Gravitational Waves: Volume 2: Astrophysics and Cosmology

835 Pages·2018·9.957 MB·English

by  Michele Maggiore

GRAVITATIONAL WAVES

Gravitational Waves
Volume 2
Astrophysics and Cosmology
Michele Maggiore
DépartementdePhysiqueThéorique
UniversitédeGenève
3
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
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Contents
Part III: Astrophysical sources of gravitational
waves 1
10 Stellar collapse 3
10.1 Historical Supernovae 4
10.2 Properties of Supernovae 10
10.2.1 SN classification 11
10.2.2 Luminosities 15
10.2.3 Rates 18
10.3 The dynamics of core collapse 21
10.3.1 Pre-SN evolution 21
10.3.2 Core collapse and neutrino-driven delayed shock 25
10.3.3 The remnant of the collapse 30
10.4 GW production by self-gravitating fluids 35
10.4.1 Energy–momentum tensor of a perfect fluid 35
10.4.2 GW production from gravitating Newtonian fluids 38
10.4.3 Quadrupole radiation from axisymmetric sources 42
10.5 GWs from stellar collapse 46
10.5.1 GWs from collapse and bounce of rotating cores 47
10.5.2 GWs from bar-mode instabilities 51
10.5.3 GWs from post-bounce convective instabilities 55
10.5.4 GWs from anisotropic neutrino emission 57
10.5.5 GWs from magneto-rotational core collapse 62
10.5.6 GWs from fragmentation during collapse 65
10.6 Complements: luminosity, color and metallicity of stars 66
Further reading 71
11 Neutron stars 74
11.1 Observations of neutron stars 74
11.1.1 The discovery of pulsars 74
11.1.2 Pulsar spindown and the P −P˙ plane 75
11.1.3 Millisecond pulsars 80
11.1.4 Pulsar demography 81
11.1.5 SGRs and magnetars 83
11.2 GW emission from neutron stars 89
11.2.1 NS normal modes 89
11.2.2 The CFS instability 99
11.2.3 GWs from post-merger NS remnants 106
11.2.4 GWs from deformed rotating NS 108
Further reading 111
vi Contents
12 Black-hole perturbation theory 115
12.1 Scalar perturbations 115
12.2 Gravitational perturbations 117
12.2.1 Zerilli tensor harmonics 118
12.2.2 The Regge–Wheeler gauge 123
12.2.3 Axial perturbations: Regge–Wheeler equation 127
12.2.4 Polar perturbations: Zerilli equation 130
12.2.5 Boundary conditions 131
12.2.6 The radiation field in the far zone 133
12.2.7 Summary 138
12.3 Black-hole quasi-normal modes 140
12.3.1 General discussion 140
12.3.2 QNMs from Laplace transform 144
12.3.3 Power-law tails 151
12.3.4 Frequency spectrum of QNMs 157
12.3.5 The physical interpretation of the QNM spectrum162
12.4 Radial infall into a black hole 164
12.4.1 The source term 165
12.4.2 Numerical integration of the Zerilli equation 166
12.4.3 Waveform and energy spectrum 167
12.5 Perturbations of rotating black holes 168
12.5.1 The Kerr metric 169
12.5.2 Null tetrads and the Newman–Penrose formalism 173
12.5.3 Teukolsky equation and QNMs of rotating BHs 177
12.6 Solved problems 180
12.1. Derivation of the Zerilli equation 180
12.2. The source term for radial infall 184
Further reading 186
13 Properties of dynamical space-times 189
13.1 The 3+1 decomposition of space-time 189
13.2 Boundary terms in the gravitational action 192
13.3 Hamiltonian formulation of GR 195
13.4 Conserved quantities for isolated systems 197
13.5 GWs and Newman–Penrose scalar 207
Further reading 209
14 GWs from compact binaries. Theory 210
14.1 Non-perturbative resummations. A simple example 212
14.2 Effective one-body action 217
14.2.1 Equivalence to a one-body problem 217
14.2.2 Conservative dynamics 227
14.2.3 Inclusion of radiation reaction 230
14.2.4 The EOB waveform 231
14.2.5 Spinning binaries 236
14.3 Numerical relativity 241
14.3.1 Numerical integration of Einstein equations 241
14.3.2 Equal-mass non-spinning BH binaries 244
Contents vii
14.3.3 Unequal-mass non-spinning BH binaries 248
14.3.4 Final BH recoil 250
14.3.5 Spinning BHs and superkicks 253
14.3.6 Astrophysical consequences of BH recoil 260
14.4 GWs from NS–NS binaries 265
14.4.1 Inspiral phase and tidal effects 265
14.4.2 Merger phase and numerical relativity 270
Further reading 273
15 GWs from compact binaries. Observations 277
15.1 GW150914. The first direct detection 278
15.1.1 Evaluation of the statistical significance 279
15.1.2 Properties of GW150914 285
15.2 Further BH–BH detections 290
15.2.1 GW151226 290
15.2.2 GW170104 293
15.2.3 GW170608 294
15.2.4 GW170814: the first three-detector observation 295
15.2.5 The population of BH–BH binaries 297
15.3 GW170817: the first NS–NS binary 299
15.3.1 GW observation 299
15.3.2 The prompt γ-ray burst 301
15.3.3 The electromagnetic counterpart 306
15.3.4 Kilonovae and r-process nucleosynthesis 309
15.3.5 The cocoon scenario 311
15.4 Tests of fundamental physics 314
15.4.1 BH quasi-normal modes 314
15.4.2 Tests of post-Newtonian gravity 316
15.4.3 Propagation and degrees of freedom of GWs 317
Further reading 322
16 Supermassive black holes 327
16.1 The central supermassive black hole in our Galaxy 327
16.2 Supermassive black-hole binaries 330
16.2.1 Formation and evolution of SMBH binaries 330
16.2.2 SMBH binaries at LISA 334
16.3 Extreme mass ratio inspirals 339
16.3.1 Formation mechanisms 339
16.3.2 EMRIs at LISA 341
16.3.3 Waveforms and the self-force approach 343
16.4 Stochastic GWs from SMBH binaries 349
16.4.1 Regime dominated by GW back-reaction 351
16.4.2 Regime dominated by three-body interactions 352
16.4.3 High-frequency regime and source discreteness 355
16.4.4 Estimates of the SMBH merger rate 357
16.4.5 Effect of the eccentricity 359
Further reading 363
viii Contents
Part IV: Cosmology and gravitational waves 367
17 Basics of FRW cosmology 369
17.1 The FRW metric 369
17.1.1 Comoving and physical coordinates 370
17.1.2 Comoving and physical momenta 371
17.2 Cosmological background equations for a single fluid 374
17.3 Multi-component fluids 377
17.4 RD–MD equilibrium, recombination and decoupling 381
17.5 Effective number of relativistic species 383
17.6 Conformal time and particle horizon 388
17.6.1 Radiation dominance 389
17.6.2 Matter dominance 391
17.6.3 Analytic formulas in RD+MD 392
17.6.4 Λ dominance 393
17.6.5 Conformal time at significant epochs 393
17.6.6 Comoving distance, angular diameter distance
and luminosity distance 395
17.7 Newtonian cosmology inside the horizon 397
17.7.1 Newtonian dynamics in expanding backgrounds 398
17.7.2 Newtonian fluid dynamics in an expanding Uni-
verse 402
Further reading 411
18 Helicity decomposition of metric perturbations 412
18.1 Perturbations around flat space 413
18.1.1 Helicity decomposition 413
18.1.2 Radiative and non-radiative degrees of freedom 418
18.2 Gauge invariance and helicity decomposition in FRW 421
18.2.1 Linearizeddiffeomorphismsandgaugeinvariance
in a curved background 421
18.2.2 Bardeen variables 422
18.3 Perturbed energy–momentum tensor 425
18.3.1 General decomposition of Tμ 426
ν
18.3.2 Perturbations of perfect fluids 428
18.3.3 Linearized energy–momentum conservation 431
18.3.4 Gauge-invariant combinations 434
Further reading 436
19 Evolution of cosmological perturbations 437
19.1 Evolution equations in the scalar sector 437
19.1.1 Single-component fluid 439
19.1.2 Multi-component fluid 441
19.1.3 Super-horizon and sub-horizon regimes 443
19.2 Initial conditions 447
19.2.1 Adiabatic and isocurvature perturbations 449
19.2.2 The variables ζ and R 451
19.3 Solutions of the equations for scalar perturbations 454
Contents ix
19.3.1 Numerical integration 454
19.3.2 Analytic solutions in RD 458
19.3.3 Analytic solutions in MD 461
19.3.4 Analytic solutions during dark-energy dominance 463
19.4 Power spectra for scalar perturbations 464
19.4.1 Definitions and conventions 464
19.4.2 The primordial power spectrum 466
19.4.3 Transfer function and growth rate 469
19.4.4 The linearly processed power spectrum 472
19.5 Tensor perturbations 474
19.5.1 Cosmological evolution 474
19.5.2 Transfer function for tensor modes 482
19.5.3 GW damping from neutrino free-streaming 486
19.5.4 The tensor power spectrum, Ω (f) and h (f) 488
gw c
19.6 Standard sirens, dark energy and modified gravity 492
19.6.1 Testing cosmological models against observations 494
19.6.2 Cosmology with standard sirens 496
19.6.3 Tensor perturbations in modified gravity 500
19.6.4 An explicit example: non-local gravity 502
Further reading 505
20 The imprint of GWs on the CMB 507
20.1 The CMB multipoles 507
20.2 Null geodesics 512
20.3 Temperature anisotropies at large angles 517
20.3.1 Photon geodesics in a perturbed FRW metric 517
20.3.2 Sachs–Wolfe, ISW and Doppler contributions 519
20.3.3 Expression of the C in terms of the Θ (k) 523
l l
20.3.4 Scalar contribution to the C 526
l
20.3.5 Tensor contribution to the C 530
l
20.3.6 Finite thickness of the LSS 538
20.3.7 The Boltzmann equation for photons 540
20.4 CMB polarization 545
20.4.1 Stokes parameters 545
20.4.2 Polarization maps. E and B modes 548
20.4.3 Polarization and tensor spherical harmonics 551
20.4.4 Generation of CMB polarization 558
20.4.5 Experimental situation 569
Further reading 571
21 Inflation and primordial perturbations 574
21.1 Inflationary cosmology 574
21.1.1 The flatness problem 574
21.1.2 The horizon problem 578
21.1.3 Single-field slow-roll inflation 580
21.1.4 Large-field and small-field inflation 584
21.1.5 Starobinsky model 590
21.2 Quantum fields in curved space 595