Hot Topics in General Relativity and Gravitation POST-NEWTONIAN MODELLING of INSPIRALLING COMPACT BINARIES Luc Blanchet Gravitation et Cosmologie ( G R ε C O ) Institut d’Astrophysique de Paris 31 juillet 2017 Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 1 / 42
Special Relativity and “ondes gravifiques” [Lorentz 1904; Poincar´ e 1905; Einstein 1905] [Fizeau 1851] [Michelson & Morley 1887] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 2 / 42
100 years of gravitational waves [Einstein 1916] ⇐ = small perturbation of Minkowski’s metric Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 3 / 42
Quadrupole moment formalism [Einstein 1918; Landau & Lifchitz 1947] Einstein quadrupole formula 1 � d E � GW � d 3 Q ij � 2 � d 3 Q ij � v = G + O 5 c 5 d t 3 d t 3 d t c Amplitude quadrupole formula 2 � 1 � d 2 Q ij � � �� TT � ij = 2 G t − D � v h TT + O + O c 4 D d t 2 D 2 c c Radiation reaction formula [Chandrasekhar & Esposito 1970; Burke & Thorne 1970] 3 5 c 5 ρ x j d 5 Q ij = − 2 G � v � 7 F reac + O i d t 5 c Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 4 / 42
The quadrupole formula works for the binary pulsar [Taylor & Weisberg 1982] � 5 / 3 1 + 73 24 e 2 + 37 � 2 πG M 96 e 4 P = − 192 π ˙ ≈ − 2 . 4 × 10 − 12 5 c 5 ν P (1 − e 2 ) 7 / 2 [Peters & Mathews 1963, Esposito & Harrison 1975, Wagoner 1975, Damour & Deruelle 1983] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 5 / 42
The radiation reaction controversy [Ehlers, Rosenblum, Goldberg & Havas 1976; Walker & Will 1980] Gravitational radiation: a review [Kip Thorne] GWs and motion of compact bodies [Thibault Damour] Methods of numerical relativity [Tsvi Piran] Interferometric detectors of GWs [Ron Drever] Optical detectors of GWs [Alain Brillet & Philippe Tourrenc] EOMs: a round table discussion [moderator: Abhay Ashtekar] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 6 / 42
Binary black-hole event GW150914 [LIGO/VIRGO collaboration 2016] Whitened H1 Strain / 10 − 21 1 . 0 6 0 . 5 3 σ noise 0 . 0 0 − 3 − 0 . 5 − 1 . 0 − 6 Whitened L1 Strain / 10 − 21 1 . 5 6 1 . 0 3 0 . 5 σ noise 0 . 0 0 − 0 . 5 Data − 3 − 1 . 0 Wavelet − 6 − 1 . 5 BBH Template 0 . 25 0 . 30 0 . 35 0 . 40 0 . 45 Time / s Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 7 / 42
Binary black-hole event GW150914 [LIGO/VIRGO collaboration 2016] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 8 / 42
Binary black-hole event GW150914 [LIGO/VIRGO collaboration 2016] GW λ ~ 2000 km merger Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 8 / 42
The quadrupole formula works also for GW150914 ! The GW frequency is given in terms of the chirp mass M = µ 3 / 5 M 2 / 5 by 1 � − 3 / 8 � 256 G M 5 / 3 f = 1 ( t f − t ) π 5 c 5 Therefore the chirp mass is directly measured as 2 � 5 � 3 / 5 c 5 Gπ 8 / 3 f − 11 / 3 ˙ M = f 96 which gives M = 30 M ⊙ thus M � 70 M ⊙ The GW amplitude is predicted to be 3 � M � 5 / 6 � 100 Mpc � � 100 Hz � − 1 / 6 h eff ∼ 4 . 1 × 10 − 22 ∼ 1 . 6 × 10 − 21 M ⊙ D f merger The distance D = 400 Mpc is measured from the signal itself 4 Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 9 / 42
Total energy radiated by GW150914 The ADM energy of space-time is constant and reads (at any time t ) 1 � t E ADM = ( m 1 + m 2 ) c 2 − Gm 1 m 2 + G d t ′ � � 2 ( t ′ ) Q (3) ij 5 c 5 2 r −∞ Initially E ADM = ( m 1 + m 2 ) c 2 while finally (at time t f ) 2 � t f E ADM = M f c 2 + G d t ′ � � 2 ( t ′ ) Q (3) ij 5 c 5 −∞ The total energy radiated in GW is 3 � t f ∆ E GW = ( m 1 + m 2 − M f ) c 2 = G � 2 ( t ′ ) = Gm 1 m 2 d t ′ � Q (3) ij 5 c 5 2 r f −∞ The total power released is 4 P GW ∼ 3 M ⊙ c 2 ∼ 10 49 W ∼ 10 − 3 c 5 0 . 2 s G Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 10 / 42
Gravitational wave BBH events [LIGO/VIRGO collaboration 2016, 2017] For BH binaries the detectors are mostly sensitive to the merger phase and a few cycles are observed before coalescence Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 11 / 42
The inspiral and merger of neutron star binaries Many physical results are expected Measurement of the equation of state of nuclear matter, formation and properties of hypermassive neutron stars, etc. Constraints on the existence of boson stars (gravitationally bound conglomerates of scalar particles [Ruffini & Bonazzola 1969] ) Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 12 / 42
Post-merger waveform of neutron star binaries [Shibata et al. , Rezzolla et al. 1990-2010s] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 13 / 42
Modelling the compact binary dynamics L m 1 CM m 2 Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 14 / 42
Modelling the compact binary dynamics J = L + S + S 1 2 L S 1 1 m S 1 2 CM m 2 Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 14 / 42
Methods to compute GW templates log 10 ( r / m ) 4 Post-Newtonian 3 −1 Theory (Compactness) 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Mass Ratio [courtesy Alexandre Le Tiec] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 15 / 42
Methods to compute GW templates [see Blanchet 2014 for a review] m 2 r log 10 ( r / m ) m 1 4 Post-Newtonian 3 −1 Theory (Compactness) 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Mass Ratio [courtesy Alexandre Le Tiec] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 15 / 42
Methods to compute GW templates [Detweiler 2008; Barack 2009] log 10 ( r / m ) 4 m 2 Post-Newtonian 3 −1 Theory (Compactness) 2 m 1 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Mass Ratio [courtesy Alexandre Le Tiec] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 15 / 42
Methods to compute GW templates log 10 ( r / m ) 4 Post-Newtonian 3 −1 Theory (Compactness) 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Mass Ratio [Caltech/Cornell/CITA collaboration] [courtesy Alexandre Le Tiec] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 15 / 42
Methods to compute GW templates [Buonanno & Damour 1998] m 2 EOB µ log 10 ( r / m ) m 1 m = m 1 + m 2 4 Post-Newtonian 3 −1 Theory (Compactness) 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Mass Ratio [courtesy Alexandre Le Tiec] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 15 / 42
The gravitational chirp of compact binaries merger phase numerical relativity inspiralling phase post-Newtonian theory ringdown phase perturbation theory Effective methods such as EOB that interpolate between the PN and NR are also very important notably for the data analysis Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 16 / 42
Why inspiralling binaries require high PN modelling [Caltech “3mn paper” 1992; Blanchet & Sch¨ afer 1993] observer i m 2 orbital plane m 1 φ � GMω � − 5 / 3 � � φ ( t ) = φ 0 − M 1 +1PN + 1 . 5PN + · · · + 3PN + · · · c 3 c 2 c 3 c 6 µ � �� � � �� � needs to be computed with 3PN precision at least quadrupole formalism Here 3PN means 5.5PN as a radiation reaction effect ! Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 17 / 42
Methods to compute PN equations of motion ADM Hamiltonian canonical formalism [Ohta et al. 1973; Sch¨ 1 afer 1985] EOM in harmonic coordinates [Damour & Deruelle 1985; Blanchet & Faye 1998, 2000] 2 Extended fluid balls [Grishchuk & Kopeikin 1986] 3 Surface-integral approach [Itoh, Futamase & Asada 2000] 4 Effective-field theory (EFT) [Goldberger & Rothstein 2006; Foffa & Sturani 2011] 5 EOM derived in a general frame for arbitrary orbits Dimensional regularization is applied for UV divergences 1 Radiation-reaction dissipative effects added separately by matching Spin effects can be computed within a pole-dipole approximation Tidal effects incorporated at leading 5PN and sub-leading 6PN orders 1 Except in the surface-integral approach Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 18 / 42
The 1PN equations of motion [Lorentz & Droste 1917; Einstein, Infeld & Hoffmann 1938] � � � d 2 r A Gm B Gm C Gm D 1 − r AB · r BD � � � = − 1 − 4 − n AB r 2 r 2 d t 2 c 2 r AC c 2 r BD AB BD B � = A C � = A D � = B � �� + 1 B − 4 v A · v B − 3 v 2 A + 2 v 2 2( v B · n AB ) 2 c 2 G 2 m B m D Gm B v AB [ n AB · (3 v B − 4 v A )] − 7 � � � + n BD c 2 r 2 c 2 r AB r 3 2 AB BD B � = A B � = A D � = B Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 19 / 42
Recommend
More recommend
