Series in Astronomy and Astrophysics An Introduction to the Science of Cosmology Derek Raine Department of Physics and Astronomy University of Leicester, UK Ted Thomas Department of Physics and Astronomy University of Leicester, UK IoP Institute of Physics Publishing Bristol and Philadelphia
Preface 1 Reconstructing time 1.1 The patterns of the stars 1.2 Structural relics 1.3 Material relics 1.4 Ethereal relics 1.5 Cosmological principles 1.6 Theories 1.7 Problems 2 Expansion 2.1 The redshift 2.2 The expanding Universe 2.3 The distance scale 2.4 The Hubble constant 2.5 The deceleration parameter 2.6 The age of the Universe 2.7 The steady-state theory 2.8 The evolving Universe 2.9 Problems 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 xi 1 1 2 4 5 6 7 9 10 10 11 14 15 16 16 17 18 19
vi 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 4.7 The x-ray background Problems 5 Relativity 5.1 Introduction 5.2 Space geometry 5.3 Relativistic geometry 5.3.1 The principle of equivalence 5.3.2 Physical relativity 5.4 Isotropic and homogeneous geometry 56 58 60 60 61 62 62 63 65
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 A 73 5.10 Time dilation 5.11 The field equations 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, A = 0, the Einstein-de Sitter model 82 5.14.2 The case/? = 0, k =+1, A =0 84 5.14.3 The case p = 0, k =-1, A = 0 86 5.15 Models with a cosmological constant 87 5.15.1 Negative A 87 5.15.2 Positive A 88 5.15.3 Positive A and critical density 88 5.15.4 The case A > 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 Models 101 6.1 The classical tests, 101 6.2 The Mattig relation 102 6.2.1 The case p = 0, A = 0 103 6.2.2 The general case p = 0, A 0 104 6.3 The angular diameter-redshift test 104
viii 6.3.1 Theory 6.3.2 Observations 6.4 The apparent magnitude-redshift test 6.4.1 Theory 6.4.2 The K-correction 6.4.3 Magnitude versus redshift: observations 6.5 The geometry of number counts: theory 6.5.1 Number counts: observations 6.5.2 The galaxy number-magnitude test 6.6 The timescale test 6.6.1 The ages of the oldest stars 6.7 The lensed quasar test 6.8 Problems with big-bang cosmology 6.8.1 The horizon problem 6.8.2 The flatness problem 6.8.3 The age problem 6.8.4 The singularity problem 6.9 Alternative cosmologies 6.10 Problems 7 Hot big bang 7.1 Introduction 7.2 Equilibrium thermodynamics 7.2.1 Evolution of temperature: relativistic particles 7.2.2 Evolution of temperature: non-relativistic particles 7.3 The plasma Universe 7.4 The matter era 7.5 The radiation era 7.5.1 Temperature and time 7.5.2 Timescales: the Gamow criterion 7.6 The era of equilibrium 7.7 The GUT era: baryogenesis 7.7.1 The strong interaction era 7.7.2 The weak interaction era: neutrinos 7.7.3 Entropy and e~ e + pair annihilation 7.8 Photon-to-baryon ratio 7.9 Nucleosynthesis 7.9.1 Weak interactions: neutron freeze-out 7.9.2 Helium 7.9.3 Light elements 7.9.4 Abundances and cosmology 7.10 The plasma era 7.10.1 Thomson scattering 7.10.2 Free-free absorption 104 106 107 107 108 110 113 114 115 118 118 119 120 120 121 122 122 123 124 128 128 130 132 132 134 135 136 136 137 138 138 139 140 140 141 142 143 144 146 146 148 148 149
ix 7.10.3 Compton scattering 7.11 Decoupling 7.12 Recombination 7.13 Last scattering 7.14 Perturbations 7.15 Appendix A. Thermal distributions 7.15.1 Chemical potentials 7.15.2 Photon energy density 7.15.3 Photon number density 7.15.4 Relativistic neutrinos 7.15.5 Relativistic electrons 7.15.6 Entropy densities 7.16 Appendix B. The Saha equation 7.17 Appendix C. Constancy of r\ 7.18 Problems Inflation 8.1 The horizon problem 8.2 The flatness problem 8.3 Origin of structure 8.4 Mechanisms, 8.4.1 Equation of motion for the inflaton field 8.4.2 Equation of state 8.4.3 Slow roll 8.5 Fluctuations 8.6 Starting inflation 8.7 Stopping inflation 8.7.1 Particle physics and inflation 8.8 Topological defects 8.9 Problems Structure 9.1 The problem of structure 9.2 Observations 9.2.1 The edge of the Universe 9.3 Surveys and catalogues 9.4 Large-scale structures 9.5 Correlations 9.5.1 Correlation functions 9.5.2 Linear distribution 9.5.3 The angular correlation function 9.5.4 Results 9.6 Bias 9.7 Growth of perturbations 9.7.1 Static background, zero pressure 150 151 151 153 153 154 154 156 157 157 158 158 159 159 160 163 164 165 165 167 168 169 170 172 172 173 175 176 176 179 179 180 181 181 182 183 183 185 185 185 187 187 188
x 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