The primordial density perturbation: cosmology, inflation and the origin of structure

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Frequently used symbols......Page 16
Preface......Page 19
1 Overview......Page 21
Units......Page 23
Reference......Page 24
Part I Relativity......Page 25
2.1 Minkowski coordinates and the relativity principle......Page 27
2.2.1 4-scalars and 4-vectors......Page 30
2.2.2 4-tensors......Page 31
2.3 Spacetime lines and geodesics......Page 32
2.4.1 Velocity field......Page 34
2.4.2 Conserved currents......Page 35
2.5.1 Energy, momentum and stress......Page 36
2.5.2 Energy and momentum conservation......Page 37
2.5.3 Angular momentum conservation......Page 38
2.6.1 Distribution function......Page 39
2.6.2 Electromagnetic current and energy–momentum tensor......Page 40
2.7.1 Liouville equation......Page 41
2.7.2 Collision term......Page 42
2.7.3 Thermal equilibrium......Page 44
Exercises......Page 45
3.1.1 Vectors and tensors......Page 46
3.1.2 Covariant derivative......Page 48
3.2 Special relativity with generic coordinates: laws of physics......Page 50
3.3.1 Metric tensor......Page 51
3.3.2 Curvature tensor......Page 52
3.3.4 Local observers......Page 53
3.4 Curved space and curved surfaces......Page 54
3.6.1 Field equation......Page 55
3.6.2 Newtonian gravity......Page 57
3.6.3 Gravitational waves......Page 58
3.7.1 Scale factor......Page 59
3.7.2 Geometry of space......Page 60
3.8 Hubble parameter and horizons......Page 62
3.9 Inflation and the Big Bang......Page 64
3.10.2 Friedmann equation......Page 65
Exercises......Page 67
References......Page 68
Part II The Universe after the first second......Page 69
4.1 Temperature and redshift......Page 71
4.2 Thermal equilibrium in the early Universe......Page 72
4.3 Baryon and lepton number......Page 74
4.4.1 Initial thermal equilibrium......Page 76
4.4.2 Neutrino decoupling and electron–positron annihilation......Page 78
4.4.3 Nucleosynthesis......Page 79
4.5 The ΛCDM model......Page 80
4.6 Evolution of the scale factor......Page 83
4.7.1 The optical depth and the visibility function......Page 85
4.7.2 Recombination and decoupling......Page 86
4.8 Neutrino mass......Page 87
Exercises......Page 88
References......Page 89
5.1.1 Epoch of horizon entry......Page 90
5.1.2 Primordial density perturbations......Page 91
5.1.3 Main features of the primordial density perturbation......Page 92
5.1.4 Other types of primordial perturbation......Page 93
5.2.1 Defining the perturbations......Page 94
5.2.2 Smoothing......Page 95
5.3.2 Separate universe assumption......Page 96
5.3.3 Strategy for this book......Page 97
5.4.1 Definition......Page 98
5.4.2 Conservation......Page 99
5.4.3 The deltaN formula......Page 100
5.5.1 Gauge transformation......Page 101
5.5.2 Primordial density perturbation......Page 102
Exercises......Page 103
Reference......Page 104
6.1 Random fields......Page 105
6.2 Fourier expansion......Page 107
6.3.1 Momentum space......Page 108
6.3.2 Position space......Page 110
6.4.1 Correlators......Page 111
6.5.2 Cosmic variance......Page 114
6.6 Spherical expansion......Page 117
6.7.1 Spectrum and spectral index......Page 119
6.7.2 Non-gaussianity......Page 120
Exercises......Page 121
References......Page 122
7.1 Free-streaming, oscillation, and collapse......Page 123
7.2.1 Newtonian cosmology......Page 125
7.2.2 Scalar and vector modes......Page 127
7.2.3 Solution for the scalar mode......Page 128
7.3 Effect of the cosmological constant......Page 129
7.4.1 Separate equations......Page 130
7.4.2 Growth and oscillation......Page 131
7.4.3 Baryon Jeans mass......Page 132
References......Page 134
8.1 Scalar, vector, and tensor modes......Page 135
8.2.1 Metric......Page 138
8.2.3 Scalar mode......Page 140
8.3.2 Evolution equations......Page 141
8.3.3 Initial conditions......Page 142
8.4 Separate fluids......Page 143
8.5 Matter density transfer function......Page 145
8.6 Acoustic oscillation......Page 147
8.7.1 Thomson scattering......Page 149
8.7.3 Acoustic oscillation at decoupling......Page 151
Exercises......Page 152
9 The matter distribution......Page 154
9.1 Smoothing......Page 155
9.2 Bottom-up structure formation......Page 156
9.3 Critical density for collapse......Page 159
9.4 Virialization......Page 162
9.5 Abundance of premature objects......Page 164
9.6.2 Galaxy distribution......Page 166
9.6.3 Inter-galactic baryons......Page 169
Exercises......Page 170
References......Page 171
10 Cosmic microwave background anisotropy......Page 172
10.1 CMB multipoles......Page 173
10.2 Spectrum of the CMB anisotropy......Page 175
10.3 Flat-sky approximation......Page 176
10.4 Scalar mode......Page 178
10.5 Sudden-decoupling approximation......Page 179
10.6.1 Height of the plateau......Page 182
10.6.2 Very large scale contribution......Page 184
10.7 Acoustic peaks and Silk damping......Page 185
10.8 Reionization......Page 187
10.9 Non-gaussianity of the CMB anisotropy......Page 188
Exercises......Page 189
References......Page 190
11.1 Perturbed Boltzmann equation......Page 191
11.2 Boltzmann hierarchy......Page 193
11.3.1 Unpolarized Thomson scattering......Page 194
11.3.2 Collision term......Page 195
11.4.1 Stokes parameters......Page 197
11.4.2 Polarized Thomson scattering......Page 198
11.5 CMB polarization......Page 200
11.6 Boltzmann hierarchy with polarization......Page 201
11.7 Initial conditions and the transfer functions......Page 204
11.8 Line-of-sight integral......Page 206
Exercises......Page 208
References......Page 209
12.1 Isocurvature modes......Page 210
12.2.2 Matter transfer function......Page 212
12.2.3 Acoustic oscillation and CMB anisotropy......Page 214
12.2.4 Super-large scale contribution......Page 215
12.3.2 Matter transfer function......Page 216
12.4 The primordial lepton number perturbation......Page 217
12.5.1 Primordial tensor perturbation......Page 219
12.5.2 Tensor Sachs–Wolfe effect......Page 220
12.5.3 Tensor Boltzmann equation and polarization......Page 221
12.6 Seeds and the vector mode......Page 224
12.7 Spatial curvature......Page 225
References......Page 227
Part III Field theory......Page 229
13.1.1 Basic concepts......Page 231
13.1.2 Effective field theory......Page 232
13.1.3 String theory and the landscape......Page 233
13.2 Action and Lagrangian......Page 234
13.2.1 Single degree of freedom......Page 235
13.2.2 degrees of freedom......Page 237
13.2.3 Field theory action......Page 238
13.3.2 Field equation......Page 239
13.4 Energy–momentum tensor......Page 240
13.5 Nearly free scalar field......Page 241
13.6.1 Canonical normalization......Page 242
13.6.3 Curved field space......Page 244
13.7 Field theory in curved spacetime......Page 245
13.8.2 Beyond Einstein gravity......Page 246
Reference......Page 247
14.1 Symmetry groups......Page 248
14.2.1 symmetry......Page 249
14.2.2 Global U(1) symmetry......Page 250
14.3.1 Global SU(2) symmetry......Page 251
14.3.2 Lie groups......Page 253
14.4 Noether’s theorem and conserved quantities......Page 254
14.5.1 Global discrete symmetry......Page 256
14.5.2 Global U(1) and SU(2) symmetry......Page 257
14.5.3 Dynamical symmetry breaking......Page 258
14.6 U(1) gauge symmetry......Page 259
14.7 SU(2) gauge symmetry......Page 261
14.8 Spontaneously broken gauge symmetry......Page 262
14.8.2 Spontaneously broken SU(2) gauge symmetry......Page 263
Exercises......Page 264
References......Page 265
15.1 Schrodinger¨ and Heisenberg pictures......Page 266
15.2 Symmetry and conserved currents......Page 268
15.2.2 Conserved quantities......Page 269
15.3 Harmonic oscillator......Page 271
15.4.1 Fourier series......Page 273
15.4.3 Quantized free complex scalar field......Page 275
15.5 Vector field......Page 276
15.6.1 Lorentz transformation and free-field Lagrangians......Page 278
15.6.2 Spin-1/2 particles......Page 279
15.7 Free scalar field with time-dependent mass......Page 281
15.8.1 Renormalization......Page 283
15.8.2 Keeping radiative corrections under control......Page 284
15.8.3 Coleman–Weinberg potential and renormalization group equations......Page 285
References......Page 287
16 The Standard Model......Page 288
16.1 Electroweak Lagrangian......Page 289
16.2 Electroweak theory: particles and interactions......Page 291
16.3 Electroweak theory with three generations......Page 293
16.4.1 Lagrangian and the QCD scale......Page 295
16.4.2 Confinement of colour......Page 296
16.4.3 Grand Unified Theory (GUT)......Page 298
16.5.2 Baryon number, lepton number and isospin......Page 299
16.5.3 Global chiral symmetries......Page 300
16.5.4 P, C and T......Page 301
16.6 Peccei–Quinn symmetry and the axion......Page 302
16.7.1 Neutrino mixing matrix......Page 304
16.7.2 Neutrino oscillation......Page 305
16.7.3 The seesaw mechanism and sterile neutrinos......Page 306
References......Page 307
17.1 The supersymmetry transformation......Page 308
17.2.1 The supersymmetric potential and superpotential......Page 309
17.3.1 Spontaneous supersymmetry breaking......Page 311
17.3.2 Soft supersymmetry breaking......Page 312
17.3.3 One-loop correction to the potential......Page 313
17.4 Supergravity......Page 314
17.5 The Minimal Supersymmetric Standard Model......Page 316
17.5.1 Unbroken supersymmetry......Page 317
17.5.2 Softly broken supersymmetry......Page 318
17.5.3 Gravitino mass......Page 319
Exercises......Page 320
References......Page 321
Part IV Inflation and the early Universe......Page 323
18.1 Inflation defined......Page 325
18.2.1 Flatness problem......Page 327
18.2.3 Unwanted relics......Page 329
18.4.1 Observable inflation......Page 331
18.4.2 How much observable inflation?......Page 332
18.5.1 Slow-roll approximation......Page 334
18.5.2 Status of the slow-roll approximation......Page 335
18.6.1 The formulation......Page 336
18.6.2 Attractor theorem......Page 337
18.7 Inflationary potentials......Page 338
Exercises......Page 342
References......Page 343
19.1 Scalar–tensor theories......Page 345
19.2 Induced gravity and variable Planck mass......Page 346
19.3 Extended inflation......Page 347
19.5 Modified gravity from the braneworld......Page 349
Exercises......Page 350
References......Page 351
20.1 Multi-field slow-roll inflation......Page 352
20.2.1 Light fields......Page 354
20.2.2 Heavy fields......Page 355
Exercise......Page 356
21.1.1 Initial reheating......Page 357
21.1.2 Second reheating......Page 358
21.2.1 Initial vacuum state......Page 360
21.2.2 Ordinary preheating (parametric resonance)......Page 361
21.2.4 Tachyonic preheating......Page 363
21.3.1 Spontaneous symmetry breaking......Page 364
21.3.2 Solitons......Page 365
21.4.1 Domain walls......Page 367
21.4.2 Cosmic strings......Page 368
21.5 Topological defects and the GUT transition......Page 370
21.6 Thermal inflation......Page 371
21.7 Moduli problem......Page 373
Exercises......Page 375
References......Page 376
22.1 Thermal equilibrium before the electroweak phase transition......Page 378
22.2 Thermal equilibrium with non-zero B - L......Page 380
22.3.1 Sakharov conditions......Page 382
22.3.2 Electroweak baryogenesis......Page 383
22.3.4 Affleck–Dine baryogenesis......Page 384
22.3.5 Spontaneous baryogenesis......Page 386
References......Page 387
23.1.1 Switching on the potential......Page 389
23.1.3 Axion CDM: no-string scenario......Page 390
23.1.4 Axion CDM: string scenario......Page 391
23.2.1 Neutralino CDM......Page 392
23.2.2 Gravitino CDM and unstable gravitinos......Page 394
23.2.3 Axino dark matter......Page 395
23.3 Supermassive CDM candidates......Page 396
23.4 Primordial black holes......Page 397
23.5.1 Cosmological constant......Page 398
23.5.2 Dark energy from a rolling scalar field......Page 399
References......Page 400
24.1 Quantum theory of a massless free scalar field during inflation......Page 402
24.1.2 Quantization......Page 403
24.1.4 Perturbing the action......Page 405
24.2 Quantum to classical transition......Page 406
24.3.1 Effect of the potential......Page 407
24.3.2 Effect of the metric perturbation......Page 409
24.4.1 Interaction Picture......Page 410
24.4.2 Non-gaussianity from the self-interaction of the field......Page 411
24.4.3 Non-gaussianity from the gravitational interaction......Page 412
24.5 Higher orders of perturbation theory......Page 413
24.6 Stochastic field evolution......Page 414
24.6.1 Fokker–Planck equations......Page 415
24.6.2 Late-time probability distribution......Page 416
24.7 Primordial tensor perturbation......Page 417
24.7.2 Non-gaussianity of the tensor perturbation......Page 418
24.7.3 Cosmic gravitational wave background......Page 419
24.8 Particle production from a perturbation created during inflation......Page 420
Exercises......Page 421
References......Page 422
25 Generating Zeta at horizon exit......Page 424
25.1 Anthropic constraints on the curvature perturbation......Page 425
25.2 Prediction of the standard paradigm for the spectrum......Page 426
25.3.1 Tensor fraction and field variation......Page 427
25.3.2 Constraints on small-field models......Page 429
25.4.1 deltaN formula......Page 430
25.4.2 Bispectrum of ζ......Page 431
25.4.3 Non-gaussianity in the squeezed limit......Page 432
25.5.1 Spectrum......Page 433
25.5.2 Higher-order graphs and higher correlators......Page 435
25.6 The standard paradigm beyond slow roll......Page 436
25.7.1 General case......Page 438
25.8 Warm inflation......Page 440
Exercises......Page 441
References......Page 442
26.1 The generic deltaN formula for ζ......Page 444
26.2 Spectrum of ζ......Page 445
26.3.1 Single field......Page 447
26.3.2 Uncorrelated non-gaussianity......Page 448
26.4 Curvaton paradigm......Page 449
26.4.1 General predictions......Page 450
26.5 Inhomogeneous decay rate......Page 452
26.6.2 End of inflation and preheating......Page 454
References......Page 455
27.1 The deltani formula......Page 457
27.2 Axion CDM isocurvature perturbation......Page 459
27.2.1 Pure axion CDM......Page 460
27.2.2 A non-gaussian axion isocurvature perturbation......Page 461
27.2.3 Bubbles of axion domain wall......Page 462
27.4 Correlated CDM or baryon isocurvature perturbation......Page 463
27.4.2 Generating CDM or baryon isocurvature from curvaton decay......Page 464
27.5 Neutrino isocurvature perturbation......Page 465
References......Page 466
28.1 Historical development......Page 467
28.2 Eternal inflation......Page 469
28.4 The eta problem......Page 471
28.4.1 The problem during inflation......Page 472
28.4.2 The problem after inflation......Page 473
28.5 Hilltop inflation......Page 474
28.6 Ledge inflation......Page 476
28.7.1 Potential with a loop contribution......Page 478
28.7.2 Mutated/smooth hybrid inflation......Page 480
28.8 D-term inflation......Page 481
28.10 Running-mass inflation......Page 483
28.11 Small-field PNGB inflation......Page 486
28.12 Modular inflation......Page 487
28.13.1 Chaotic inflation......Page 488
28.13.2 Natural inflation......Page 489
28.13.3 N-flation......Page 490
References......Page 491
29 Perspective......Page 495
Appendix A: Spherical functions......Page 497
References......Page 501
Appendix B: Constants and parameters......Page 502
Index......Page 504