Gravitational waves from compact objects inspiralling into massive black holes
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1 Gravitational waves from compact objects inspiralling into massive black holes Éanna Flanagan, Cornell University American Physical Society Meeting Tampa, Florida, 16 April 2005
2 Outline Extreme mass-ratio inspirals as sources for LISA: capture mechanism, estimates of detection rates Science payoffs from these systems: physics & astrophysics Theoretical challenge: high accuracy computation of strong-field orbital evolution & waveforms (generic orbits, Kerr) Flux-balancing approach Gravitational self-force approach Recent developments: adiabatic waveforms now computable using result due to Mino, sufficient for signal detection.
3 Orbits about Sgr A* (3 x 10 M ) 6 o Ghez et al 2003
4 Capture of compact objects Stars in stellar cusp kicked into loss cone through multibody scattering, captured into tight elliptical orbit Main sequence stars tidally disrupted, compact objects (white dwarfs, neutron stars, black holes) survive Orbit gradually decays; orbits during last year, inside LISA waveband (Finn & Thorne 2000), in the relativistic regime v/c 1. Orbit gradually circularizes due to gravitational wave emission, but still substantial eccentricity at innermost stable orbit (Cutler, Kennefick & Poisson 1994)
5 Signal seen by LISA 10 c n log h, h , , , , yr r ~ = ~ 3.26 N cyc = 460, / f, Hz.999 m = 2 1 mo , day 2,410 a =.999 r ~ isco = 1.18 Circular, equatorial inspiral of 1 M object into a 10 6 M black hole at 1 Gpc (Finn & Thorne 2000).
6 Event Rates From Gair et. al. (2003), based on estimates of black hole mass density using M σ relation, and using Freitag s (2001) simulation of stellar population/dynamics of Milky Way.
7 Event Rates (cont) (Gair et. al. 2003).
8 Richness of orbital dynamics equatorial plane horizon spin axis r = min pm 1 + e inclination angle (ι) r = pm max 1 e Boyer Lindquist frequencies (Hz) f φ f θ f r M = 10 6 M sun a/m = eccentricity = 1/2 inclination = 45 degrees 10 1 minimum radius, or periholion (M)
9 Science Goals LISA should observe 1000 inspirals of compact objects (µ 1 10M ) into massive black holes (M 10 6 M ) (Gair et al. 2003). Last 1 yr of inspiral will contain M/µ 10 5 cycles of waveform. Many scientific payoffs: 1. Measure BH masses and spins to accuracy 10 4 (Poisson 1994, Barack & Cutler 2004); constrain growth history (mergers versus accretion) of the black holes (Hughes & Blandford 2003). 2. Learn about central parsec of galactic nuclei from measured event rate and distribution of inspiralling objects masses. 3. Precise test of general relativity in the strong field regime. Measure multipole moments of central object (Ryan 1995,1997), unambiguous identification as BH
10 Theoretical Challenge Achieving these science goals will require templates with phase errors 1 over entire signal of 1 yr; fractional phase accuracy Signal detection will require templates with phase errors 1 over 3 weeks (Gair et. al. 2004). Required: (a 1). eccentric, inclined orbits about rapidly spinning black holes We currently lack the theoretical tools to compute such accurate templates. However there has been much progress recently. Foundation for all computational methods: µ/m 1. Gravitational field of compact object treated as linear perturbation of Kerr spacetime.
11 Current methods of computing templates Use of post-newtonian methods: Yields crude waveforms that have been already used to roughly scope out LISA s ability ability to detect inspiral events (Gair et al. 2004) and to measure the waveform s parameters (Barack & Cutler 2004). Use of conservation laws: Use fluxes of energy and z-component of angular momentum to infinity and down the black hole to infer adiabatic evolution of orbit; restricted to equatorial and circular orbits (Cutler, Kennefick & Poisson 1994; Shibata 1994; Glampedakis & Kennefick 2002; Hughes 2000,2001). Direct computation of the self-force: Formal expression known (Mino, Sasaki & Tanaka 1997, Quinn & Wald 1997). Finding a practical computational scheme in Kerr is difficult; much work over last few years. Time-domain numerical simulations: Numerically integrate the Teukolsky equation as a 2+1 PDE in the time domain, regularizing the point particle source. (Krivan et al. 1997, Burko & Khanna 2002, Scheel et al. 2003, Martel 2003, Lopez-Aleman et al. 2003, Pazos-Avalos & Lousto 2004). Accuracy 10% so far.
12 Conservation law method Orbits characterized by energy E, angular momentum L z, Carter constant Q. Use Teukolsky-Sasaki-Nakamura black hole perturbation formalism to compute de/dt and dl z /dt, infer gradual evolution of orbit in special cases where Q evolution not needed. Summary: a > 0 e > 0 ι > 0 evolution Cutler et al. (1994) Shibata (1993) Shibata (1994) Hughes (2001) Glampedakis et al. (2002) Drasco et al. (2005)
13 Gravitational Self Force Analogous to Abraham-Lorentz-Dirac force for charged particles ma = qe + 2q 2 ȧ/c 3. Meaning of self force: consider an isolated body of mass M and size L moving in a vacuum gravitational field characterized by a lengthscale L ext.
14 Gravitational Self Force (cont) Complications arise in curved spacetime:
15 Gravitational Self Force (cont) General result is (Mino, Sasaki and Tanaka, 1997; Quinn and Wald 1997) F α (τ) = 1 2 µ(gαβ + u α β) [ β h tail λσ λh tail βσ ] σh tail βλ, τ(x) ε h tail αβ = µ lim dτ G ret αβ,α ε 0 β [ x, (τ )u z(τ)]uα β (τ ) = h ret αβ hsing αβ. Substantial efforts have been made to turn this into a practical computational scheme using mode decompositions (eg Barack et al 2002). Not yet completely successful. Significant simplifications arise when one makes use of adiabaticity of inspirals (Mino 2003).
16 Adiabatic waveforms Orbital periods M; radiation reaction time M 2 /µ = M/ε, where ε = µ/m. Orbital evolution governed by d 2 x α dτ + dx β dx γ 2 Γα βγ dτ dτ = ε [ a α 0,diss + ] [ aα 0,cons + ε 2 a α 1,diss + ] aα 1,cons + O(ε 3 ), where a 0 = P (h ret h sing ), and a 1 is unknown. Orbital [ phase can be expanded using two-time expansion as Φ(t) = Φ0 (t, εt) + εφ 1 (t, εt) + O(ε 2 ) ]. 1 ε Adiabatic waveforms are those obtained using leading order orbit Φ 0. Instantaneously accurate to O(ε), cumulative phase errors O(1). Orbits characterized by E = ξ α u α, L z = ψ α u α, Q = K αβ u α u β. Sufficient to know Ė, L z, Q. Conservation-law waveforms are adiabatic. Mino (2003) has shown how to obtain adiabatic waveforms for generic orbits: Q = 2ε K αβ u α a β 0 = 2ε K αβu α a β 0 where a 0 = P (h ret h adv )/2.
17 Mino s Argument Generic bound orbits in Kerr have the property that they are arbitrarily close to geodesics for which there exists an isometry of the form t t 0 t, ϕ ϕ 0 ϕ under which the geodesic is invariant. Obtain K αβ u α a β 0 [hret αβ ] = K αβ u α a α 0 [hadv αβ, ] where ξ = /( t) and brackets mean a time average. Now average these two terms using a 0 [h] = P (h h sing ); the singular parts cancel.
18 Accuracy of adiabatic waveforms In slow-motion limit v/c 1, post-adiabatic correction terms can be identified in post-newtonian expressions. Using post-3.5-newtonian waveforms to compute the phase error, minimized over time of arrival and initial phase, for a 10M, 10 6 M inspiral gives: (Finn and Thorne 2000) Adiabatic waveforms are likely good enough for signal detection [phase coherence requirement 3 weeks (Gair et al. 2004)], but not for parameter extraction (phase coherence requirement entire signal).
19 Implementing Mino s prescription Use expansion of Gal tsov (1992) of radiative Greens function in terms of modes. Use radiation gauge (expression for < Q > is gauge invariant). Evaluate time average using integral over torus in phase space, obtain (Drasco, Flanagan & Hughes 2005): Ė = Λ Z H Λ 2 + Z Λ 2 L z = Λ ω mkn m [ Z H Λ 2 + Z Λ 2] Q = Λ ( Z H Λ ) Z H Λ + ( Z Λ ) Z Λ, where Λ = (l, m, k, n). Amplitudes ZΛ H, Z Λ computed by existing Drasco-Hughes code. New amplitudes Z Λ H, Z Λ have to be added to code. Both sets of amplitudes are given by integrals over tori in phase space.
20 Conclusions There are no obstacles remaining to prevent computation of adiabatic inspiral waveforms for generic orbits. These waveforms will likely be accurate enough for signal detection with LISA. Analysis of detected signals and parameter extraction will require going beyond this approximation and using the local self-force. This remains a significant challenge.
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