of all neutrino data (solar, at mospheric and other) in terms of 3-neutrino oscillations can be found in reference 5 . To disentangle the SMA and LMA solutions experiments as Kamland and Borexino are needed. In the case of Borexino rates for the different solutions are shown on table 1.
*full list at http://almibx.mi.infn.it/html/collaboration.html tfor the Borexino collaboration
227
228 Table 1. For a given 7 Be flux and for a 100 ton detector we expect the following rates per day. SSM
totalrates withSuperK
msw — SMA
msw — LMA
msw — LOW
12+16
32+7
26
-.5
-8
55
beforeSNO afterSNO
Vacuum 38+15 -10
36+5
32+7
-4
-3
32+9 -4
35+5
32+3
32+5
-3
-3
-4
2. Experimental setup Borexino is installed in Hall C at LNGS (Laboratorio Nazionale di GranSasso) deep underground with an overburden of 3500 meters water equiv alent. Borexino is an unsegmented liquid detector featuring 300 tonnes of well shielded ultrapure scintillator viewed by 2200 photomultipliers. The detector core is a transparent spherical vessel (Nylon Sphere, 100 micron thick). 8.5m of diameter, filled with 300 tonnes of liquid scintillator and surrounded by 1000 tonnes of a high-purity buffer liquid. The installation is completed the 2200 Photo-multipliers are mounted on the stainless steel sphere looking inside, as well as the 210 phototubes looking outside to be used as a muon veto. Inside the Stainless Steel Sphere (sss) two nylon vessels are to be mounted: the outer one playing the role of Rn barrier- 1000 tons of pseudocumene buffer and the inner vessel con taining 300 tons of liquid scintillator. The scintillator mixture is made out of pseudocumene (PC) and 1.5 g/1 of 2,5-diphenyloxazole (PPO), while the buffer liquid is pure PC alone (with the possible addition of a light quencher, dimethylphtalate (DMP). The Borexino design is based on the concept of a graded shield of progressively lower intrinsic radioactivity as one approaches the sensitive volume of the detector; this culminates in the use of 200 tons of the low background scintillator to shield the 100 tons innermost Fiducial Volume (see fig 1.) 3. Physics Goals In Borexino we are mainly interested in the observation of the higher energy 7 Be neutrinos, which is a monochromatic line at 863 keV. The detection reaction is the electroweak elastic scattering on electrons; the recoil electron spectrum will have a maximum of 664 keV kinetic energy. The detection threshold for observing such process will be of 250 keV, so that a rate
229
Figure 1.
Borexino Set-up.
of 46 events/day is predicted by the Standard Solar Models in the 100 tons Borexino fiducial volume such a rate has to be compared to those for different solutions as it is shown (Table 1). The feasibility of this program depends critically on the expected back ground in the 250/700 keV energy window 3 . The general strategy for background reduction is shortly described in the Conunting Test Facility paragraph. The 7 Be neutrino line is predicted by all standard solar models to be the second most important neutrino production reaction (after the basic pp reaction) in the sun. The flux of 7 Be neutrinos is predicted much more accurately (uncertainty less than 10%) and is about a 1000 times larger than the 8 B neutrino flux that is measured by SuperKamiokande and SNO. Also, since 7 Be decay produces only neutrino lines, the theoretical predictions of neutrino oscillations are more unique for 7 Be.than for the 8 B neutrinos.
230
3 -
i ^
(M
0.6
0.8
1
12
electron energy (MeV) Figure 2. Number of events in arbitrary units as a function of electron energy expected for beryllium line, background and total.
which have a broad energy spectrum (0-15 MeV). BOREXINO will be the only solar neutrino experiment in the next decade that will measure the solar neutrino flux below 5 MeV for a spe cific solar neutrino producing reaction. Standard solar models predict that 7 Be contribution to the observed gallium rate should be about 35 SNU and about 1.2 SNU in the chlorine experiment. In both cases, the predicted rate from 7 Be is about one-half the total observed rate. There are many possible combinations of explanations for the observed rates in the radiochemical experiments if neutrino oscillations are assumed to occur, but only BOREXINO can provide a measurement of the specific 7 Be contribution In summary, the solar 7 Be neutrino line intensity and its time (seasonal, day-night) variations will be the main goal of the Borexino experimental
231
program. In addition to 7 Be neutrino physics, other goals under study by the collaboration (potential Borexino goals) are: (1) More solar neutrino studies: 8 B and CNO studies. 8 B neutrinos have a spectrum extending up to 15 MeV; they will produce about 1 event/day signal in Borexino above 4 MeV, in a region where the background is expected to be of the same magnitude. An ex cess production of CNO-cycle neutrinos can also possibly be seen in Borexino. (2) Neutrino magnetic moment can be studied using a Y- 90 Sr or a 5 1 Cr source. An indicative sensitivity of 1 0 - 1 1 Bohr magnetons is ex pected. (3) Antineutrino searches can be performed through the Reines-Cowan detection reaction. This can be used to search for neutrino / an tineutrino transitions in the case of solar neutrinos or to a measure ment of the Earthly emitted neutrinos. (4) The signal expected from type-II supernova neutrino is of the order of 30 events on 12 C and leave Borexino with the possibility to detect a 3xl0 5 3 erg supernova at 10 kpc. It is also possible to detect vx through NC proton scattering (6) And possibility is given to contribute to the Supernova Network for real-time alarm to the astronomical community (SNEWS 3.1. Counting
Test Facility and low
radioactivity.
The background level requirements for Borexino are : Uranium, Thorium at the level of 10^ 16 g/g in the inner vessel where is the scintillator and 10-100 times more in the outer vessel where buffer liquid is found. Because the sensitivity of germanium counter measurements is rather lim ited around 10 _ 1 1 g/g we have been using a specific detector called the Counting Test Facility (CTF). The Counting Test Facility (CTF) has been built to test the plant in their final configuration before to fill the Borexino vessels with the very high level of radiopurity in the scintillator. The liquid scintillator is contained in a nylon sphere (lm radius, 500 mi crons thickness) viewed by 100 PMT's. the central sphere is surrounded by another nylon vessel and immersed in 1000 ton of ultrapure water. The water tank is a cylinder 11m in diameter, 10 m high. The nylon sphere acts as a radon barrier and let us hope to have very good results. 4 The CTF up
232 t o now has been used to measure the radiopurity of scintillator in particular the radioactive nuclides from U and T h families, as well as 1 4 C of which the activity has been derived below 250 keV. T h e measured value expressed as 14 C / 1 2 C yielded 2 x l 0 ~ 1 8 for P C . To measure t h e U r a n i u m - T h o r i u m rates t h e Bismuth-Polonium method is used. We count the delayed coincidences 2 i 4 B i . 2 i 4 p 0 w i t h a i i f e t i m e 0 f 236MS and 2 1 2 B i - 2 1 2 P o with a lifetime of .433/xs, respectively on t h e basis of secular equilibrium we are able t o de termine t h e content in Uranium-Thorium from t h e rates of what is called long and short Bi-Po. For P C , which is t h e scintillator we plan t o use, radiopurity thorium-uranium equivalent has been measured and was about 4.4xlCT 1 6 g/g. 4.
Conclusions
T h e C T F is currently taking d a t a and as first results it is going t o provide t h e Borexino collaboration with t h e information needed t o get t h e best purification processes before filling t h e detector. Borexino is on it's way and will probably b e ready to take d a t a about middle of 2003. Acknowledgment s I want t o t h a n k t h e Faro meeting organisers and I would like t o express my gratitude t o Alessandra Di Credico with whom I discussed about this presentation. References 1. Till A. Kirsten, Solar neutrino experiments: results and implications, Reviews of Modern Physics, Vol 71, NO. 4, July 1999. 2. SNO collaboration, Q.R Ahmad et al., Direct evidence for neutrino flavor transformation frm neutral-current interactions in the Sudbury Neutrino Ob servatory, Phys. Rev. Lett. 89,(2002) 3. BOREXINO collaboration,G.Alimonti et al.,Science and technology ofBorexino: a real time detector for low energy solar neutrinos, Astropart. P/M/S.16,(2002),205.
4. BOREXINO collaboration,C. Arpesella et al., Measurements of extremely low radioactivity levels in BOREXINO, Astropart. Phys.l8,(2002) 1-25. 5. M.C Gonzalez-Garcia, Yosef Nir, Developments in Neutrino physics hepph/0202058 6. John F. Beacom, W.M Farr, Petr Vogel, Detection of Supernova neutrinos by neutrino-proton elastic scattet'mg,hep-ph/0205220 7. J.N Bahcall, M.C. Gonzalez-Garcia, and C. Pena-Garay, Robust signatures of solar neutrino pscillation solutions, J. High Energy Phys. 04, 007 (2002).
T E S T I N G N E U T R I N O P A R A M E T E R S AT F U T U R E ACCELERATORS*
J. C. ROMAO Departamento de Fisica, Instituto Superior Tecnico Av. Rovisco Pais 1,1049-001 Lisboa, Portugal E-mail: Jorge, romao Qist. utl.pt
The simplest unified extension of the Minimal Supersymmetric Standard Model with bilinear It-Parity violation provides a predictive scheme for neutrino masses which can account for the observed atmospheric and solar neutrino anomalies. De spite the smallness of neutrino masses R-parity violation is observable at present and future high-energy colliders, providing an unambiguous cross-check of the model.
1. Introduction The announcement of high statistics atmospheric neutrino data by the SuperKamiokande collaboration * has confirmed the deficit of muon neutri nos, especially at small zenith angles, opening a new era in neutrino physics. Although there may be alternative solutions of the atmospheric neutrino anomaly 2 it is fair to say that the simplest interpretation of the data is in terms of v^ to vT flavor oscillations with maximal mixing. This excludes a large mixing among vT and ve x, in agreement also with the CHOOZ reactor data 3 . On the other hand the persistent disagreement between solar neutrino data and theoretical expectations 4 has been a long-standing problem in physics. Recent solar neutrino data 5 are consistent with both vacuum oscillations and MSW conversions. In the latter case one can have either the small or the large mixing angle solutions, with the latter being clearly preferred 2 . Many attempts have appeared in the literature to explain the data. Here we review recent results 6 obtained in a model 7 which is a simple extension of the MSSM with bilinear R-parity violation (BRpV). This model, despite *This work is supported by the European Commission RTN network HPRN-CT-200000148.
233
234
being a minimal extension of the MSSM, can explain the solar and atmo spheric neutrino data. Its most attractive feature is that it gives definite predictions for accelerator physics for the same range of parameters that explain the neutrino data. 2. Bilinear R-Parity Violation 2.1. The
Model
Since BRpV SUSY has been discussed in the literature several times 6 ' 7 , 8 we will repeat only the main features of the model here. We will follow the notation of 6 . The simplest bilinear IjLp model (we call it the $ p MSSM) is characterized by three additional terms in the superpotential W
where
WMSSM
= WMSSM
+ W$p
(1)
is the ordinary superpotential of the MSSM and W$p = eiUHu.
(2)
These bilinear terms, together with the corresponding terms in the soft SUSY breaking part of the Lagrangian, £soft — C<soft + Bi€iLiHu
(3)
define the minimal model, which we will adopt throughout this paper. The appearance of the lepton number violating terms in Eq. (3) leads in general to non-zero vacuum expectation values for the scalar neutrinos (z>j), called Vi in the rest of this paper, in addition to the VEVs vu and VD of the MSSM Higgs fields H° and H°. Together with the bilinear parameters e, the Vi induce mixing between various particles which in the MSSM are distinguished (only) by lepton number (or R-parity). Mixing between the neutrinos and the neutralinos of the MSSM generates a non-zero mass for one specific linear superposition of the three neutrino flavor states of the model at tree-level while 1-loop corrections provide mass for the remaining two neutrino states, see 6 . 3. Neutrino Masses and Mixings 3.1. Tree Level Neutral
Fermion
Mass
Matrix
0T
In the basis ip = ( —i\', — %}?,H\,H.\,ve,vil^vT) mass terms in the Lagrangian are given by Cm = -1-(^)TMN^
+ h.c.
the neutral fermions
(4)
235
where the neutralino/neutrino mass matrix is Mxa mT
MN
m
(5)
0
with 0
Mi 1
V
\g'vu
M2
\gvd
%gvd
0
-/i
-fj,
0
0
Mxo =
-\g'va
r
-\gvu
;
1
»
~29 d
-\gvu
\g'vu
-i
ax m =
(6)
a.2 a-3
where a^ = (— \g'vi, |j,0,ej). This neutralino/neutrino mass matrix is diagonalized by Af* MNN~ 3.2. Approximate
1
= diag(m x o, mxo, mxo, mxo, m„, ,mv^,mV3) Diagonalization
(7)
at Tree Level
If the $ p parameters are small, then we can block-diagonalize MN approx imately to the form diag(m e //,-M x o)
= —m ■ -M„0 m
Ml92 + M2g'2 4det(Mxo)
I A? A e-a-fi e A u Aa. eA i\.r \ e
AeAM
T
A^ A^Ar
\^AeAT AMAr
A
2 T
(8) )
The matrices N and Vv diagonalize Mxo and m e / / N*MxoN^
= diag(mxq)
;
mv = Tr(meff)
=
VjmeffVv
=
diag(0,0,mv),
(9)
where
3.3. Approximate
Mxg2 + M2g'2 2 |A| 4det(Mxo)
Formulas for
(10)
1-Loop
3.3.1. The masses Looking at the numerical results 6 we found that the most important con tribution came from the bottom-sbottom loop. To gain an analytical un derstanding of the results we expanded the exact results in the small ftp parameters. The result is 9 M„~c0
'AiAj | A2Ai ,A 3 Ai
AiA 2 A2A2 A3A2
AiA3\ / e i e j eie 2 e^sX A 2 A 3 j + cj £261 £262 £263 A3A3/ \ e 3 e i e3e2 e 3 e 3 /
(11)
236
where Ml92 + M2g'2 4 det(M x o)
3 1677^
.
h\ ixz
,2
rn
Diagonalization of the mass matrix gives 9 mVl=0 m
(13) m Sln 2 ^" ^/^2 l 0 § ^T 16-7T12 * m22 2 Mig +M2g' -2
,r.o |A
(14)
The formula for m ^ is the tree-level formula that we used to fix the scale of the atmospheric neutrinos by choosing |A|. Details of the derivation can be found in Ref. 9 where the second most important contribution, coming from the loop with charged Higgs/charged leptons, is also discussed. 3.3.2. The mixings The atmospheric angle is easily obtained in terms of the ratio A2/A3. For the solar angle in the same approximation we also get a simple formula 9 , 2e2 ■ (£2 + £3r that is also in very good agreement with the exact result. tan 2 0sol =
(16)
4. Results for the Solar and Atmospheric Neutrinos 4.1.
The
masses
The BRpV model produces a hierarchical mass spectrum for almost all choices of parameters. The largest mass can be estimated by the tree level value using Eq. (15). Correct A m 2 t m can be easily obtained by an appro priate choice of |A|. The mass scale for the solar neutrinos is generated at 1-loop level and therefore depends in a complicated way in the model parameters. However, in most cases the result of Eq. (14) is a good ap proximation and there is no problem in having both A m 2 i m and &m2solar set to the correct scales. 4.2.
The
mixings
Now we turn to the discussion of the mixing angles. We have found that if e 2 /|A|
237
results and the flavor composition of the 3rd mass eigenstate is approxi mately given by Ua3 « A a /|A|
(17)
As the atmospheric and reactor neutrino data tell us that v^ —> vT oscilla tions are preferred over i/M -> ve, we conclude that Ae < AM ~ A r
(18)
are required for BRpV to fit the data. This is sown in Fig. 1 a). We cannot get so easily maximal mixing for solar neutrinos, because in this case Ue3 would be too large contradicting the CHOOZ result as shown in Fig. 1 b).
Figure 1. a) Atmospheric angle as a function of | A p | / \ / A | + A?, b) t/g3 as a function of \Ae\/JA^ + A?. We have then two scenarios. In the first one, that we call the mSUGRA case, we have universal boundary conditions of the soft SUSY breaking terms. In this case we can show 6 that
r"r
^
Then from Fig. 1 b) and the CHOOZ constraint on U%3, we conclude that both ratios in Eq. (19) have to be small. Then from Fig. 2 we conclude that the only possibility is the small angle mixing solution for the solar neutrino problem. In the second scenario, which we call the MSSM case, we consider non-universal boundary conditions of the soft SUSY breaking terms. We
238
have shown that even a very small deviation from universality of the soft parameters at the GUT scale relaxes this constraint. In this case (20)
- * T -
Then we can have at the same time small J723 determined by A e /Ap as in Fig. 1 b) and large tan 2 (# so j a? .) determined by e e /e M as in Fig. 2 b).
Figure 2.
Solar angle as function of: a) | A e | / i / A 2 + A2- ; b)
ee/e^.
5. Probing Neutrino Mixing via Neutralino Decays If R-parity is broken, the neutralino is unstable and it will decay through the following channels: Xi ~> vivjv\t, Viqq, Vilt l^, ifqq', fij- It was showna in Ref. 10 , that the neutralino decays well inside the detectors and that the visible decay channels are quite large. This was fully discussed in Ref. 10 where it was shown that the ratios |A»/Aj| and \(-i/ej\ were very important in the choice of solutions for the neutrino mixing angles. What is exciting now, is that these ratios can be measured in accelerator experiments. In the left panel of Fig. 3 we show the ratio of branching ratios for semileptonic LSP decays into muons and taus: BR{\ —> /J.q'q)/BR(x —> rq'q) as function of tan 2 0atm- We can see that there is a strong correlation. The spread in this figure can in fact be explained by the fact that we do not know the SUSY parameters. This is illustrated in the right panel where we a
T h e relation of the neutrino parameters to the decays of the neutralino has also been considered in Ref. u .
239 considered that SUSY was already discovered with the following values for the parameters, M 2 = 120 GeV, fj, = 500 GeV, tan /3 = 5,m 0 = 500 GeV, A = - 5 0 0 GeV (21)
10
pq
10"
10~
510" 2 10"
0.5 1
tair
Figure 3. Ratios of semileptonic branching ratios as functions of tan 9atm- On the left for random SUSY values and on the right for the SUSY point of Eq. (21)
6. Probing Neutrino Mixing via Charged Lepton Decays If R-parity is broken the lightest supersymmetric particle (LSP) will decay. If the LSP decays then cosmological and astrophysical constraints on its nature no longer apply. Thus, within R-parity violating SUSY a priori any superparticle could be the LSP. We have studied 12 the case where a charged scalar lepton, most probably the scalar tau, is the LSP. The production and decays of f, as well as the decays of e and ft, and demonstrate that also for the case of charged sleptons as LSPs neutrino physics leads to definite predictions of various decay properties. This is shown in Figs. 4 and 5 7. Conclusions The Bilinear R-Parity Violation Model is a simple extension of the MSSM that leads to a very rich phenomenology. We have calculated the one-loop corrected masses and mixings for the neutrinos in a completely consistent way, including the RG equations and correctly minimizing the potential. We have shown that it is possible to get easily maximal mixing for the atmospheric neutrinos and both small and large angle MSW.
240
Figure 4. a)e + e —> 11 production cross section at a Linear Collider A/S = 0.8TeV, b) Charged slepton decay length at a linear collider with i / i = 0.8TeV.
Figure 5. Ratios of branching ratios for scalar tau decays versus tan #©. The left panel shown all data points, the right one refers only to data points with £2/63 restricted to the range [0.9,1.1].
We emphasize that the LSP decays inside the detectors, thus leading to a very different phenomenology than the MSSM. The LSP can be either the lightest neutralino, like in the MSSM, or a charged particle, must probably the lightest stau. In both cases we have shown that ratios of the branching ratios of the LSP can be correlated with the neutrino parameters. If the model is to explain solar and atmospheric neutrino problems many signals will arise at future colliders. These will probe the neutrino mixing parameters. Thus the model is easily falsifiable!
241
References 1. S. Fukuda et al. [Super-Kamiokande Collaboration], of Super-Kamiokande-I data," Phys. Lett. B 539 (2002) 179 [arXiv:hep-ex/0205075]. Phys. Rev. Lett. 8 1 , 1562 (1998) [arXiv:hep-ex/9807003]. 2. M. Maltoni, T. Schwetz, M. A. Tortola and J. W. Valle, atmospheric data," arXiv:hep-ph/0207227, Phys. Rev. D in press; M. C. Gonzalez-Garcia, M. Mal toni, C. Pena-Garay and J. W. F. Valle, Phys. Rev. D 63 (2001) 033005 [hep-ph/0009350]; 3. M. Apollonio et al, CHOOZ Coll., Phys. Lett. B 466 (1999) 415; F. Boehm et al, Palo Verde Coll., Phys. Rev. D 64 (2001) 112001 4. J. N. Bahcall, S. Basu and M. H. Pinsonneault, Phys. Lett. B 433, (1998) 1. 5. Q. R. Ahmad et al. [SNO Collaboration], constraints on neutrino mixing pa rameters," Phys. Rev. Lett. 89 (2002) 011302 [arXiv:nucl-ex/0204009]. 6. J. C. Romao, M. A. Diaz, M. Hirsch, W. Porod and J. W. Valle, Phys. Rev. D 61, 071703 (2000) [arXiv:hep-ph/9907499]; M. Hirsch, M. A. Diaz, W. Porod, J. C. Romao and J. W. Valle, Phys. Rev. D 62, 113008 (2000) [Erratum-ibid. D 65, 119901 (2002)] [arXiv:hep-ph/0004115]. 7. M. A. Diaz, J. C. Romao and J. W. Valle, Nucl. Phys. B 524, 23 (1998) [arXiv:hep-ph/9706315]. 8. F. de Campos, M. A. Garcia-Jareno, A. S. Joshipura, J. Rosiek and J. W. Valle, Nucl. Phys. B 451 (1995) 3 [arXiv:hep-ph/9502237]. A. G. Akeroyd, M. A. Diaz, J. Ferrandis, M. A. Garcia-Jareno and J. W. Valle, Nucl. Phys. B 529, 3 (1998) [arXiv:hep-ph/9707395]. T. Banks, Y. Grossman, E. Nardi and Y. Nir, Phys. Rev. D 52, 5319 (1995) [arXiv:hep-ph/9505248]. 9. M. Hirsch, M. A. Diaz, W. Porod, J. W. F. Valle and J. C. Romao, work in preparation. 10. W. Porod, M. Hirsch, J. Romao and J. W. Valle, Phys. Rev. D 6 3 (2001) 115004, [arXiv:hep-ph/0011248]. 11. B. Mukhopadhyaya, S. Roy and F. Vissani, Phys. Lett. B 443 (1998) 191, [arXiv:hep-ph/9808265]. 12. M. Hirsch, W. Porod, J. C. Romao and J. W. Valle, Phys. Rev. D. in press, [arXiv:hep-ph/0207334].
This page is intentionally left blank
Gravitational Waves and Tests of General Relativity
This page is intentionally left blank
RELATIVISTIC R-MODES IN SLOWLY ROTATING N E U T R O N STARS
SHIJUN YOSHIDA Centro Multidisciplinar de Astrofisica - CENTRA, Departamento de Fisica, Institute Superior Tecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: [email protected] UMIN LEE Astronomical
Institute, Tohoku University, Sendai E-mail: [email protected]
980-8578,
Japan
We investigate the properties of relativistic r-modes of slowly rotating neutron stars by using a relativistic version of the Cowling approximation. In our formalism, we take into account the influence of the Coriolis-like force on the stellar oscillations, but ignore the effects of the centrifugal-like force. For three neutron star models, we calculated the fundamental r-modes with I' = m = 2. We find that the oscil lation frequency
1. Introduction It was Andersson 1 and Friedman & Morsink2 who realized that the rmodes in rotating stars are unstable against gravitational radiation reac tions. Since then, a large number of papers have been published to explore the possible importance of the instability in neutron stars. For recent stud ies on the r-mode instability, see reviews by, e.g., Andersson & Kokkotas. 3 So far, most of the studies on the r-mode instability in neutron stars have been done within the framework of Newtonian dynamics. Since the rela tivistic factor can be as large as GM/c?R ~ 0.2 for neutron stars, where M and R are respectively the mass and the radius, however, the relativistic effects on the r-modes are essential.
245
246
From time to time, the effects of general relativity on the r-modes in neutron stars have been discussed in the context of the r-mode instabil ity. In the slow rotation approximation, Kojima 4 derived a second-order ordinary differential equation governing the relativistic r-modes, expanding the linearized Einstein equation to the first order of the rotation frequency of the star fi and assuming that the toroidal component of the displace ment vector is dominant and that the oscillation frequency is comparable to 0 . Kojima 4 showed that this equation has a singular property and allows a continuous part in the frequency spectrum of the r-modes. Re cently, Lockitch, Andersson, & Friedman 5 found that both discrete regular r-modes and continuous singular r-modes are allowed in Kojima's equation for uniform-density stars and suggested that the discrete regular r-modes are a relativistic counterpart of the Newtonian r-modes. Yoshida6 showed that discrete regular r-mode solutions to Kojima's equation exist only for some restricted ranges of the polytropic index and the relativistic factor for polytropic models, and that regular r-mode solutions do not exist for the typical ranges of the parameters appropriate for neutron stars. It may be instructive to turn our attention to a difference in mathe matical property between the Newtonian r-modes and the relativistic rmodes associated with Kojima's equation. In the case of the Newtonian r-modes, if we employ a perturbative method for r-modes in which the an gular frequency fi is regarded as a small expanding parameter, the radial eigenfunctions of the order of fi can be determined by solving a differential equation derived from the terms of the order of O 3 , which brings about the couplings between the oscillations and the buoyant force in the interior (e.g., [7]). In other words, there is no differential equation of the order of Q that determines the radial eigenfunction of the Newtonian r-modes. On the other hand, in the case of the general relativistic r-modes derived from Kojima's equation, we do not have to take account of the rotational effects of the order of Cl3 to obtain the radial eigenfunctions of the order of fi. That is, the eigenfrequency and eigenfunction of the r-modes are both determined by a differential equation (i.e., Kojima's equation) derived from the terms of the order of Q. We think that this is an essential difference be tween the Newtonian r-modes and the relativistic r-modes associated with Kojima's equation. Considering that Kojima's equation can give no rela tivistic counterpart of the Newtonian r-mode, it is tempting to assume that some terms representing certain physical processes are missing in Kojima's original equation. On the analogy of the r-modes in Newtonian dynam ics, we think that the buoyant force in the interior plays an essential role
247
in obtaining a relativistic counterpart of the Newtonian r-modes and that the terms due to the buoyant force will appear when the rotational effects higher than the first order of 0, are included. In this paper, we calculate relativistic r-modes by taking account of the effects of the buoyant force in a relativistic version of the Cowling approximation, in which all the metric perturbations are omitted. In this paper, we use units in which c = G = 1, where c and G denote the velocity of light and the gravitational constant, respectively. 2. Formulation We consider slowly and uniformly rotating relativistic stars in equilibrium. If we take account of the rotational effects up to the first order of 0 , the geometry in the stars can be described by the following line element (see, e.g., [8]): ds2 = ga0dxadxP
= - e2v^dt2
+ e2X^dr2
+ v2dB2 + r2 sin2 Ody2
- 2ui(r) r2 sin2 Odtdip.
(1)
The fluid four-velocity in a rotating star is given by ua = 7 (r, 6) (ta + {l<pa)= e~v^
(ta + Q <pa),
(2)
where ta and <pa stand for the timelike and rotational Killing vectors, re spectively. In order to simplify the problem, we employ a relativistic version of the Cowling approximation, in which all the metric perturbations are omit ted in the pulsation equations. 9 The relativistic Cowling approximation is accurate enough for the / - and p-modes. 10 ' 11 The relativistic Cowling approximation is a good approximation for determining the oscillation fre quency for oscillation modes in which the fluid motions are dominating over the metric fluctuations. Therefore, it is justified to employ the relativis tic Cowling approximation for the r-modes, for which the fluid motion is dominating. Since we are interested in pulsations of stationary rotating stars, we can assume that all the perturbed quantities have time and azimuthal de pendence given by e " r t + i m *', where m is an integer and a is a frequency measured by an inertial observer at spatial infinity. In this study, the adiabatic condition is adapted for the pulsation. Under those assumptions, we can obtain the perturbed energy equation, -VahC)
+ -^(6p
+ <;ttVap) = 0,
(3)
248
and the perturbed momentum equation,
\p + p p + pj where p and p mean respectively the pressure and mass-energy density, and Aa is the relativistic Schwarzschild discriminant, defined by Aa = —— V a / o - — Vap, p +p Tp and r is the adiabatic index, defined as
r=
p
-±r(d/)
(5)
.
(6)
P \dPJad a Here, ( is the Lagrangian displacement, 5p stands for the Eulerian change of the pressure, q% = 5^ + uaup, and a is the frequency defined in the corotating frame as a = a + mQ,. On the analogy between general relativity and Newtonian gravity, the second term on the left hand side of equation (4) is interpreted as a rel ativistic counterpart of the Coriolis force. In our formulation, the terms due to the Coriolis-like force are included in the perturbation equations, but the terms due to the centrifugal-like force, which are proportional to Cl2/(GM/R3), are all ignored. In the Newtonian theory of oscillations, this approximation is justified for low-frequency modes satisfying the conditions 2 3 |20/CT| > 1 and Q, /(GM/R ) < l. 12 - 13 In this study, the eigenfunctions are expanded in terms of spherical harmonic functions Ylm(6, tp) with different values of I for a given m as follows: OO
c
P
P
+v
^
^
OO
=^6Ui(r)Yr(9>V>)eM,
l>\m\
l,V>\m\
C = r £
S^r)^™(0,
, (7)
l>\m\
K
r
J
l,l'>\m\
where I = \m\ + 2k + 1 and /' = / - 1, where k = 0, 1, 2 • • -,14 Due to this expansion, we obtain an infinite system of coupled ordinary differential equations for the expanded coefficients. The details of our basic equations
249
are given in [15]. For numerical calculations, the infinite set of ordinary differential equations are truncated to be a finite set by discarding all the expanding coefficients associated with / larger than lmaK, the value of which is determined so that the eigenfrequency and the eigenfunctions are well converged as / m a x increases. 16 3. r-Modes of Neutron Star Models The neutron star models that we use in this paper are taken from the evolutionary sequences for cooling neutron stars calculated by Richardson et al., 17 in which the envelope structure is constructed by following Gudmundsson, Pethick & Epstein. 18 These models are composed of a fluid core, a solid crust and a surface fluid ocean. In order to avoid the complexity in the modal properties of relativistic r-modes brought about by the existence of the solid crust in the models, 13 we treat the whole interior of the models as a fluid in the following modal analysis. Table 1. Neutron Star Models Model NS05T7 NS05T8 NS13T8
M(MQ) 0.503 0.503 1.326
GM/(c2R)
R (km)
p c (g cm 3 )
TC(K)
9.839
9.44 x 10 1 4
1.03 x 10 7
7.54 x 1 0 " 2
9.785
9.44 x 10
14
9.76 x 10 7
7.59 x 1 0 - 2
3.63 x 10
15
8
2.49 X 1 0 - 1
7.853
1.05 x 10
We computed frequency spectra of r-modes for the neutron star models called NS05T7, NS05T8, and NS13T8. 19 The physical properties such as the total mass M, the radius R, the central density pc, the central temperature Tc, and the relativistic factor GM/c2R are summarized in Table 1. In Figure 1, scaled eigenfrequencies K = CT/O of the r-modes of the three neutron star models are given as functions of Q = 0,/^/GM/R3 for m = 2 mode. Here only the fundamental r-modes with V = m are considered because they are most important for the r-mode instability of neutron stars. From this figure, we can see that the scaled eigenfrequency K is almost constant as fi varies. In other words, the relation o ~ K,O$1 is a good approximation for the fundamental r-modes with I' — m, where KQ is a constant. Comparing the two frequency curves, which nearly overlap each other, for the models NS05T7 and NS05T8, it is found that the detailed interior structure of the stars, such as the temperature distribution T(r), does not strongly affect the frequency of the fundamental r-modes with I' — m. This modal property is the same as that found for the fundamental /' =
250
1 ' ' ' ' 1' ' ' ' 0.6
-
-
-
0.2
NS05T7 - ._ NS05T8 - _NS13T8
l'=m=2 r—mode -
, , , ! , , , , 0.2 0.1 1
i
t
i
i
1
0.3
i
i
i
i
0.4
n Figure 1. Scaled frequencies K = a/Q of the fundamental r-modes in the neutron star models NS05T7, NS05T8, and NS13T8, plotted as functions of fi = Sl/fGM/fl 3 ) 1 / 2 for I' = m = 2. Note that the two r-mode frequency curves for the models NS05T7 and NS05T8 overlap each other almost completely.
m r-modes in Newtonian dynamics. 20 On the other hand, comparing the frequency curves for the models NS05T7 (NS05T8) and NS13T8, we note that the r-mode frequency of relativistic stars is strongly dependent on the relativistic factor GM/c2R of the models. This is because the values of the effective rotation frequency Q = Q—OJ in the interior are strongly influenced by the relativistic factor. Similar behavior of the GM/c2R dependence of the r-mode frequency has been found in the analysis of Kojima's equation. 6 Our numerical procedure shows that the r-modes obtained here are isolated and discrete eigenmodes. 4. Conclusion We have investigated the properties of relativistic r-modes in slowly rotating neutron stars in the relativistic Cowling approximation by taking account of higher order effects of rotation than the first order of O. In our formalism, only the influence of the Coriolis-like force on the oscillations is taken into account, and no effects of the centrifugal-like force are considered. We obtain the fundamental r-modes associated with /' = m = 2 for three
251
neutron star models. We find that the fundamental r-mode frequencies are given in a good approximation by a K> KQ£1. The proportional coefficient Ko is only weakly dependent on Q, but strongly depends on the relativistic parameter GM/c2R. All the r-modes obtained in this paper are discrete modes with distinct regular eigenfunctions. Our results suggest that the appearance of singular r-mode solutions can be avoided by extending the original Kojima's equation so that terms due to the buoyant force in the stellar interior are included. As discussed recently by Yoshida & Putamase, 21 we believe that the answer to the ques tion of whether r-mode oscillations in uniformly rotating relativistic stars show true singular behavior may be given by solving the forth-order ordi nary differential equation derived by Kojima & Hosonuma 22 for r-modes. Verification of this possibility remains as a future study. References 1. 2. 3. 4. 5.
N. Andersson, Astrophys. J. 502, 708 (1998). J. L. Friedman and S. M. Morsink, Astrophys. J. 502, 714 (1998). N. Andersson and K. D. Kokkotas, Int. J. Mod. Phys. D10, 381 (2001). Y. Kojima, Mon. Not. R. Astron. Soc. 293, 49 (1998). K. H. Lockitch, N. Andersson and J. L. Friedman, Phys. Rev. D 63, 024019 (2001). 6. S. Yoshida, Astrophys. J. 558, 263 (2001). 7. H. Saio, Astrophys. J. 256, 717 (1982). 8. K. S. Thorne, in General Relativity and Cosmology, edited by R. K. Sachs (Academic Press, New York, 1971), p. 237. 9. P. N. McDermott, H. M. Van Horn and J. F. Scholl, Astrophys. J. 268, 837 (1983). 10. L. Lindblom and R. J. Splinter, Astrophys. J. 348, 198 (1990). 11. S. Yoshida and Y. Kojima, Mon. Not. R. Astron. Soc. 289 , 117 (1997). 12. U. Lee and H. Saio, Mon. Not. R. Astron. Soc. 221, 365 (1986). 13. S. Yoshida and U. Lee, Astrophys. J. 546, 1121 (2001). 14. K. S. Thorne, Rev. Mod. Phys. 52, 299 (1980). 15. S. Yoshida and U. Lee, Astrophys. J. 567, 1112 (2002). 16. S. Yoshida and U. Lee, Astrophys. J. 529, 997 (2000). 17. M. B. Richardson, H. M. Van Horn, K. F. RatclifF and R. C. Malone, Astro phys. J. 255, 624 (1982). 18. E. H. Gudmundsson, C. J. Pethick and R. I. Epstein, Astrophys. J. 272, 286 (1983). 19. P. N. McDermott, H. M. Van Horn and C. J. Hansen, Astrophys. J. 325, 725 (1988). 20. S. Yoshida and U. Lee, Astrophys. J. Suppl. 129, 353 (2000). 21. S. Yoshida and T. Futamase, Phys. Rev. D 64, 123001 (2001). 22. Y. Kojima and M. Hosonuma, Phys. Rev. D 62, 044006 (2000).
COLLISION OF HIGHLY RELATIVISTIC PARTICLES W I T H BLACK HOLES: T H E GRAVITATIONAL R A D I A T I O N GENERATED
VITOR CARDOSO CENTRA,
Departamento de Fisica, Instituto Superior Av. Rovisco Pais 1, 1096 Lisboa, Portugal, E-mail: [email protected]
CENTRA,
Departamento de Fisica, Instituto Superior Av. Rovisco Pats 1, 1096 Lisboa, Portugal, E-mail: [email protected]
Tecnico,
J O S E P. S. L E M O S Tecnico,
We consider the high-speed collision of a particle with a black hole, and compute the gravitational wave spectra and total radiated energy, for both rotating and non-rotating holes. Some interesting properties of the energy spectrum characterize these processes, such as universal zero frequency limit and a power-law dependence on the multipole index. Making the extrapolation to the two black hole collision at the speed of light case, we conclude that this is the most efficient gravitational wave generator in the Universe.
1. Introduction Gravitational effects propagate at the speed of light, according to General Relativity, and so the production of gravitational waves in spacetime, is inevitable whenever masses accelerate. This was recognized by Einstein himself long ago, through the quadrupole formula, expressing the rate of loss of energy in a system to gravitational waves, as a function of its energy momentum tensor. The recognition that there are extremely powerful astrophysical sources out there in the Universe led one to the understanding that gravitational waves may play an important role in the evolution of such systems 1'2. Thus the direct observation of gravitational waves would open up a new window to the Universe, a window from which one could look directly into the heart of matter. This has motivated a world wide effort not only to understand the basic properties of gravitational waves 1,2,3 ] but
252
253
also to develop gravitational wave detectors, some of them already operat ing 4 ' 5 . Among the most powerful sources of gravitational waves are black hole-black hole collisions, but these may be rare events in the Universe. Recently, however, the hypothesis that black holes could be produced at the Large Hadron Collider (LHC) at CERN has been put forward 6 , in so called TeV-scale gravity scenarios. In such scenarios, the hierarchy problem is solved by postulating the existence of n extra dimensions, sub-millimeter sized, such that the 4 + n Planck scale is equal to the weak scale ~ ITeV. If TeV-scale gravity is indeed correct, then one could manufacture black holes at the LHC. The ability to produce black holes would change the status of black hole collisions from a rare event into a human controlled one. This calls for accurate predictions of gravitational wave spectra and gravitational energy emitted during black hole formation from the high speed encounter of two particles. This investigation was first carried out by D'Eath and Payne 7 by doing a perturbation expansion around the Aichelburg-Sexl metric, describing a boosted Schwarzschild black hole. Their computation was only valid for non-rotating black holes and it seems quite difficult to extend their methods to include for spinning black holes. Here we shall use perturbation methods as first developed by Regge and Wheeler 8 and Zerilli 9 to study these events, as was done quite recently by Cardoso and Lemos 10 . This approach allows one to also study spinning black holes, which is of great importance, since if one forms black holes at the LHC, they will most probably be rotating ones (the chance for having a zero impact parameter is vanishingly small).
2. Basic Formalism Since the mathematical formalism for this problem has been thoroughly exploited over the years, we will just outline the procedure. Treating the particle as a perturbation, we write the metric functions for this spacetime, black hole + infalling particle, as
ffa6(**) = ffi°V) + M**) >
(!)
where the metric gab{x") is the background metric, (given by some known solution of Einstein's equations), which we now specialize to the Schwarzschild metric fir2
ds2 = -f(r)di2
+ -jfT+
r2(d92 + sin2 Odtf),
(2)
254
where f(r) = 1 — 2M/r. Also, hab{xl/) is a small perturbation, induced by the test particle, which is described by the stress energy tensor = -^YNjdX54(x--z(X))^^.
T^
(3)
Here, zv is the trajectory of the particle along the world-line, parametrized by an affine parameter A (the proper time in the case of a massive parti cle) , /J is the mass of the particle and S4 is the four-dimensional Dirac delta function. To proceed, we decompose Einstein's equations Gab = 8nTah in tensorial spherical harmonics and specialize to the Regge-Wheeler 8 gauge. For our case, in which the particle falls straight in, only even parity pertur bations survive. Finally, following Zerilli's 9 prescription, we can combine the metric functions hah and their derivatives to arrive at a wavefunction Z (a function of the time t and radial r coordinates only) whose evolution can be followed by the wave equation d
2
^'
0
+ K 2 - V(r)] Z(w,r) = (1 - 2M/r)S,
(4)
Here, the I—dependent potential V is given by _ -f M [ 2ff2 ( a + 1 ) r 3 + 6 ( j 2 r 2 M + ISffrM2 + 18M 3 ] - > ~~ r 3 (3M + a r ) 2 '
V{ r
(5)
an(1 where a = *^ e t o r t ° i s e coordinate r* is defined as ^- = f(r). 2 The passage from the time variable t to the frequency u has been achieved through a Fourier transform, Z(ui,r) = f 9 J 1 / 2 J ^ elutZ(t,r)dt. For highly relativistic particles the source term S is 10
4Jtie-iur~y(4l
+ 2)1/*a
w(3M + ar)2
'
[ J
where 7 is the Lorentz factor. To get the energy spectra, we use dE du
1 (l + 2)\ - c^|Z(W,r)|2,
32TT (Z -
2)!
and to reconstructe the wavefunction Z(t,r) transform
(7)
one uses the inverse Fourier
To find Z(w,r) from the differential equation (4), one uses a Green's func tion technique with appropriate boundary conditions (the reader is referred
255
to Cardoso and Lemos 10 and references therein). The procedure just out lined is valid only for non-rotating black holes. If the hole is rotating a different approach is needed. A sucessful formalism is due to Teukolsky 11 , further exploited by Sasaki and Nakamura 12 . We have used 10 the Teukolsky-Sasaki-Nakamura formalism to handle collisions between parti cles and spinning black holes. In the following we shall only present the results. 3. Numerical Results 3.1. Non Rotating
Black
Holes
The results for the wavefunction Z(t,r) for non-rotating black holes as a function of the retarded time u = £ — r« are shown in Fig.l, for the first radiatable multipoles, ^ = 2,3 and 4. As expected from the work of Ruffini 13
Z 1=2
z
1.6 0.8 0 -0.8 -40
-20
0
20
40
-20
0
20
40
-20
0 u/M
20
40
0.2 0 i=3 - 0 . 2 -0.4 -40
0.1 Z 1=4
Q
-0.1 -40
Figure 1. Waveforms for the three lowest radiatable multipoles, for a massless parti cle falling from infinity into a Schwarzschild black hole. Here, the wavefunction Z is measured in units of /J.'j.
and Ferrari and Ruffini 14 , the wavefunction is not zero at very early times, reflecting the fact that the particle begins to fall with non zero velocity. At late times, the I = 2 (for example) signal is dominated by quasinormal ringing with frequency w ~ 0.35/M, the lowest quasinormal frequency for this spacetime 15 . The energy spectra is shown in Figure 2, for the four
256
0.3
0.9
Figure 2. The energy spectra for the three lowest radiative multipoles, for a massless particle falling from infinity into a Schwarzschild black hole.
lowest radiative multipoles. First, as expected from Smarr's work 16 , the spectra is flat, up to a certain critical value of the frequency, after which it rapidly decreases to zero. This (/-dependent) critical frequency is well approximated, for each /-pole, by the fundamental quasinormal frequency. In Table 1, we list the zero frequency limit (ZFL) for the first ten lowest radiative multipoles. Table 1. The zero frequency limit (ZFL) for the ten lowest radiative multipoles. 1
ZFL(X-^T-)
I
Z F L f x ^ - ")
2
0.265
7
0.0068
3
0.075
0.0043
4 5 6
0.032
8 9 10 11
0.0166 0.01
0.003 0.0023 0.0017
For high values of the angular quantum number I, a good fit to our numerical data is 2-25
(dEA
= -JTV
2 2
(9)
7
We therefore estimate the zero ZFL as (iMl)
- M
I dw )ui=0 ~
=U
(^Ml)
1 4- 12,25 2 2
Z-W = 2 l du )u=o\ 2
2
= 0.424V7
+
2 12 2 ^
'
(10)
257
To calculate the total energy radiated to infinity, we proceed as follows: as we said, the spectra goes as 2.25/Z3 as long as w < W/QJV, where W/QJV is the lowest quasinormal frequency for that Z-pole. For to > UIIQN, dE/du ~ 0 (in fact, our numerical data shows that dE/dtu ~ e -27auM , with a a factor of order unity, for ui > W/QAT). NOW, from the work of Ferrari and Mashhoon 17 and Schutz and Will 18 , one knows that for large /, U>IQN ~ X^UM ■ Therefore, for large I the energy radiated to infinity in each multipole is A E
_ 2.25(i + 1/2)
i - ^ m p
M
V
jjT'
(11)
and an estimate to the total energy radiated is then AE = ] T AE, = 0.262
3.2. The general case of rotating
M
(12)
holes
Recent studies 10 on high energy collisions of point particles with black holes point to the existence of some characteristic features of these pro cesses, to wit: (i) The spectra and waveform largely depend upon the lowest quasinormal frequency of the spacetime under consideration; (ii) there is a (non-vanishing) zero frequency limit (ZFL) for the spectra, ^ , and it seems to be independent of the spin of the colliding particles (for low-energy collisions the ZFL is zero); (hi) The energy radiated in each multipole has a power-law dependence (rather than exponential for low-energy collisions). Table 2. Power-law dependence of the energy radiated in each multipole I, here shown for some values of o, the rotation parameter. We write AEi for the energy emitted for each I and ABtot for the total energy radiated away. a M
AE, = c x l~-b c b ^ t o t ^
0.999
0.61
1.666
0.69
0.8
0.446
1.856
0.36
0.5
0.375
1.88
0.29
0
0.4
2
0.26
This study reinforces all these aspects. The existence of a non-vanishing ZFL is evident, but the most important in this regard is that the ZFL is
258
exactly the same, whether the black hole is spinning or not, or whether the particle is falling along the equator or along the symmetry axis. In fact, our numerical results show that, up to the numerical error of about 1% the ZFL is given by Table 1 (the exact value is given by Smarr 1 6 ) , and this holds for highly relativistic particles falling along the equator, along the symmetry axis, or simply falling into Schwarzschild black holes. The /-dependence of the energy radiated is a power-law; in fact for large I we find (see Table 2) A
^ = ° - 6 1 ] ^ r 6 6 6 ' a = 0-999M-
( 13 )
Such a power-law dependence seems to be universal for high energy colli sions. Together with the universality of the ZFL this is one of the most important results borne out of our numerical studies. The exponent of I in (13) depends on the rotation parameter. As a decreases, the expo nent increases monotonically, until it reaches the Schwarzschild value of 2 {AEi ~ ^ ) which was also found for particles falling along the symmetry axis of a Kerr hole. In Table 2 we show the values of the exponent, as well as the total energy radiated, for some values of the rotation parameter a. This power-law dependence and our numerical results allow us to infer that the total energy radiated is 2
AEtot
2
= 0 . 6 9 ^ - ^ , a = 0.999M.
(14)
This represents a considerable enhancement of the total radiated energy, in relation to the Schwarzschild case 10 (for which AEtot = 0.26^^^) or even to the infall along the symmetry axis 10 (for which AEtot = O.Sl1^-). 2
2
Again, the energy carried by the / = 2 mode (AEi--2 — 0 . 2 ^ ^ , o = 0.999M) is much less than the total radiated energy. Let us now consider, using these results, the collision at nearly the speed of light between a Schwarzschild and a Kerr black hole, along the equatorial plane. We have argued in previous papers 10 that the naive extrapolation jU. —> M may give sensible results, so let's pursue that idea here. We obtain an efficiency of 34.5% for gravitational wave generation, a remarkable in crease relative to the Schwarzschild-Schwarzschild collision. Now, the Area Theorem gives an upper limit of 38.7% so we may conclude with two re marks: (a) these perturbative methods pass the Area Theorem test; (b) should these results be correct (and everything points in that direction), we are facing the most energetic events in the Universe, with the amazing fraction of 34.5% of the rest mass beeing converted into gravitational waves.
259 Acknowledgments This work was partially funded by Fundagao p a r a a Ciencia e Tecnologia (FCT) through project P E S O / P R O / 2 0 0 0 / 4 0 1 4 . V.C. also acknowledges finantial support from F C T through P R A X I S XXI programme.
References 1. B. F. Schutz and F. Ricci, in Gravitational^ Waves, by I. Ciufolini et al (Edi tors), (Institute of Physics Publishing, Bristol, 2001). 2. N. Andersson, this volume. 3. K. S. Thome, Rev. Mod. Phys, 52, 299 (1980); 4. K. Danzmann et al., in Gravitational Wave Experiments, eds. E. Coccia, G. Pizzella and F. Ronga (World Scientific, Singapore, 1995). 5. A. Abramovici et al., Science 256, 325 (1992). 6. S. Dimopoulos, and G. Landsberg, Phys. Rev. Lett. 87,161602 (2001);M. Cavaglia, hep-ph/0210296; 7. P. D. D'Eath and P. N. Payne, Phys. Rev. D46, 658 (1992); Phys. Rev. D46, 675 (1992); Phys. Rev. D46, 694 (1992). 8. T. Regge, J. A. Wheeler, Phys. Rev. 108, 1063 (1957). 9. F. Zeril\i,Phys. Rev. Lett. 24, 737(1970); Phys. Rev. D 2 , 2141 (1970). 10. V. Cardoso and J. P. S. Lemos, Phys. Lett. B 538, 1 (2002); V. Cardoso and J. P. S. Lemos, Gen. Rel. Gravitation, in press, (2002); gr-qc/0207009; V. Cardoso and J. P. S. Lemos, submitted, gr-qc/0211094. 11. S. A. Teukolsky, Astrophys. J. 185, 635 (1973); 12. T. Nakamura and M. Sasaki, Phys. Lett. 89A, 68 (1982); M. Sasaki and T. Nakamura, Phys. Lett. 87A, 85 (1981); 13. R. Ruffini, Phys. Rev. D 7, 972 (1973). 14. V. Ferrari and R. Ruffini, Phys. Lett. B 98, 381 (1981). 15. S. Chandrasekhar, and S. Detweiler, Proc. R. Soc. London A 344, 441 (1975); 16. L. Smarr, Phys. Rev. D 15, 2069 (1977). 17. V. Ferrari and B. Mashhoon, Phys. Rev. Lett. 52, 1361 (1984). 18. B. F. Schutz and C. M. Will, Ap. J. 291, L33 (1985).
PAIR OF ACCELERATED BLACK HOLES I N A N A N T I - D E SITTER B A C K G R O U N D : T H E A D S C-METRIC
OSCAR J. C. DIAS CENTRA,
Departamento de Fisica, Instituto Superior Av. Rovisco Pais 1, 1096 Lisboa, Portugal, E-mail: [email protected]
CENTRA,
Departamento de Fisica, Instituto Superior Av. Rovisco Pais 1, 1096 Lisboa, Portugal, E-mail: [email protected]
Tecnico,
J O S E P. S. L E M O S Tecnico,
The anti-de Sitter C-metric (AdS C-metric) is characterized by a quite interesting new feature when compared with the C-metric in flat or de Sitter backgrounds. Indeed, contrarily to what happens in these two last exact solutions, the AdS Cmetric only describes a pair of accelerated black holes if the acceleration parameter satisfies A > l/£, where £ is the cosmological length. The two black holes cannot interact gravitationally and their acceleration is totally provided by the pressure exerted by a strut that pushes the black holes apart. Our analysis is based on the study of the causal structure, on the description of the solution in the AdS 4hyperboloid in a 5D Minkowski spacetime, and on the physics of the strut. We also comment the cases A — \jt and A < l/l that represent a single accelerated black hole in the AdS background. The C-metric is an exact solution which includes a radiative term and describes the final state of quantum pair creation of black holes.
1. Introduction The original C-metric has been found by Levi-Civita in his studies between 1917 and 1919. During the following decades, many authors have rediscov ered it and studied its mathematical properties. In 1963 Ehlers and Kundt 1 have classified degenerated static vacuum fields and put this Levi-Civita solution into the C slot of the table they constructed. From then onwards this solution has been called C-metric. This spacetime is stationary, axially symmetric, Petrov type D, and is an exact solution which includes a radia tive term. Although the C-metric had been studied from a mathematical point of view along the years, its physical interpretation remained unknown
260
261
until 1970 when Kinnersley and Walker 2 , in a pathbreaking work, have shown that the solution describes two uniformly accelerated black holes in opposite directions. Indeed, they noticed that the original solution was geodesically incomplete, and by defining new suitable coordinates they have analytically extended it and studied its causal structure. The solution has a conical singularity in one of its angular poles that was interpreted by them as due to the presence of a strut in between pushing the black holes away, or as two strings from infinity pulling in each one of the black holes. The strut or the strings lie along the symmetry axis and cause the accel eration of the black hole pair. This work also included for the first time the charged version of the C-metric. In an important development, Ernst in 1976 3 , through the employment of an appropriate transformation, has removed all the conical singularities of the charged C-metric by appending an external electromagnetic field. In this new exact Ernst solution the ac celeration of the pair of oppositely charged black holes is provided by the Lorentz force associated to the external field. The asymptotic properties of the C-metric were analyzed by Ashtekar and Dray 4 who have explicitly shown that gravitational radiation is emitted by this solution, and more over that the causal diagrams drawn in 2 were not quite accurate. The issue of physical interpretation of the C-metric has been recovered by Bonnor 5 , but now following a different approach. He transformed the C-metric into the Weyl form in which the solution represents a finite line source (that corresponds to the horizon of the black hole), a semi-infinite line mass (cor responding to a horizon associated with uniform accelerated motion) and a strut keeping the line sources apart. By applying a further transformation that enlarges this solution, Bonnor confirmed the physical interpretation given in 2 . The issue of the radiative properties of the C-metric has been recovered by Bicak 6 and Pravda and Pravdova 7 . In what concerns the cosmological C-metric, it has been introduced by Plebahski and Demiahski in 1976 8 , and the de Sitter (dS) case (A > 0) has been analyzed by Podolsky and Griffiths 9 and by Dias and Lemos 10 , whereas the anti-de Sitter (AdS) case (A < 0) has been studied, in special instances, by Emparan, Horowitz and Myers u and by Podolsky 12 . In general the C-metric (either flat, dS or AdS) describes a pair of accelerated black holes. Indeed, in the flat and dS backgrounds this is always the case. However, in an AdS background the situation is not so simple and depends on the relation between the acceleration A of the black holes and the cosmological length L Since the AdS C-metric presents such peculiar features it deserves a careful analysis. One can divide the study into three
262
cases, namely, A < l/l, A = l/l and A > l/l. The A < l/l case was the one analyzed by Podolsky 12 , and the A = l/l case has been investigated by Emparan, Horowitz and Myers n , which has acquired an important role since the authors have shown that, in the context of a lower dimensional Randall-Sundrum model, it describes the final state of gravitational collapse on the brane-world. Both cases, A < l/l and A = l/l, represent one single accelerated black hole. The A > l/l case has been analyzed by Dias and Lemos in 13 and describes a pair of accelerated black holes. It has been applied, in addition to the flat and dS cases, in pair creation of black holes 14 (see 15 for a review). In this contribution we briefly review the properties of the AdS C-metric. We analyze the causal structure of the solution, the description of the solu tion in the AdS 4-hyperboloid, and we comment on the strut's physics. For a fuller set of references on the C-metric, as well as for a detailed analysis of the AdS C-metric we ask the reader to see 13 . 2. General Properties In this section, we will briefly mention some general properties of the AdS C-metric that are well established. For details we ask the reader to see, e.g., 2 ' 1 3 . The AdS C-metric, i.e., the C-metric with positive cosmological constant A, has been obtained by Plebanski and Demiariski 8 . For zero rotation and zero NUT parameter, the gravitational field of the AdS Cmetric is given by (see 13 ) ds2 = [A(x + y)}-2(-Fdt2
+ F^dy2
+ G^dx2
+ Gdz2) ,
(1)
where F{y) = ( ^
- l) + y2 - 2mAy3 +
G(x) = 1 - x2 - 2mAx3 - q2A2x4 ,
q2A2y\ (2)
and the non-zero components of the electromagnetic vector potential, A^dx^, are given by At = —ey and Az = gx. This solution depends on four parameters namely, the cosmological length I2 = 3/|A|, A > 0 which is the acceleration of the black holes, and m and q which are interpreted as the ADM mass and electromagnetic charge of the non-accelerated black holes, respectively. In general, q2 = e2 + g2 with e and g being the electric and magnetic charges, respectively. The coordinates used in Eq. (1) hide the physical interpretation of the solution. To understand the physical properties of the solution we will
263
introduce progressively new coordinates more suitable to this propose, fol lowing the approach of Kinnersley and Walker 2 and Ashtekar and Dray 4 . Although the alternative approach of Bonnor 5 simplifies in a way the in terpretation, we cannot use it here since the cosmological constant prevents such a coordinate transformation into the Weyl form. We start by defining a coordinate r as r = [A(x + J/)] _1 which is inter preted as a radial coordinate. Indeed, calculating a curvature invariant of the metric, namely the Kretschmann scalar Rp.vafiWva®, we conclude that it is equal to 2A/£2 when the mass m and charge q are both zero. Moreover, when at least one of these parameters is not zero, the curvature invariant diverges at r = 0, revealing the presence of a curvature singularity. Finally, when we take the limit r —> oo, the curvature invariant approaches the expected value for a spacetime which is asymptotically AdS. We consider only the values of A, A, m, and q for which G{x) has at least two real roots, xs and xn (say) and we demand that x belongs to the range [xs, xn] where G{x) > 0. By doing this we guarantee that the metric has the correct signature (—I- + + ) [see Eq. (1)] and that the angular surfaces S (t =const and r =const) are compact. In these angular surfaces we now define two new coordinates, 8 = I " G~ll2dx,
= Z/K,
(3)
where 4> ranges between [0, 2n] and K is an arbitrary positive constant which will be discussed soon. The coordinate 8 ranges between the north pole, 8 = 8n = 0, and the south pole, 9 = 9S (not necessarily at n). When A = 0 or when both m = 0 and q = 0, Eq. (3) gives x = cos 8, G = 1 — x2 = sin2 8 and if we use the freedom to put K = 1, the metric restricted to E is given by do1 = r2(d92 + sin2 8dtp2)- This implies that in this case the angular surface is a sphere and justifies the label given to the new angular coordinates defined in Eq. (3). When m ^ 0 and q ^ 0, if we draw a small circle around the north or south pole, as the radius goes to zero, the limit circunference/radius is not 2TT. Indeed there is a deficit angle at the poles given by 5n/s = 27r[l — (/c/2)|da;G|a;n/a]. The value of K can be chosen in order to avoid a conical singularity at one of the poles but we cannot remove simultaneously the conical singularities at both poles (for a more detailed analysis see 1 3 ) . When A = 0, the general AdS C-metric, Eq. (1), reduces to the usual AdS black holes. Therefore, the parameters m and q are, respectively, the ADM mass and ADM electromagnetic charge of the non-accelerated black holes.
264
3.
Causal Structure
In this section we will briefly sketch the causal structure of the massive un charged (m > 0, q = 0) AdS C-metric. For a full and detailed analysis that leads to the Carter-Penrose diagrams that will be sketched here, as well as for the causal structure of the charged solutions, we ask the reader to see 13 . These causal diagrams will allow us to clearly identify the presence of two AdS black holes and to conclude that they cannot interact gravitationally. We first notice that, contrarily to what happens in the A > 0 background 2 10 ' where the causal structure and physical nature of the corresponding Cmetric is independent of the relation between the acceleration A and £, in the A < 0 case we must distinguish and analyze separately the cases A > l/£, A = l/£ and A < l/£ (see 1 3 ) . Here we will consider only the case A > l/£ and m > 0, q = 0 that describes a pair of uncharged accelerated black holes. For an analysis of the A
(a) South
Equator
Figure 1. Carter-Penrose diagrams of the A > l/t,mA < 3 - 3 / 2 , and q — 0 AdS C-metric. Case (a) describes the solution seen from the vicinity of the south pole, case (b) applies to the equatorial vicinity, and case (c) describes the solution seen from the vicinity of the north pole. The zigzag line represents a curvature singularity, an accelerated horizon is represented by TA, the Schwarzschild-like horizon is sketched as r+. r = 0 corresponds to y = +00 and r = +00 (/) corresponds to y = —x.
265
on the value of y (—x < y), the choice of the Kruskal coordinates (and therefore the Carter-Penrose diagrams) depends on the angular direction x we are looking at. In Fig. 1, three relevant cases are sketched.
4.
Physical Interpretation
The parameter A that is found in the AdS C-metric is interpreted as being an acceleration and the AdS C-metric with A > 1/1 describes a pair of black holes accelerating away from each other in an AdS background, while the AdS C-metric with A < l/£ describes a single accelerated black hole. In this section we will justify this statement. We first interpret case 1. Massless uncharged solution (m = 0, q = 0), which is the simplest, and then with the acquired knowledge we interpret case 2. Massive uncharged solution (in > 0, q = 0). Massless uncharged solution (m = 0, q — 0): It is useful to interpret the solution following two complementary descriptions, the 4-Dimensional one and the 5-Dimensional. This is done in detail in 13 . The AdS spacetime can be represented as the 4-hyperboloid, -(z0)2 + (z1)2 + (z2)2 + (z3)2 - (z4)2 = -I2, in the 5D Minkowski (with two timelike coordinates) embedding spacetime, ds2 = -(dz0)2 + (dz1)2 + (dz2)2 + (dz3)2 - (dz4)2 (see Fig. 2). Now, the massless uncharged AdS C-metric (m = 0, q = 0) is an AdS spacetime in disguise and there is a coordinate transformation 13 that defines an em bedding of the massless uncharged AdS C-metric into the 4-hyperboloid. Moreover 13 , when A > l/£ the origin of the radial coordinate moves in the 5D Minkowski describing two hyperbolic lines that lye on the AdS hyper boloid and that result from the intersection of this hyperboloid surface and the z 4 =constant> £ plane (see Fig. 2.(a)). Thus, the origin is subjected to a uniform acceleration, and consequently moves along a hyperbolic worldline in the 5D embedding space, describing a Rindler-like motion [see Fig. 2.(a)] that resembles the well-known hyperbolic trajectory, X2 — T2 = a~2, of an accelerated observer in Minkowski space. These two hyperbolas approach asymptotically the Rindler-like acceleration horizon (r^), so called because it is is absent when A = 0 and present even when A > l/£, m = 0 and 5 = 0. The global description on the AdS hyperboloid of the AdS C-metric origin when the acceleration A varies from +00 to zero is the following. When A = +00 the origin of the solution is represented in the hyperboloid by two mutual perpendicular straight null lines at 45° that result from the intersection of the hyperboloid surface and the z4 = £ plane (see Figs. 2 and 3). When A belongs to ]!/£, +oo[, the origin of the solution is represented
266
hyperbolic lines
^
A>l/£ Figure 2. (a) AdS 4-hyperboloid embedded in the 5D Minkowski spacetime with two timelike coordinates, z° and z 4 . The directions z2 and z 3 are suppressed. The two hyperbolic lines lying on the AdS hyperboloid result from the intersection of the hyperboloid surface with the z 4 = c o n s t a n t > £ plane. They describe the motion of the origin of the AdS C-metric with A > l/l. (b) The origin of the AdS C-metric with A < l/l moves in the hyperboloid along the circle with z 1 = c o n s t a n t < 0. When A = 0 this circle is at the plane z1 = 0 and has a radius I.
by two hyperbolic lines lying on the AdS hyperboloid and result from the intersection of hyperboloid and the z 4 =constant> £ plane (see Fig. 2.(a)). As the acceleration approaches the value A = l/£ the separation between the two hyperbolic lines increases. When A = l/£ the separation between the two hyperbolic lines becomes infinite and they collapse into two half circles which, on identifying the ends of the AdS hyperboloid at both infini ties, yields one full circle in the z° — zA plane at infinite z1. At this point we see that the value A = l/£ sets a transition stage between A > l/£ and A < l/l. When A belongs to ]0, l/£[ the origin of the solution is described again by a circle in the z° — z4 plane but now at a constant z1 < 0. As the acceleration approaches the value A = 0, the radius of this circle decreases and when A = 0 the circle has a radius with value £ and is at z1 = 0 (see Fig. 2.(b)) and we recover the usual AdS solution whose origin is at rest. Massive uncharged solution (m > 0, q = 0): When we add a mass to the solution the Carter-Penrose diagrams sketched in Figs. l.(b)-(c) allow us to conclude that each of these two simple hyperbolas r = 0 is replaced by the more complex structure that represents a Schwarzschild black hole with its spacelike curvature singularity and its horizon [these are represented by r + in the left and right regions of Figs. l.(b)-(c)]. So, the two accelerating points r = 0 have been replaced by two Schwarzschild black holes that approach asymptotically the Rindler-like acceleration horizon [represented by TA in the middle region of Figs. l.(b)-(c)]. The above description is
267
Figure 3. Schematic diagram representing the 5D hyperbolic motion of two uniformly accelerating massive charged black holes approaching asymptotically the Rindler-like accelerated horizon (/IA)- The inner and outer charged horizons are represented by h— and h+. The strut that connects the two black holes is represented by the zigzag lines. The north pole direction is represented by N and the south pole direction by S.
schematically sketched in Fig. 3 for the massive charged AdS C-metric. The Carter-Penrose diagram of case (a) of Fig. 1 indicates that an observer that is looking through an angular direction which is in the vicinity of the south pole does not see the acceleration horizon and notices the presence of a single black hole. This is in agreement with Fig. 3. Indeed, in this schematic figure, coordinates z° and z1 can be seen as Kruskal coordinates and we conclude that an observer, initially located at infinity (z1 = oo) and moving inwards into the black hole along the south pole, passes through the black hole horizons and hits eventually its curvature singularity. Therefore, he never has the opportunity of getting in contact with the acceleration horizon and with the second black hole. This is no longer true for an observer that moves into the black hole along an angular direction that is in the vicinity of the north pole. In Fig. 3 this observer would be between the two black holes, at one of the points of the z° < 0 semi-axis (say) and moving into the black hole. Clearly, this observer passes through the acceleration horizon before crossing the black hole horizons and hitting its curvature singularity. This description agrees with cases (b) and (c) of Fig. 1 which describe the solution along an angular direction which includes the equatorial plane [case (b)] as well as the north pole [case (c)]. Source of acceleration and radiative properties: Now, the value of the arbitrary parameter K introduced in Eq. (3) can be chosen in order to avoid a conical singularity at the south pole (6S = 0), leaving a conical singularity at the north pole (Sn < 0). This is associated to a strut that joins the two black holes along their north poles and provides their acceleration 13 . This
268
strut satisfies the relation p = —fj, > 0, where p and /J, are respectively its pressure and its mass density 13 . Alternatively, we can choose K such that avoids the deficit angle at the north pole (5n = 0) and leaves a conical singularity at the south pole (Ss > 0). This option leads to the presence of a string (with p — —\i < 0) that connects the two black holes along their south poles, and furnishes the acceleration. The C-metric is an exact solution that emits gravitational and electro magnetic radiation. In the flat background the Bondi news functions have been explicitly calculated in 4>6'7. In dS background these calculations have not been carried yet, in fact dS still lacks a peeling theorem. At this point a remark is relevant. From the analysis of the CarterPenrose diagrams of the AdS C-metric we conclude that the two black holes cannot interact gravitationally. For example, looking into Fig. l.(b) we conclude that a null ray sent from the vicinity of one of the black holes can never cross the acceleration horizon (r^) into the other black hole. So, if the two black holes cannot communicate through a null ray they cannot interact gravitationally. The black holes accelerate away from each other due only to the pressure of the strut. 5. Conclusions The C-metric solution for generic A has been used 14 ' 15 to describe the final state of the quantum process of pair creation of black holes, that once cre ated accelerate apart by an external field. In this context, we expect that the black hole pair created in the AdS background must have an accelera tion A > ljL Indeed, the AdS background is globally contracting with an acceleration precisely equal to l/£. Therefore, a pair of virtual black holes created in this background can only become real if the black hole acceler ation is greater than the contracting acceleration of the AdS background, otherwise, the annihilation is inevitable. The quantum process that might create the pair would be the gravitational analogue of the Schwinger pair production of charged particles in an external electromagnetic field. This would be one possible scenario to create two exactly equal black holes with the same acceleration that are described by the AdS C-metric solution with A > l/£ described in this contribution. References 1. J. Ehlers, W. Kundt, in: Gravitation: an introduction to current research, (ed. L. Witten, Wiley, New York, London, 1962).
269 2. W. Kinnersley, M. Walker, Phys. Rev. D 2, 1359 (1970). 3. F. J. Ernst, J. Math. Phys. 17, 515 (1976). 4. A. Ashtekar, T. Dray, Comm. Phys. 79, 581 (1981). 5. W. B. Bonnor, Gen. Rel. Grav. 15, 535 (1983). 6. J. Bicak, Proc. Roy. Soc. A 302, 201 (1968). 7. V. Pravda, A. Pravdova, Czech. J. Phys. 50, 333 (2000). 8. J. F. Plebariski, M. Demiariski, Annals of Phys. 98, 98 (1976). 9. J. Podolsky, J.B. Griffiths, Phys. Rev. D 63, 024006 (2001). 10. O. J. C. Dias, J. P. S. Lemos, Pair of accelerated black holes in a de Sitter background: the dS C-metric, Phys. Rev. D, in press; hep-th/0301046. 11. R. Emparan, G. T. Horowitz, R. C. Myers, JHEP 0001 007 (2000). 12. J. Podolsky, Czech. J. Phys. 52, 1 (2002). 13. O. J. C. Dias, J. P. S. Lemos, Pair of accelerated black holes in an anti-de Sitter background: the AdS C-metric, Phys. Rev. D, in press; hep-th/0210065. 14. H. F. Dowker, J. P. Gauntlett, D. A. Kastor, J. Traschen, Phys. Rev. D 49, 2909 (1994); S. W. Hawking, S. F. Ross, Phys. Rev. Lett. 75, 3382 (1995); R. Emparan, Phys. Rev. Lett. 75, 3386 (1995); R. B. Mann, S. F. Ross, Phys. Rev. D 52, 2254 (1995); R. Bousso, S. W. Hawking, Phys. Rev. D 54, 6312 (1996); R. B. Mann, Class. Quantum Grav. 14, L109 (1997). 15. O. J. C. Dias, Pair creation of particles and black holes in external fields, in Astronomy and Astrophysics: Recent developments, edited by J. P. S. Lemos, A. Mourao et al (World Scientific, Singapore, 2001); gr-qc/0011092.
TIMING THE PSR J2016+1947 BINARY SYSTEM: T E S T I N G T H E F U N D A M E N T A L A S S U M P T I O N OF GENERAL RELATIVITY
P A U L O C. F R E I R E Arecibo Observatory, HC OS Box 53995, Arecibo, Puerto Rico PR 00612, USA E-mail: [email protected] J O S E A. N A V A R R O Schlumberger, Cambridge Research High Cross, Madingley Road, Cambridge, CBS OEL, UK E-mail: [email protected]
Ltd.,
STUART B. ANDERSON 610 Milikan (bldg 32), Caltech 18-34, Pasadena CA 91125-3400, USA E-mail: [email protected]
Pulsars, and in particular pulsars in binary systems, have been used over the last 25 years to test some of the most fundamental predictions of gravitational theo ries. A good example was the precise measurement, by Russel Hulse and Joseph Taylor, at the Arecibo Observatory, of the motion of P S R B1913+16, a member of a double neutron-star system. These measurements showed that gravitational waves are not mere theoretical artifacts, and helped to confirm General Relativ ity as a valid theory of gravitation. An even more fundamental prediction of this theory is the Strong Equivalence Principle (SEP). This states that two objects under the influence of the same gravitational field will always have the same ac celeration, independent of their peculiar chemical compositions and gravitational binding energies. In this work, we describe how the measurement of the motion of the PSR J2016+1947 binary system (now being undertaken at Arecibo) can be used to place stringent constraints on SEP violation. These limits can be used to test alternative theories of gravitation, specially the bi-tensor bi-scalar theories, which are among the very few viable alternatives to General Relativity.
270
271
1. The Strong Equivalence Principle The Strong Equivalence Principle (SEP) is the most fundamental property of general relativity (GR). Like the Weak Equivalence principle (WEP), that led Einstein to elaborate GR, it requires the universality of free fall: acceleration of any object in an external gravitational field is independent of size or chemical composition of the object. However, SEP also requires the same accelerations under external fields for objects that have significantly different gravitational binding energies 8 . All theories of gravitation that describe gravity cLS
Effect
If the assumption of SEP is wrong, as postulated in many alternative the ories of gravitation (i.e., if |A| = |1 — mi/mG\ ^ 0, where mj and ma are the inertial and gravitational masses of an object) the accelerations in the same external field of two objects will not be exactly equal. The difference is not due to different chemical compositions (which would indicate WEP violation), but to the different gravitational binding energies. The effect will be similar to that of a neutral atom under a strong electric field, which causes different accelerations on the nucleus and electrons. The net effect is a polarization of the atom, which can be routinely detected in the laboratory through a detailed spectroscopical analysis. In atomic physics, this is known as the "Stark" effect, the resulting polarization is more intense for the electrons that lie further from the nucleus. The existence of an equivalent "Nordtvedt" effect for an astronomical binary is similarly due to the hypothetical difference of accelerations of the components. This causes a "polarization" of the binary, which is an increase in the eccentricity of the system along the direction of the external field 8 . As for an atom, the Nordtvedt effect becomes more pronounced when the separation between the two components is wider, as the external
272
field becomes more relevant compared to the effect of each component on the other.
2. Testing SEP using the Nordtvedt effect In the remainder of this paper, we will deal exclusively with SEP tests that use the Nordtvedt effect. However, we call the reader's attention to the fact that the existence of SEP-violating scalar fields predicted by the BransDicke and other theories of gravitation can also be tested using measure ments of compact double neutron-star binary systems, like PSR B1913+16 and PSR B1534+12. They have yielded negative results, i.e., no scalar fields have ever been detected with the present precision limits. A good review of the tests of gravitational theories made with pulsars can be found in Esposito-Farese (1999) and Bell (1999).
2.1. Weak Field
tests
The Earth and the Moon have different self-gravitational energies, due to their very different masses and gravitational fields. Do both of them fall with exactly the same acceleration in the gravitational field of the Sun? If not, we should be able to observe a small "polarization" of the orbit. The Brans-Dicke theory predicts a small violation of SEP, and therefore an observable Nordtvedt effect for the Earth-Moon system. In 1969 and 1971, the Apollo 11, 14 and 15 missions placed silica laser retro-reflectors at the Sea of Tranquility, Fra Mauro and Hadley Rille. In 1973, the Soviet lunar rover Lunakhod II placed a French-built set of retroreflectors near the Le Monnier crater. Since then, the Lunar Laser Ranging (LLR) program has been carried out at three locations: McDonald Obser vatory (Texas), Mt. Haleakala (Maui) and CERGA, in Grasse (France). The precision of each distance measurement has improved from about 2030 cm a few weeks after Apollo 11 to about 2-3 cm today. The results of 24 years of ranging are discussed in Williams, Newhall and Dickey (1996). Among other things, they have found that the fractional difference of the accelerations of the Earth and the Moon in the gravitational field of the Sun are smaller than 5 x 10~ 13 . If we attribute the hypothetical difference in accelerations to the difference in the self-gravitational energy of the two
273
objects a , we will have:
^ -
1
= A =
^XM?'
(1)
where UQ is the gravitational binding energy and rjw is a Post-Keplerian parameter that measures deviation from GR (it is zero if GR is true). The similarity of the Earth and the Moon's acceleration in the Sun's gravita tional field measured by the LLR experiment implies 77 = —0.0007± 0.0010, which is entirely consistent with GR. 2.2. Strong
Field
tests
The measurement of any given Post-Keplerian parameter like rj contains both a weak field contribution r/w and a strong field contribution rjs 5 : T] = i\w + Vs{ci + c2 + ...) + ...
(2)
where the c* are the compactness of the bodies involved: Ci
= M&-
(3)
Is there a strong-field component of rf! If so, GR is not the right de scription of gravitation. The tensor-bi-scalar theories of gravitation 4 are among the very few viable alternatives to General Relativity. They pre dict virtually the same results in the weak-field limit probed by weak-field experiments like LLR, they are therefore untestable in the Solar System. However, they predict a non-zero r/s for strong fields. A pulsar - white dwarf system, with the Galaxy generating the external field, provides the ideal laboratory to make such a measurement. Pulsars have very large gravitational binding energies of about —15% of the total mass (i.e., c\ ~ —0.15 - but this depends on the equation of state for cold matter at high densities). Their rotation is extremely regular and can be readily measured at radio wavelengths from their lighthouse-type radio pulses. This enables an extremely precise measurement of their motion and therefore of their orbital period and eccentricity. The white dwarf companion has, comparatively, a negligible gravitational binding energy, about 104 times smaller than for the neutron star. If there is any strongfield component of the acceleration, it should be felt just by the pulsar, a
We have to assume that this is indeed the case, as the precision of the W E P test in lab oratory experiments is only about 10~ 1 2 , see Will's (1993) description of the Braginsky and Panov experiment at the university of Moscow. There is a proposed space mission (STEP) that could improve the precision of the W E P test to one part in 10 1 7
274
hence the difference in acceleration compared to the white dwarf and the associated Nordtvedt effect. If the system has a large orbital separation (therefore, a large orbital period), it will be more sensitive to the Nordtvedt effect: as with atoms, the polarization is larger if the two components are well separated. If the system is formed with an orbit of small eccentricity, its orbit should retain that low eccentricity if the SEP is true. If SEP is violated, then, because of the Nordtvedt effect, the binary will inevitably become more and more eccentric as time passes. The effect cannot accumulate much more than a fraction of a Galactic orbital period, because the direction of the external field (the Galaxy) changes by 2w on that time-scale. On the other hand, if the pulsar was formed only a few million years ago, then there was simply no time for the Nordtvedt effect to affect the system. Wex (1997) shows that for an unambiguous interpretation of the binary's eccentricity, the age of the system must be considerably larger than its Galactic orbital period. Therefore, a pulsar - white dwarf binary with a long orbital period, a low eccentricity and a characteristic age larger than one Galactic orbit is the ideal probe of SEP violations in the strong-field domain.
3. The discovery of P S R J 2 0 1 6 + 1 9 4 7 A newly discovered 65-ms pulsar, PSR J2016+1947, is a member of a pul sar - white dwarf binary. It was discovered in a 430-MHz survey of the Intermediate Galactic Latitudes carried out at the Arecibo Observatory in the early 1990's with the 430-MHz Carriage House line feed. The details of this survey, and a list of the pulsars found in it, will be published elsewhere. This pulsar could only be confirmed after the Arecibo upgrade in an observation made in December 1997 with the same receiver and an analog filterbank, the Penn State Pulsar Machine, as a back-end. It has been observed from the end of 1997 to the end of 1999. From those observations, we have determined a minimum companion mass of 0.29 M© (assuming for the pulsar a mass of 1.35 M Q ) and an orbital period of 635 days. This is the third longest known orbital period for this type of system; the longest are those of PSR B0820+02 (1232 days 2 ) and PSR J0407+16 (680 days, now being timed at Arecibo 1 ) . The first system has not been significantly recycled by interaction with the progenitor of the companion white dwarf. The much shorter rotational periods of PSR J2016+1947 (65 ms) and of PSR J0407+16 (25 ms) are suggestive of extensive recycling.
275
It is not clear whether we have genuine phase connection, as there is a large gap in the observations in 1998 (see Fig. 1). The best solution, despite the possible rotation count ambiguities, yields the following parameters: Name a S P P DM PB X
To e U)
mc
PSR J2016+1947 20ft 16 m 57.43(2)s +19° 47' 51.5(8)" 64.9403881791(7) m 34 c m - 3 pc 635.0255(18) days 150.77419(18) s 50913.8(3) 0.001478(9) 95.79(14)° 0.29 M Q
A second full orbit must be observed to eliminate any possible rotation ambiguities. The timing parameters could be well outside the error mar gins given in the above table. We note that even if we were sure of having achieved phase connection, the pulsar's period derivative (P) has not yet been measured because of its strong correlation with the orbital parameters (orbital period PB, projected semi-major axis of the orbit of the pulsar in light seconds x = a/ sin i/c, time of passage through periastron TQ , eccen tricity e and longitude of periastron w). This correlation will disappear with the timing of a full second orbit. The determination of P will allow the determination of the characteristic age (r c = P/2P) of the pulsar. This is very important for constraining formation scenarios. If r c this is of the order of several times 109 years, then it will be clear that the pulsar has been recycled by accretion from the gas envelope of the companion star in the later phases of its evolution. The confirmation of the present value for the eccentricity is also very important. The theories that describe the recycling of pulsars postulate that the neutron star spin-up is due to accretion of matter from a companion star in the latter stages of its evolution, it is this companion that later becomes the white dwarf. Such theories predict the order of magnitude of the eccentricity as a function of the orbital period of the binary. For 635 days, the eccentricity should be of the order of 1 x 10~ 3 7 . The value derived so far is in excellent agreement with the prediction.
276
1998
2000
Figure 1. Timing residuals (predicted arrival time of a pulse at the telescope minus the prediction of our best model) of PSR J2016+1947 as a function or date. The unit of the vertical axis is one 65-ms rotation. The very small r.m.s. makes us believe that phase connection might have been achieved, but there is still a large gap in the d a t a between early 1998 and 1999, so up to three phase rotations are allowed by the fitting in t h a t interval. Times of periastron are marked with vertical dotted lines, there is no d a t a for practically 180 degrees after periastron.
4. Comparison of P S R J2016+1947 with other binary systems as a Nordtvedt laboratory. The figure-of-merit for a test of SEP violation is -Pg/e 2 . Among all the known binaries, this number is highest for the PSR B1800—27 binary sys tem. This system is inadequate for such tests, as the pulsar's age (r c = 300 Myr) is of the order of a single Galactic orbit. As mentioned in Sec.2.2, for an unambiguous interpretation of the low eccentricity of a binary system, the system's age has to be significantly larger than one Galactic orbit. Of all the previously known binary pulsars that pass the large r c cri terion, PSR J1713+0747 (r c = 8.5 Gyr) is the system for which P | / e is largest: 6.14 x 107 days 2 . If the large r c and low eccentricity of PSR J2016+1947 are con firmed, then its P | / e is 2.73 x 108 days 2 , or a factor of 4.5 better than
277 P S R J1713+0747 for S E P tests. T h e eccentricity of the P S R J0407+16 binary system is still unknown, t h a t system could be even better for this purpose. Using all the binary systems t h a t pass the large age criterion, a value of | A | < 0.004 was obtained 9 . Assuming again t h a t the pulsar's compactness is 0.15, this implies rj < 0.027. Given the more stringent limits for rjw from LLR, this implies rjs < 0.03. T h e inclusion of P S R J2016+1947 in t h a t ensemble of binary systems would enable a limit for A between 0.001 and 0.002. An interesting aspect of these limits is t h a t they will improve with the mere addition of binary systems to the ensemble being used, even if the individual Pg/e are not out standing 9 . Therefore, adding systems like P S R J2016+1947 and possibly P S R J0407+16 to this analysis is j u s t a step; the limits on A will improve geometrically with every new wide, low-eccentricity binary discovered. 5.
Conclusion
The upper limits of | A | can be used to impose fundamental constraints to any alternative theories of gravitation, in particular the bi-tensor biscalar theories. For a detailed discussion of this, see Esposito-Farese (1999). However, the determination of the equivalence of inertial and gravitational mass, particularly when very strong fields and very large self-gravitational energies are involved, is an i m p o r t a n t measurement in itself. It will forever stay as a fundamental constraint to our understanding of gravitation. References 1. The Arecibo Observatory Newsletter, March 2002 issue, (http://www.naic.edu/about/newslett/aonews.htm) 2. Arzoumanian, Z. 1995, PhD thesis, Princeton University 3. Bell, J. 1999, Pulsar Timing, General Relativity and the Internal Structure of Neutron Stars, 31 4. Esposito-Farese, G. 1999, Pulsar Timing, General Relativity and the Internal Structure of Neutron Stars, 13 5. Damour, T. and Esposito-Farese, G. 1992, Phys. Rev. D, 46, 4128 6. Hawking, S. W. fc Ellis, G. F. R. 1975, The Large Scale Structure of SpaceTime, 1975, Cambridge University Press. 7. Phinney, E. S. 1992, Phil. Trans. Roy. Soc. A, 341, 39 8. Will, C. M. 1993 Theory and Experiment in Gravitational Physics, Revised Edition, Cambridge University Press. 9. Wex, N. 1997, AfeA, 317, 976 10. Williams, J. G., Newhall, X. X., & Dickey, J. O. 1996, Phys. Rev. D, 53, 6730
This page is intentionally left blank
Supernovae and Dark Matter
This page is intentionally left blank
SUPERNOVAE A N D D A R K E N E R G Y
ARIEL GOOBAR Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden E-mail: [email protected] Observational cosmology is currently experiencing a revolution. The measurements of magnitude vs redshift of very distant type la supernovae indicate that the ex pansion rate of the universe is increasing. This acceleration requires the existence of a Dark Energy that overcomes the gravitational self-attraction of matter, such as the vacuum energy density associated with the cosmological constant (A). The prospects for improving our knowledge of the two most fundamental missing pieces of cosmology, the Dark Matter and Dark Energy, are discussed.
1. Introduction In the Standard Model of cosmology the Universe started with a Big Bang. The expansion of an isotropic and homogeneous Universe is described by the Friedmann-Lemaitre-Robertson-Walker model (or FLRW model, for short). The free parameters of the FLRW model are the energy contributions from radiation, matter and vacuum fluctuations. At the present epoch, the energy density in the form of radiation pTa& can be neglected in comparison with the matter density pm, and the Friedmann equation for the Hubble parameter (H) becomes: TT2
HS
fa\2
8TTG
+
A
k
UJ = - r ^ 3 - ^
(1)
where a is the growing scale factor of the Universe and fc=-l,0 or 1 represent the three possible geometries for the Universe: open, flat or closed. Thus the expansion rate of the Universe depends on the matter density, the cosmological constant (A = 8irGpvac) and the geometry of the Universe. It is also customary to rewrite equation (1) so that it instead contains the fractional energy density contributions at the present epoch (z = 0). We thus introduce the definitions: 8nG 0 „ A _ -k ^M = ^7722KPmi ° A A -= 7TFJ2 3i?0 ™' 3# 0 2 ft^K _— agffg
281
282
There are only two independent contributions to the energy density since in the FLEW model: ClM + nA + ClK = l
(2)
The time evolution of the scale factor a and thus the fate of the uni verse as determined by the two independent cosmological parameters. A large cosmological constant, for example, leads to rapid "inflation" of the universe. The deceleration parameter (at z=0), qo, is defined as:
thus, a negative value of qo implies that the rate of expansion of the Universe is increasing, i.e. the expansion is accelerating. In the next section we generalize the discussion as to also include con tribution from any arbitrary energy form characterized by the the relation between its pressure and density. 2. Cosmological parameters from "standard candles" A source of known strength, a standard candle can be used to measure relative distances to provide information on the cosmological parameters (see Gooobar & Perlmutter) 8 . If QM and Clx denote the present-day en ergy density parameters of ordinary matter CIM{Z) and a "dark energy" component Qx(z), respectively. The "dark energy" is characterized by the equation of state parameter, w(z), where px = w-px- For the specific case of the cosmological constant, w — —1, i.e. PA = — PAThe apparent magnitude m of a supernova at redshift z is then given by m(z)=M
+ 5\ogw[d'L(z)],
(4)
M = 25 + M + 5logw(c/H0),
(5)
where M is the absolute magnitude of the supernova, and d'L = H0 dj, is the Ho-independent luminosity distance, where H0 is the Hubble parameter*. Hence, the intercept M. contains the "nuisance" parameters M and H0 that apply equally to all magnitude measurements (in this section we do a
In the expression for M., the units of c and HQ are km s respectively.
1
and km s
l
Mpc
1
,
283
not consider possible evolutionary effects M — M{z)). The if 0 -independent luminosity distance d'L is given by ' (1 +z)7±=sm(S=ShI),
d'L = l(i + z)i, (l+z)^smh(y/ShI),
ilk < 0
nfc = o
nk = i-fim-nx,
(7)
Jo H'{z'Y H'(z) = H(z)/H0
(8)
=
V(i + zf nm + f(z) nx + (i + zy nk,
[ r^i+^') dz l + z'
f(z) = exp 33/
(6)
J7 fc >0
(9) (10)
As the measurements are performed through broad-band niters one has to correct for the fact that different parts of the supernova spectrum are detected depending on the redshift z of the source. For example, at a redshift z ~ 0.5 the light captured with a red (R) filter at a telescope at Earth originates from the blue (B) part of the spectrum. This so called "K-correction" is preferentially done using blue (B) absolute magnitudes in the resframe and V,R,I niters for the observation of supernovae with increasing redshift, as shown in (Kim, Goobar & Perlmutter) 11 . 3. Current results Two collaborations, the SCP 14 - 21 and the High-Z team 1 9 ' 1 7 ' 6 , have been searching for high-redshift Type la supernovae with the aim to measure cosmological parameters. Both groups find that the data is consistent with the existence of some "Dark Energy" form that is accelerating the rate of expansion of the universe at present, i.e. qo < 0, as shown in Figure 1. The Hubble diagram for 42 supernovae found by the SCP along with 18 low-z supernovae from the Calan/Tololo Supernova Survey 10 indicates that supernovae at z ~ 0.5 are 0.2-0.5 magnitudes too faint to be consistent with an open or flat universe with A = 0. The error bars in figure 1 represent the photometric uncertainty with 0.17 magnitudes of intrinsic dispersion of SN la magnitudes that remain after applying the width-luminosity correction add in quadrature. The theoretical curves for a universe with no cosmological constant are shown as red (open) and green (flat) lines. The blue line shows the best fitcosmology for which the total mass-energy density ^ M + £1A = 1- The best
284
fit value for the mass density in a flat universe is
(QA
= 1—
^M 1 *):
nflat _ r> 90+0.08 +0.05 i'M — u-^°-0.08 -0.04'
where the first uncertainty is statistical and the second due to known systematics. The details of the estimation of systematic errors such as from extinction, Mamlquist bias and brightness evolution of type la supernovae can be found in (Perlmutter et al) 1 4 . The supernova results are in good agreement with what is found from a varity of independent techniques. The CMB anisotropies at scales 1° or smaller as measured by the BOOMERANG, MAXIMA and DASI collaborations 4,5 ' 16 give a firm constrain on the geometry of the universe indicating that the sum of all energy densities, i.e. QM + ^X must be unity with only 5 % unceratinty. Constraints on the matter density A M from cluster abundances (Bahcall & Fan) 1 and (Carlberg et al.) 3 and largescale structure (Peacock et al) 13 has left cosmology with a concordance model with QM » 0.3 and f&A » 0.7. 4. How much better can we do with Type l a SNe? Figure 2 shows the degenaracy in the CL-region of the A M — ^ A param eter space defined by observations at single redshifts, ranging from z=0.2 to 1.8, assuming an accuracy of Am = 0.02 mag in the measured mean. In figure 2, a hypothetical data-set including supernovae at z=0.2-1.8 is used to demonstrate how the major axis of the confidence region could be dramatically shrunk. Clearly, enlarging the redshift range of the followed supernovae has the potential of refining our understanding of the cosmological parameters. 5. The next generation of S N experiments In Figure 2 we demonstrated how the accuracy in the magnitude-redshift method increases as supernovae at higher redshifts are added to the sam ple. In particular, at redshifts above z ~ 1 one can study the transition from acceleration to deceleration as the mass density term contribution, enhanced by the the shrinking volume as (1 + z)3, overtakes the effect of OA, as shown in Figure 3. Several projects with the aim to discover thousends of high-z supernovavae are being proposed. One of the most interesting ones is the SNAP satellite 15 , a 2-m telescope equipped with an optical and NIR mosaic cam era with a field of view of ~ 0.6 square deg.
285
i
i O.Bx
i 0.7>:
i O.Cx
i O.ax
Scale of the Universe
[relative to today's scale] Figure 1. Hubble diagram for 42 high-redshift Type la supernovae from the Su pernova Cosmology Project , and 18 low-redshift Type la supernovae from the Calan/Tololo Supernova Survey , plotted along with 3 combinations of cosmological parameters, two of which have J2A = 0. The best-fit flat cosmology is ( O M J ^ A ) = (0.28,0.72). Yellow data points indicate that the central value is in the go < 0 region (acceleration) while the red points are in the go > 0 region (deceleration). Clearly, the data favours an accelerating universe. Courtesy of S. Perlmutter2.
In addition of having the capability of discovering about 2500 SNe a year up to a redshift z ~ 2, the design of the SNAP satellite also includes an integral field spectrograph. This will allow for detailed spectoscopic studies of the supernovae and their host galaxies. Thus, systematic uncertainties on the measured supernova brightnesses are supposed to stay below 0.02
286 1cr bands at each redshift f o r A m = 0 . 0 2 mag
1a bands at each redshift for A m = 0 . 0 2 mag
Figure 2. Left:68 % CL-regions in the VlM—V.\ parameter space defined by each redshift bin (Az = 0.2) assuming a total uncertainty in the mean brightness of A m = 0.02 / b i n . Right:The bands are superimposed. The resulting CL region is defined by the common area.
I <J
-
A dominated ..... Open Fiat (A-O)
CD 0.6
:
0 -0.2
:
"\,
*"
-0.+ -0,6 -G.S
: :
: _1
" " " " " ■ - ■ - . ,
, ,,0.2
0A
0.6
0.8
Figure 3. Differential magnitude for three cosmologies, nM,nA=(0.3,0.7) (solid line), (0.2,0) (dashed line) and (1,0) (dotted line), compared with an empty universe, ( J I M J ^ A ) = ( 0 , 0 ) (horizontal, dash-dotted line).
mag in which case one can expect to measure A M and OA simultaneously to about 2% and 5% respectively, as shown in Figure 4.
287 [email protected])
c
CD T>
o 11]
E O
CO
1_
d
o u
s O O
E W
8.
>
1
2
mass density Figure 4. Target uncertainty for the SNAP satellite experiment (small ellipses) com pared to the published results in 14 .
6. The quintessence alternative The exciting results from the SCP and High-Z teams suggest that the method can be used to further improve our knowledge of cosmological parameters with Type la supernovae. While the existence of an energy form with negative pressure is strongly supported by the present data, it is not clear that the "Dark Energy" really is identical with the cosmologi cal constant. Alternative solutions have been proposed. E.g. Steinhardt 20 suggests that the effect might be caused by a different type of matter char acterized by an equation of state p = w(z)p, where w > —1, as shown in equation 10. The "quintessence" models do not suffer from the two fun damental problems of the cosmological constant: a) a value of OA ~ 0.7 is about 122 orders of magnitude from the naive theoretical calculation(l)
288
b) It seems somewhat unnatural that we happen to live in a time when " A ~ 2 since this ratio depends on the third power of the redshift. For 9 instance, epoch of radiation decouDline decoupling ^stance, at the eDoch i, ~ 10 . In "quintessence" models, the "Dark Energy" density tracks the development of the leading energy term making both comparable. Figure 5 indicates the accuracy to which the effective equation-of-state parameter w can be measured in a flat universe using supernovae ranging from z=0.2 to 1.8, assuming an accuracy of Am = 0.02 mag in the measured mean. 1 c bands at each redshift for Am=0.02 mag
1 cr bands at each redshift for £m=0.02 mag
Figure 5. Left:68 % CL-regions in the W — QM parameter space defined by each redshift bin (Az = 0.2) assuming a total uncertainty in the mean brightness of Am = 0.02 /bin and a flat universe. Right:The bands are superimposed. The resulting CL region is defined by the common area.
The situation becomes more complicated once we try to measure the time evolution of the equation of state parameter. Assming a linear expan sion, w(z) = WQ + w\ ■ z, is sufficient for the small redshift range z < 2, one additional parameter has to be considered. Figure 6(a) (from Goliath et al 7 ) shows what the fit of simulated data corresponding to one year of the SNAP satellite. The accuracy on the estimate of the nature of the dark energy will depend on independent knowledge, especially, of the HM from e.g. weak lensing measurements. The SNAP satellite, with is large field of view, will also provide extremely accurate measurements of cosmic shear. In addition, dedicated low-z supernova searches will be required in order to bound the intercept of the Hubble diagram, M.
289 1.01
■ ....Oo.-0.0S A!7m=0.1 exact .«
. -0.6-
Figure 6. (a) Left: 68.3 % confidence regions for (wo,wi) in the one-year SNAP sce nario. The elongated ellipses correspond to the assumption of exact knowledge of fim: the dash-dot-dot-dotted line is with exact M and the long-dashed line corresponds to no knowledge of M. The larger, non-elliptic regions assume prior knowledge of H m : the dash-dotted line assumes that ftm is known with a Gaussian prior for which aam -prior = 0-05; the short-dashed line assumes the same prior and exact knowledge of M; finally, the solid line is with fim confined to the interval Qm ± 0.1 and exact knowledge of M. (b) Right:68.3 % CL region of H M — WQ fit from lensed supernovae in the SNAP 3-yeardata. The dark (green) region shows the smaller confidence region that would result if h would be exactly known from independent measurements. The dashed line shows the expected statistical uncertainty from a 3 year SNAP data sample of Type la SNe.
Strongly gravitationally lensed SNe could be detected in large num bers in SNAP, probably on the order of several hundred (Goobar et al) 9 . Time-delay measurements of lensed SNe are potentially interesting as they provide independent measurements of cosmological parameters, mainly Ho, but also the energy density fractions and the equation of state of dark en ergy. The results are independent of, and would therefore complement the Type la program, as shown in Figure 6(b). 7. The Nature of Dark Matter With Type la supernovae it may be also possible to shed light on the nature of Dark Matter. Gravitational lensing in the inhomogeneous path that the beam of high-z supernovae follow from the source to us, affects the dispersion of the data points in the Hubble diagram. Thus, with a large sample of high-z supernovae, it is possible to measure the fraction of compact objects in the universe from the residuals of the Hubble diagram. While the compact objects are likely to be of astrophysical nature, e.g. faint stars or black holes, a smooth Dark Matter component would indicate that the missing mass is in the form of particles, such as the lightest stable supersymmetric particles. In (Mortsell, Goobar & Bergstrom) 12 we used
290 Monte-Carlo simulations to show that with one year of SNAP data, the fraction of compact objects can be measured with 5% absolute precision. 8. Summary and conclusions Observational cosmology is arguably one of the most exciting fields in physics at the moment. Techniques developed during the last years have provided new and unexpected results: the energy density of the universe seems to be dominated by the Einstein's cosmological constant (A), or pos sibly some even more exotic form of dark energy. Within the next decade, several measurement techniques are likely to provide conclusive evidence for the nature of the energy form that is currently causing the universe to expand at an accelerated rate, and for ther nature of the dark matter, opening a new era of precission observational cosmology. Acknowledgments: I am very grateful to Ana Mourao for inviting me to the Algarve. References 1. Bahcall, N. A and Fan, X., ApJ. 504, 1, 1998. 2. Bahcall, N. A, Ostriker, J. P., Perlmutter, S. and Steinhardt, P. J. 1999, Science, 284, 1481. 3. Carlberg, R. G. et al., ApJ. 516, 552, 1998. 4. de Bernardis, P. et aJ., Nature 404, 955, 2000. 5. Balbi, A. et al., ApJ, 545,1, 2000; erratum: ApJ, 558, 145, 2001. 6. Garnavich, P., et al. 1998, ApJ, 493, L53 7. Goliath, G., Amanullah, R., Astier, P., Goobar, A., and Pain, R., A&A, 380, 6, 2001. 8. Goobar A. and Perlmutter S., ApJ, 450, 14, 1995. 9. Goobar, A., Mortsell, E., Amanullah, R., & Nugent, P., 2002, A&A, 393, 25 10. Hamuy, M.,Phillips, M. M.,Maza, J.,Suntzeff, N. B.,Schommer, R. A. and Aviles, R., AJ, 112, 2391, 1996. 11. Kim, A., Goobar, A. and Perlmutter, S., PASP, 108, 190, 1996. 12. Mortsell, E., Goobar,A. and Bergstrom, L., ApJ, 559, 53, 2001. 13. Peacock, J. A. et al, Nature, 410, 169, 2001. 14. Perlmutter, S. et al., ApJ. 517, 565, 1999. 15. Perlmutter, S. et al., The SNAP Science Proposal, http://snap.lbl.gov . 16. Pryke, C. et al., pre-print astro-ph/0104490 (2001). 17. Riess, A. G., et al., AJ, 116, 1009, 1998. 18. Riess, A. G., Press, W. H. and Kirshner, R. P. ApJ, 473, 88, 1996. 19. Schmidt, B. P. et al. 1998, ApJ, 507, 46. 20. Steinhardt, P. J., Proc. of the Nobel Symposium "Particle Physics and the Universe", L. Bergstrom, P. Carlson and C. Fransson (eds.), T85, 177, 2000. 21. Sullivan, M. et al, 2002, MNR in press, astro-ph/0211444
C U R R E N T STATUS OF TYPE IA SUPERNOVAE THEORY A N D THEIR ROLE IN C O S M O L O G Y
S. B L I N N I K O V ITEP, 117218, Moscow, Russia E-mail: [email protected]
Thermonuclear supernovae are valuable for cosmology but their physics is not yet fully understood. Modeling the development and propagation of nuclear flame is complicated by numerous instabilities. The predictions of their light curves still involve some simplifying assumptions. In spite of great progress in recent years, a number of issues remains unsolved both in flame physics and light curve modeling.
1. Introduction Supernovae of type la (SNe la) are important for cosmology (better to say, for cosmography) due to their brightness. They are not standard can dles, but can be used for measuring distances with the help of the peak luminosity - decline rate correlation, established by Yu.P. Pskovskii 19 and M.M. Phillips 18 (see the review 8 ) . To exclude systematic effects in link ing the observed light of distant SNe la to the parameters of cosmological models, one has to understand the nature of supernova outbursts and to build accurate algorithms for predicting their emission. This involves: 1) understanding the progenitors of SNe la; 2) the birth of thermonuclear flame and it accelerated propagation leading to explosion; 3) light curve and spectra modeling. When the full understanding will be achieved, one can try to evaluate the importance of evolution effects in using supernovae as distance indica tors. In spite of great progress in recent years, a number of issues remains unsolved both in flame physics and light curve modeling. I point out some problems which seem most important to me. 2. Progenitors Mechanical Equilibrium and evolution of stars is easily understood from the virial theorem for a star: 3fPdV = —U , where P is the pressure,
291
292
V is volume, and U is the gravitational energy of a star. Crude esti mates V ~ R3, M ~ R3p, U GM2/R in the virial equation give 2 3 3 P ~ GM l p^l . For an ideal classical gas, the equation of state P = TZpT implies 1ZT ~ GM2/3p1/3, with TZ the gas constant. This is already enough to understand evolution massive stars! While a star loses energy and con tracts, its internal temperature T grows. If the losses are balanced by a nuclear energy release, then the contraction stops and thermal equilibrium is established: nuclear heating power L+ = radiative cooling (luminosity) L~. The rate of thermonuclear heating scales as (avo) ~ exp— ( G J G / T ) 1 / 3 due to the Gamow's peak: the chances to penetrate the Coulomb barrier for fast nuclei grow, but the tail of Maxwell distribution goes down. Here ao depends strongly on nuclei charges Z»: ota oc Z\Z\, thus high-Z ions can fuse only at high T. Small perturbations of T produce huge variations in L+ since, normally, T
293
plode because they can merge due to emission of gravitation waves (doubledegenerate, or DD scenario u ) . If one star in the binary is alive, the white dwarf can accrete its lost mass and reach an instability (single-degenerate, or SD scenario 2 5 , 5 ) . It is unclear which scenario is most important, there are strong arguments 14 from chemical evolution that only SD is the viable one. On the other hand, it seems that DD can produce a richer variety of SN la events. Moreover, discoveries of intergalactic SNe la 2 ' 7 can be ex plained more naturally, because a DD system may evaporate from a galaxy. It is quite likely that both scenarios are being played, but their relative role may change in young and old galaxies. If so, a systematic trend may appear in SNe la properties with the age of Universe, and this may have important consequences for cosmology.
3. Thermonuclear flames After merging in DD scenario, or after the white dwarf accretes large amount of material in SD case, the explosive instability develops. In prin ciple, combustion can propagate either in the form of a supersonic detona tion1 wave, or as a subsonic deflagration12'16 (flame). In detonation, the unburned fuel is ignited by a shock front propagating ahead of the burn ing zone itself. In deflagration, the ignition is governed by heat and active reactant transport, i.e. by thermal conduction and diffusion. Most likely, the runaway starts as a laminar flame propagating due to thermal conduction. In terrestrial flames, the 'fusion' of molecules goes with the rate: (crvo) ~ exp(—Ea/lZT), - the Arrhenius law of chemical burning. Here Ea is activation energy. The parameter, showing the strong T-dependence of the heating Ze = d\og(av0)/dlogT ~ Ea/1ZT is called the Zeldovich number in the theory of chemical flames. For them typi cally Ze ~ 1 0 . . . 20. The classical theory 28 predicts the flame speed vt ~ Z e - V ^ r / w C T b ) ] 1 / 2 , with 'rreac(2n) °c exp[£?a/(7?.T)]. In SNe, for nu clear flames, Treac (T) oc expfaJ/V^T 1 / 3 )], and, Ze = <91og(0)/<91ogT ~ Q J /(3T 1 / 3 ), which has values very similar to terrestrial chemical flames. A big difference with chemical flames is the ratio of heat conduction and mass diffusion, the Lewis number, Le = {VTW)I{VDID)Le~ 1 in 7 laboratory gaseous flames, while Le~ 10 in thermonuclear SNe, since heat is transported by relativistic electrons, VT ~ c, and there is almost no diffusion, IT ~5> ID- Nevertheless, the modern computations 24 follow the old theory 28 closely. The conductive flame propagates in a presupernova with Vf which is too slow to produce an energetic explosion: the ratio of
294 Vf to sound speed, i.e. the Mach number, Ma, is very small (see Table 1). The star has enough time to expand, to cool down, and the burning dies completely. So an acceleration of the flame is necessary in order to explain the SN phenomenon. This is the main problem in current research of SNe la hydrodynamics. Table 1.
Flame speed Vf and width If in C + O 2 4 ,
p 10 9 gcc
Vf km/s
If cm
Ap/p
Ma
6 1 0.1
214 36 2.3
1.8 x 1CT6 2.9 x 1 0 " 4 2.7 x 1(T 2
0.10 0.19 0.43
2 x lO-2 4 x lO"3 4 x 10~ 4
There is a rich variety of instabilities that can severely distort the shape of a laminar flame. The Rayleigh-Taylor (RT) instability governs the cor rugation of the front on the largest scales. On the smallest scales the flame is controlled by the Landau-Darrieus (LD) instability. RT, LD instabilities and turbulence make computations difficult, but without them a star would not explode. All these instabilities were considered already by L.Landau 15 as a means to accelerate the flame. Because of instabilities, the flame surface becomes wrinkled and its area grows as S a Ra , with average radius R and a > 2, i.e. faster than S oc R2. In other words the surface becomes 'fractal'. The exponent a is actually the fractal dimension. The effective flame speed is determined 26 by the ratio of the maximum scale of the instability to the minimum one: ves = ff(A m a x /A m i n ) I ) F _ 2 . When the vorticity is not important it is possible to study in detail the non-linear stage of LD instability and to find the fractal dimension 3 . A similar dependence of the flame fractal dimension on the density jump across the front was found in SPH simulations of the flame subject to RT instability 4 . The fractal description is good for LD while it remains mild, because it operates in a star on the scales from the flame thickness (a tiny fraction of a cm) up to ~ 1 km. For the RT instability, A max /A m i n is very uncer tain and the fractal dimension is uncertain too. So a direct 3D numerical simulation is necessary. The same is true for a low density regime of LD when it is strongly coupled to turbulence (generated on the front, or cas cading from large RT vortices). A great progress is achieved here in several groups 9 ' 20 ' 21 ' 13 , The problem of simulating 3D turbulent deflagrations has
295
two aspects: the representation of the thin, propagating surface separating hot and cold material, and the prescription of the local velocity Vf of this surface as a function of the large-scale flow with a crude numerical resolu tion > 1 km. One solution to this problem is sketched in 20 ; for a different approach see 13 . In spite of the progress this problem cannot be treated as completely solved, and even ID approach may give interesting results, especially for unusual SNe la 6 .
4. Light curves of SNe la Given a hydrodynamic structure of SN ejecta, one can compute a light curve which should be compared with observations. There are several effects in SNe physics which lead to difficulties in the light curve modeling of any type of SNe. For instance, an account should be taken correctly for deposition of gamma photons produced in decays of radioactive isotopes, mostly 56 Ni and 56 Co. To find this one has to solve the transfer equation for gamma photons together with hydrodynamical equations. Full system of equations should involve also radiative transfer equations in the range from soft X-rays to infrared for the expanding medium. There are millions of spectral lines that form SN spectra, and it is not a trivial problem to find a convenient way how to treat them even in the static case. The expansion makes the problem much more difficult to solve: hundreds or even thousands of lines give their input into emission and absorption at each frequency. Currently, powerful codes appear aimed to attack a full 3D timedependent problem of SN la light 10 . Yet there are some basic questions, like averaging the line opacity in expanding media, that remain controversial. In our current work with E.I.Sorokina we predict the broad-band UBVI and bolometric light curves of SNe la, using our ID-hydro code which models multi-group time-dependent non-equilibrium radiative transfer in side SN ejecta. We employ our new corrected treatment for line opacity 22 in the expanding medium, which is important especially in UV and IR bands. The results are compared with the observed light curves. It seems that classical ID thermonuclear supernova models, e.g. the deflagration W7 1 7 one and the delayed detonation DD4 27 one, produce the light curves fitting the observations not so good as the recent 3D deflagration model MR computed at MPA 20 . We believe that the main feature of the latter model which allows us to get the correct flux during the first month, is strong mixing that moves the material enriched with radioactive 56 Ni to the outermost layers of SN ejecta.
296
0
20
40
60
—i—i—i—|—i—i—i—|—i—i—i—I
0
20
40
60
0
20
40
60
I—i—i—i—|—i—i—i—|—i—i—i—
0
20
40
60
t, days Figure 1. UBVI light curves for the 3D (MR; solid) and ID (W7; dashed) models. Crosses, stars and triangles show the light curves for three observed SNe la.
Fig. 1 demonstrates that in spite of quite different structure of the old W7 model and the new MR one their light curves are similar in many details. Moreover the new model behaves better in U and B bands. Unfor tunately, the bolometric light curve for MR model is somewhat too slow. The ejecta must expand with a higher speed to let photons to diffuse out faster. 5. Conclusions There are several points which require attention for applying SNe la in cos mology: progenitors may be different in younger galaxies; burning regimes may change with the age of Universe 23 . The physical understanding of the Pskovskii-Phillips is not yet achieved. In the flame modeling the new 3D SN la model 20 is very appealing. Yet it is not a final one: a detailed post-processing of nucleosynthesis is not yet checked in the light curve calculation.
297 T h e SN light curve modeling still has a lot of physics t o be added, such as a 3D time-dependent radiative transfer, including as much as possible of N L T E effects 10 , which are especially essential for SNe la. All this will improve our understanding of thermonuclear supernovae and their role in cosmology. I am grateful to A n a Mourao for support. My work in Russia is partly funded by R F B R (grant 02-02-16500). References 1. W.D.Arnett, Ap.Sp.Sci. 5, 180 (1969). 2. O. Bartunov, Outlying Supernovae - Myth or Reality? UCSB Workshop on SNe, h t t p : / / w w w . s a i . m s u . s u / ~ m e g e r a / s n / o u t s n / (1997). 3. S.I.Blinnikov, RV.Sasorov, Phys.Rev. E53 4827 (1996). 4. E.Bravo, D.Garcia-Senz, ApJ 450, L17 (1995). 5. A. Bragaglia et a l , ApJLett 365, L13 (1990). 6. N.V. Dunina-Barkovskaya et al., Astron.Letters 27, 353 (2001). 7. A.Gal-Yam et al., astro-ph/0211334 (2002). 8. Leibundgut, B.Astr.Ap.Rev. 10, 179 (2000). 9. W. Hillebrandt, J. C.Niemeyer, Ann. Rev. Astron. Ap. 38, 191 (2000). 10. P. Hoflich, Workshop on Stellar Atmosphere Modeling, Eds: I. Hubeny et a l , astro-ph/0207103 (2002). 11. Iben, I. J. et al., ApJ 475, 291 (1997). 12. Ivanova L.N. et a l , Space Sci. 31, 497 (1974). 13. A. Khokhlov, e-print astro-ph/0008463 (2000). 14. C. Kobayashi et al., ApJLett 503, L155 (1998). 15. L.D. Landau, Acta Physicochim. USSR 19, 77 (1944) 16. Nomoto K. et al., Ap.Space Sci. 39, L37 (1976). 17. K. Nomoto et al., ApJ 286, 644 (1984). 18. Phillips M.M., ApJ, 413, L105 (1993). 19. Pskovskii Yu.P., Sov.Astronomy 21, 675 (1977). 20. M. Reinecke et al. Astr.Ap. 386, 936 (2002). 21. F.K. Ropke, et al. Proc. 11th Workshop Nuclear Astrophysics, 2002, W.Hillebrandt and E. Miiller (Eds.). MPA/P13 (2002), p. 41. 22. E.I. Sorokina, S.I. Blinnikov: Proc. 11th Workshop Nuclear Astrophysics, 2002, W.Hillebrandt and E. Miiller (Eds.). MPA/P13 (2002), p.57. 23. E.I. Sorokina et al., Astron. Letters 26, 67 (2000) 24. F.X. Timmes, S.E. Woosley, ApJ 396, 649 (1992). 25. Whelan, J. and I. J. Iben 1973.ApJ 186, 1007-1014. 26. S.E.Woosley, in: Supernovae, ed. A. G. Petschek, A & A library, 1990, p. 182 27. S.E. Woosley, T.A. Weaver. In: Supernovae, ed. by J. Audouze et al., Elsevier Science Publishers, Amsterdam (1994), p.63 28. Zeldovich Ya.B. & Frank-Kamenetsky D.A., Ada Physicochim. USSR 9, 341 (1938)
I N T E N S I V E S U P E R N O V A E SEARCHES
K. S C H A H M A N E C H E L.P.N.H.E.,
Universit Paris 6 et 7, 4, place Jussieu 75252 Paris cedex 05, FRANCE E-mail: [email protected]
The arrival of large imagers like Megacam at the Canada-France-Hawaii Telescope running in a survey mode, make it possible to detect and follow a large amount of high redshift supernovae on the same instrument. Using Megacam during 5 years around 1000 Type la SNe should be detected, z ranging from 0.3 to 1.2, and more than 500 will be usable for measuring the cosmological parameters via the Hubble diagram. They should allow the first leasurement of the cosmic state equation parameter w. A first test run, using the CFH 12k calera in a "rolling mode" has been done last spring at the CFHT. Detection procedure and preliminary results are presented in this article.
1. Current results At the end of the last decade, two international collaborations detected and followed-up enough distant SNe la to reconstruct the expansion evolution of the Universe back to about 10% of the age of the universe 1'2. To every body's surprise, the results of the two teams, favored a model indicating that the universe is currently accelerating. This suggests that some kind of unknown, repulsive dark energy is now driving the expansion. The two teams came to similar conclusion, using a different set of distant SNe la, but by large the same set of nearby supernovae, discovered in the early nineties by the Calan- Tololo Supernova Search 3 . Figure 1 shows the (£lm,flx) and (£lm,iv) constraints published by the Supernova Cosmology Project (SCP). It is important to note that to build the confidence contours in the (^lm,^x) plane, it was necessary to make assumptions on the nature of the dark energy. A value of w = —1 was chosen, corresponding to a vacuum energy model where X = A is called the cosmological constant. Similarly, the (Q,m,w) contours have been deter mined for a flat universe. It is also clearly visible that only two quantities are well constrained: (7 Qm + flx) in Figure l.a, and (fixiux) in Figure l.b.
298
299
Figure 1. Constraints in the ( n m , H x ) an(A (^m,wx) planes, published by the Su pernova Cosmology Project. The current SN la datasets do not allow to constrain the cosmological parameters independently, but rather the sum 7f2 m + &x a n d the product flmwx-
It is possible however to reduce significantly the (fi m ,fix) degeneracy by increasing the redshift range of the dataset : the 7 factor, i.e. the orienta tion of the error ellipsis is indeed a function of the SN sample redshift lever arm. Breaking the flm — w degeneracy is almost impossible even by going very far in redshift (z ~ 2). However, combining the D,m — w constraints with an independent determination of fim, such as the one provided by the weak lensing surveys, will allow to determine wxIt is seen in Figure 1 that the Einstein-De Sitter model, a matterdominated flat universe (fl m = 1), is strongly excluded. If we assume that the universe is flat, as suggested by the recent CMB results, then a low value of fiTO ~ 0.3 is favored. The other important result is that fl\ > 0 at a 99% confidence level. For a flat universe, the favored result is fim ~ 0.28 and 0 A ~ 0.72. The statistical uncertainties are still the largest contribution to the total error budget. Nevertheless multiplying by 10 the number of supernovae will require at the same time a better understanding of systematics. The next step is then to collect a large amount of SNe la at high redshift and also at low redshift (cf Figure 2 ). This is the goal of the satellite project SNAP (SuperNova Acceleration
300
Figure 2. The two Hubble's diagram published by the two collaborations : only 18 lowz and 42 high-z SNe for the SCP and 27 low-z and 16 high-z for the High-z SN Search Team.
Probe) which should be launched around 2010. However an important step toward this will be achieved in the next 5-6 years with two major ground-based projects : the Nearby Supernova Factory (SN Factory) and the Supernova program at the CFHT Legacy Survey (SNLS). 2. The Nearby Supernova Factory To achieve a precision of a few percent in the measurement of the cosmological parameters, we have to improve our understanding of the intrinsic prop erties of SNe la. The SN Factory aims at detecting ~ 400 nearby SNe la at a redshift z ~ 0.05 and collect for each of these events very high quality spectrophotometric follow-up data. The detection will be done using images taken by the two 1.2-m telescopes operated by the Near-Earth-AsteroidTracking (NEAT) searching for asteroids. The SN Factory is expected to run all year long and detect several supernovae every clear night. To observe these events, a dedicated Integral Field Spectrometer (SNIFS) is being built in France and will be mounted on the UH-2.2-m telescope in Hawaii. This instrument will produce precise spectrophotometry of supernova? and from this it will be possible to obtain a synthetic broadband photometry and spectra of the supernova and its host galaxy. The SN Factory will start taking data at the end of 2003. In 5 years, it
301 The time series of spectra is a "CAT Scan" of the Supernova
Figure 3. The Supernovae Integral Field Spectrometer (SNIFS), will allow us to deter mine the empirical properties of the SNe la (standardization, rest-frame U band, ...), but also to study the Physics of these objects and constrain the models.
will increase the current number of photometrically well-observed nearby SNe la by a factor of 10 and the number of spectroscopically well observed SNe la by a factor of 40. Empirical studies of the SNe la characteristics with a precision never achieved before, will become possible due to the sample size, its homogeneity and the wide range of observables including SNe la luminosities, colors, spectral line shapes and host galaxy spectral properties. The Nearby Supernova Factory will therefore be able to improve the standardization techniques, and thus the precision of the cosmological measurements. This dataset will also tell us how to correct for the various foreground effects such as the dust absorption of SN la light. Finally, with SNIFS high spectral resolution of about 3 A in the wavelength range [3200 — 10000] A , it will be possible to reconstruct SNe la velocity lines with very high accuracy and thus constrain SNe la theoretical models (cf Figure 3).
302
3. The supernova program of the CFHT Legacy survey The primary goal of the Supernova program at the CFHT Legacy Survey (SNLS) is to achieve first characterization of the dark energy, through the measurement of wx • This project aims at increasing the number of SNe la detected and followed-up in the redshift range 0.3 < z < 0.9 by a factor of ten. Such a dataset will permit one to target a precision of ±0.1 in the measurement of the cosmological parameter w. The project will use the MegaCam imager, a wide-field camera currently being mounted on the prime focus of the CFHT 3.6-m telescope, located in Hawaii. This instrument, made of a mosaic of 40 thinned 2Kx4.5K CCDs will cover about 1 square degree on the sky, with a pixel resolution of 0.18". The whole CFHT Legacy Survey project consists in 3 different surveys, requiring more than 450 nights of observation during 5 years. The supernova part of the project will implement a continuous detection and follow-up on the same 4 square degrees in the four bands: g,r, i and z. This original detection and observation strategy will allow us to gather a homogeneous of 2000 SNe sample. The expected peak brightness resolution in each instrumental band is shown on Figure 4.
Figure 4.
The expected resolution in each instrumental filter in function of the redshift.
303
If we restrain this sample to the SNe la having each observation between -10 days before max until +15 after max with a signal to noise ratio greater than 10, and if we consider only the ones in the redshift range 0.3 < z < 0.9 (rest-frame B-band and extinction addressed with CFHT only data), the total sample of well measured SNe la will be ~ 700 (~ 100 per 0.1 redshift bin). Such a high number of SNe la will need a large spectroscopic program. To identify the SNe, determine precisely the redshift, we estimated the spectroscopic time budget to 120 hours on a 8 m class telescope per semester (10 hours per field and per lunation).
5 years CFHLS + 200 nearby ~ -0.5 ■ _
0.55
-0.55
-0.6 0.65 -0.7 0.75
- \^\ N\\\ \\ V»
-0.S 0.85 -0.9 0.95
-
\m
m 1\
Figure 5. On the left, precision achieved after 5 years of SNLS + 200 nearby SNe la (SN Factory). On the right, same plots but with a prior on the matter density (coming, for example, from the coming weak-lensing programs).
If we combine the SN Factory low-z sample and the .STVLS'dataset, it will possible to reach a precision of ±0.05 on the 0,m and 0.x measurements. In addition, the weak-lensing program, also performed at the CFHT, will pro duce an independent measurement of Q,m. Combining this result and the
304
(fi m , w) constraints coming from the SN la studies will provide a measure ment of Wx with a typical uncertainty on wx oWx ~ ±0.1, as it's shown on Figure 5. 4. The spring 2002 "rolling search" at C F H T This new searching strategy has been tested in spring 2002 (March, April and May 2002) at CFHT, in collaboration with the SCP. The imager was the "CFH 12k" camera which field of view represents one third of the future MegaCam. 25 h of observation were spread of the three month with two epochs for spectroscopy : April and May at four different telescopes (VLT, Keck, Gemini, Subaru). Two different fields were followed during the all time with 30' exposure in I band and 10' in R band at each epoch. Unfortunately, due to poor weather and instrument scheduling no data were gather during more than three weeks (from March 19 to April 12). This is a huge gap for the new kind of search strategy ("rolling search"). Moreover all the April spectroscopic time was lost. Despite these weather problems, and taking into account the small amount of telescope time (25 hours in total), we could manage to observe the two fields at 11 epochs : March 16-19, April 12-17-20-22 and May 38-11-17-20. Three SNe la were discovered with redshift of 0.26, 0.28 and 0.45.
Figure 6. Discovery of a SN la at a redshift of 0.28 by image subtraction. The first image is a deep reference image obtained by stacking images taken in 2000 and 2001, and giving a total exposure time of 10 hours. The second image is a stack of all images taken during this 2002 run. And the last vignette is the convolved subtraction where the supernova candidate is obvious.
Figures 6,7 and 8 show an example of a candidate found during the campaign. On the Figure 6, you can see the deep reference image, the stack of all the observations of the spring run and the subtraction : the host galaxy and the supernova are obvious. Figure 7 shows the time sequence of that supernova : the observations are taken at each 11 epochs. And the Figure 8 shows the "on-line" light-curve. The gap is due to 3 weeks of poor
305
Figure 7. Time series of that supernova (11 subtractions corresponding to each epoch) : March 16-19, April 12-17-20-22 and May 3-8-11-17-20.
fJ.fJCWW)
Figure 8. "On the fly" light-curve of the same SN la : approximate instrumental mag nitude versus the Julian date (day 0 corresponding to the discovery day of the SN).
weather and that why the peak is missing, but one can even see the second bump in the light-curve. This test run showed the feasibility of this new kind of "rolling search" where an "on the fly" preliminary light-curve of each variable object is build. 5. Conclusion In the coming years, these intensive searches of supernovae will tell us much more on supernovae Physics and should allow us to use these objects to measure the cosmological parameters with a few percent uncertainties. References 1. P e r l m u t t e r , S. et al. , Astrophysical Journal 5 1 7 , 565 (1999). 2. S c h m i d t , B.P. et al. , Astrophysical Journal 5 0 7 , 46 (1998). 3. H a m u y , M. et al. , Astrophysical Journal 1 0 9 , 1 (1995).
P R O B I N G T H E D A R K M A T T E R W I T H I N T H E SOLAR INTERIOR *
I. P. L O P E S Instituto
Centro Multidisciplinar de Astrofisica, Superior Tecnico, Universidade Tecnica de Lisboa Av. Rovisco Pais,1049-001 Lisbon, Portugal Department of Physics, University of Oxford Deans Williamson Building, Keble Road, Oxford 0X1 3RH, United Kingdom E-mail: [email protected]
The existence of dark matter in the solar neighbourhood can be tested in the framework of stellar evolution theory by using the new results of helioseismology and solar neutrinos. If weakly interacting massive particles accumulate in the centre of the Sun, they can provide an additional mechanism for transferring energy from the solar core to the outer regions. The presence of these particles produces a change in the local luminosity of the Sun of the same order of magnitude or larger that the difference between the Sun model's and the seismic data. In this paper I briefly present the difficulties in modelling the Sun, and argue in favour of using seismic data and neutrino measurements to constrain the super-symmetric particles candidates to the non-baryonic component of dark matter. Moreover, I discuss about how future helioseismological experiments have the potential for exploring most, if not all, of the parameter space of the weakly interacting massive particles accessible to the current and planned direct experiments for dark matter detection.
1. Introduction Observational solar physics is probing the history and dynamics of the Sun with ever-increasing precision, provided by well-calibrated instruments on board of international solar missions from the European Space Agency (ESA) and National Aeronautics and Space Administration (NASA), as well as ever-increasing networks of solar observatories on Earth. This high "This work is supported by a grant from the Portuguese Pundagao para a Ciencia e a Tecnologia.
306
307
flow of observational data has made the Sun not only, the most well known star, defining a standard for stellar astrophysics, but also a laboratory for plasma and particle physics. The sun's interior has become a new labo ratory for 'experimental research' that permits to study the solar plasma and the high-energy particles under conditions that are not attainable on Earth. In the last two decades, the flow of observational data, motivated the astrophysical community to study the evolution of the Sun in detail, in or der to put some constraints on its age, chemical composition, microscopic physics, such as the equation of state and radiative transfer, and the dy namic processes going on at the solar surface and interior. Their goal was also, of course, to understand the origin of the fifty percent deficit of neutri nos produced in the solar core, and measured on the ever increasing number of Earth detectors. During these years on the solar neutrino physics front, the surprise on the theoretical predictions was twofold: firstly, the predic tions were in disagreement with the neutrino experiments, namely the first chlorine experiment developed by Davis and collaborators, and secondly, the predictions of neutrino fluxes differed significantly among various the oreticians, even with rather similar hypotheses. Since the early 1980s, people working on stellar evolution produced a reference solar model along the framework of the classical stellar evolution (Turck-Chieze & Lopes 1993) . This activity was encouraged by the detec tion and interpretation of a large number of acoustic modes, creating the basis for a new research field in solar physics, the so-called helioseismology. The network of seismology observations around the Earth, as well the as the Space Mission Solar and Heliospheric Observatory (SoHO), a satellite built in Europe by ESA, has changed significantly our understanding of the solar interior. SoHO carries twelve sets of instruments, provided by European and American investigators, that were launched into space by NASA on the 2nd December 1995. Three of these instruments were exclu sively for seismic observations, monitoring velocity variations of the solar surface and the global luminosity variation. Ever since its flow of seismic data is contributing to establish a well defined understanding of the phys ical mechanisms operating in the solar interior, the solar model became a well-established reference (Turck-Chieze et al. 2001), the so-called solar standard model, that is now unanimously accepted by all the astrophysical community. The theoretical predictions of solar neutrinos by different authors were in agreement from the late 80's mainly due to the progress in solar modelling
308
provided by helioseismology, but the puzzle about the origin of the fifth percent deficit on the neutrino fluxes detections had remained a problem for the beginning of the new century. Finally, in the year 2001, the long standing puzzle of the origin of the solar neutrino deficit was solved. The Sudbury Neutrino Observatory (SNO) has measured the SB solar neutrinos and the once exotic explanation, that the deficit of neutrinos measured on Earth is due to the transformation of the Sun's electron-type neutrinos into other active flavours (Bahcall 1989), became the most appealing theory. Indeed, the SNO data obtained established a direct indication of the nonelectron flavour component in the solar neutrino flux and yielded the first unequivocal determination of the total flux of 8B neutrinos produced by the Sun (Ahmad et al. 2001; Bahcall 2001). The neutrinos with electronic flavour produced in the proton-proton reactions in the nuclear region of the Sun's interior during its trip towards the Earth detectors, have their flavour changed to the r or the (i flavour. The SNO measurement of 8B neutrino fluxes leads to the prediction of the central temperature of the solar core in agreement with the temperature of the standard solar model: the difference is much less than 0.5%. In the beginning of this new century, the physics of the solar interior is now constrained by two powerful probes: the seismic waves of the Sun and the solar neutrino fluxes. The Sun became a reasonably well-known star, which can be used to constrain the fundamental physics, such as the value of Newton's gravitational constant (Lopes &; Silk 2003) and used for the research of new particles in cosmology. In particular, we use it to study some of the most exciting problems of theoretical astrophysics and modern cosmology: the origin and nature of dark matter. The effect of some dark matter candidates in the core of the Sun, it is at least of the same order of magnitude as the microscopic physics and dynamical processes that are now being discussed in the framework of stellar evolution, most of them in the light of the most recent results of helioseismology and solar neutrinos physics. In this brief introduction to this new field of research, we did not concern ourselves with particular particle physics models, but considered a generic case, for dark matter particles interacting weakly with baryons.
2. Solar Seismology and Inversions The Sun is optically thick, so the data we got comes from observations of the surface layers. But the Sun is acoustically transparent; waves penetrate deep into the interior, providing information about the internal structure
309 Single Dopplergram Minus 45 Images Average
Figure 1. This velocity image known as Dopplergram, reveals the surface motions associated with sound waves travelling through the Sun's interior. The small scale light and dark regions represent the up and down motions of the hot gas near the Sun's surface. The pattern falls off towards the limb because the acoustic waves are primarily radial.
and probing the physical processes that govern its evolution (Gough et al. 1996). The acoustic modes are very likely stochastically excited by turbu lence in the layer immediately below the photosphere and damped by ra diative losses. The lifetime of these modes ranges from a few hours to more than several months. Several processes contribute to the observed velocity field of the solar surface fluctuations: differential rotation, convective flows, circulation and the oscillations. The oscillatory motion or oscillations con sists of the incoherent superposition of roughly 107 pressure-gravity modes with temporal cyclic frequencies of about 2 to 4 mHz (i.e., with periods of about 5 minutes) and a maximum velocity amplitude of about 20 cm/s per mode, typically, a velocity amplitude of 1 cm/s and an associated relative brightness variation of about 1 0 - 7 . In particular, global modes have life times longer than their travel time around the Sun (typically 5 days). In fig. 1 it is shown the velocity field component for the acoustic oscillations. The Dopplergram is a velocity image which reveals the surface motions as sociated with sound waves travelling through the Sun's interior (figure 1). Through a Fourier analysis of Dopplergrams it is possible to obtain the dis persion relation of the stationary pressure (acoustic) waves. The so-called spatial-temporal spectrum of the solar surface MDI/SOI (cf. Fig. 2). The acoustic-gravity modes are thus able to be sensitive to the average ther-
310
modynamic stratification of the solar interior. However, their sensitivity to the solar structure, their degree of penetration on the solar interior depends on their nature and the geometrical properties of their ray paths. Three
Figure 2. MDI/SOI Spatial-temporal spectrum: This represents the observed frequencies for acoustic waves of a certain degree 1, and radial order n: The ridges in the diagram correspond to different values n, that increase with the frequency, and the location of the ridges in the spectrum are used to infer properties of the solar structure. kinds of waves are investigated by helioseismologists: acoustic waves, for which pressure is the restoring force (these waves generate p modes), grav ity waves for which buoyancy is the restoring force (these waves generate g modes), and surface gravity waves (these waves generate f modes). On the current experiments only acoustic and surface gravity waves have been successfully observed in the Sun's surface. Acoustic oscillations have proved to be very useful in probing the solar interior (Christensen-Dalsgaard et al. 1996). The data from the SoHO experiments allowed to improve the accuracy of the radial distribution of the sound speed, of the density and of the rate of differential rotation. The square of sound speed is now extracted with a precision higher than 1 0 - 3 . This is due to the high precision obtained in the measurement of spectral peaks < 10~ 5 , and the large quantity of data: nearly 3500 acoustic modes relevant for the internal structure of the Sun. Solar acoustic modes have provided key points in the modelling of the solar structure down to 0.4 Ro (80% solar mass). Unfortunately, gravity modes have not yet being observed. Gravity modes are trapped in the radiative region and are largely
311
influenced by the physics of the nuclear region. The gravity modes are influenced as much as 75% in the region below 0.2R. Their sensitivity to the physical processes on the solar core is as much as 10 times greater than the acoustic modes. Their observation is crucial to determine the density and mass distribution in the core. Their definitive observation could constrain the physics of the solar core. 3. The Seismic Sun and The Solar Standard Model The standard stellar evolution of the Sun assumes that the star is in hydro static equilibrium, is spherically symmetric and that the effects of rotation and magnetic fields are negligible. The present structure of the Sun is ob tained by evolving a initial star from the pre-main sequence, around 0.05 Gyr from the ZAMS, until its present age, 4.6 Gyr. The present solar model is obtained by choosing the initial Helium content, as well as the mixing length parameter of convection that best predicts the present solar lumi nosity and radius. The helioseismolgy requires that the present structure of the Sun, including the global quantities, should be determined with a very high precision.
.002"
001 "
^rA
lw **
001 "
002"
.003-
0.0
02
0.4
0.6
0.6
1.0
solar radius
Figure 3. The relative differences between the square of the sound speed of the standard solar model. The curve with error bars represents the relative differences between the squared sound speed in the Sun (as inverted from solar seismic data) and in a standard solar model (Kosovichev et al. 1999). The horizontal bars show the spatial resolution and the vertical bars are error estimates. The thermodynamic structure in the interior of the Sun, namely in the
312
nuclear region, is presently known with a precision of much less than a few per cent. This level of accuracy in constraining the solar interior has been achieved by a systematic study of the differences between the acoustic spectrum obtained from helioseismology experiments and the theoretical spectrum. Presently, this difference is less than WfiHz for almost all of the 3000 modes that probe the interior of the Sun (Gough et al. 1996). In particular, this has reduced the difference between the square of the sound speed observed and the theoretical one. In fig. 3, we illustrate the square of the sound speed as inferred for the present Sun by using the data of Global Oscillations at Low Frequency (GOLF) and Michelson Doppler Imager (MDI) experiments. The average mean hydrostatic structure of the Sun has been obtained by an optimally localized averaging inversion method by Kosovichev (1999). The method for independently determining the radial dependencies of the sound speed and density also yields the radial dependence of the first adiabatic index or the chemical composition. All the different methods of inversion are extremely sensitive to the quality of the frequency measurements, and very accurate seismic data for the low-degree acoustic modes is necessary for a precise inversion. Figure 3 suggest that the Sun's interior needs some important improve ments: at the surface, to improve the interaction between the acoustic waves, convection and radiation in the super-adiabatic region; furthermore, a possible asymmetry of the background state due to rotation needs to be corrected; at the transition region between convection and radiative region, the so-called tachocline region, a better treatment of microscopic diffusion and turbulence is needed; at the intermediate region, improvements on the determination of chemical abundance and opacities. The nuclear region seems to be more problematic. This region is probed by as many as 120 acoustic modes that are significantly influenced by the turbulence, non-adiabatic effects and magnetic field perturbations at the surface layers (Lopes & Gough 2001). Nevertheless, the long duration of continuous measurements has reduced the uncertainty related with dynam ics of the outer layers. The modes that penetrate the solar core are weakly sensitive to its structure. They spend less than 5% of their travel time in the nuclear region. The solar core has an impact in the absolute values of frequencies of low-degree acoustic modes of as much as 0.5 Hz. Fortunately, the solar neutrino fluxes can be used to constrain the central temperature, and at first order we can claim that we understand the nuclear physics of the solar core. Indeed, the high accuracy in inversion obtained from the acoustic modes
313
is good enough to contrain the physics of the solar interior. Naturally, there are still open questions that need to be answered, but these remain at a second level on the physics of the solar modelling. In particular, our knowledge of the abundance of several elements linked to the sound speed inversions, allow us to constrain their abundance, test some of the nuclear reaction rates and disentangle between the different physical processes that could be responsible by the present solar core. However, a much better answer about the structure and dynamics of the core is expected when the new seismology of gravity modes becomes well established. In particular, the impact of g-modes is important for the inversion of the density profile. It seems clear that the next step on constraining the very central region of the Sun, will be only through a significant increase on the number of low-degree modes, namely, the gravity modes. Even if some important and significant questions remain open, for seis mologists to model the interior of the Sun, the solar standard model differs from the Sun's seismic model, by less than 0.2% on the square of sound speed.
4. Evolution of the Sun within the presence of D a r k M a t t e r The dynamical behaviour of various astronomical objects, from galaxies to galaxy clusters and to large-scale structure, in the observed universe can only be understood if the dominant component of the mean matter density is dark, amounting to flm = 0.27±0.1. Constraints from primordial nucleosynthesis of the light elements provide a compelling measure of the mean baryon abundance, fi(, = 0.044 ± 0.02. The bulk of the dark matter is consequently non-baryonic, and the existence of particles that interact with ordinary matter on the scale of the weak force, the so-called Weakly Interacting Massive Particles (WIMPs), provides one of the best-motivated candidates, arising from the lightest stable particle predicted by SUSY, for solving this problem. The gravitational field created by the Sun is able to capture to its inte rior, the WIMPS that have velocities smaller that the escape velocity from the Sun's surface. Once captured, the WIMPs are thermalised within the solar core and remain on Keplerian orbits around the solar center, inter acting through elastic scattering with solar nuclei, such as hydrogen and helium, thereby providing an alternative mechanism of energy transport other than radiation. The heat transport is optimized for as ~ crc when the WIMP scale height is roughly equal to its mean free path. ac is a
314
natural geometrical scattering cross-section, depending on the proton mass and the radius and mass of the star, ac — 8 x 10~ 36 cm 2 . In order to be effective in heat transport, the WIMPs must have mean scattering crosssection per baryon in the range of 10 - 4 3 cm 2 < as < 10 _ 3 3 cm 2 , depending upon the annihilation cross-section and mass of the WIMP. The transport of energy by WIMPs falls rapidly outside this range, and it becomes very difficult to test this effect on the solar structure against the solar seismic data. At higher cross-sections the energy is transported locally and the conductivity falls as ac/crs. At lower cross-sections the conductivity falls as osjoc and in addition only a fraction are captured by the Sun (Lopes, Silk and Hansen 2002; Lopes, Bertone, Silk 2002). The net result is a nearly flat temperature distribution (Lopes and Silk 2002), leading to an isother mal core (Fig. 4). Consequently, the central temperature is reduced. This reduction of temperature has two main consequences: since central pres sure support must be maintained, the central density is increased in the WIMP-accreting models, and since less hydrogen is burnt at the centre of the Sun, the central helium abundance and the central molecular weight are smaller than in standard solar models. The increase of the central density and hydrogen partially offset the effect of lowering the central temperature in the central production of energy. In fact, this is the reason why minor changes are required to the initial helium abundance and the mixing-length parameter in order to produce a solar model of the Sun with the observed luminosity and solar radius (Lopes, Silk and Hansen 2002). This readily leads to a balance between the temperature, T, and the molecular weight, ju, in the core, leading to the peculiar profile of the square of the sound speed, c2 oc T/fi, and the density, p oc /z/T, within the solar interior. This seems to be the case for most of the WIMP-accreting solar models. The presence of WIMPs in the solar core leads to a significant difference be tween the radial profile of the square of the sound speed of the standard solar model and the square of the sound speed for WIMP-accreting solar models and the Sun. This difference is larger that the difference between the inverted square of the sound speed and the square sound speed of the solar standard model. The WIMP-accreting solar models are computed in a way similar to the standard solar model, the only difference being the existence of an alternative mechanism of transport of energy supplemented by the presence of WIMPs. Starting at 0.05 Gyr from the ZAMS, with a standard primordial chemical composition the present solar luminosity and radius is reached at its present age 4.6 Gyr by readjusting Yjn and the ctin. The luminosity in the core of the Sun is presently known with a precision
315
15.6
V
\ \ 00
0.02
0.04
0.06
solar radius
Figure 4. This figure represents the variation of the radial distribution of the temperature on the Sun's nuclear region, for the solar standard model and solar models with different concentration of WIMPs in the solar core. The different curves correspond to the following scalar scattering cross-sections: 10 cm2 (gray continuous curve), 10 -36 cm 2 (black dashed curve) and 1 0 - 4 0 cm2 (gray dashed curve). The standard evolution of the Sun is represented by the black continuous curve. The evolution of the Sun occurs within an halo of WIMPs of mass of 100 GeV and annihilation cross-section of 1 0 - 3 4 cm3 /sec. of one part in 1 0 - 3 . In the coming years, it is very likely that the new seismic data available from the SOHO experiments, will allow us to obtain a seismic model of the Sun with an accuracy of 10~ 5 . In such conditions, the Sun can and should be used as an excellent probe for dark matter in our own galaxy. In Fig. 5, we compute the ratio of the WIMP luminosity against the Sun's luminosity produced in the inner core of 5% of the solar radius. A significant region of the <7geot — m x plot shows changes in the solar luminosity of the order of 10~ 3 . This order of magnitude on the luminosity produced in the solar core can be tested through seismological data. 5. The Future of the Solar internal structure The observed Sun represents a powerful tool that can be used to constrain the dark matter particle parameters within the expected experimental val ues. Although important questions still remain that need to be addressed regarding the structure of the Sun, such as the asymmetric macroscopic motions in the core, the dynamical effects in the nuclear reaction rates and the chemical abundances in the nuclear region (Turck-Chieze et al. 2001), seismic analysis has been a powerful diagnostic tool for the Sun's interior.
316
This has led us to constrain the WIMP parameters based on the proposi tion that the present standard solar model is an accurate approximation of the observed Sun.
Figure 5. This figure illustrates the relative variation of the core luminosity of the Sun, produced in the solar interior by the presence of WIMPs, as a function of WIMPs mass and scattering cross-section. Annihilation cross section is {crav) — 10~36 cm . The iso-curves represent the decimal logarithm of the ratio of the luminosity of WIMPs against the Sun's luminosity in the inner core of 5% of the solar radius. The computation was made using the structure of the present Sun (*0 ~ 4.6 Gyr) which was obtained by using an evolution code that assumes the evolution of the Sun under a standard evolution. The solid curves represent the current experimental bounds placed by DAMA (Bernabei et al. 2000; black grey solid line) and CDMS (Schnee 1999; black solid line), and the future projects of CDMS (black dashed line), GENIUS (Baudis 1998; black gray dashed line) and CRESST (Bravin et al. 19999; light grey dashed line).
The WIMP-accreting solar models are not certainly an unequivocal con straint to the WIMP parameters. If we had modified some of the solar parameters within the error bars, like for example the age of the Sun, we would probably have found other solutions. However, the general results, as far as the WIMPs are concerned, would be qualitatively the same. It should be noticed that the resolution in the inner core is relatively poor; about 0.05 Ro, which is insufficient to detect small discontinuities, prob ably due to the WIMPs isothermal core. This certainly justifies further searching for gravity modes and better measurement of the solar neutrino fluxes. After 20 years of efforts, the acoustic modes have revealed signif-
317
icant macroscopic motions in the solar interior. These processes start to be included in the evolution of the Sun. Some of these, like microscopic diffusion, already belong to our accepted view of the Suns evolution, the so-called Standard Solar Model. The role of rotation and magnetic field only now start to be successfully probed by Helioseismology. Continuing the search of gravity modes would be the best probe to disentangle different physical processes present in the solar nuclear region which can be missed otherwise. In the last decade the aim of research in this field has gradually changed from a direct check of the central temperature and proton-proton nuclear energy generation, to a double goal: a detailed verification of the solar nuclear plasma and a possible test of the neutrino oscillation models. The Sun appears today as a natural laboratory of nuclear, plasma, atomic physics and particle physics, that gives access to a range of neutrino oscil lation parameters which is not accessible in the Earth laboratories. Finally, it is worth noticing that this study of dark matter candidates can be extended to other stars, apart the Sun. The high density of dark matter towards the centre of the Galaxy, which can be as much as one million times the density in the solar neighbourhood, can produce signifi cant changes in a star evolving in these dense dark matter regions. Massive stars are not affected but the low-mass stars are expected to be shifted into the red region of the Hertzsprung-Russell diagram (Salati & Silk 1989). Therefore, astroseismology constitutes a new window for the research of dark matter. Theoretically, the low degree modes are the most sensitive to the conditions of the nuclear region, the only region where dark mat ter can produce substantial deviations from the standard evolution theory. Indeed, these dark matter stellar models have oscillation modes with fre quencies spacing which are markedly different from those of other stellar models, depending upon the concentration of dark matter in their cores. The concentration of WIMPs within the stars and the nature of these par ticles can be better known if, by using seismic diagnostics, we could infer the structure and evolution changes produced by the presence of dark mat ter. We will be able to test this theoretical work using the new generation of satellites dedicated to seismology, such as Eddington. This satellite will carry a state-of-the-art experiment designed to make fundamental measure ments of unprecedented quality, enabling us to probe a large variety of stars from their cores to their transition regions. Studies of the solar and stellar physics, particle physics and cosmology may have much to gain from each other.
318
References 1. Q. R. Ahmad et al, Phy. Rev. Let., submitted (2001) 2. J. N. Bahcall, Neutrino Astrophysics, Cambridge University Press, Cam bridge, 1989. 3. J. N. Bahcall, Nature, 412, 29, 2001. 4. R. Bernabei et al. [DAMA Collaboration] 2000, Phys. Lett. B 480, 23. 5. M Bravin et al. (CRESST CoUab.) 1999, hep-ex/9904005. 6. L. Baudis et al. (GENIUS CoUab.) 1998, Phys. Rep. 307, 301. 7. J. Christensen-Dalsgaard et al. 1996, Science, 272, 1286. 8. D. O. Gough et al, 1996, Sci, 272, 1296 9. I. P. Lopes, D. O. Gough 2001, MNRAS 322, 473. 10. I. P. Lopes, J. Silk, S. H. Hansen, 2002, MNRAS, 331, 361-368 [astroph/0111530]. 11. I. P. Lopes, G. Bertone, J. Silk, 2002, MNRAS, 337, 1179-1184. 12. I. P. Lopes and J. Silk, 2002, Physical Review Letters, 88, 151303. [astroph/0112390]. 13. I. P. Lopes, J. Silk 2003, accepted for publication in MNRAS. 14. A. G. Kosovichev, 1999, J. Comp. Appl. Math, 109, 1. 15. P. Salati and J. Silk, Astrophys. J. 338 (1989) 24. 16. R. Schnee [for the CDMS CoUab.], talk presented at Inner Space/Outer Space II," Fermilab, May 1999. 17. S. Turck-Chieze, I. Lopes, 1993, Astrophys. J., 408, 347 18. S. Turck-Chieze et al, 2001 , Astrophys. J., 555, L69.
LIST OF PARTICIPANTS
Maria da Conceicao Abreu Universidade do Algarve FCT - Dept. de Fisica Campus de Gambelas 8000-117 Faro Portugal [email protected]
Luisa Arruda LIP Av. Elias Garcia, 14 - 1° 1000-146 Lisboa Portugal [email protected] Pedro Assis LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected]
Pedro Abreu LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected] Nils Andersson Dept. of Mathematics University of Southampton, Highfield Southampton S0171BJ United Kingdom [email protected]
Fernando Barao LIP Av. Elias Garcia, 14 - 1° 1000-146 Lisboa Portugal [email protected] Alain de Bellefon CNRS/IN2P3 College de France - Physique Corpusculaire F75231 - Paris Cedex 05 France [email protected]
Sofia Andringa LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected]
Daniel Bertrand Brussels Free University Faculte des Sciences CP230 Boulevard du Triomphe, B-1050 Brussels Belgium [email protected]
Nuno Anjos LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected] Anna Anzalone IASF - Palermo/CNR Via Ugo la Malfa 153 90146 Palermo Italy [email protected]
Alain Blanchard LAOMP 14 Av. Edouard Bellin, 31000 Toulouse France [email protected]
319
320 Sergey Blinnikov ITEP 117218 B. Cheremushk, 25 Moscow Russia [email protected]
Giacomo D'Ali IASF - Palermo/CNR Via Ugo la Malfa 153 90146 Palermo Italy [email protected]
Joao Borges LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected]
Jorge Dias de Deus 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]
Vitor Cardoso 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]
Oscar Dias 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]
Pedro Castelo 1ST Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]
Catarina Espfrito Santo LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected]
Nuno Castro LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal Nuno. castro @ cern. ch
Sebastien Fabbro 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]
Osvaldo Catalano IASF - Palermo/CNR Via Ugo la Malfa 153 90146 Palermo Italy [email protected]
Manuel Fiolhais Universidade de Coimbra Departamento de Fisica 3004-516 Coimbra Portugal [email protected]
321 Paulo Freire Arecibo Observatory HC 3 Box 53995 Arecibo, Puerto Rico 00612 USA [email protected] Norman Glendenning Nuclear Science Division MS 70-319 Lawrence Berkeley Laboratory Berkeley California 94720 USA [email protected] Ariel Goobar University of Stockholm Dept. of Physics SCFAB SE-106 91 Stokholm Sweden [email protected]
Michael Hasinoff CERN/University of British Columbia CH-1211 Geneva 23 Switzerland [email protected] Alfredo Barbosa Henriques 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected] Alexander Krasnitz Universidade do Algarve FCT - CENTRA Campus de Gambelas 8000-117 Faro Portugal [email protected]
Patricia Goncalves LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected]
Ralf Lehnert Universidade do Algarve FCT - CENTRA Campus de Gambelas 8000-117 Faro Portugal [email protected]
Gino Gugliotta IASF - Palermo/CNR Via Ugo la Malfa 153 90146 Palermo Italy gugliotta @ pa.iasf.cnr.it
Jose Sande Lemos 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]
John Hargreaves Royal Holloway, Univ. of London Egham, Surrey TW20 OEX United Kingdom [email protected]
Peter Levai 5 CSIKI - HEGYEK Street Budapeste, 1118 Hungary [email protected]
322 Ilidio Lopes 1ST - CENTRA/Oxford Av. Rovisco Pais 1049-001 Lisboa Portugal lopes @ astro.ox.ac.uk
Dmitry Naumov LAPP Chemin de Bellevue - BPllO 74941 Annecy-le-Vieux Cedex France [email protected]
Maria Concetta Maccarone IASF - Palermo/CNR Via Ugo la Malfa 153 90146 Palermo Italy [email protected]
Rui Neves Universidade do Algarve FCT - CENTRA Campus de Gambelas 8000-117 Faro Portugal [email protected]
Jose Maneira Queen's University Physics Department - Stirling Hall Kingston, Ontario Canada K7L 437 [email protected] Matthew Mewes Indiana University Physics Department Bloomington, IN 47405 USA mmewes @ indiana.edu Sandra Moreno LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected] Ana Maria Mourao 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal amourao @ aries.ist.utl.pt
Antonio Onofre LIP Av. Elias Garcia, 14 - 1° 1000-146 Lisboa Portugal antonio.onofre @ cern.ch Catarina Ortigao LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected] Rodrigo Pascoal 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected] Nils Pickert Physikalisches Institut II Erwin - Rommel - Str. 1 D91058 Erlangen Germany [email protected]
323 Mario Pimenta LIP Av. Elias Garcia, 1 4 - 1° 1000-146 Lisboa Portugal [email protected]
Kyan Schahmaneche LPNHE, Paris 7 University 4 place Jussieu 75252 Paris cedex 05 France ky an @ Ipnhep.in2p3. fr
Eric Plagnol CDF 11, place M. Berthelot 75005 Paris France
Joao Seixas 1ST - CFIF Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]
Robertus Potting Universidade do Algarve FCT - CENTRA Campus de Gambelas 8000-117 Faro Portugal [email protected] Jorge Romao 1ST - CFIF Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected] Paulo M. Sa Universidade do Algarve FCT - CENTRA Campus de Gambelas 8000-117 Faro Portugal [email protected] Livio Scarsi IASF - Palermo/CNR Via Ugo la Malfa 153 90146 Palermo Italy [email protected]
Yuli Shabelski Petersburg Nuclear Physics Inst. Gatchina, St. Petersburg 188300 Russia [email protected] Adriana Silva Universidade do Porto Faculdade de Ciencias Portugal [email protected] Pedro Silva LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected] George Smoot University of California Lawrence Berkeley Laboratory Astrophysics Group 366 Le Conte Hall, Berkeley CA 94720 USA smoot @ physics .berkeley.edu
324 Peter Sonderegger LIP/CERN Av. Elias Garcia, 14 - 1° 1000-146 Lisboa Portugal Peter, sonderegger @ cern. ch
Miguel Vazquez-Mozo CERN - Th CH-1211, Geneva 23 Switzerland [email protected]
Manfred Steuer CERN-EP (MIT-LNS) CH-1211 Geneva 23 Switzerland [email protected]
Hector de Vega LPTHE - University Paris VI Tour 16, l er . et., 4 Place Jussieu 75252 Paris cedex 05 France [email protected]
Bernardo Tome LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected]
Filipe Veloso LIP Av. Elias Garcia, 1 4 - 1 ° 1000-146 Lisboa Portugal [email protected]
Brigitte Tome Universidade do Algarve FCT - CENTRA Campus de Gambelas 8000-117 Faro Portugal [email protected]
Gabriele Veneziano CERN - Th CH-1211, Geneva 23 Switzerland gabriele. veneziano @cern.ch
Piero Vallania INFN - Torino Via Pietro Giuria 1, 10125 Torino Italy [email protected] Joao Varela LIP Av. Elias Garcia, 14 - 1° 1000-146 Lisboa Portugal joao. varela @ cern.ch
Shijun Yoshida 1ST - CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected] Enrique Zas University of Santiago Compostela Dep. Fisica Particulas E-15706 Santiago de Compostela Spain [email protected]
T h e Fourth International W o r k s h o p o n N e w W o r l d s in A s t r o p a r t i c l e P h y s i c s w a s t h e latest in the biennial s e r i e s , held in Faro, Portugal. The program included both invited a n d contributed talks. Each of the s e s s i o n s o p e n e d with
NEW WORLDS IN !
a p e d a g o g i c a l o v e r v i e w of the current state of the respective field. T h e following topics w e r e c o v e r e d : cosmological parameters; neutrino physics and astrophysics; gravitational w a v e s ; b e y o n d standard models: strings; cosmic rays: origin, propagation and i n t e r a c t i o n ; matter under e x t r e m e
I conditions; supernovae and dark Proceedings of the Fourth International Workshop
matter.
ISBN 981-238-584-3
14/M/n/. worldscientific.com 5377 he
9 l 789812"385840 l
