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重力波放出の反作用

Takahiro Tanaka (Kyoto university)

BH重力波Gravitational

waves

素粒子・天文合同研究会「初期宇宙の解明と新たな自然像」

2

Gravitation wave

detectors

LISA⇒DECIGO/BBO

TAMA300　⇒ LCGT

LIGO⇒adv LIGO

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Various sources of gravitational waves

Earth-based Interferometer• Binary

coalescence– -ray bursts

• Spinning NS– LMXB– SN remnant

• GW background• SN formation

– high kick velocitytaken from Cutler & Thorne (2002)

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Various sources of gravitational waves

Space Interferometer

taken from Cutler & Thorne (2002)

• Guaranteed binary sources– WD-WD– AM CVn– LMXB

• Supermassive BH– merger – formation from

a super massive star

• Stochastic BG– WD binary noise– primordial

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Rough estimate of GW frequencies and amplitude

• Frequency

• Amplitude of continuous sources

• Amplitude of inspiraling binary sources

1

221222

10010

Mpcr

MM

vr

GMvr

GMRrQG

h太陽

3

1

53

Hz10 vMM

GMv

太陽

vR

2vR

GM

16/55/35/26/7

521

6/72/13/1

100Hz1010

/~

Mpcr

MMf

rGM

rddEG

fh

太陽

3/13/2 GM

RGM

E

Gh

E22

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ＤＥＣＩＧＯ

• space interferometer with shorter arm length

• high frequencies ⇒ more cycles acceleration of the universe better angular resolution 1deg ⇒ 1min

川村さんの学会トラペより

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Binary coalescence• Inspiral phase (large separation)

Merging phase - numerical relativity, ray burst

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, gr-qc/0411129 )

for parameter extraction

Accurate prediction of the wave form is requested

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Do we need to predict accurate wave form ？

We know how higher expansion goes.

fiefAfh 6/7

Template (prediction of waveform)

uuftf cc 16

4

11

331

743

9

201

128

32 3/2

3

5

MM

DA

L

,,20

1 52536/5

3

3vOfMu 1PN 1.5PNfor circular orbit

though we need to know template to exclude unphysical parameter region from search to some

extent .

⇒Only for detection, higher order templates may not be necessary?

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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)parameter normalized( error 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

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　相対論のテストという観点では

しかし、 Spin があると・・・

bound from solar system

current bound: Cassini > 4×104

Future LATOR mission > 4×108

－ 1 PN の振動数依存性

Scalar-tensor 理論

uubuf 16

4

11

331

743

9

201

128

3 3/23/23/5

2

1b

LISA で 1.4M◎+400M◎ の場合： > 4×105

　　　 DECIGO はもっとすごいはず

(Berti et al, gr-qc/0411129 )

> 4×104

ふたつは見ている効果が違う　スカラー波の放出 vs PN correction

また、コンパクト星は大きな scalar charge を持つ可能性もある。

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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 waveform

Red ○ means determination based on balance argument

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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)

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Teukolsky formalism

TgL sss 4

TT ss Teukolsky equation

0ss L

First we solve homogeneous equation

mnmnC 2

Newman-Penrose quantities

tisss eZrR

,

Angular harmonic function

02 rRr

projection of Weyl curvature

2nd order differential operator

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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

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Leading order wave form

2 0 OOdt

dEorbit

dtdf

Energy balance argument is sufficient.

dtdE

dtdE orbitGW

Wave form for quasi-circular orbits, for example.

dfdE

dtdE

dtdf orbitorbit

2 geodesic OOdf

dEorbit

leading order

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We need to directly evaluate the self-force acting on the particle, but…

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. 　　　　　

gwEE Kerr conserved quantities E, Lz

⇔ Killing vector Q ⇔ Killing vector×

In total, conservation law holds.

GWGW tdE

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§3 Adiabatic approximation for Q

We evaluate the radiative field instead of the retarded field.

Self-force is computed from the radiative field, and it determines long time average of the change rate of E,Lz,Q.

2advretrad hhh

radhFdQu

Q

TDD T

TT

2

11lim

differs from energy balance argument.

For E and Lz the results are consistent with the balance argument. (Gal’tsov ’82)

For Q, it has been proven that the estimate by using the radiative field gives the correct long time average. (Mino ’03)

Key point: Under the transformation

every geodesic is transformed back into itself.

• Radiative field does not have divergence at the location of the particle.

aa ,,,,,, rtrt

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• A remarkable property of the Kerr geodesic equations is

with

• Only discrete Fourier components arise

• Skipping many interesting details, we just present the final formula:

aLarEar

LaEaddt

2222

2sin

nnddmddt rr

nnm

r //1,

rRddr

2

2

dd /drd

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

aLarEaL

aEdd

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2sin

ddt

ttt r

Periodic functions of periods11 2,2 r

Simplified dQ/dt formula(Sago, Tanaka, Hikida & Nakano PTPL(’05))

nrnmml

mlrr Z

ndtdLraP

dtdErPar

dtdQ

,,,

2

,,

22

222

aLarErP 22

222 cosar 22 2 aMrr

amplitude of partial wave evaluated at

infinity

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Summary up to here

uuhhhuugf ;;;2

1

TxdgW s

outs

ups

ss

'1 4

ssh *

fuK

ddQ

2

nrnmlml

mlrr Z

ndtdLraP

dtdErPar

dtdQ

,,,,

2

,,

22

222

Basically this part is Z

simplified

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Second order waveform

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

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)(z

Electro-magnetism (DeWitt & Brehme (1960))

cap1

cap2

tube

§４ Self-force in curved space

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2

c

ee

m

Abraham-Lorentz-Dirac

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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)

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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

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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

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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.

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)])([)]([(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

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can be expanded in terms of

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

S-part (Skip)

)(,)( 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

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)])([)]([(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 we can neglect this term as long as the gauge transformation R stays small.

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 RGH

SGH

)(xhFxhF

zx

We do not know how to evaluate

decompose it as

)]([ RGH xhF

RR

GH 2 h

Force in the hybrid-gauge

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What is the remaining problem?Basically, we understand how to compute the self-force in the hybrid-gauge.But actual computation is still limited to simple 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 self-force and the second order source term simultaneously.

RRRSR GTG 1211122 8,2 hhhhh

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Conclusion

Adiabatic radiation reaction for the Carter constant is now almost ready to compute.

We need to evaluate the second order source term and the self-force as a pair.

2 0 OOdt

dEorbit

2 geodesic OOdf

dEorbit

second order leading order

The leading order self-force is also almost ready to compute.

However, due to gauge degrees of freedom, it might be the case that only the secular change of constants of motion in the self-force has physical meaning. If so, there is no merit in computing the self-force directly before the second order perturbation is solved.