Radiation Reaction in Captures of Compact Objects by

Massive Black Holes Leor Barack

(UT Brownsville)

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• 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)

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• 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.

410~m/m S/S, M/M, δδδ

Basic physics

Astro-physics

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Capture of a CO by a MBH: mechanism

CO kicked into “loss cone” through multibody scattering

Orbit possibly still highly eccentric (0.4 e0.7) when entering LISA band

105-106 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)

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Signal Output (at 1 Gpc)

LB & Cutler (2003)

m/M=10/107m/M=10/106

LSO:e=0.3=0.165 mHz5.1M<r<9.4M

LSO-10 yr:e=0.324=0.151 mHz5.2M<r<10.2M

LSO-10 yr:e=0.77=0.23 mHz6.2M<r<47.5M

LSO:e=0.3=1.65 mHz5.1M<r<9.4M

SNR~55 (last year)

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Timescale for RR effect (10 M/106 M)

)year(~)day(~min)15(~ obsRRorb TTT

“adiabatic” evolution over Torb

RR effect important!

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Need high-accuracy theoretical modeling

• The theoretical challenge: Accurately model orbital evolution and subsequent waveform for

1-100 M CO inspiraling onto 106-108 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

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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”)

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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

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• 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

},,{ QLE z

},,{ QLE z lm

4

Q

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Local (self-)force approach: basic idea

• Perturbation h ( m) exerts a “self force” F ( m2):

• 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)

)()(0 2mFmma

acqqFam ext

)(32)( 22

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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

curvadvretretRad Hhhhh 2/)(

BH +

tidal pert.

Perturbed background

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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.

)(

)()()()(

0tail

taildirectfull ][

xxFF

FFhmF

self

xxxx

from Scattering

inside light- cone

from propagation

along light cone

dire

cttai

l

xx0

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Practical calculation scheme: the mode-sum method

• LB & Ori 2000, 2003 (scalar,EM) • LB 2001 (gravitational)

l

ll

zxxx FFF )()( directfull

)0(lim Multipole contributions finite at

the particle, llFF ll )(, directfull

l

l

l

l lCBlAFlCBlAFF // )0()0( directfull

• “Regularization parameters” A, B, C, D derived analytically by local analysis:

D

LB, Mino, Nakano, Ori, & Sasaki (2002) - equatorial orbit in Schwarzschild

LB & Ori (2003) - generic orbit in Kerr

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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 using RW-Zerilli-Moncrief)

Construct the “full force” modes at the particle

Apply mode-sum formula:

lmlm40 ,

lmh

m

lml hFfull

DlCBlAFFl

l full

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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)

)(2/11416

15 4222

22 lOlrME

r

EmF rl

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• 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

uuRuugmFF

hh

xx

;;

:

C

DlCBlAFFl

l gaugeanygaugeany full

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• 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)

DlCBlAFFl

l ~~~~RWradiation/teintermedia full

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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 2nd-order perturbation theory. Perhaps Radiative fields could solve problem of regularization at 2nd order?

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Signal from low-mass MS star at Sgr A*

Barack & Cutler (2003), following Freitag (2003)

M=2.6106 Mm=0.06 MD=8 kpce(LSO)=0.02e(LSO-1Myr)=0.43e(LSO-10Myr)=0.80

- 105 yrs - 106 yrs - 2106 yrs - 5106 yrs - 107 yrs to plunge