Table Of Content

GRAVITATIONAL WAVES Gravitational Waves Volume 2 Astrophysics and Cosmology Michele Maggiore DépartementdePhysiqueThéorique UniversitédeGenève 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©MicheleMaggiore2018 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2018 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2017943894 SetISBN 978–0–19–875528–9 Volume1 978–0–19–857074–5 Volume2 978–0–19–857089–9 onlyavailableaspartofaset DOI10.1093/oso/9780198570899.001 Printedandboundby CPILitho(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. 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