1

Gravitational waves from Extreme mass ratio inspirals

Gravitational Radiation Reaction Problem

Takahiro Tanaka (Kyoto university)

BH重力波Gravitational

waves

2

• 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– γ- ray burst

• Stochastic background– GWs from the early universe– Unresolved foreground

Various sources of gravitational waves

3

Inspiraling binariesIn 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 γ-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

4

• Inspiral phase (large separation)

Merging phase - numerical relativity recent progress in handling BHs Ringing tail - quasi-normal oscillation

of BH

for precision test of general relativity

Clean system

Negligible effect of internal structure

(Cutler et al, PRL 70 2984(1993))

for detection

(Berti et al, PRD 71:084025,2005)

for parameter extraction

Accurate prediction of the wave form is requested

5

• LISA sources 0.003-0.03Hz　　　 → merger to white dwarfs (=0.6M◎),

neutron stars (=1.4M◎) BHs (=10M◎ ,~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

Extreme mass ratio inspirals (EMRI)

◎◎ MMM 65 10510~

BH

X

M

GW

(Gair et al, CGQ 21 S1595 (2004))

although still very uncertain. (Amaro-Seoane et al, astro-ph/0703495)

6

• High-precision determination of orbital parameters • maps of strong field region of spacetime

– Central BH will be rotating: a~0.9M

• ≪M Radiation reaction is weak　 Large number of cycles N

before plunge in the strong field region

Roughly speaking,

difference in the number of cycle

N>1 is detectable.

BH重力波

M

7

Probably clean system

yr101010

105.41

2

1

612

EddM

m

MM

M

◎◎

•Interaction with accretion disk

(Narayan, ApJ, 536, 663 (2000))

df

obs

t

T

f

f

df

obsobsobs t

TfTTfN

Frequency shift due to interaction

Change in number of cycles

,assuming almost spherical accretion (ADAF)

satellite

reldf mG

vt

2

3

log4

obs. period ~1yr

8

Theoretical prediction of Wave form

We know how higher expansion proceeds.

fiefAfh 6/7

Template in Fourier space

uuftf cc 16

4

11

331

743

9

201

128

32 3/23/5

MM

DA

L

,,20

1 52536/5

3

3vOfMu 1.5PNfor quasi-circular orbit

⇒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). observational error

in parameter estimate

∝ signal to noise ratio

1PN

c.f.

9

Test of GR

Scalar-tensor type

uuuf g 16

3

128

9

55

756

37151

128

3 3/23/23/5

Current constraint on dipole radiation: BD ＞ 140, (600) 4U 1820-30 （ NS-WD in NGC6624)

Dipole radiation ＝ － 1 PN

(Will & Zaglauer, ApJ 346 366 (1989))

3vOfMu

Effect of modified gravity theory

Mass of graviton

BD 1

LISA 1.4M◎NS+400M◎BH: BD > 2×104

Decigo1.4M◎NS+10M◎BH ： BD >5×109 ?　　　

(Berti & Will, PRD71 084025(2005))

daM

gg 2

2

2

LISA 107M◎BH+107M◎BH: graviton compton wavelength g > 1kpc

(Berti & Will, PRD71 084025(2005))Constraint from future observation:

Constraint from future observation:

10

Black hole perturbation GTG 8g

21 hhgg BH

v/c can be O(1)

BH重力波

M≫

11 8 GTG h:master equation

Linear perturbation

11 4 TgL

Gravitationalwaves

Regge-Wheeler formalism (Schwarzschild)Teukolsky formalism (Kerr) Mano-Takasugi-Suzuki’s method (systematic PN expansion)

xTeYrRxdgW

xZx tiin

s

upup ,'1 4

11

Teukolsky formalism

TgL 4

TT Teukolsky equation

First we solve homogeneous equation tieYrR

,

Angular harmonic function 02 rRr

2nd order differential operator

hih 2

1~

2Z

dt

dEat r →∞

insr

upss RRW

Construct solution using Green fn. method.

Wronskian

0L

projected Weyl curvature

2Z

m

dt

dLz

: energy loss rate

: angular momentum loss rate

,,m

12

Leading order wave form

2 0 OOdt

dEorbit

dt

df

Energy balance argument is sufficient.

dt

dE

dt

dE orbitGW

Wave form for quasi-circular orbits, for example.

df

dE

dt

dE

dt

df orbitorbit

2 geodesic OOdf

dEorbit

leading order

self-forceeffect

13

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

Radiation reaction for General orbits in Kerr black hole background

Radiation reaction to the Carter constantSchwarzschild “constants of motion” E, Li

⇔ Killing vector Conserved current for GW corresponding to Killing vector

exists. 　　　　　

gworbit EE Kerr conserved quantities E, Lz

⇔ Killing vector Q ⇔ Killing vector×

In total, conservation law holds.

GWGW tdE

14

Adiabatic approximation for Q,

• orbital period << 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. 2advretrad hhh

radhFdQu

Q

T

T

TT

2

11lim

which differs from energy balance argument.

At the lowest order, we assume that the trajectory of a particle is given by a geodesic specified by E,Lz,Q.

. . .

:radiative field(Mino Phys. Rev. D67 084027 (’03))

Radiative field is not divergent at the location of the

particle.

Regularization of the self-force is unnecessary!

15

Simplified dQ/dt formula

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

uuhhhuugf ;;;2

1

fuK

d

dQ2

(Sago, Tanaka, Hikida, Nakano, Prog. Theor. Phys. 114 509(’05))

Killing tensor associated with Q

sh * Complicated operation is necessary for metric reconstruction from the master variable.

nrnmlml

mlrr Z

n

dt

dLraP

dt

dErPar

dt

dQ,

,,,

2

,,

22

222

aLarErP 22

nnm rr

nnm

r ,

Only discrete Fourier components exist

after several non-trivial manipulations

• We arrived at an extremely simple formula:

uuKQ

22 2 aMrr

16

(Ganz, Hikida, Nakano, Sago, Tanaka, Prog. Theor. Phys. (’07))

Use of systematic PN expansion of BH perturbation.Small eccentricity expansionGeneral inclination

17

Summary

Adiabatic radiation reaction for the Carter constant has been computed.

2 0 OOdt

dEorbit

2 geodesic OOdf

dEorbit

second order leading order

Direct computation of the self-force at O() is also almost ready in principle.

However, to go to the second order, we also need to evaluate the second order self-force.

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.

18

Summary up to here

uuhhhuugf ;;;2

1

TxdgW s

outs

ups

ss

'1 4

sssh *

fuK

ddQ

2

nrnmlml

mlrr Z

ndtdLraP

dtdErPar

dtdQ

,,,,

,,

22

222

Basically this part is Z

simplified

19

Second order wave form

2 0 OOdt

dEorbit

dfdE

dtdE

dtdf orbitorbit

2 geodesic OOdf

dEorbit

second order

leading order

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.

the leading order self-force

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Higher order in Post-Teukolsky formalism hhh ,8 2 GTG

1 2

1 2

h h h Perturbed Einstein equation

expansion

2nd order perturbation 21122 8, GTG hhh

22 4 TgL : post-Teukolsky equation

(1)construct metric perturbation h from (1) (2) derive T(2) taking into account the self-force

11 8 GTG h:Teukolsky equation

linear perturbation

11 4 TgL

21

)(z

Electro-magnetism (DeWitt & Brehme (1960))

cap1

cap2

tube

§４ Self-force in curved space

22

2

c

ee

m

Abraham-Lorentz-Dirac

22

tail-term

'',' zuzxvdexF tail

Tail part of the self-force

zxvzxuzxG ret ,,,

geodesic along distance2

1, zx

Retarded Green function in Lorenz gauge

x

)(z

curvature scattering

tail

direct

direct part (S-part) tail part (R-part)

23

Matched asymptotic expansion

Extension to the gravitational case

(Mino et al. PRD 55(1997)3457, see also Quinn and Wald PRD 60 (1999) 064009)

Extension is formally non-trivial.

１） equivalence principle e=m ２） non-linearity

22

200

c

em

emmm

mass renormalization

near the particle ） small BH()+perturbations |x|/(GM)<< 1

far from the particle ） background BH(M) ＋ perturbation G/|x| << 1

matching region

24

Gravitational self-force

'',' zTzxvdxhR

Tail part of the metric perturbations

Rhg E.O.M. with self-force = geodesic motion on

zxvzxuzxG ret ,,, Retarded Green function in harmonic gauge

direct part (S-part) tail part (R-part)

x

)(z

curvature scattering

tail

direct

(MiSaTaQuWa equation)

Extension of its derivation is non-trivial, but the result is a trivial extension.

25

)])([)]([(lim][)(

xFxFF Sfull

zx

R

Since we don’t 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

)]([)]([ , )]([)]([ xFxFxFxhF SSfullfull

Coincidence limit can be taken before summation over

)]([)]([lim)]([ xhFxhFhF Sfull

zx

R

Decomposition into spherical harmonics Ym modes

cos11

0l

l

l

Prr

r

rr finite value in the limit r→r0

26

can be expanded in terms of

・ S-part is determined by local expansion near the particle.

S-part

)(,)( eqeq xuxzfR

abcda

dcb

),,,()( RTzx

: spatial distance between x and zx

)(z

)(ret x

)(eq x

・ Mode decomposition formulae (Barack and Ori (’02), Mino Nakano & Sasaki (’02))

DLCBLAF /(S)

, 21L

0 DC

{

where

27

)])([)]([(lim)]([lim SH

fullH

)(

R

)(xhFxhFxhF

zxzx

We usually evaluate full- and S- parts in different gauges.

But it is just a matter of gauge, so is it so serious?

Gauge problem

cannot be evaluted directly

in harmonic gauge (H)

)])([)]([)]([(lim fullGH

SH

fullG

)(xhFxhFxhF

zx

gauge transformation connecting two

gauges )]([lim full

GH)(

xhFzx

is divergent in general.

can be computed in a convenient gauge

(G).

)])([)]([(lim SH

fullG

)(xhFxhF

zx

cannot be evaluated without error.

also diverges.

)]([lim fullGH

)(xhF

zx

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 (x→x+) to be small.

28

29

Hybrid gauge method (Mino-Barack-Ori?)

HGHHG )()( hxhxh

RHh

gauge transformation

)()()( SH

fullH

RH xhxhxh

)()( SH

fullHGH

fullG xhhxh

RHGH

SH

SHGH

fullG )()( hxhhxh

stays finite ⇒

)()()( SH

SHRWH

fullRW

RHyb xhhxhxh

We can compute the self-force by using

RHGH h also automatically stays

finite if it is determined by local value of . (T.T.)

RHh

RHybh

A similar but slightly different idea was proposed by Ori.

30

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.

2nd order perturbation 21122 8, GTG hhh

22 4 TgL : post-Teukolsky equation

Both terms on the right hand side are gauge dependent.

but T (2) in total must be gauge independent.

regularization ？

What we want to know is the second order wave form

We need the regularized self-force and the regularized second order source term simultaneously.

RRRSR GTG 1211122 8,2 hhhhh

31

Post-Newton approx. ⇔ BH perturbation• Post-Newton approx.

v < c

• Black hole perturbation m1 >>m2

v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11

0 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○μ1 ○ ○ ○ ○ ○μ2 ○ ○ ○μ3 ○μ4

BH pertur- bation

post-Newton

Post Teukolsky

○ : done

§ ２　 Methods to predict wave form

Red ○ means determination based on balance argument

32

Standard post-Newtonian approximation

GTG 8g hgg

0, h

hGThflat 16□

n

nnhGh

Post-Minkowski expansion (B+C)vacuum solution

Post-Newtonian expansion (A+B)

hhGTh tflat 216△

it v slow motionr

c

GMcv

21/

Asource

B C

33

Green function method

hih 2

1~2

2

22

2)(2

4

r

ddt

Ed out

at r →∞

up

down in out

Boundary condi. for homogeneous modes

insr

upss RRW

Construct solution with source by using Green function.

Wronskian

xTeZrRxdgxW

sti

sin

sup

ss

s ,'

1 4

34

For E and Lz the results are consistent with the balance argument. (shown by Gal’tsov ’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.

aa ,,,,,, rtrt

xxG retadv 4/ □ 0radG□

35

Chrzanowski (‘75) '','4 zTxxGxdgxh ret

s

''1

', rrxxW

xxG outs

ups

ss

rets

Mode function for

metric perturbation

Assume factorized form of Green function.

Txxdgx

WxhD out

sup

sss

ss ''1 4

sss D

Compute ψ following the definition.

TxdgW s

outs

ups

ss

'1 4 Calculation using Green

function for

TxdgTxxdg s

outss

outs ''' 44

compariso

n

Metric re-construction in Kerr case

Further, using the Starobinsky identity, one can also determines .

outsss

outs x *'

*s is obtained from s by integration

by parts.

since the relation holds for arbitrary T

36

Constants of motion for geodesics in Kerr

← definition of Killing tensor

37

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

zx

uKd

dQ

~

zx

td

dE

2

,,,,

ml

mlZd

dE

← amplitude of the partial wave

2

,,,,

ml

mlZm

d

dL

dTZ

zxml ,,

zxml TuK

i

dZ

,,ˆ

,,,,,,

ˆml

mlml ZZd

dQ

just –i after mode decomposition

with

38

Further reduction • A remarkable property of the Kerr geodesic

equations is

with

• Only discrete Fourier components arise

• In general for a double-periodic function

aLarEar

LaEad

dt

2222

2sin

nnddmddt rr

nnm

r //1,

rRd

dr

2

2

d

d /drd

By using , r- and -oscillations can be solved independently.

aLarEaL

aEd

d

22

2sin

d

dtttt r

Periodic functions of periods 11 2,2 r

ggfddggfd

T rr

rr

T

T

r

T

r

,2

,2

1lim

11 2

0

2

02

39

dI jrr ,, , , j

rj IIrr Def.

Final expression for dQ/dt in adiabatic approximation

nrnmlml

mlrr Z

n

dt

dLraP

dt

dErPar

dt

dQ,

,,,

2

,,

22

222

After integration by parts using the relation in the previous slide,

aLarErP 22

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}.

dI jrr ,, , , j

rj IIrr

,,

d

ddt

ItItt jr

jr ...

40

41

42

43

2 0 OOdt

dEorbit

2 geodesic OOdf

dEorbit

second order leading order

44

Probably clean system

yr105.4log4 1

6102

3

mm

M

mG

vt

satellite

reldf

•Interaction with accretion disk

(Narayan, ApJ, 536, 663 (2000))

210m典型的な値としては

dftT

ff

観測期間

dfdf tTN

tTf

TfN

2

相互作用によるfrequency の変化

cycle 数の変化に焼きなおすと

rvr

M24

： almost spherical 　　 accretion (ADAF)

)1.0(yr 105.4~4

7

p

T

Edds Gm

c

M

Mt

Krel vv

Kr vv EddMmM

solsatellite Mmm 110

solMMM 6610

1.0

45

Test of GR

Scalar-tensor type の重力理論の変更

uuuf g 16

3

128

9

55

756

37151

128

3 3/23/23/5M

BD

ss

64

5 221

NS 同士では同じ scalar charge をもっているので４重極放射が leading になってしまう。その場合、

双極子放射からの BD に対する制限は 4U 1820-30 （ NS-WD in globular cluster NGC6624) から BD ＞ 140, (600) が得られている。

双極子放射＝－ 1 PN の振動数依存性

]

1221 ≪ss

(Will & Zaglauer, ApJ 346 366 (1989))

(Berti & Will, PRD71 084025(2005))

3vOfMu

46

Parameter estimate における error =10

number of cycles in LISA band for BH-NS systems

-1BD -1

BD -1BD -1

BD

他の全てのparameter が与えられている場合スピンが無視

できるとした場合

スピンも観測から決定されるべき parameter のひとつと考えた場合

47

Spin を考慮するとがあると・・・

bound from Solar system current bound: 　　 Cassini BD> 2×104

Future LATOR mission　　　　　　 BD> 4×108

LISA で 1.4M◎+400M◎ の場合： BD > 4×105

　　　 DECIGO はもっとすごいはず

BD> 2×104

しかし、見ている効果が違う　スカラー波の放出 vs PN correction　　　スカラー場の non-linear interaction ⇒ 　コンパクト星が大きな scalar charge を持つ可能性

重力波では大した制限が得られないのではないかと思うかも知れない。

(Plowman & Hellings, CQG 23 309(’06) )

48

uuuf g 16

3

128

9

55

756

37151

128

3 3/23/23/5

graviton が mass を持っている効果

重力の伝播速度の変更

f

DfcfDtf

gphase 2

22

222

2

2

11

21

fmk

fcg

phase

massive graviton の phase velocity

振動数に依存した位相のずれ

(Berti & Will, PRD71 084025(2005) より )

number of cycles in LISA band for BH-BH systems

2

2

gg

DM

2adD

49

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

For large or small , higher order coefficients can be important.

For TAMA best sensitivity,

1 ierror due to noise

errors coming from ignorance of higher order coefficientsare 　　 @3PN ~10-2/ @4.5PN ~10-4/

Wide band observation is favored to determine parameters

⇒ Multi band observation will require more accurate template

ortho-normalized parameters

50

Gravitation wave

detectors

LISA⇒DECIGO/BBO

TAMA300 CLIO　⇒ LCGT

LIGO⇒adv LIGO

VIRGO, GEO