1. Radiation Reaction in Captures of Compact Objects by Massive Black Holes Leor Barack (UT Brownsville)

2. 2 Extreme mass-ratio inspirals as sources for LISA: Capture mechanism, estimates of S/N and detection rates Theoretical challenge: high accuracy computation of strong-field orbital evolution & consequent waveforms (generic orbits, Kerr) Energy-momentum balance approach Self-force approach: –issues of principle (regularization, gauge,….) –development of practical calculation methods –implementation of calculation methods Summary Over the last 6 years ~70 papers, 6 dedicated conferences (“Capra meetings”: Capra Ranch ‘98, Dublin ‘99, Caltech ‘00, Potsdam ‘01, Penn State ‘02, Kyoto ‘03, Brownsville ‘04)

3. 3 Accurate mapping of the MBH’s strong field; test of “no hair” theorem [Kerr has only 2 independent mass multipoles; LISA could accurately measure at least 3-5 (Ryan 1997) ] Test of alternative theories of gravity (scalar, YM) … Precise measurement of MBHs masses and spins (LB & Cutler 2003) : Decide on SMBH growth mechanism (Hughes & Blandford 2002) … Scientific payoffs  December 2001: LIST decides that LISA noise floor is to be determined by the requirement of detecting CO inspirals. Basic physics Astro- physics

4. 4 Capture of a CO by a MBH: mechanism CO kicked into “loss cone” through multibody scattering Orbit possibly still highly eccentric (0.4  e  0.7) when entering LISA band 10 5 -10 6 orbits during last few years, inside LISA band, deep in highly- relativistic regime Last Stable Orbit: GW signal cuts off. 1 Myr to LSO (f ~ 0.1 mHz) 10 yrs to LSO (f ~ 1 mHz) LSO Event Rate: ~10 -8 /yr/galaxy (for stellar-mass BHs) LISA Detection Rate: ~ 1 to few per year out to 1 Gpc (very uncertain!) (Hils & Bender 1995, Sigurdson & Rees 1997, Freitag 2003)

5. 5 Signal Output (at 1 Gpc) LB & Cutler (2003) m/M=10/10 7 m/M=10/10 6 LSO: e=0.3 =0.165 mHz 5.1M<r<9.4M LSO-10 yr: e=0.324 =0.151 mHz 5.2M<r<10.2M LSO-10 yr: e=0.77 =0.23 mHz 6.2M<r<47.5M LSO: e=0.3 =1.65 mHz 5.1M<r<9.4M SNR~55 (last year)

6. 6 Timescale for RR effect ( 10 M  /10 6 M  ) “adiabatic” evolution over T orb RR effect important!

7. 7 Need high-accuracy theoretical modeling The theoretical challenge: Accurately model orbital evolution and subsequent waveform for  1-100 M  CO inspiraling onto 10 6 -10 8 M  MBH (  m/M=10 -4 -10 -8 )  Generic orbits (inclined, eccentric)  Kerr MBH [neglect CO’s spin (?) – Hartl; Burko; LB & Cutler 2003 ] How accurate? To assure     1/yr  need fractional accuracy of     Accurate modeling crucial for detection:  Data analysis difficult (huge parameter space, long & complicated waveforms)  Will have to use sub-optimal methods (stacking, hierarchical)  With SNR~30-50, Captures might be just marginally detectable

8. 8 Possible approaches PN methods well developed, but, in our case, not useful! (most S/N comes from strong-field region, v/c~1) Full Numerical Relativity: Recent 1st attempt (Bishop et al., 2003). Computed one orbit in Schwarzschild. Expansion in m/M: assuming point particle on a fixed background and using black hole perturbation theory (Teukolsky-Sasaki-Nakamura formalism). Two variants:  conservation law  local force (“self force”)

9. 9 Tanaka, Shibata, Sasaki, Tagoshi, & Nakamura (1993) Cutler, Kennefick, & Poisson (1994) Mino, Tanaka, Shibata, Sasaki, Tagoshi, Nakamura (1998) Kennefick (1998) Finn & Thorne (2000) Hughes (2000, 2001) Glampedakis & Kennefick (2002) Conservation-law method Schwarzschild  circular, equatorial  circular, inclined  eccentric, equatorial

10. 10 Assume adiabaticity: orbit is approximately geodesic along a few cycles Solve for with geodesic orbit as a source (use Teukolsky/Sasaki-Nakamura to reduce PDEs to ODEs) From fluxes at infinity and through the horizon infer Evolve orbit adiabatically Failure of method: 1.Cannot calculate for generic (eccentric, inclined) orbits in Kerr. 2.Does not account for conservative piece of force: Cannot describe the (conservative) evolution of the 3 orbital phases. Conservation law method: basic idea

11. 11 Local (self-)force approach: basic idea Perturbation h  (  m) exerts a “self force” F  (  m 2 ): Analogous to Abraham-Lorentz-Dirac EOM for electric charge in flat space: Given F , can  directly integrate EOM, or  obtain momentary rate-of-change of all COMs Generalization to curved spacetime: –DeWitt & Brehme (1960) - EM case –Mino, Sasaki, & Tanaka (1997) – Gravitational case (based on Retarded field) –Quinn (2000) – Scalar field case –Detweiler & Whiting (2003) – All cases (based on Radiative field)

12. 12 Gravitational self-force: regularization methods Mino, Sasaki, & Tanaka (1997): Detweiler & Whiting (2003): (1) Hadamard expansion+ world-tube integration+ local conservation (3) Radiative field: (2) Matched asymptotic expansions Alternative methods:  Comparison axiom (Quinn & Wald 1997)  Zeta function (Lousto 2000)  Extended object (Ori & Rosenthal 2002)  power expansion (Mino, Nakano, & Sasaki 2002)  Massive field approach (Rosenthal 2003)  Kerr symmetry (Mino 2003) All consistent with Mino-Sasaki-Tanaka BH + tidal pert. Perturbed background

13. 13 The gravitational self-force Has only “tail” contribution, no “light-cone” contribution: To make it practical: need to formulate a subtraction scheme at the level of individual multipole modes. from Scattering inside light- cone from propagation along light cone direct tail xx0x0 

14. 14 Practical calculation scheme: the mode-sum method LB & Ori 2000, 2003 (scalar,EM) LB 2001 (gravitational) Multipole contributions finite at the particle, “ Regularization parameters” A a, B a, C a, D a derived analytically by local analysis: D  LB, Mino, Nakano, Ori, & Sasaki (2002) - equatorial orbit in Schwarzschild  LB & Ori (2003) - generic orbit in Kerr

15. 15 Summary of prescription for computing the local self-force  Solve (numerically) Teukolsky-Sasaki-Nakamura equations, with a geodesic source  get  Construct metric perturbation using Wald-Chrzanowski-Ori (In Schwarzschild can solve directly for h ab using RW-Zerilli-Moncrief)  Construct the “full force” modes at the particle  Apply mode-sum formula:

16. 16 Implementation Static particle in Schwarzschild (Burko 2000) Circular orbit in Schwarzschild (Burko 2000) Radial infall in Schwarzschild (LB & Burko 2000) Static particle in Kerr (Burko & Liu 2001) Radial trajectories in Schwarzschild (LB & Lousto 2002) Circular orbit in Schwarzschild (LB & Lousto, in progress) Circular orbit in Kerr (LB & Lousto, in progress) Scalar force Gravitational force  Local analysis can be pushed to higher order in 1/l, giving analytic approximation for the self force. For example: Radial infall in Schwarzschild (LB & Lousto 2002)

17. 17 Local SF is gauge-dependent: Gauge-invariants (e.g., orbit-integrated ) can be derived from F in whatever gauge. Mode-sum formula originally formulated in “harmonic” gauge, but is “gauge invariant” for all regular gauges (LB & Ori 2001) : Problem: Standard BH perturbation theory gives h in “radiation” (or RW) gauges. These gauges are pathological in the presence of a particle ( h is singular along a null ray). The gauge problem

18. 18 We devised an “intermediate” gauge, which is both  regular (h singular only at the particle)  simply related to the radiation/RW gauge Then: The “Gauge-corrected” parameters recently calculated for the radiation gauge (generic orbits in Kerr) and for the RW gauge (circular orbits in Schwarzschild ). F (intermediate) can now be used to calculate the long-term, gauge-invariant orbital evolution Gauge problem solved (LB & Ori 2003)

19. 19 Ongoing work & Alternative ideas Contribution from conservative modes (l=0,1) (Detweiler, Poisson, Ori) Mode-sum method combined with PN expansion (Nakano, Sago, Sasaki…) Analytic approximation through large l expansion (Barack, Lousto) Analytic solution to the homogeneous Teukolsky equation through the Mano- Takasugi expansion (Mino) Adiabatic evolution based on Kerr symmetry & the Radiative Green’s function (Mino 2003) Thoughts on 2 nd -order perturbation theory. Perhaps Radiative fields could solve problem of regularization at 2 nd order?

20. 20 Signal from low-mass MS star at Sgr A* Barack & Cutler (2003), following Freitag (2003) M=2.6  10 6 M  m=0.06 M  D=8 kpc e(LSO)=0.02 e(LSO-1Myr)=0.43 e(LSO-10Myr)=0.80  - 10 5 yrs  - 10 6 yrs  - 2  10 6 yrs  - 5  10 6 yrs  - 10 7 yrs to plunge