radiation reaction in captures of compact objects by massive black holes leor barack (ut...
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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