Self Force Calculations for Binary Black Hole Inspirals Sam R. - PowerPoint PPT Presentation

Self Force Calculations for Binary Black Hole Inspirals Sam R. Dolan University of Southampton EPSRC Post-doctoral Fellow BritGrav 2012 @ Southampton, 3rd-4th April 2012.

Talk Outline Motivation: Black holes, astrophysics and the 2-body problem in relativity. Orbital resonances on Kerr spacetime: a key challenge. Self-Force on Kerr: with m -mode regularization and 2+1D evolution Progress: Circular orbits on Schw., first results on Kerr. Low multipoles: Energy, angular momentum and centre-of-mass. Problem: Gauge-mode instabilities and their mitigation. Conclusion .

Motivation: Astrophysics I Supermassive BHs in galactic centres: Figure: Orbits in Central Arcsec Figure: Eisenhauer et al. , (Credit: Keck/UCLA) Astrophys. J. 628 , 246 (2005)

Motivation: Astrophysics II ‘Cusp’ population of BH and neutron stars in vicinity of SM BH.

Motivation: Astrophysics III Strong but indirect evidence for existence of Gravitational Waves: Figure: Three decades of data from the Hulse-Taylor binary pulsar.

Motivation: Astrophysics IV Bodies in orbit emit GWs First GW detection possible within five years 2015: Newly-upgraded ground-based detectors 2025: Space-based mission: eLISA Key aim: to map spacetime near event horizons Birth of new field: Multimessenger astronomy

Motivation: LISA?

Motivation: eLISA Rescoping exercise for ESA mission “The new [LISA] configuration should detect thousands of galactic binaries, tens of (super)massive black hole mergers out to a redshift of z=10 and tens of extreme mass ratio inspirals out to a redshift of 1.5 during its two year mission.” Karsten Danzmann, Aug 2011.

Motivation: the general 2-body problem in relativity

Motivation: the general 2-body problem in relativity Effective One-Body (EOB) model (Damour et al. ) provides a possible analytic fitting framework

Gravitational Self Force Test bodies ( µ = 0) follow geodesics on background spacetime Compact bodies ( µ � = 0) are deflected away from test-body geodesics by effect of a ‘self-force’ O ( µ 2 ) V � Μ � 1 � 0.9 50r � M 0 10 20 30 40

Gravitational Self Force Mass ratio : M ≫ µ with η ≡ µ/M ∼ 10 − 4 − 10 − 6 . Perturbation theory : split into black-hole background + perturbation g µν = ˜ g µν + h µν Back-reaction : h µν ∼ O ( µ ) generates back-reaction at O ( µ 2 ) Self force w.r.t. background spacetime, F self ∼ O ( µ 2 ), leading to α self-acceleration a α ∼ O ( µ ). Key steps : Regularization and gauge.

Gravitational Self Force: Dissipative and Conservative Dissipative part F diss ⇒ secular loss of energy and angular α momentum. Conservative part F cons ⇒ shift in orbital parameters, periodic. α Conservative and dissipative parts of perturbation 1 = 1 h R h R ret + h R h ret + h adv − 2 h S � � � � = cons adv 2 2 1 = 1 h R h R ret − h R � � = 2 ( h ret − h adv ) diss adv 2 Dissipative part does not need regularization, get from (e.g.) energy balance arguments. Conservative part requires careful regularization.

Application: Resonances on Kerr (I) Two distinct timescales: τ orb ∼ M ≪ τ rad ∼ M/η Second-order GSF needed for x ∼ O ( η 0 ), as x ∼ ( ηa 0 + η 2 a 1 ) t 2 where t rad ∼ 1 /η . Two-timescale expansion using action-angle variables [Hinderer & Flanagan (2010)] : E/µ, L z /µ, Q/µ 2 � � Action : ‘constants’ of motion : J ν = Angle : ‘phase’ variables q α = ( q t , q r , q θ , q φ ). Frequencies ω α ( J ) = ( ω r , ω θ , ω φ ) Generic orbits on Kerr are ergodic (space-filling) q r → q r + 2 π as orbit goes r = r min → r max → r min with period τ r = 2 π/ω r . Isometries of Kerr ⇒ ( q t , q φ ) ‘irrelevant’, ( q r , q θ ) ‘relevant’ params

Application: Resonances on Kerr (II) 1. Geodesic approximation ( η = 0): dq α = ω α ( J ) dτ dJ ν = 0 dτ Solution : q α ( τ, η = 0) = ω α τ (1) J ν ( τ, η = 0) = const. (2) Timescale : unchanging

Application: Resonances on Kerr (III) 2. Adiabatic approximation: dq α = ω α ( J ) dτ dJ ν � � G (1) = η ν ( q r , q θ , J ) dτ Solution : η − 1 ˆ q α ( τ, η ) = q ( ητ ) ˆ J ν ( τ, η ) = J ( ητ ) Timescale : τ rad.reac. ∼ η − 1

Application: Resonances on Kerr (IV) 3. Post-adiabatic approximation: dq α ω α ( J ) + ηg (1) α ( q r , q θ , J ) + O ( η 2 ) = dτ dJ ν ηG (1) ν ( q r , q θ , J ) + η 2 G (2) ν ( q r , q θ , J ) + O ( η 3 ) . = dτ Two timescales : ∼ η − 1 (secular) and ∼ 1 (oscillatory).

Application: Resonances on Kerr (V) Key question: Is adiabatic approximation justified? Consider Fourier decomposition G (1) G (1) � νk r ,k θ ( J ) e i ( k r q r + k θ q θ ) ν ( q r , q θ , J ) = k r ,k θ ω r τ 2 + . . . , q θ = ω θ τ + ˙ ω θ τ 2 + . . . and q r = ω r τ + ˙ ω θ ) τ 2 + . . . k r q r + k θ q θ = ( k r ω r + k θ ω θ ) τ + ( k r ˙ ω r + k θ ˙ Cannot neglect higher Fourier components when resonance condition is satisfied: k r ω r + k θ ω θ = 0 ⇒ ω r /ω θ = integer ratio

Application: Resonances on Kerr (VI) ω θ ) τ 2 ∼ 1, i.e. Duration of resonance set by ( k r ˙ ω r + k θ ˙ τ res ∼ 1 / √ pη where p ≡ | k r | + | k θ | . Net change in ‘constants’ of motion is � ∆ J ∼ η/p Net change in phase is ∆ q ∼ 1 / √ ηp Need to compute full 1st-order and dissipative part of 2nd-order GSF on Kerr. Without complete knowledge, a resonance effectively resets the phase.

Application: Resonances on Kerr (VII) Credit: Hinderer & Flanagan, arXiv:1009.4923.

Gravitational Self Force: Formulation Linearized Einstein Eqs: Ten linear second-order equations with δ -fn source: h αβ ] = − 16 πT µν ∝ δ 4 [ x − z ( τ )] � ¯ β ν ¯ h αβ + ˆ B µν [¯ h µν + 2 R α µ Gauge choice: Lorenz-gauge ¯ h ; ν µν = 0 gives ‘symmetric’ singularity h µν ∼ u µ u ν /r , and ˆ B µν = 0 ⇒ hyperbolic wave equations. Regularization: split into ‘S’ and ‘R’ h µν = h ( S ) µν + ¯ h ( R ) µν [Symmetric/Singular + Radiative/Regular parts] h ( R ) � ¯ h ( R ) β ν ¯ αβ = S eff µν + 2 R α µ µν Self-force: found from gradient of regularized perturbation self = k αβµν ∇ β ¯ h ( R ) F α µν

Gravitational Self Force: Formulation Schw. ⇒ separability of equations ⇒ l -mode regularization ⇒ easy! decompose ¯ h ab in tensor spherical harmonics Y lm ( i ) ab use Lorenz gauge ∇ b ¯ h ab = 0 with gauge constraint damping solve 1+1D in time domain, or ODEs in freq. domain apply l -mode regularization: ∞ � F self � F ℓ, ret � = − A ( l + 1 / 2) − B − C/ ( l + 1 / 2) − D µ µ ℓ =0

Gravitational Self Force: Formulation Kerr ⇒ lack of separability . . . hard choices . . . Teukolksy variables Ψ 0 , Ψ 4 . . . spin-weighted spheroidal harmonics . . . metric reconstruction in radiation gauge [Chrzanowski ’77] → Lorenz gauge? l = 0 , 1 modes? Hertz potential approach under development by Friedman et al. tensor spheroidal harmonics . . . [don’t exist?] Full 3+1D approach . . . expensive! m -mode + 2+1D evolution . . . practical compromise. Proof-of-principle for m -mode recently established with scalar-field toy model for circular orbits on Kerr [Dolan & Barack 2011] ∞ ∞ Φ ( m ) = q∂ r Φ ( m ) � R e imϕ , F ( m ) � F ( m ) Φ R = R , F µ = µ µ m = −∞ m = −∞

Gravitational Self Force: Formulation Linearized equations: ∆ L ¯ h ab ≡ ∇ c ∇ c ¯ h ab + 2 R ca d b ¯ h cd + g ab Z c ; c − Z a ; b − Z b ; a = − 16 πT ab where Z b ≡ ∇ a ¯ h ab Mixed hyperbolic-elliptic type equations. Impose Lorenz gauge constraints Z a = 0 ⇒ � Z a = 0. Z4 system: add constraints to linearized equations ∆ L ¯ h ab → ∆ L ¯ h ab + Z a ; b + Z b ; a − g ab Z c ; c How to enforce constraints? Gauge-constraint damping [Gundlach et al. ’05] ∇ c ∇ c ¯ h ab + 2 R ca d b ¯ h cd + n a Z b + n b Z a = − 16 πT ab .

GSF on Kerr m -mode decomposition: ¯ h ab = α ab ( r, θ ) u ab ( r, θ, t ) e imφ , (no sum) 10 wave equations: � sc u ab + M ab ( u cd,t , u cd,r ∗ , u cd,θ , u cd ) = S ab

2+1D Wave Equations (Schw.) f � sc u ab + M ab ( ˙ u cd,t , u cd,r ∗ , u cd,θ , u cd ) = 0 + 2 f 2 ( u 22 + u 33 ) 2 r 2 ( ˙ ` ´ 2 u 01 − u ′ 00 ) + u 00 − u 11 + 4 f ( u 00 − u 11 ) M 00 = r 4 r 3 r 3 − 2 f 2 (cos θu 02 + imu 03 ) − 2 f 2 ( u 01 + ∂ θ u 02 ) + 2( ˙ u 00 + ˙ u 11 − 2 u ′ 01 ) M 01 = r 2 sin θ r 2 r 2 − f 2 u 02 + 2( ˙ u 12 − u ′ 02 ) − f ( u 02 + 2 im cos θu 03 ) + f [(4 + r ) u 02 + 2 r∂ θ u 01 ] M 02 = r 2 sin 2 θ r 2 r 3 r 2 − f 2 u 03 u 13 − u ′ − f ( u 03 − 2 im cos θu 02 ) + 2 fimu 01 + 2( ˙ 03 ) + f (4 + r ) u 03 = M 03 r 2 sin 2 θ r 2 sin θ r 2 r 3 r 2 − 4 f 2 (cos θu 12 + imu 13 ) + 2[2 r 2 ( ˙ u 01 − u ′ 11 ) + u 11 − u 00 ] − 4 f ( u 00 − u 11 ) M 11 = r 2 sin θ r 4 r 3 − 2 f 2 (2 ru 11 + u 22 + u 33 + 2 r∂ θ u 12 ) + 2 f 3 ( u 22 + u 33 ) r 3 r 2 − 2 f 2 [cos θ ( u 22 − u 33 ) + imu 23 ] − f ( u 12 + 2 im cos θu 13 ) + 2( ˙ u 02 − u ′ 12 ) M 12 = r 2 sin 2 θ r 2 sin θ r 2 − f 2 (5 u 12 + 2 ∂ θ u 22 ) + f [(4 + r ) u 12 + 2 r∂ θ u 11 ] r 3 r 2