Elements of String Cosmology

Contents......Page 4
Acknowledgements......Page 9
Notation, units and conventions......Page 10
Reference......Page 11
1.1.1 Einstein equations......Page 12
1.1.2 Homogeneity and isotropy......Page 13
1.1.3 Perfect fluid sources......Page 18
1.1.4 Thermal equilibrium......Page 25
1.2.1 Standard kinematic problems......Page 31
1.2.2 de Sitter inflation......Page 36
1.2.3 Slow-roll inflation......Page 37
1.2.4 Initial singularity......Page 41
References......Page 46
2 The basic string cosmology equations......Page 48
2.1 Tree-level equations......Page 49
2.2 The Einstein-frame representation......Page 56
2.3 First-order α' corrections......Page 63
2.3.1 Higher-order gravi-dilaton equations......Page 67
Appendix 2A Differential forms in a Riemannian manifold......Page 75
References......Page 81
3 Conformal invariance and string effective actions......Page 82
3.1 Strings in curved backgrounds and conformal anomalies......Page 83
3.2 Higher-curvature and higher-genus expansion......Page 95
Appendix 3A The massless multiplet of the bosonic string in Minkowski space......Page 102
3A.1 Classical closed string......Page 104
3A.2 Classical open string......Page 106
3A.3 Quantization......Page 108
3A.4 Open string spectrum......Page 111
3A.5 Closed string spectrum......Page 113
Appendix 3B Superstring models and effective actions......Page 115
3B.1 Fermionic boundary conditions......Page 119
3B.2 Classical constraints......Page 120
3B.3 Quantization......Page 122
3B.4 Type IIA and type IIB superstrings......Page 128
3B.5 Type I superstring......Page 132
3B.6 Heterotic superstrings......Page 135
References......Page 141
4 Duality symmetries and cosmological solutions......Page 143
4.1 Scale-factor duality and the pre-big bang scenario......Page 145
4.2 Duality with matter sources......Page 153
4.2.1 General integration of the lowest-order equations......Page 159
4.2.2 Asymptotic limits......Page 164
4.3 Global O(d, d) symmetry......Page 168
4.3.1 O(d, d) symmetry and matter sources......Page 171
4.3.2 General integration of the matrix equations......Page 177
4.3.3 Non-trivial solutions via duality transformations......Page 179
4.3.4 Non-Abelian duality......Page 182
Appendix 4A A non-local, general-covariant dilaton potential......Page 185
Appendix 4B Examples of regular and self-dual solutions......Page 192
References......Page 203
5 Inflationary kinematics......Page 205
5.1 Four different types of inflation......Page 206
5.2 Dynamical equivalence of super-inflation and accelerated contraction......Page 209
5.3 Horizons and kinematics......Page 213
References......Page 220
6 The string phase......Page 221
6.1 High-curvature fixed points of the string cosmology equations......Page 224
6.2 Strong coupling corrections and the curvature "bounce"......Page 234
6.3 String gas cosmology......Page 242
Appendix 6A Birth of the Universe in quantum string cosmology......Page 250
6A.1 Tunneling from the string perturbative vacuum......Page 254
6A.2 Operator ordering......Page 255
6A.3 Scattering of the wave function in minisuperspace......Page 257
6A.4 Antitunneling from the string perturbative vacuum......Page 259
References......Page 261
7 The cosmic background of relic gravitational waves......Page 264
7.1 Propagation of tensor perturbations......Page 265
7.1.1 Frame independence......Page 271
7.1.2 Canonical normalization......Page 273
7.2 Parametric amplification and spectral distribution......Page 277
7.2.1 Spectral amplitude......Page 282
7.2.2 Graviton production and spectral energy density......Page 287
7.2.3 Matching conditions......Page 292
7.3 Expected relic gravitons from inflation......Page 298
7.3.1 Phenomenological bounds on the graviton background......Page 305
7.3.2 Primordial gravitons from pre-big bang inflation......Page 310
7.4 Sensitivities and cross-correlation of gravitational detectors......Page 321
7.4.1 Correlated response of two detectors......Page 327
Appendix 7A Higher-derivative corrections to the tensor perturbation equations......Page 334
References......Page 342
8 Scalar perturbations and the anisotropy of the CMB radiation......Page 345
8.1 Scalar perturbations in a cosmological background......Page 346
8.1.1 Gauge-invariant variables......Page 351
8.1.2 Scalar perturbation equations......Page 359
8.1.3 Scalar field source......Page 362
8.1.4 Perfect fluid source......Page 365
8.1.5 Generalized comoving gauges......Page 370
8.1.6 Frame transformations......Page 372
8.2 The anisotropy spectrum of the CMB radiation......Page 373
8.2.1 Primordial spectrum and slow-roll inflation......Page 376
8.2.2 "Conservation" of the Bardeen spectrum......Page 381
8.2.3 Sachs–Wolfe effect......Page 383
8.2.4 Adiabatic initial conditions......Page 388
8.2.5 Isocurvature initial conditions......Page 390
8.2.6 The angular power spectrum......Page 391
8.3 Adiabatic metric perturbations from the string theory axion......Page 400
8.3.1 The curvaton mechanism......Page 405
8.3.2 Normalization of the Bardeen spectrum......Page 414
Appendix 8A Photon–dilaton interactions and cosmic magnetic fields......Page 419
Appendix 8B Seeds for the CMB anisotropy......Page 427
References......Page 435
9 Dilaton phenomenology......Page 439
9.1 Spectral intensity of a massive dilaton background......Page 440
9.1.1 Dilaton mass and couplings......Page 448
9.1.2 Light and weakly coupled dilatons......Page 455
9.2 Interaction with gravitational antennas......Page 456
9.2.1 Pattern functions for interferometric detectors......Page 460
9.2.2 Signal-to-noise ratio......Page 463
9.2.3 Non-relativistic backgrounds......Page 467
9.2.4 Signal enhancement for flat spectra......Page 471
9.3 Dilaton dark energy and late-time acceleration......Page 474
9.3.1 Saturation of the loop corrections......Page 476
9.3.2 The dragging and freezing regimes......Page 479
9.3.3 A numerical example......Page 483
Appendix 9A The common mode of interferometric detectors......Page 489
References......Page 493
10 Elements of brane cosmology......Page 495
10.1 Effective gravity on the brane......Page 498
10.1.1 Covariant projection on the brane......Page 501
10.2 Warped geometry and localization of gravity......Page 507
10.2.1 Short-range corrections......Page 517
10.3 Brane-world cosmology......Page 519
10.3.1 Inflation on the brane......Page 524
10.3.2 Induced gravity on the brane......Page 530
10.4 Ekpyrotic and cyclic scenario......Page 534
10.5 Brane–antibrane inflation......Page 544
References......Page 556
Index......Page 559