What I did in grad school. Marc Favata

Transcription

1 What I did in grad school Marc Favata B-exam June 1, 006

2 Kicking Black Holes Crushing Neutron Stars and the adiabatic approximation in extreme-mass-ratio inspirals

3 How black holes get their kicks: The Gravitational Radiation Rocket Effect recoil Work based on: Favata, Hughes, & Holz, ApJL 607, L5, astro-ph/ Merritt, Milosavljevic, Favata, Hughes, & Holz, ApJL 607, L9, astro-ph/040057

4 Astrophysical Motivations Gravitational wave recoil: a GR application for astrophysics, NOT gravitational wave detection. Galaxies: Ejection of black holes from galaxies: [700 - few 1000km/s for large galaxies, 5-00 km/s for dwarf galaxies] Wandering of BHs not ejected. Smearing of central density cusp. Formation of SMBHs through hierarchical mergers Globular clusters: 1. Ejection from cluster: V esc ~ (3-100) km/s. Formation of IMBHs: seed holes with M d 50 M susceptible to ejection. [Gültekin, Miller, Hamilton, astro-ph/04053]

5 Understanding Gravitational Radiation Recoil: GW momentum flux: [Wiseman, PRD 46, 1517]

6 Understanding Gravitational Radiation Recoil: 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]

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 m q = m

8 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

9 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

10 Famous Moments in Recoil History

11 Foundations: History of Recoil Calculations Bonnor & Rotenberg (1961); Papapetrou (196); Peres (196): expressions for dp/dt Campbell & Morgan (1971); Dionysiou (1974); Booth (1974); Thorne (1980): generalizations Recoil from gravitational collapse: Bekenstein (1973): [upper limit of 300 km/s] Moncrief (1979): [recoil ~ 5 km/s ] Recoil from binaries: Post-Newtonian: Fitchett (1983): quasi-newtonian calculation; highly uncertain as high as 0,000 km/s Pietila et.al (1995): includes.5pn rad. reaction in Fitchett s calc; slightly larger values Wiseman (199): full extends Fitchett s calc. to 1PN order in dp/dt [.5PN Eqs. of motion] Kidder (1995): includes spin-orbit contribution to dp/dt Perturbation Theory Fitchett & Detweiler (1984): BH perturbation (a/m=0, no radiation reaction, circular orbits) Oohara & Nakamura (1983): plunge from infinity into Schwarzschild [~75 km/s] Kojima & Nakamura (1984): extension to Kerr Numerical Relativity and the Head-on Collision: Nakamura & Haugan (1983): Kerr radial in-fall along symmetry axis [~ 5 km/s] Andrade & Price (1997): head-on; close-limit approx.; highly uncertain [1- several 100 km/s] Anninos & Brandt (1998): head-on; full numerical [~9 km/s] Lousto & Price (004): BH perturbation theory; [~ 5 km/s] Brandt & Anninos (1999): BH distorted by axisymmetric Brill waves [~ km/s]

12 What we did: Extended BH perturbation theory work of Fitchett & Detweiler to BHs that are: spinning (but point-mass is nonspinning) inspiralling due to radiation reaction Estimated recoil due to final plunge from the last stable orbit (used more realistic orbits than Nakamura et. al, but neglecting some relativistic effects) Extended Fitchett s analytical computations to spinning holes (but neglected radiation reaction and all other post- Newtonian effects)

13 Our approach: circular, equatorial Kerr orbits Adiabatic inspiral: use BH perturbation theory to compute momentum flux up to the ISCO. this is an exact computation in the test mass limit. Plunge into the horizon: use Kerr geodesic that plunges from the ISCO use two different approximations for the momentum flux dp/dt Ringdown: Split the coalescence into 3 phases: ignore; its contribution to the total recoil is small; this was confirmed by Damour & Gopakumar (006) a/ M = 0, η=0.1

14 Main Approximation: small mass ratio, q = m 1 /m << 1 The only way to correctly compute the recoil is with numerical relativity but perturbation theory can be more useful than you might think. mass ratio of BH binaries vs. redshift Astrophysical Motivation: many BH coalescences will have mass ratios ~ 0.1 these can be treated with BH perturbation theory with modest accuracy [fig. from Volonteri, Haardt,& Madau; ApJ 58, 559 (003) ]

15 Physical motivation for small mass ratio assumption: effective-one-body treatment: (m 1,m ) (M,µ) S S 1 m 1 S = am ɶ µ m q = m m 1 η = µ M M = m1+ m perturbative calculations of the head-on collision agree with numerical relativity when scaled to higher mass-ratios: de dt For head-on collision of two (non-spinning) BHs, the scaling law In q 1 limit, q. q η de dt η, produces agreement with full numerical relativity. [Smarr (1978)]

16 Physical motivation for small mass ratio assumption: effective-one-body treatment: (m 1,m ) (M,µ) S S 1 m 1 S = am ɶ µ m q = m m 1 η = µ M M = m1+ m perturbative calculations of the head-on collision agree with numerical relativity when scaled to higher mass-ratios: Post-Newtonian studies have recently shown that this scaling is accurate (Blanchet, Qusailah, & Will; Damour & Gopakumar) A similar scaling holds for the momentum flux (Fitchett & Detweiler): q dp dt q q dp dt f q f q q q f q j When 1,. To scale-up to large, we j use ( ). ( ) for small, ( = 0) = 0. (the scaling is more complicated for spinning bodies)

17 Calculations I: BH perturbation theory Solve Teukolsky equation for Y 4 to get momentum flux up to the ISCO: [using code developed by Hughes(000)] Ψ = 1 imφ dω R () r S ( θ; aω) e e 4 4 lmω ( r ia cos θ ) lm 1 H 1 = Zlmk Slm a mk e e = hɺɺ + ihɺɺ r r lmk + lm iωt imφ iωmk ( t r* ) ( θ; ω ) ( ) (as ) 1. pick a geodesic orbit with E, L z. Solve Teukolsky equation for this geodesic. j dpgw H = FZ [ lmk(), t ωmk ()] t dt 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.

18 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

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

20 Results III: final recoil up to ISCO V kick,isco f( q) M = 4 km/s fmax risco ( r / M) isco η = 0.1 [a convenient fitting function] Recoil depends strongly on the ISCO radius ( large spins should be excluded due to finite-size effects. )

21 Calculations II: Recoil from plunge (lower limit): Use a ``semi-relativistic or ``hybrid method to compute recoil from ISCO to plunge into the horizon (circular-equatorial Kerr orbits) Match plunging geodesic onto adiabatic inspiral just before ISCO. Use orbit [x(t), y(t)] to compute Newtonian-order multipole moments: I STF STF jk = [ µ xj() txk()] t, I jki = [ µ xj() txk() txi()] t, S = [ µ x ()[ t x() t v()] t ] jk k j Plug into lowest order multipole expansion of 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 Truncate when: STF r = r horizon + µ

22 Calculations II: Recoil from plunge (upper limit): Perform a quick and dirty over-estimate of the momentum flux Again, match plunging geodesic onto adiabatic inspiral near ISCO From BH perturbation code, the momentum flux follows a power-law in radius up to the ISCO: Extrapolate power-law into plunge region; stop power-law at 3M and use: Integrate dp GW /dt to get kick dp dt GW iϕ () t x y 1 [ ɺ ɺ GW GW ] α = e P + ip dt d PGW = const, r 3M dτ dt r Truncate when: r = r horizon + µ

23 Limits on final recoil Large uncertainties, especially for retrograde orbits rapid, prograde case is more certain (dominated by inspiral recoil) V kick = 10 km/s bisects shaded region. For a/m=0, scaling to q=0.38 gives: V up,max = 465 km/s V low,max = 54 km/s upper and lower limits on total kick velocity: η = 0.1 (Large effective spins excluded because of finite-size effects)

24 summary of main results: first group to examine recoil from realistic orbits into spinning BHs made clear the importance of the plunge in determining the final kick performed BH perturbation calculation of recoil up to ISCO - recoil reduced relative to Newtonian estimates strong-field effects important - kick of ~ few km/s for large ISCO radius; up to ~ 00 km/s for moderately large prograde inspiral final kick still uncertain due to modeling of plunge phase Summary of kick values: V kick d 100 km/s likely; V kick ~ few 100 km/s not unexpected; largest possible kicks have V kick d 500 km/s. Ejection of BHs from large galaxies is very unlikely!

25 recent recoil calculations: Since our paper was published, there has been much recent progress in kick computations from post-newtonian theory and numerical relativity: Post-Newtonian Recoil Calculations (all for non-spinning holes): Blanchet, Qusailah, & Will (005): 1. Computed momentum flux to PN order (for circular orbits). Used this flux formula to perform a calculation analogous to our lower-limit calculation 3. Most of the finite-mass ratio effects are contained in Fitchett s function f(q), justifying our scaling-up procedure 4. Their ISCO recoil ( km/s) is a bit higher than our exact result (16 km/s) [neglect of 3PN effects?] 5. They find maximum kicks of 50 ± 50 km/s 6. Although they use PN fluxes, they assume circular orbits and the wrong Kepler s law during the plunge to simply their expressions. (Our lower-limit calculation does not). This overestimates the recoil.

26 recent recoil calculations: Since our paper was published, there has been much recent progress in kick computations from post-newtonian theory and numerical relativity: Post-Newtonian Recoil Calculations (all for non-spinning holes): Damour & Gopakumar (006): 1. Used effective-one-body (EOB) approach: models dynamics on a deformed Schwarzschild metric. Corrected the PN momentum flux of Blanchet et. al. 3. Their ISCO recoil (using Pade approximants) agrees more closely with our exact result. 4. Also confirm that additional finite-mass ratio effects are small (< 8%) 5. Show analytically that the kick is dominated by the peak in dp j /dt that occurs during the plunge 6. Show that the ringdown makes a relatively small contribution ( < 15% of total) 7. Their best-bet estimate is 74 km/s (but acknowledge that uncertainty remains) 8. Their quasi-newtonian estimate (throwing away PN corrections to momentum flux) agrees very well (< 7%) with our lower-limit calculation (as it should).

27 recent recoil calculations: Since our paper was published, there has been much recent progress in kick computations from post-newtonian theory and numerical relativity: Numerical Relativity (all for initially non-spinning holes): UTB group [Campenlli (005)]: 1. Use Lazarus approach (full GR plus close-limit approximation). Recoils are highly uncertain. examples: 77 ± 160 km/s (rescaled from q=0.5); 17 ± 95 km/s (rescaled from q=0.83); Penn State group [Herrmann, Shoemaker, & Laguna (006)] 1. full GR using moving-puncture method. max recoil of 118 km/s (rescaled from q=0.85) Goddard group [Baker et. al (006)] 1. full GR using moving-puncture method; most accurate simulations available.. max recoil of 163 km/s (rescaled from q=0.67)

28 summary of recent recoil calculations: Recoil calculations from selected groups (non-spinning holes) all estimates remain within our upper and lower bounds estimates seem to be converging to a range of ~70 00 km/s Nakamura Favata UTB Blanchet PennState Damour, Goddard