Rencontres du Vietnam Hot Topics in General Relativity & Gravitation POST-NEWTONIAN THEORY VERSUS BLACK HOLE PERTURBATIONS Luc Blanchet Gravitation et Cosmologie ( G R ε C O ) Institut d’Astrophysique de Paris 28 mars 2015 Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 1 / 29
World-wide network of gravitational wave detectors A Global Network of Interferometers A Global Network of Interferometers LIGO Hanford 4 & 2 km GEO Hannover 600 m Kagra Japan 3 km LIGO South Indigo Virgo Cascina 3 km LIGO Livingston 4 km The network will observe the GWs in the high-frequency band 10 Hz � f � 10 3 Hz Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 2 / 29
Space-based laser interferometric detector eLISA eLISA will observe the GWs in the low-frequency band 10 − 4 Hz � f � 10 − 1 Hz Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 3 / 29
The inspiral and merger of compact binaries Neutron stars spiral and coalesce Black holes spiral and coalesce Neutron star ( M = 1 . 4 M ⊙ ) events will be detected by ground-based 1 detectors LIGO/VIRGO Stellar size black hole ( 5 M ⊙ � M � 20 M ⊙ ) events will also be detected by 2 ground-based detectors Supermassive black hole ( 10 5 M ⊙ � M � 10 8 M ⊙ ) events will be detected 3 by the space-based detector eLISA Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 4 / 29
Extreme mass ratio inspirals (EMRI) for eLISA A neutron star or a stellar black hole follows a highly relativistic orbit around a supermassive black hole. The gravitational waves generated by the orbital motion are computed using black hole perturbation theory Observations of EMRIs will permit to test the no-hair theorem for black holes, i.e. to verify that the central black hole is described by the Kerr geometry Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 5 / 29
Modelling the compact binary inspiral J = L + S + S 1 2 L S 1 1 m S 1 2 CM m 2 Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 6 / 29
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 Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 7 / 29
Methods to compute GW templates 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 Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 7 / 29
Methods to compute GW templates 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 Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 7 / 29
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] Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 7 / 29
The gravitational chirp of compact binaries merger phase inspiralling phase ringdown phase innermost circular orbit r = 6M Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 8 / 29
The gravitational chirp of compact binaries merger phase numerical relativity inspiralling phase post-Newtonian theory ringdown phase perturbation theory innermost circular orbit r = 6M Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 8 / 29
Inspiralling binaries require high-order PN modelling [Cutler, Flanagan, Poisson & Thorne 1992; Blanchet & Sch¨ afer 1993] observer i m 2 orbital plane m 1 φ ascending node � 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 result of the quadrupole formalism (sufficient for the binary pulsar) Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 9 / 29
Short History of the PN Approximation EQUATIONS OF MOTION RADIATION FIELD 1PN equations of motion [Lorentz & 1918 Einstein quadrupole formula Droste 1917; Einstein, Infeld & Hoffmann 1938] 1940 Landau-Lifchitz formula Radiation-reaction controvercy [Ehlers 1960 Peters-Mathews formula et al 1979; Walker & Will 1982] EW multipole moments [Thorne 1980] 2.5PN equations of motion and GR BD moments and wave generation prediction for the binary pulsar formalism [BD 1989; B 1995, 1998] [Damour & Deruelle 1982; Damour 1983] 1PN orbital phasing [Wagoner & Will The “3mn” Caltech paper [Cutler, 1976; BS 1989] Flanagan, Poisson & Thorne 1993] 2PN waveform [BDIWW 1995] 3.5PN equations of motion [Jaranowski 3.5PN phasing and 3PN waveform & Sch¨ afer 1999; BF 2001; ABF 2002; BI 2003; [BFIJ 2003; BFIS 2007] Itoh & Futamase 2003; Foffa & Sturani 2011] Ambiguity parameters resolved [BI Ambiguity parameters resolved [DJS 2004; BDEI 2004, 2005] 2001; BDE 2003] 4.5PN (?) 4PN [DJS, BBBFM] Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 10 / 29
4PN equations of motion of compact binaries d v i d t = − Gm 2 1 n i 12 r 2 12 1PN Lorentz-Droste-EIH term � �� � �� 5 G 2 m 1 m 2 � � + 4 G 2 m 2 1 2 n i + + · · · 12 + · · · r 3 r 3 c 2 12 12 � 1 � + 1 + 1 + 1 + 1 1 c 4 [ · · · ] c 5 [ · · · ] c 6 [ · · · ] c 7 [ · · · ] + c 8 [ · · · ] + O c 9 � �� � � �� � � �� � � �� � � �� � 2PN 2.5PN 3PN 3.5PN 4PN radiation reaction radiation reaction conservative & radiation tail [Jaranowski & Sch¨ afer 1999; Damour, Jaranowski & Sch¨ afer 2001] ADM Hamiltonian [Blanchet & Faye 2000; de Andrade, Blanchet & Faye 2001] Harmonic equations of motion 3 PN [Itoh, Futamase & Asada 2001; Itoh & Futamase 2003] Surface integral method [Foffa & Sturani 2011] Effective field theory � [Jaranowski & Sch¨ afer 2013; Damour, Jaranowski & Sch¨ afer 2014] ADM Hamiltonian 4 PN [See the talk of Laura Bernard in this meeting] Harmonic Lagrangian Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 11 / 29
3.5PN energy flux of compact binaries (4.5PN?) [Blanchet, Faye, Iyer & Joguet 2002] 1.5PN tail � � � F GW = − 32 c 5 � �� � − 1247 336 − 35 5 G ν 2 x 5 4 πx 3 / 2 1 + 12 ν x + � � − 44711 9072 + 9271 504 ν + 65 x 2 + [ · · · ] x 5 / 2 18 ν 2 + � �� � 2.5PN tail x 5 �� � [ · · · ] x 3 + [ · · · ] x 7 / 2 + [ · · · ] x 4 + [ · · · ] x 9 / 2 + + O � �� � � �� � � �� � � �� � 3.5PN tail 3PN 4PN (?) 4.5PN (?) includes a tail-of-tail The orbital frequency and phase for quasi-circular orbits are deduced from an energy balance argument d E d t = −F GW Spin contributions are also known to high order [Boh´ e, Marsat, Faye & Blanchet 2013] Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 12 / 29
General problem of the gravitational perturbation A particle is moving on a background u µ u µ = f µ = 0 space-time m 2 Its own stress-energy tensor modifies the background gravitational field Because of the “back-reaction” the motion f µ of the particle deviates from a background m 1 geodesic hence the appearance of a gravitational self force (GSF) The self acceleration of the particle is proportional to its mass � m 1 � u µ D¯ = f µ = O d τ m 2 The self force is computed by numerical methods [Sago, Barack & Detweiler 2008] Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 13 / 29
Common regime of validity of GSF and PN m 2 r log 10 ( r / m ) m 1 4 Post-Newtonian Theory Post-Newtonian 3 −1 & Theory (Compactness) Perturbation Theory 2 Numerical Perturbation 1 Relativity Theory 0 log 10 ( m 2 / m 1 ) 0 1 2 3 4 Mass Ratio Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 14 / 29
Why and how comparing PN and GSF predictions? Both the PN and GSF approaches use a self-field regularization for point particles followed by a renormalization. However, the prescription are very different GSF theory is based on a prescription for the Green function G R that is at 1 once regular and causal [Detweiler & Whiting 2003] PN theory uses dimensional regularization and it was shown that subtle issues 2 appear at the 3PN order due to the appearance of poles ∝ ( d − 3) − 1 How can we make a meaningful comparison? Restrict attention to the conservative part (circular orbits) of the dynamics 1 Find a gauge-invariant observable computable in both formalisms 2 Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 15 / 29
Circular orbit means Helical Killing symmetry µ u 1 µ K µ µ 1 K K light cylinder time space particle's trajectories Luc Blanchet (IAP) PN theory vs BH perturbations Rencontres de Moriond 16 / 29
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