GRAVITATIONAL RECOIL OF BINARY BLACK HOLES Luc Blanchet Gravitation - - PowerPoint PPT Presentation

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GRAVITATIONAL RECOIL OF BINARY BLACK HOLES Luc Blanchet Gravitation et Cosmologie ( G R C O ) CNRS / Institut dAstrophysique de Paris 23 novembre 2006 Based on Gravitational recoil of inspiralling black-hole binaries to

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GRAVITATIONAL RECOIL OF BINARY BLACK HOLES Luc Blanchet

Gravitation et Cosmologie (GRεCO) 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 (GRεCO) Gravitational recoil From geometry to numerics 1 / 19

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Digest on the history of gravitational recoil

1

General formalisms

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]

2

Core collapse to BH

Vrecoil 300km/s (PN calculation) [Bekenstein 1973] Perturbation of Oppenheimer-Snyder collapse to BH [Moncrief 1979]

3

Compact binary systems

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 (GRεCO) Gravitational recoil From geometry to numerics 2 / 19

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Recent calculations in the case of compact binaries

1

Analytical or semi-analytical

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]

2

Numerical

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 (GRεCO) Gravitational recoil From geometry to numerics 3 / 19

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Flux of linear momentum

1

Use stress-energy tensor of GWs T GW

µν =

1 32π ∂µhTT

ij ∂νhTT ij

2

Derive the linear momentum loss as surface integral at infinity dP i dt GW = −r2

  • dΩ ni T GW

00

General expression in terms of radiative moments UL and VL [Thorne 1980] dP i dt GW =

+∞

  • ℓ=2

1 c2ℓ+3

  • αℓ U (1)

iL U (1) L + βℓ εijk U (1) jL−1 V (1) kL−1 + γℓ

c2 V (1)

iL V (1) L

  • Note that the multipolar order (ℓ) scales with the PN order (c−1)

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 4 / 19

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Linear momentum flux at Newtonian order

The radiative moments UL, VL reduce to the source multipole moments UL = I(ℓ)

L + O

1 c3

  • VL

= J(ℓ)

L + O

1 c3

  • The source moments IL, JL take on their usual Newtonian expressions

IL =

  • d3x ρ ˆ

xL + O 1 c2

  • JL

= εkliℓ

  • d3x ρ vk ˆ

xL−1l + O 1 c2

  • The “Newtonian” linear momentum flux takes the expression

dP i dt GW = 1 c7 2 63I(4)

ijkI(3) jk + 16

45εijk I(4)

jk J(3) kl

  • corresponds to a 3.5PN radiation reaction effect

+O 1 c9

  • Luc Blanchet (GRεCO)

Gravitational recoil From geometry to numerics 5 / 19

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Radiation-reaction calculation of the recoil

To 3.5PN order the radiation reaction force is electromagnetic-like with both scalar Vreac and vectorial Ai

reac potentials [Blanchet & Damour 1984]

In a certain gauge the radiation reaction potentials are [Blanchet 1997] Vreac = − 1 5c5 xijI(5)

ij + 1

c7 1 189xijkI(7)

ijk + 1

70x2 xijI(7)

ij

  • + O

1 c9

  • Ai

reac

= 1 21c7 ˆ xijkI(6)

ijk +

4 45c7 εijkxjlJ(5)

kl + O

1 c9

  • The total recoil force (integrated over the source) is

F i

reac = − 1

c7 2 63I(4)

ijkI(3) jk + 16

45εijk I(4)

jk J(3) kl

  • agrees with the Newtonian flux calculation

+O 1 c9

  • Luc Blanchet (GRεCO)

Gravitational recoil From geometry to numerics 6 / 19

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Gravitational recoil of BH binaries (Newtonian order)

The linear momentum ejection is in the direction of the lighter mass’ velocity

v

2

m

1

m2

smaller mass

V

recoil motion center−of−mass

v

1

larger mass momentum ejection

In the Newtonian approximation [with f(η) ≡ η2√1 − 4η] Vrecoil = 20 km/s 6M r 4 f(η) fmax = 1500 km/s 2M r 4 f(η) fmax

[Fitchett 1983]

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 (GRεCO) Gravitational recoil From geometry to numerics 7 / 19

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Linear momentum flux to 2PN order

1

We need to include higher-order radiative moments dP i dt GW ∼ 1 c7

  • U (1)

ijk U (1) jk + εijk U (1) jl V (1) kl

  • +

1 c9

  • U (1)

ijkl U (1) jkl + εijk U (1) jlm V (1) klm + V (1) ijk V (1) jk

  • +

1 c11

  • U (1)

ijklm U (1) jklm + εijk U (1) jlmn V (1) klmn + V (1) ijkl V (1) jkl

  • 2

To 2PN order the tail contributions are Uij = I(2)

ij + 2G m

c3 t

−∞

dτ I(4)

ij (τ)

  • ln

t − τ 2

  • + 11

12

  • ,

Uijk = I(3)

ijk + 2G m

c3 t

−∞

dτ I(5)

ijk(τ)

  • ln

t − τ 2

  • + 97

60

  • ,

Vij = J(2)

ij + 2G m

c3 t

−∞

dτ J(4)

ij (τ)

  • ln

t − τ 2

  • + 7

6

  • Luc Blanchet (GRεCO)

Gravitational recoil From geometry to numerics 8 / 19

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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] Iij = η m

  • xij
  • 1 + γ
  • − 1

42 − 13 14η

  • + γ2
  • − 461

1512 − 18395 1512 η − 241 1512η2

  • + r2vij

11 21 − 11 7 η + γ 1607 378 − 1681 378 η + 229 378η2

  • ,

Iijk = −η δm

  • xijk
  • 1 − γη − γ2

139 330 + 11923 660 η + 29 110η2

  • +r2 xivjk
  • 1 − 2η − γ
  • −1066

165 + 1433 330 η − 21 55η2

  • ,

Jij = −η δm

  • εabixjavb
  • 1 + γ

67 28 − 2 7η

  • +γ2

13 9 − 4651 252 η − 1 168η2

  • where η ≡ µ/m (mass ratio) and γ ≡ m/r (PN parameter)

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 9 / 19

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Result for the 2PN linear momentum [Blanchet, Qusailah & Will 2005]

dP i dt GW = −464 105 f(η) x11/2

  • 1 +

1PN

  • −452

87 − 1139 522 η

  • x +

tail

  • 309

58 π x3/2 +

  • −71345

22968 + 36761 2088 η + 147101 68904 η2

  • x2
  • 2PN
  • ˆ

λi The recoil of the center-of-mass follows from integrating dP i

recoil

dt = − dP i dt GW The recoil velocity V i

recoil can be obtained analytically in the adiabatic

approximation (up to the ISCO)

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 10 / 19

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Recoil velocity at the ISCO

Table: Recoil velocity (km s−1) at the ISCO defined by xISCO = 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 (GRεCO) Gravitational recoil From geometry to numerics 11 / 19

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Estimate of the recoil accumulated during the plunge

We make a number of simplifying assumptions

1

The plunge is approximated as that of a test particle of mass µ moving on a geodesic of the Schwarzschild metric of a BH of mass m

2

The 2PN linear momentum flux is integrated on that orbit (y ≡ m/r) ∆V i

plunge = L

horizon

ISCO

1 m ω dP i dt

  • dy
  • E2 − (1 − 2y)(1 + L2 y2)

M plunging geodesic

  • f Schwarzschild

ISCO

E and L are the constant energy and angular momentum of the Schwarzschild plunging orbit

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 12 / 19

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Matching to the circular orbit at the ISCO

1

We evolve a circular orbit at the ISCO (where x = 1/6) piecewise to a new

  • rbit using energy and angular momentum balance equations

dE dt = −32 5 η mx5

ISCO

dL dt = 1 ωISCO dE dt

2

We discretize these relations around the ISCO values over a fraction of orbital period α P (where 0 < α < 1) E = EISCO − 64π 5 η α x7/2

ISCO

L = LISCO − 64π 5 η α x2

ISCO

3

We check that the results are insensitive to the value of α below 0.1

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 13 / 19

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Estimation of the recoil up to coalescence at r = 2m

[Blanchet, Qusailah & Will 2005]

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 14 / 19

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Brownsville group [Campanelli & Lousto 2005]

For the mass ratio η = 0.24 corresponding to m2/m1 = 0.66 the final kick is around ∼ 200 km/s but with large error bars

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 15 / 19

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Goddard group [Baker, Centrella, Choi, Koppitz, van Meter & Miller 2006]

⇐ = 2PN peak For the mass ratio η = 0.24 (corresponding to m2/m1 = 0.66) Kick at the maximum is ∼ 170 km/s Final kick is ∼ 105 km/s We note that the kick at the maximum is in rather good agreement with the 2PN calculation for this mass ratio (namely ∼ 160 km/s)

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 16 / 19

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Jena group [Gonzalez, Sperhake, Bruegmann, Hannam & Husa 2006]

100 200 300 t (MADM) 50 100 150 200 250 300 v (km/s)

h1 h2 h3

100 200 300 400 500 t (MADM) 50 100 150 200 250 300 v (km/s)

r0 = 6.0 M r0 = 7.0 M r0 = 8.0 M

⇐ = 2PN peak For the mass ratio η = 0.195 (m2/m1 = 0.36) Kick at the maximum is ∼ 250 km/s Final kick is ∼ 175 km/s Again the kick at the maximum is in good agreement with the 2PN calculation for this mass ratio (namely ∼ 250 km/s)

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 17 / 19

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Summary of comparisons

100 200 300 km/s =0.24 η q=0.66

(peak)

Favata et al (2004) Brownsville (2005) Jena (2006) BQW (2005) DG (2006) Goddard (2006) 100 200 300 km/s η=0.2 q=0.36

(peak)

Favata et al (2004) Brownsville (2005) Jena (2006) BQW (2005) DG (2006) Goddard (2006)

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 18 / 19

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Conclusions

The gravitational recoil is likely to have important astrophysical consequences in models for massive BH formation involving successive mergers from smaller BH seeds The computation of the recoil at 2PN order gives a maximal contribution of 22 km/s up to the ISCO (probably very accurate) For a mass ratio of 0.36 the recoil up to the BH coalescence at r = 2m is estimated at ∼ 250 km/s using some approximation in the plunge phase Recent progresses in numerical relativity confirm this estimate but show a subsequent decrease of the recoil presumably due to the ring-down phase to the value ∼ 175 km/s The braking of the recoil velocity in the ring-down phase should be better understood theoretically

Luc Blanchet (GRεCO) Gravitational recoil From geometry to numerics 19 / 19