Hot Topics in General Relativity and Gravitation FIRST LAW OF COMPACT BINARY MECHANICS AT 4PN ORDER 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 / 41
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 2 / 41
Modelling the compact binary dynamics L m 1 CM m 2 Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 3 / 41
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 3 / 41
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 4 / 41
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 4 / 41
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 4 / 41
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 4 / 41
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 5 / 41
Comparisons between PN and GSF COMPARISONS BETWEEN THE PN AND GRAVITATIONAL SELF-FORCES Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 6 / 41
Comparisons between PN and GSF Problem of the gravitational self-force (GSF) [Mino, Sasaki & Tanaka 1997; Quinn & Wald 1997; Detweiler & Whiting 2003] A particle is moving on a background µ µ a = F space-time of a massive black hole GSF µ Its stress-energy tensor modifies the a = 0 background gravitational field M m Because of the back-reaction the motion of the particle deviates from a background geodesic hence the gravitational self force � m � a µ = F µ ¯ GSF = O M The GSF is computed to high accuracy by numerical methods [Sago, Barack & Detweiler 2008; Shah, Friedmann & Whiting 2014] analytical ones [Mano, Susuki & Takasugi 1996; Bini & Damour 2013, 2014] Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 7 / 41
Comparisons between PN and GSF Common regime of validity of GSF and PN m 2 r log 10 ( r / m ) m 1 4 Post-Newtonian Theory Post-Newtonian −1 3 & 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 ( G R ε C O ) PN modelling of ICBs HTGRG 8 / 41
Comparisons between PN and GSF Why and how comparing PN and GSF predictions? Both the PN and SF approaches use a self-field regularization for point particles followed by a renormalization. However, the prescription are very different SF theory is based on a prescription for the Green’s function G R based on 1 Hadamard’s elementary solution [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 ( G R ε C O ) PN modelling of ICBs HTGRG 9 / 41
Comparisons between PN and GSF Circular orbit means Helical Killing symmetry µ u 1 µ K µ µ 1 K K light cylinder particle's trajectories Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 10 / 41
Comparisons between PN and GSF Looking at the conservative part of the dynamics + + J J I 0 standing waves I 0 at infinity no incoming radiation condition - - J J Situation with the HKV Physical situation Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 11 / 41
Comparisons between PN and GSF The redshift observable [Detweiler 2008] For exactly circular orbits the geometry admits a 1 helical Killing vector with particle K µ ∂ µ = ∂ t + Ω ∂ ϕ R Ω The four-velocity of the particle is tangent to the 2 Killing vector hence K µ 1 = z 1 u µ 1 k µ 2π Ω This z 1 is the Killing energy of the particle 3 u µ associated with the HKV and can also be viewed as a redshift factor For eccentric orbits one considers the averaged 4 redshift [Barack & Sago 2011] time � P � z 1 � = 1 space black hole d t z 1 ( t ) space P 0 Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 12 / 41
Comparisons between PN and GSF Post-Newtonian calculation of the redshift factor In a coordinate system such that K µ ∂ µ = ∂ t + ω ∂ ϕ we have v 1 y 1 � � 1 / 2 v µ 1 v ν z 1 = 1 1 = − ( g µν ) 1 r 12 u t c 2 1 � �� � y 2 regularized metric v 2 One needs a self-field regularization Hadamard’s partie finie regularization is extremely useful in practical calculations but yields (UV and IR) ambiguity parameters at high PN orders Dimensional regularization is an extremely powerful regularization which seems to be free of ambiguities at any PN order Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 13 / 41
Comparisons between PN and GSF High-order PN result for the redshift factor [Blanchet, Detweiler, Le Tiec & Whiting 2010, 2011] The redshift factor of particle 1 through 3PN order and augmented by 4PN and 5PN logarithmic terms is 1PN 2PN 3PN � 3 � √ 4 − 3 1 − 4 ν − ν � �� � � �� � � �� � [ · · · ] x 2 + [ · · · ] x 3 + u t [ · · · ] x 4 = 1 + x + 1 4 2 � � � � x 5 + x 6 + O � x 7 � + · · · + [ · · · ] ν ln x · · · + [ · · · ] ν ln x � �� � � �� � 4PN log 5PN log � Gm Ω � 3 / 2 where we pose ν = m 1 m 2 and x = m 2 c 3 Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 14 / 41
Comparisons between PN and GSF High-order PN result for the redshift factor [Blanchet, Detweiler, Le Tiec & Whiting 2010, 2011] We re-expand in the small mass-ratio limit q = m 1 /m 2 ≪ 1 so that u T = u T + q 2 u T Schw + q u T + O ( q 3 ) SF PSF � �� � � �� � self-force post-self-force � 3 / 2 we find � Gm 2 Ω Posing y = c 3 3PN � �� � � � − 121 + 41 − y − 2 y 2 − 5 y 3 + u T 32 π 2 y 4 = SF 3 � � � � a 4 + 64 a 5 − 956 y 5 y 6 + o ( y 6 ) + 5 ln y + 105 ln y � �� � � �� � 4PN 5PN Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 15 / 41
Comparisons between PN and GSF High-order PN fit to the numerical self-force � Gm 2 Ω � 2 / 3 Numerical SF data is fitted with a PN series in y = c 3 � [ a n PN + b n PN ln y + · · · ] y n +1 z 1 = a The 3PN prediction agrees with the SF value with 7 significant digits 3PN value SF fit 32 π 2 = − 27 . 6879026 · · · a 3PN = − 121 3 + 41 − 27 . 6879034 ± 0 . 0000004 Logarithmic coefficients b 4PN and b 5PN also perfectly agree Post-Newtonian coefficients are measured up to 7PN order a 4PN − 114 . 34747(5) a 5PN − 245 . 53(1) a 6PN − 695(2) b 6PN +339 . 3(5) a 7PN − 5837(16) Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 16 / 41
Comparisons between PN and GSF Further developments 4PN coefficient known analytically by GSF calculation [Bini & Damour 2013] 1 a 4PN = − 1157 + 677 512 π 2 − 256 5 ln 2 − 128 5 γ E 15 and agrees with numerical value [Blanchet, Detweiler, Le Tiec & Whiting 2011] Super-high precision analytical and numerical GSF calculations of the redshift 2 factor up to 10PN order, including a previously unexpected existence of half-integral PN terms starting at 5.5PN order [Shah, Friedman & Whiting 2013] Half-integral conservative PN terms [Blanchet, Faye & Whiting 2013, 2014] 3 a 5.5PN = − 13696 a 6.5PN = 81077 a 7.5PN = 82561159 525 π , 3675 π , 467775 π Luc Blanchet ( G R ε C O ) PN modelling of ICBs HTGRG 17 / 41
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