An Introduction to Mathematical Cosmology

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the first edition......Page 11
Preface to the second edition......Page 13
Dedication......Page 15
1 Some basic concepts and an overview of cosmology......Page 17
2.1 Summary of general relativity......Page 28
2.2.1 Killing vectors......Page 34
2.2.2 Tensor densities......Page 37
2.2.3 Gauss and Stokes theorems......Page 40
2.2.4 The action principle for gravitation......Page 44
2.2.5 Some further topics......Page 48
3.1 A simple derivation of the Robertson–Walker metric......Page 53
3.2 Some geometric properties of the Robertson–Walker metric......Page 58
3.3 Some kinematic properties of the Robertson–Walker metric......Page 61
3.4 The Einstein equations for the Robertson–Walker metric......Page 67
3.5 Rigorous derivation of the Robertson–Walker metric......Page 69
4.1 Introduction......Page 76
4.2 Exact solution for zero pressure......Page 80
4.3 Solution for pure radiation......Page 83
4.5 Exact solution connecting radiation and matter eras......Page 84
4.6 The red-shift versus distance relation......Page 87
4.7 Particle and event horizons......Page 89
5.1 Introduction......Page 92
5.2 Measurement of H0......Page 93
5.3 Measurement of q0......Page 96
5.4 Further remarks about observational cosmology......Page 101
Appendix to Chapter 5......Page 106
6.1 Introduction......Page 110
(ii) Lemaître model (Lemaître, 1927, 1931)......Page 112
(iii) Eddington–Lemaître model......Page 113
6.2 Further remarks about the cosmological constant......Page 114
6.3 Limits on the cosmological constant......Page 116
6.4.1 Introduction......Page 118
6.4.2 An exact solution with cosmological constant......Page 120
6.4.3 Restriction of parameter space......Page 123
7.1 Introduction......Page 128
7.2 Homogeneous cosmologies......Page 129
7.3 Some results of general relativistic hydrodynamics......Page 131
7.4 Defenition of singularities......Page 134
7.5 An example of a singularity theorem......Page 136
7.6 An anisotropic model......Page 137
7.7 The oscillatory approach to singularities......Page 138
7.8 A singularity-free universe?......Page 142
8.1 Introduction......Page 144
(i) First frame......Page 145
(iv) Fourth frame......Page 146
(v) Fifth frame......Page 147
(vi) Sixth frame......Page 148
8.2 The very early universe......Page 151
8.3 Equations in the early universe......Page 158
8.4 Black-body radiation and the temperature of the early universe......Page 159
8.5 Evolution of the mass-energy density......Page 164
(i) 1012K>T>5.5 x109K......Page 165
(ii) 5.5 x109K>T>109K......Page 166
(iii) T<109K......Page 167
8.6 Nucleosynthesis in the early universe......Page 169
8.7 Further remarks about helium and deuterium......Page 175
8.8 Neutrino types and masses......Page 180
9.1 Introduction......Page 182
9.2 Inflationary models – qualitative discussion......Page 183
9.3 Inflationary models – quantitative description......Page 190
9.4 An exact inflationary solution......Page 194
9.5 Further remarks on inflation......Page 196
9.6 More inflationary solutions......Page 199
Appendix to Chapter 9......Page 202
10.1 Introduction......Page 205
10.2 Hamiltonian formalism......Page 207
10.3 The Schrödinger functional equation for a scalar field......Page 211
10.4 A functional differrential equation......Page 213
10.6 The free electromagnetic field......Page 215
10.7 The Wheeler–De Witt equation......Page 217
10.8 Path integrals......Page 218
10.9 Conformal fluctuations......Page 222
10.10 Further remarks about quantum cosmology......Page 225
11.2 Three ways for a star to die......Page 227
11.3 Galactic and supergalactic black holes......Page 229
11.4 Black-hole evaporation......Page 231
11.5 Slow and subtle changes......Page 232
11.6 A collapsing universe......Page 234
A2 Neutrino types......Page 236
A3 A critique of the standard model......Page 238
A4 An accelerating universe?......Page 239
A5 Particle physics and quantum field theory......Page 240
A6 Cosmic background radiation......Page 243
A8 Galactic distribution......Page 244
A10 Cosmic book of phenomena......Page 245
A11 Cosmic strings......Page 247
A12 Topological structures......Page 248
A13 Extended inflation......Page 249
A14 Quantum cosmology......Page 250
Bibliography......Page 254
Index......Page 263