Contents
Preface xi
1 Reconstructing time 1
1.1 The patterns of the stars 1
1.2 Structural relics 2
1.3 Material relics 4
1.4 Ethereal relics 5
1.5 Cosmological principles 6
1.6 Theories 7
1.7 Problems 9
2 Expansion 10
2.1 The redshift 10
2.2 The expanding Universe 11
2.3 The distance scale 14
2.4 The Hubble constant 15
2.5 The deceleration parameter 16
2.6 The age of the Universe 16
2.7 The steady-state theory 17
2.8 The evolving Universe 18
2.9 Problems 19
3 Matter 21
3.1 The mean mass density of the Universe 21
3.1.1 The critical density 21
3.1.2 The density parameter 21
3.1.3 Contributions to the density 22
3.2 Determining the matter density 23
3.3 The mean luminosity density 24
3.3.1 Comoving volume 24
3.3.2 Luminosity function 25
3.3.3 Luminosity density 25
3.4 The mass-to-luminosity ratios of galaxies 25
3.4.1 Rotation curves 26
vi Contents
3.4.2 Elliptical galaxies 28
3.5 The virial theorem 28
3.6 The mass-to-luminosity ratios of rich clusters 28
3.6.1 Virial masses of clusters 29
3.7 Baryonic matter 30
3.8 Intracluster gas 31
3.9 The gravitational lensing method 32
3.10 The intercluster medium 33
3.11 The non-baryonic dark matter 33
3.12 Dark matter candidates 34
3.12.1 Massive neutrinos? 34
3.12.2 Axions? 35
3.12.3 Neutralinos? 36
3.13 The search for WIMPS 36
3.14 Antimatter 38
3.15 Appendix. Derivation of the virial theorem 39
3.16 Problems 39
4 Radiation 41
4.1 Sources of background radiation 41
4.1.1 The radio background 41
4.1.2 Infrared background 43
4.1.3 Optical background 43
4.1.4 Other backgrounds 44
4.2 The microwave background 45
4.2.1 Isotropy 45
4.3 The hot big bang 47
4.3.1 The cosmic radiation background in the steady-state theory 48
4.4 Radiation and expansion 49
4.4.1 Redshift and expansion 49
4.4.2 Evolution of the Planck spectrum 50
4.4.3 Evolution of energy density 51
4.4.4 Entropy of radiation 52
4.5 Nevertheless it moves 53
4.5.1 Measurements of motion 54
4.6 The x-ray background 56
4.7 Problems 58
5 Relativity 60
5.1 Introduction 60
5.2 Space geometry 61
5.3 Relativistic geometry 62
5.3.1 The principle of equivalence 62
5.3.2 Physical relativity 63
5.4 Isotropic and homogeneous geometry 65
Contents vii
5.4.1 Homogeneity of the 2-sphere 66
5.4.2 Homogeneity of the metric 67
5.4.3 Uniqueness of the space metric 67
5.4.4 Uniqueness of the spacetime metric 68
5.5 Other forms of the metric 68
5.5.1 A radial coordinate related to area 69
5.5.2 A radial coordinate related to proper distance 69
5.6 Open and closed spaces 70
5.7 Fundamental (or comoving) observers 70
5.8 Redshift 71
5.9 The velocity–distance law 73
5.10 Time dilation 74
5.11 The field equations 74
5.11.1 Equations of state 75
5.11.2 The cosmological constant 75
5.11.3 The critical density 76
5.12 The dust Universe 78
5.12.1 Evolution of the density parameter 79
5.12.2 Evolution of the Hubble parameter 79
5.13 The relationship between redshift and time 80
5.13.1 Newtonian interpretation 81
5.14 Explicit solutions 82
5.14.1 p = 0, k = 0, = 0, the Einstein–de Sitter model 82
5.14.2 The case p = 0, k = +1, =0 84
5.14.3 The case p = 0, k = −1, =0 86
5.15 Models with a cosmological constant 87
5.15.1 Negative 87
5.15.2 Positive 88
5.15.3 Positive and critical density 88
5.15.4 The case > 0, k = +1 89
5.16 The radiation Universe 90
5.16.1 The relation between temperature and time 91
5.17 Light propagation in an expanding Universe 92
5.18 The Hubble sphere 93
5.19 The particle horizon 95
5.20 Alternative equations of state 96
5.21 Problems 97
6 Models 101
6.1 The classical tests 101
6.2 The Mattig relation 102
6.2.1 The case p = 0, = 0 103
6.2.2 The general case p = 0, = 0 104
6.3 The angular diameter–redshift test 104
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6.3.1 Theory 104
6.3.2 Observations 106
6.4 The apparent magnitude–redshift test 107
6.4.1 Theory 107
6.4.2 The K-correction 108
6.4.3 Magnitude versus redshift: observations 110
6.5 The geometry of number counts: theory 113
6.5.1 Number counts: observations 114
6.5.2 The galaxy number-magnitude test 115
6.6 The timescale test 118
6.6.1 The ages of the oldest stars 118
6.7 The lensed quasar test 119
6.8 Problems with big-bang cosmology 120
6.8.1 The horizon problem 120
6.8.2 The flatness problem 121
6.8.3 The age problem 122
6.8.4 The singularity problem 122
6.9 Alternative cosmologies 123
6.10 Problems 124
7 Hot big bang 128
7.1 Introduction 128
7.2 Equilibrium thermodynamics 130
7.2.1 Evolution of temperature: relativistic particles 132
7.2.2 Evolution of temperature: non-relativistic particles 132
7.3 The plasma Universe 134
7.4 The matter era 135
7.5 The radiation era 136
7.5.1 Temperature and time 136
7.5.2 Timescales: the Gamow criterion 137
7.6 The era of equilibrium 138
7.7 The GUT era: baryogenesis 138
7.7.1 The strong interaction era 139
7.7.2 The weak interaction era: neutrinos 140
7.7.3 Entropy and e− − e+ pair annihilation 140
7.8 Photon-to-baryon ratio 141
7.9 Nucleosynthesis 142
7.9.1 Weak interactions: neutron freeze-out 143
7.9.2 Helium 144
7.9.3 Light elements 146
7.9.4 Abundances and cosmology 146
7.10 The plasma era 148
7.10.1 Thomson scattering 148
7.10.2 Free–free absorption 149
Contents ix
7.10.3 Compton scattering 150
7.11 Decoupling 151
7.12 Recombination 151
7.13 Last scattering 153
7.14 Perturbations 153
7.15 Appendix A. Thermal distributions 154
7.15.1 Chemical potentials 154
7.15.2 Photon energy density 156
7.15.3 Photon number density 157
7.15.4 Relativistic neutrinos 157
7.15.5 Relativistic electrons 158
7.15.6 Entropy densities 158
7.16 Appendix B. The Saha equation 159
7.17 Appendix C. Constancy of η 159
7.18 Problems 160
8 Inflation 163
8.1 The horizon problem 164
8.2 The flatness problem 165
8.3 Origin of structure 165
8.4 Mechanisms 167
8.4.1 Equation of motion for the inflaton field 168
8.4.2 Equation of state 169
8.4.3 Slow roll 170
8.5 Fluctuations 172
8.6 Starting inflation 172
8.7 Stopping inflation 173
8.7.1 Particle physics and inflation 175
8.8 Topological defects 176
8.9 Problems 176
9 Structure 179
9.1 The problem of structure 179
9.2 Observations 180
9.2.1 The edge of the Universe 181
9.3 Surveys and catalogues 181
9.4 Large-scale structures 182
9.5 Correlations 183
9.5.1 Correlation functions 183
9.5.2 Linear distribution 185
9.5.3 The angular correlation function 185
9.5.4 Results 185
9.6 Bias 187
9.7 Growth of perturbations 187
9.7.1 Static background, zero pressure 188
x Contents
9.7.2 Expanding background 189
9.8 The Jeans’ mass 190
9.9 Adiabatic perturbations 192
9.10 Isocurvature (isothermal) perturbations 193
9.11 Superhorizon size perturbations 194
9.12 Dissipation 194
9.13 The spectrum of fluctuations 194
9.14 Structure formation in baryonic models 196
9.15 Dark matter models 197
9.15.1 Growth of fluctuations in dark matter models 197
9.16 Observations of the microwave background 198
9.17 Appendix A 200
9.18 Appendix B 202
9.19 Problems 203
10 Epilogue 205
10.1 Homogeneous anisotropy 205
10.1.1 Kasner solution 206
10.2 Growing modes 207
10.3 The rotating Universe 208
10.4 The arrow of time 208
Reference material 210
Constants 210
Useful quantities 210
Formulae 211
Symbols 212
References 213
Index 217
Preface
In this book we have attempted to present cosmology to undergraduate students
of physics without assuming a background in astrophysics. We have aimed at a
level between introductory texts and advanced monographs. Students who want
to know about cosmology without a detailed understanding are well served by
the popular literature. Graduate students and researchers are equally well served
by some excellent monographs, some of which are referred to in the text. In
setting our sights somewhere between the two we have aimed to provide as much
insight as possible into contemporary cosmology for students with a background
in physics, and hence to provide a bridge to the graduate literature. Chapters 1 to
4 are introductory. Chapter 7 gives the main results of the hot big-bang theory.
These could provide a shorter course on the standard theory, although we would
recommend including part of chapter 5, and also the later sections of chapter 6 on
the problems of the standard theory, and some of chapter 8, where we introduce
the current best buy approach to a resolution of these problems, the inflation
model. Chapters 5 and 6 offer an introduction to relativistic cosmology and to the
classical observational tests. This material does not assume any prior knowledge
of relativity: we provide the minimum background as required. Chapters 1 to 4
and some of 5 and 6 would provide a short course in relativistic cosmology. Most
of chapter 5 is a necessary prerequisite for an understanding of the inflationary
model in chapter 8. In chapter 9 we discuss the problem of the origin of structure
and the correspondingly more detailed tests of relativistic models. Chapter 10
introduces some general issues raised by expansion and isotropy. We are grateful
to our referees for suggesting improvements in the content and presentation.
We set out to write this book with the intention that it should be an updated
edition of The Isotropic Universe published by one of us in 1984. However, as
we began to discard larger and larger quantities of the original material it became
obvious that to update the earlier work appropriately required a change in the
structure and viewpoint as well as the content. This is reflected in the change of
title, which is itself an indication of how far the subject has progressed. Indeed, it
would illuminate the present research paradigm better to speak of the Anisotropic
Universe, since it is now the minor departures from exact isotropy that we expect
to use in order to test the details of current theories. The change of title is at least
in part a blessing: while we have met many people who think of the ‘expanding
xi
xii Preface
Universe’ as the Universe, only more exciting, we have not come across anyone
who feels similarly towards the ‘isotropic Universe’.
We have also taken the opportunity to rewrite the basic material in order to
appeal to the changed audience that is now the typical undergraduate student of
physics. So no longer do we assume a working knowledge of Fourier transforms,
partial differentiation, tensor notation or a desire to explore the tangentialmaterial
of the foundations of the general theory of relativity. In a sense this is counter
to the tenor of the subject, which has progressed by assimilation of new ideas
from condensed matter and particle physics that are even more esoteric and
mathematical than those we are discarding. Consequently, these are ideas we can
only touch on, and we have had to be content to quote results in various places as
signposts to further study.
Nevertheless, our aim has been to provide as much insight as possible into
contemporary cosmology for students with a background in physics. A word of
explanation about our approach to the astrophysical background might be helpful.
Rather than include detours to explain astrophysical terms we have tried to make
them as self-explanatory as required for our purposes from the context in which
they appear. To take one example. The reader will not find a definition of an
elliptical galaxy but, from the context in which the term is first used, it should be
obvious that it describes a morphological class of some sort, which distinguishes
these from other types of galaxy. That is all the reader needs to know about this
aspect of astrophysics when we come to determinations of mass density later in
the book.
A final hurdle for some students will be the mathematics content. To help
we have provided some problems, often with hints for solutions. We have tried to
avoid where possible constructions of the form ‘using equations . . . it is readily
seen that’. Nevertheless, although the mathematics in this book is not in itself
difficult, putting it together is not straightforward. You will need to work at it. As
you do so we have the following mission for you.
It is sometimes argued, even by at least one Nobel Laureate, that
cosmologists should be directed away from their pursuit of grandiose selftitillation
at the taxpayers’ expense to more useful endeavours (which is usually
intended to mean biology or engineering). You cannot counter this argument by
reporting the contents of popular articles—this is where the uninformed views
come from in the first place. Instead, as you work through the technical details of
this book, take a moment to stand back and marvel at the fact that you, a more or
less modest student of physics, can use these tools to begin to grasp for yourself
a vision of the birth of a whole Universe. And in those times of dark plagues and
enmities, remember that vision, and let it be known.
D J Raine
E G Thomas
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