The Gravitational Radiation Rocket Effect. Marc Favata Cornell University GR17, Dublin, July 2004

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1 The Gravitational Radiation Rocket Effect recoil Marc Favata Cornell University GR17, Dublin, July 004 Favata, Hughes, & Holz, ApJL 607, L5, astro-ph/ Merritt, Milosavljevic, Favata, Hughes, & Holz, ApJL 607, L9, astro-ph/040057

2 astro-ph/ Astrophysical Motivation - Ejection/displacement of BH/BH binary from various stellar systems. [km/s] - Escape likely for globular clusters, dwarf galaxies, high-z halos. - Implications for IMBH and SMBH formation. fig. from Merritt, et.al, (004)

3 The gravitational radiation recoil/rocket effect: GW momentum flux: j dpgw d jab d 16 d pa d ab qa = I I + ε I S 4 3 jpq dt 63 dt dt 45 dt dt [fig. from Wiseman, PRD 46, 1517]

4 Gravitational radiation recoil/rocket effect: GW momentum flux: j dpgw d jab d 16 d pa d ab qa = I I + ε I S 4 3 jpq dt 63 dt dt 45 dt dt kick for Newtonian circular binary [Fitchett 1983]: V kick 4 f( q) Gm ( 1+ m)/ c = 1480 km/s f r f ( q) = q (1 q) 5 (1 + q) max term. q = m m 1 f max = f q ( max = 0.38) If system is symmetric (m 1 =m ), recoil is zero (for non-spinning holes). [fig. from Wiseman, PRD 46, 1517]

5 Previous work: Foundations: Bonnor & Rotenberg (1961); Papapetrou (196); Peres (196). Recoil from gravitational collapse: Bekenstein (1973) [upper limit of 300 km/s] Moncrief (1979) [recoil ~ 5 km/s ] Recoil from binaries: Fitchett (1983): quasi-newtonian calculation Fitchett & Detweiler (1984): BH perturbation (a/m=0, no radiation reaction, circular orbits) Oohara & Nakamura (198): plunge from infinity into Schwarzschild [10-40 km/s]; Nakamura & Haugan (1983): Kerr radial in-fall along symmetry axis [~ 5 km/s] Redmount & Rees (1989): astrophysical implications of GW recoil Wiseman (199): Post-Newtonian calc. [1-3 km/s at r = 9M tot ] Kidder (1995): spin effects on recoil Numerical Relativity: Anninos & Brandt (1998): head-on collision [10-0 km/s] Brandt & Anninos (1999): axisymmetric Brill waves [~150 km/s]

6 Recoil relies on symmetry breaking Lowest order quasi-newtonian calculation gives (circular orbits) [Fitchett (1983)]: V kick 4 f( q) Gm ( 1+ m)/ c = 1480 km/s f r max term. If system is symmetric (m 1 =m ), recoil is zero (for non-spinning holes). f max q (1 q) f ( q) = 5 (1 + q) = f q = 0.38) ( max m q = m

7 Recoil relies on symmetry breaking Lowest order quasi-newtonian calculation gives (circular orbits) [Fitchett (1983)]: V kick 4 f( q) Gm ( 1+ m)/ c = 1480 km/s f r max term. If system is symmetric (m 1 =m ), recoil is zero (for non-spinning holes). f max q (1 q) f ( q) = 5 (1 + q) = f q = 0.38) ( max q = m m Spin-orbit corrections to Fitchett s formula (circular binary) [Kidder 1995]: [symmetry broken even for q=1] V kick 4 9/ ( ) SO(, 1, ) 1480 km/s f q M 883 km/s f qa ɶɶ a M = + f r f r max term. SO,max term. [valid for non-precessing binary, spins aligned/anti-aligned] f = q ( aɶɶ aq)/(1 + q ) 5 SO 1

8 Our approach: circular, equatorial Kerr orbits Adiabatic inspiral: use BH perturbation theory (test mass limit) to compute momentum flux up to the ISCO. Plunge into the horizon: approximate scheme to calculate orbit and radiation Scaling functions used to extrapolate to higher mass ratios. a/ M = 0, η=0.1

9 mass ratio of BH binaries vs. redshift only way to really compute the recoil is with numerical relativity but perturbation theory can be more useful than you might think. [fig. from Volonteri, Haardt,& Madau; ApJ 58, 559 (003) ]

10 Accumulated recoil for a/m=0.8 η=0.1 orbit momentum vector Why isn the kick zero for circular orbits? 1. radiation reaction means orbits are not exactly circular.. final orbit before horizon is not closed, so momentum can t cancel. center of mass accumulated recoil

11 Scaling to larger q effective-one-body treatment: S (m 1,m ) (M,µ) S 1 m 1 S = am ɶ µ m q = m m 1 η = µ M M = m1+ m

12 Scaling to larger q effective-one-body treatment: S (m 1,m ) (M,µ) S 1 m 1 S = am ɶ µ m q = m m 1 η = µ M M = m1+ m dp dt j When q 1, q. To scale-up to large q, we dp dt j use f( q). f( q) q for small q, f( q= 0) = 0.

13 BH perturbation theory (test mass limit): Solve Teukolsky equation for Ψ 4 to get momentum flux: [using code developed by Hughes PRD 61, PRD 64] Ψ = 1 imφ dω R () r S ( θ; aω) e e 4 4 lmω ( r ia cos θ ) lm 1 H 1 = Zlmk Slm θ aωmk e e = hɺɺ + ihɺɺ r r lmk + 1. pick a geodesic orbit with E, L z lm iωt imφ iωmk ( t r* ) ( ; ) ( ) (as ) j dpgw H = FZ [ lmk(), t ωmk ()] t dt. Solve Teukolsky equation for this geodesic. 3. Compute GW fluxes de/dt and dl z /dt to infinity and down the horizon. 4. Update E, L z for the orbit and generate an inspiral trajectory up to the ISCO 5. Use calculated quantities to compute dp j /dt along the orbit.

14 Results I: Center of mass velocity for circular, equatorial orbit up to ISCO. [Schwarzschild, a/m=0] [reduced mass ratio=0.1] V MAX = 4.7 km/s Agrees well with Fitchett

15 Results II: Center of mass velocity for circular, equatorial orbit up to ISCO. [Kerr a/m=0.99, ignoring finite size effects ] [ mass ratio=0.1 ] V MAX = 57 km/s Kick reduced by gravitational redshift wave scattering

16 ISCO recoil vs. effective-spin V kick,isco f( q) M = 4 km/s fmax risco ( r / M) isco η = 0.1 [a convenient fitting function] [ large effective spins should be excluded due to finite-size effects. ]

17 Recoil from plunge: Approximate methods to compute recoil from near ISCO through final plunge for circular, equatorial orbits in Kerr. Match plunging geodesic orbit onto adiabatic inspiral just before ISCO. Compute momentum flux: (approximate lower limit) Use orbit [x(t), y(t)] to compute Newtonianorder multipole moments: I = [ µ x () tx ()] t, I = [ µ x () tx () tx ()] t, S = [ µ x ()[ t x() t v()] t ] STF STF STF jk j k jki j k i jk k j (approximate upper limit) Extrapolate perturbation results past ISCO: Truncate when: j dpgw d jab d 16 d pa d ab qa = I I + ε I S 4 3 jpq dt 63 dt dt 45 dt dt α iϕ () t x y B/ r, r 3M e [ Pɺ > GW + ipɺ GW ] = dτ / dt (const), r 3M r = r horizon + µ

18 Limits on final recoil plunge & final kick still uncertain - averaging over spins and mass ratios: <V upper limit > = 60 km/s <V lower limit > = 38 km/s

19 summary of results: accurate calculation of recoil up to ISCO (for small mass ratios) - reduced relative to Newtonian estimates strong-field effects important - few km/s for large ISCO radius; up to a few 100 km/s for large prograde inspiral plunge & final kick still uncertain V kick t 100 km/s likely; V kick ~ few 100 km/s not unexpected; largest possible kicks have V kick d 500 km/s.

20 Consequences of radiation recoil

21 Application: IMBH growth GW recoil (along with 3-body ejection) makes IMBH growth through hierarchical mergers in globular clusters less likely (unless initial seed mass is t 150 M ) IMBHs growth from cluster core collapse, accretion, or collapse of Pop III stars more likely Miller & Colbert (003), astro-ph/ van der Marel (003), astro-ph/ Gultekin, Miller, Hamilton (004), astro-ph/04053

22 Application: BH displacement dynamical friction returns BH (and bounded stars) in ~ yrs (~3-5 times longer for triaxial galaxies) nuclear density profile lowered off-nuclear AGN activity at medium-high redshift? fig. from Madau & Quataert, ApJL 606, L0, astro-ph/ Merritt, et.al, ApJL 607, L9, astro-ph/040057

23 Application: SMBH growth SMBHs grow through mergers + accretion; DM halo escape velocities smaller at high redshift; GW recoil makes it hard to confine low mass seeds in mergers at zt 10 [Haiman, astro-ph/ ]: Madau & Quataert, ApJL 606, L0, astro-ph/ Merritt, et.al, ApJL 607, L9, astro-ph/ Volonteri, Haardt, Madau, ApJ 58, 559 (003) growth of M~ SMBH in z=6.43 quasar SDSS J puts limits on typical recoils < 64 km/s OR implies super-eddington accretion [but Haiman ignores mass ratio dependence of GW recoil] Yoo & Miralda-Escudé, astro-ph/040617: using our formulas for mass ratio dependence, find that super-eddington accretion is NOT necessary to achieve present BH mass (even for high kick velocities).

24 Application: displacement of X-shaped radio lobes Core of radio galaxy NGC 36 BH/BH merger realigns jet axis Recoil may displace jets from center of X highly speculative!! Merritt & Ekers, Science 97, 1310 (00) Merritt, et.al, ApJL 607, L9, astro-ph/040057

25 Future work: extend to circular, inclined orbits explore spin-orbit interactions with post-newtonian eqns getting the plunge right effect on gravitational wave signal