GRAVITATIONAL RECOIL OF BINARY BLACK HOLES Luc Blanchet Gravitation et Cosmologie ( G R ε C O ) CNRS / Institut d’Astrophysique de Paris 23 novembre 2006 Based on Gravitational recoil of inspiralling black-hole binaries to second-post-Newtonian order [Blanchet, Qusailah & Will 2005] Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 1 / 19
Digest on the history of gravitational recoil General formalisms 1 Near-zone computation of recoil in linearized gravity [Peres 1958] Flux computations of recoil as interaction between quadrupole and octupole moments [Bonnor & Rotenberg 1961, Papapetrou 1971] General multipole expansion ( ∀ ℓ ≥ 2) of linear momentum flux [Thorne 1080] Radiation-reaction computation of recoil and linear momentum balance equation [Blanchet 1996] Core collapse to BH 2 V recoil � 300km / s (PN calculation) [Bekenstein 1973] Perturbation of Oppenheimer-Snyder collapse to BH [Moncrief 1979] Compact binary systems 3 Recoil for point-mass binaries in Newtonian approximation [Fitchett 1983] Recoil for particle around Kerr BH (perturbation theory) [Fitchett & Detweiler 1984] Particle falling on symmetric axis of Kerr [Nakamura & Haugan 1983] 1PN calculation to the recoil from point-mass binaries [Wiseman 1992] Contributions of spins (PN calculation) [Kidder 1995] Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 2 / 19
Recent calculations in the case of compact binaries Analytical or semi-analytical 1 Perturbation calculation ( µ ≪ M ) of recoil during final plunge of two BH [Favata, Hughes & Holz 2004] 2PN calculation and estimate of the contribution of the plunge phase [Blanchet, Qusailah & Will 2005] (this work) Application of the effective-one-body (EOB) approach [Damour & Gopakumar 2006] Numerical 2 Perturbation/full numerical (Lazarus code) [Campanelli & Lousto 2004] Binary BH grand challenge [Baker, Centrella, Choi, Koppitz, van Meter & Miller 2006] Binary BH grand challenge [Gonzalez, Sperhake, Bruegmann, Hannam & Husa 2006] Close limit approximation [Sopuerta, Yunes & Laguna 2006] Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 3 / 19
Flux of linear momentum Use stress-energy tensor of GWs 1 1 T GW 32 π � ∂ µ h TT ij ∂ ν h TT µν = ij � Derive the linear momentum loss as surface integral at infinity 2 � dP i � GW � d Ω n i T GW = − r 2 00 dt General expression in terms of radiative moments U L and V L [Thorne 1980] � dP i � GW + ∞ 1 � kL − 1 + γ ℓ � � α ℓ U (1) iL U (1) L + β ℓ ε ijk U (1) jL − 1 V (1) c 2 V (1) iL V (1) = L c 2 ℓ +3 dt ℓ =2 Note that the multipolar order ( ℓ ) scales with the PN order ( c − 1 ) Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 4 / 19
Linear momentum flux at Newtonian order The radiative moments U L , V L reduce to the source multipole moments � 1 � I ( ℓ ) U L = L + O c 3 � 1 � J ( ℓ ) V L = L + O c 3 The source moments I L , J L take on their usual Newtonian expressions � 1 � � d 3 x ρ ˆ I L = x L + O c 2 � 1 � � d 3 x ρ v k ˆ J L = ε kl � i ℓ x L − 1 � l + O c 2 The “Newtonian” linear momentum flux takes the expression � 2 � 1 � dP i � GW � � 1 jk + 16 63 I (4) ijk I (3) 45 ε ijk I (4) jk J (3) = + O kl c 7 c 9 dt � �� � corresponds to a 3.5PN radiation reaction effect Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 5 / 19
Radiation-reaction calculation of the recoil To 3.5PN order the radiation reaction force is electromagnetic-like with both scalar V reac and vectorial A i reac potentials [Blanchet & Damour 1984] In a certain gauge the radiation reaction potentials are [Blanchet 1997] � 1 � 1 � � − 1 ij + 1 ijk + 1 70 x 2 x ij I (7) 5 c 5 x ij I (5) 189 x ijk I (7) V reac = + O ij c 7 c 9 � 1 � 1 4 x ijk I (6) 45 c 7 ε ijk x jl J (5) A i = 21 c 7 ˆ ijk + kl + O reac c 9 The total recoil force (integrated over the source) is � 2 � 1 � � reac = − 1 jk + 16 63 I (4) ijk I (3) 45 ε ijk I (4) jk J (3) F i + O kl c 7 c 9 � �� � agrees with the Newtonian flux calculation Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 6 / 19
Gravitational recoil of BH binaries (Newtonian order) The linear momentum ejection is in the direction of the lighter mass’ velocity smaller mass m 2 center−of−mass motion v v 2 V recoil 1 m 1 larger mass momentum ejection In the Newtonian approximation [with f ( η ) ≡ η 2 √ 1 − 4 η ] � 4 f ( η ) � 6 M V recoil = 20 km / s r f max � 4 f ( η ) � 2 M = 1500 km / s [Fitchett 1983] r f max Very interesting result which shows the astrophysical relevance of GW recoil but illustrates the fact that the recoil is mainly generated in the strong field region Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 7 / 19
Linear momentum flux to 2PN order We need to include higher-order radiative moments 1 � GW � dP i 1 � � U (1) ijk U (1) jk + ε ijk U (1) jl V (1) ∼ kl dt c 7 1 � � U (1) ijkl U (1) jkl + ε ijk U (1) jlm V (1) klm + V (1) ijk V (1) + jk c 9 � � 1 U (1) ijklm U (1) jklm + ε ijk U (1) jlmn V (1) klmn + V (1) ijkl V (1) + c 11 jkl To 2PN order the tail contributions are 2 � t � � t − τ � � ij + 2 G m + 11 I (2) dτ I (4) U ij = ij ( τ ) ln , c 3 2 12 −∞ � t � � t − τ � � ijk + 2 G m + 97 I (3) dτ I (5) U ijk = ijk ( τ ) ln , c 3 2 60 −∞ � t � � t − τ � � ij + 2 G m + 7 J (2) dτ J (4) V ij = ij ( τ ) ln c 3 2 6 −∞ Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 8 / 19
Application to compact binaries in circular orbits All the required source multipole moments in the case of compact binaries on circular orbits are known [Blanchet, Iyer & Joguet 2002, Arun, Blanchet, Iyer & Qusailah 2004] � � � � � �� − 1 42 − 13 − 461 1512 − 18395 1512 η − 241 x � ij � + γ 2 1512 η 2 I ij = η m 1 + γ 14 η � 11 � 1607 ��� 21 − 11 378 − 1681 378 η + 229 + r 2 v � ij � 378 η 2 7 η + γ , � � � 139 �� 330 + 11923 660 η + 29 x � ijk � 1 − γη − γ 2 110 η 2 I ijk = − η δm � � ��� − 1066 165 + 1433 330 η − 21 + r 2 x � i v jk � 55 η 2 1 − 2 η − γ , � � � 67 � 28 − 2 ε ab � i x j � a v b J ij = − η δm 1 + γ 7 η � 13 ��� 9 − 4651 1 + γ 2 168 η 2 252 η − where η ≡ µ/m (mass ratio) and γ ≡ m/r (PN parameter) Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 9 / 19
Result for the 2PN linear momentum [Blanchet, Qusailah & Will 2005] 1PN tail � �� � � �� � � GW � dP i � � � − 464 − 452 87 − 1139 309 105 f ( η ) x 11 / 2 58 π x 3 / 2 = 1 + 522 η x + dt � � � − 71345 22968 + 36761 2088 η + 147101 ˆ 68904 η 2 x 2 λ i + � �� � 2PN The recoil of the center-of-mass follows from integrating � dP i � GW dP i recoil = − dt dt The recoil velocity V i recoil can be obtained analytically in the adiabatic approximation (up to the ISCO) Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 10 / 19
Recoil velocity at the ISCO Table: Recoil velocity (km s − 1 ) at the ISCO defined by x ISCO = 1 / 6. η = µ/m 0.05 0.1 0.15 0.2 0.24 Newtonian 2.29 7.92 14.56 18.30 11.78 N + 1PN 0.27 0.77 1.16 1.12 0.55 N + 1PN + 1.5PN (tail) 2.87 9.80 17.74 21.96 13.97 N + 1PN + 1.5PN + 2PN 2.73 9.51 17.57 22.22 14.38 Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 11 / 19
Estimate of the recoil accumulated during the plunge We make a number of simplifying assumptions The plunge is approximated as that of a test particle of mass µ moving on a 1 geodesic of the Schwarzschild metric of a BH of mass m The 2PN linear momentum flux is integrated on that orbit ( y ≡ m/r ) 2 � 1 � horizon � dP i dy ∆ V i plunge = L � E 2 − (1 − 2 y )(1 + L 2 y 2 ) m ω dt ISCO ISCO E and L are the constant energy and angular momentum of the Schwarzschild M plunging orbit plunging geodesic of Schwarzschild Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 12 / 19
Matching to the circular orbit at the ISCO We evolve a circular orbit at the ISCO (where x = 1 / 6) piecewise to a new 1 orbit using energy and angular momentum balance equations dE − 32 η mx 5 = ISCO dt 5 dL 1 dE = dt ω ISCO dt We discretize these relations around the ISCO values over a fraction of orbital 2 period α P (where 0 < α < 1) E ISCO − 64 π 5 η α x 7 / 2 E = ISCO L ISCO − 64 π η α x 2 L = ISCO 5 We check that the results are insensitive to the value of α below 0 . 1 3 Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 13 / 19
Estimation of the recoil up to coalescence at r = 2 m [Blanchet, Qusailah & Will 2005] Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 14 / 19
Brownsville group [Campanelli & Lousto 2005] For the mass ratio η = 0 . 24 corresponding to m 2 /m 1 = 0 . 66 the final kick is around ∼ 200 km / s but with large error bars Luc Blanchet ( G R ε C O ) Gravitational recoil From geometry to numerics 15 / 19
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