Gravitational%20waves%20from%20Extreme%20mass%20ratio%20inspirals

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Gravitational waves from Extreme mass ratio
inspirals Gravitational Radiation Reaction
Problem
Gravitational waves
Takahiro Tanaka (Kyoto university)
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Various sources of gravitational waves

• Inspiraling binaries
• (Semi-) periodic sources
• Binaries with large separation (long before
  coalescence)
• a large catalogue for binaries with various mass
  parameters with distance information
• Pulsars
• Sources correlated with optical counter part
• supernovae
• ?- ray burst
• Stochastic background
• GWs from the early universe
• Unresolved foreground

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Inspiraling binaries

• In general, binary inspirals bring information
  about
• Event rate
• Binary parameters
• Test of GR
• Stellar mass BH/NS
• Target of ground based detectors
• NS equation of state
• Possible correlation with short ?-ray burst
• primordial BH binaries (BHMACHO)
• Massive/intermediate mass BH binaries
• Formation history of central super massive BH
• Extreme (intermidiate) mass-ratio inspirals
  (EMRI)
• Probe of BH geometry

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• Inspiral phase (large separation)
• Merging phase - numerical relativity
• recent progress in handling BHs
• Ringing tail - quasi-normal oscillation of BH

Clean system
(Cutler et al, PRL 70 2984(1993))
Negligible effect of internal structure
Accurate prediction of the wave form is requested

• for detection

• for parameter extraction

• for precision test of general relativity

(Berti et al, PRD 71084025,2005)
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Extreme mass ratio inspirals (EMRI)

• LISA sources 0.003-0.03Hz
• ? merger to
• white dwarfs (m0.6M?),
• neutron stars (m1.4M?)
• BHs (m10M?,100M?)
• Formation scenario
• star cluster is formed
• large angle scattering encounter put the body
  into a
• highly eccentric orbit
• Capture and circularization due to gravitational
  radiation reaction last three years
  eccentricity reduces 1-e ?O(1)
• Event rate
• a few 102 events for 3 year observation by LISA

m
X
BH
M
GW
(Gair et al, CGQ 21 S1595 (2004))
although still very uncertain.
(Amaro-Seoane et al, astro-ph/0703495)
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• mM Radiation reaction is weak

Large number of cycles N before plunge in the
strong field region
m
M
Roughly speaking, difference in the number of
cycle DNgt1 is detectable.

• High-precision determination of orbital
  parameters
• maps of strong field region of spacetime
• Central BH will be rotating a0.9M

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Probably clean system
(Narayan, ApJ, 536, 663 (2000))

• Interaction with accretion disk

,assuming almost spherical accretion (ADAF)
Frequency shift due to interaction
Change in number of cycles
obs. period 1yr
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Theoretical prediction of Wave form
Template in Fourier space
1.5PN for quasi-circular
orbit
1PN

• We know how higher expansion proceeds.

?Only for detection, higher order template
may not be necessary?

• We need higher order accurate template
• for precise measurement of parameters
  (or test of GR).

c.f.
observational error in parameter estimate
? signal to noise ratio
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• Test of GR

Effect of modified gravity theory
Scalar-tensor type
Mass of graviton
Dipole radiation -1 PN
Current constraint on dipole radiation
wBDgt140, (600) 4U 1820-30(NS-WD in NGC6624)
Constraint from future observation
LISA- 107M?BH107M?BH graviton compton
wavelength lg gt 1kpc
(Will Zaglauer, ApJ 346 366 (1989))
(Berti Will, PRD71 084025(2005))
Constraint from future observation
LISA- 1.4M?NS400M?BH wBD gt 2104
(Berti Will, PRD71 084025(2005))
Decigo-1.4M?NS10M?BH wBD gt5109 ?
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Black hole perturbation

• Mm

• v/c can be O(1)

Gravitational waves
Linear perturbation
master equation
Regge-Wheeler formalism (Schwarzschild) Teukolsky
formalism (Kerr)
Mano-Takasugi-Suzukis method (systematic PN
expansion)
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Teukolsky formalism
Teukolsky equation
2nd order differential operator
projected Weyl curvature
First we solve homogeneous equation
Angular harmonic function
Construct solution using Green fn. method.
Wronskian
at r ?8
energy loss rate
angular momentum loss rate
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Leading order wave form
Energy balance argument is sufficient.
Wave form for quasi-circular orbits, for
example.
leading order
self-force effect
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Radiation reaction for General orbits in Kerr
black hole background

• Radiation reaction to the Carter constant

Schwarzschild constants of motion E, Li ?
Killing vector Conserved current for GW
corresponding to Killing vector exists.
In total, conservation law holds.
Kerr conserved quantities E, Lz ? Killing
vector
Q ? Killing vector

• We need to directly evaluate the self-force
  acting on the particle, but it is divergent in a
  naïve sense.

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Adiabatic approximation for Q,
which differs from energy balance argument.

• orbital period ltlt timescale of radiation reaction
• It was proven that we can compute the self-force
  using the radiative field, instead of the
  retarded field, to calculated the long time
  average of E,Lz,Q.

. . .
(Mino Phys. Rev. D67 084027 (03))
radiative field
At the lowest order, we assume that the
trajectory of a particle is given by a geodesic
specified by E,Lz,Q.
Radiative field is not divergent at the location
of the particle.
Regularization of the self-force is unnecessary!
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Simplified dQ/dt formula
(Sago, Tanaka, Hikida, Nakano, Prog. Theor. Phys.
114 509(05))

• Self-force f a is explicitly expressed in terms
  of hmn as

Killing tensor associated with Q
Complicated operation is necessary for metric
reconstruction from the master variable.
after several non-trivial manipulations

• We arrived at an extremely simple formula

Only discrete Fourier components exist
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• Use of systematic PN expansion of BH
  perturbation.
• Small eccentricity expansion
• General inclination

(Ganz, Hikida, Nakano, Sago, Tanaka, Prog. Theor.
Phys. (07))
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Summary
Among various sources of GWs, E(I)MRI is the best
for the test of GR.
For high-precision test of GR, we need accurate
theoretical prediction of the wave form.
Adiabatic radiation reaction for the Carter
constant has been computed.
leading order
second order
Direct computation of the self-force at O(m) is
also almost ready in principle.
However, to go to the second order, we also need
to evaluate the second order self-force.
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Summary up to here
Basically this part is ZL
simplified
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Second order wave form
leading order
second order
the leading order self-force
To go to the next-leading order approximation for
the wave form, we need to know at least the
next-leading order correction to the energy loss
late (post-Teukolski formalism) as well as the
leading order self-force. Kerr case is more
difficult since balance argument is not enough.
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Higher order in m Post-Teukolsky formalism
Perturbed Einstein equation
expansion
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4 Self-force in curved space
Abraham-Lorentz-Dirac
Electro-magnetism (DeWitt Brehme (1960))
cap1
tube
cap2
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tail-term
Retarded Green function in Lorenz gauge
direct part (S-part)
tail part (R-part)
Tail part of the self-force
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Extension to the gravitational case
mass renormalization
Extension is formally non-trivial. 1)equivalence
principle em 2)non-linearity

• Matched asymptotic expansion

(Mino et al. PRD 55(1997)3457, see also
Quinn and Wald PRD 60
(1999) 064009)
matching region
near the particle) small BH(m)perturbations
x/(GM)ltlt 1
far from the particle) background
BH(M)perturbation Gm /x ltlt 1
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Gravitational self-force
Extension of its derivation is non-trivial, but
the result is a trivial extension.
Retarded Green function in harmonic gauge
direct part (S-part)
tail part (R-part)
Tail part of the metric perturbations
E.O.M. with self-force geodesic motion on
(MiSaTaQuWa equation)
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• Since we dont know the way of direct
  computation of the tail (R-part), we compute

Both terms on the r.h.s. diverge ? regularization
is needed

• Mode sum regularization

Decomposition into spherical harmonics Ym modes
Coincidence limit can be taken before summation
over
finite value in the limit r?r0
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S-part
S-part is determined by local expansion near the
particle.
can be expanded in terms of

Mode decomposition formulae (Barack and Ori
(02), Mino Nakano Sasaki (02))
where
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Gauge problem
We usually evaluate full- and S- parts in
different gauges.
cannot be evaluted directly in harmonic gauge (H)
gauge transformation connecting two gauges
can be computed in a convenient gauge (G).
is divergent in general.
also diverges.
cannot be evaluated without error.
But it is just a matter of gauge, so is it so
serious?
The perturbed trajectory in the perturbed
spacetime is gauge invariant. But coordinate
representation of the trajectory depends on the
gauge. Only the secular evolution of the orbit
may be physically relevant. Then we only need to
keep the gauge parameter xm (xm?xmxm) to be
small.
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Hybrid gauge method (Mino-Barack-Ori?)
gauge transformation
stays finite ?
also automatically stays
finite if it is determined by local value of
. (T.T.)
A similar but slightly different idea was
proposed by Ori.
We can compute the self-force by using
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What is the remaining problem?
Basically, we know how to compute the self-force
in the hybrid-gauge. But actual computation is
still limited to particular cases.
numerical approach straight forward?
(Burko-Barack-Ori) but many parameters, harder
accuracy control? analytic approach can take
advantage of (Hikida et al. 04)
Mano-Takasugi-Suzuki method.
What we want to know is the second order wave form
2nd order perturbation
Both terms on the right hand side are gauge
dependent.
post-Teukolsky equation
but T (2) in total must be gauge independent.
regularization
?
We need the regularized self-force and
the regularized second order source term
simultaneously.
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2 Methods to predict wave form
Post-Newton approx. ? BH perturbation

• Post-Newton approx.
• v lt c

• Black hole perturbation
• m1 gtgtm2

v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11
µ0 ? ? ? ? ? ? ? ? ? ? ?
µ1 ? ? ? ? ?
µ2 ? ? ?
µ3 ?
µ4
BH pertur- bation
Post Teukolsky
post-Newton
? done
Red ? means determination based on balance
argument
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Standard post-Newtonian approximation
C
B
A source
l
Post-Minkowski expansion (BC)
vacuum solution
Post-Newtonian expansion (AB)
slow motion
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Green function method
Boundary condi. for homogeneous modes
up
down
in
out
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• For E and Lz the results are consistent with the
  balance argument. (shown by Galtsov 82)
• For Q, it has been proven that the estimate by
  using the radiative field gives the correct long
  time average. (shown by Mino 03)
• Key point Under the transformation
• every geodesic is transformed
  into itself.
• Radiative field does not have divergence at the
  location of the particle.
• Divergent part is common for both retarded and
  advanced fields.
• Remark Radiative Green function is source free.
  

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Metric re-construction in Kerr case

• Chrzanowski (75)

Mode function for metric perturbation
Assume factorized form of Green function.
Compute ? following the definition.
comparison
Calculation using Green function for y
since the relation holds for arbitrary T
by integration by parts.
is obtained from
Further, using the Starobinsky identity, one can
also determine zs .
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Constants of motion for geodesics in Kerr
? definition of Killing tensor
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Hint similarity between expressions for dE/dt
and dQ/dt

• Energy loss can be also evaluated from the
  self-force.
• Formula obtained by the energy balance argument
• dQ/dt formula is expected to be given by

just iw after mode decomposition
? amplitude of the partial wave
with
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Further reduction

• A remarkable property of the Kerr geodesic
  equations is
• 
  with
• Only discrete Fourier components arise
• In general for a double-periodic function

By using l, r- and q -oscillations can be solved
independently.
Periodic functions of periods
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Final expression for dQ/dt in adiabatic
approximation
After integration by parts using the relation in
the previous slide,
This expression is similar to and as
easy to evaluate as dE/dt and dL/dt.
Recently numerical evaluation of dE/dt has been
performed for generic orbits. (Hughes
et al. (2005))
Analytic evaluation of dE/dt, dL/dt and dQ/dt
has been done for generic orbits.
(Sago et al. PTP 115 873(2006) )
secular evolution of orbits
Solve EOM for given constants of motion, I j
E,L,Q.
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Probably clean system
(Narayan, ApJ, 536, 663 (2000))

• Interaction with accretion disk

?????????
almost spherical accretion (ADAF)
???????frequency???
cycle???????????
????
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• Test of GR

(Berti Will, PRD71 084025(2005))
Scalar-tensor type ????????
?????-1 PN???????
NS??????scalar charge????????4?????leading????????
?????
????????wBD???????4U 1820-30(NS-WD in globular
cluster NGC6624) ??wBDgt140, (600)????????
(Will Zaglauer, ApJ 346 366 (1989))
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?????parameter??????????
Parameter estimate???? error r 10
??????????????
???????????????parameter??????????
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LISA? 1.4M?400M???? wBD gt 4105
DECIGO?????????
Spin??????????
wBD gt 2104
bound from Solar system

• current bound
• Cassini wBD gt 2104
• Future LATOR mission
• wBD gt 4108

(Plowman Hellings, CQG 23 309(06) )
????????????????????????????????
????????????? ???????? vs PN correction
??????non-linear interaction ?
??????????scalar charge??????
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??????????
(Berti Will, PRD71 084025(2005)??)
massive graviton?phase velocity
?????????????
graviton?mass????????
number of cycles in LISA band for BH-BH systems
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• We need higher order accurate template
• for precise measurement of parameters
  (or test of GR).

error due to noise
ortho-normalized parameters
For TAMA best sensitivity,
errors coming from ignorance of higher order
coefficients are _at_3PN 10-2/h , _at_4.5PN
10-4/h
For large r or small h m/M , higher
order coefficients can be important.
Wide band observation is favored to determine
parameters
? Multi band observation will require more
accurate template
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Gravitation wave detectors
TAMA300 CLIO ?LCGT
LISA ?DECIGO/BBO
LIGO?adv LIGO
VIRGO, GEO