1

How to test the Einstein gravity using gravitational

waves

Takahiro Tanaka (YITP, Kyoto university)

BH重力波Gravitational

waves

Based on the work in collaboration with R. Fujita, S. Isoyama, H. Nakano, N. Sago

(Berti et al, gr-qc/0504017 )

2

Binary coalescence• Inspiral phase (large separation)

Merging phase - numerical relativity 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 for parameter extraction

Accurate prediction of the wave form is requested

3

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

MM

DA

L

,,20

1 52536/5

3

3vOfMu 1PN 1.5PNfor circular orbit

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

But we need higher order accurate template for the test of GR.

Chern-Simons Modified Gravity

(Yunes & Spergel, arXiv:0810.5541): J0737-3039(double pulsar)

Right handed and left handed gravitational waves are magnified differently during propagation, depending on the frequencies.

The evolution of background scalar field is hard to detect.cm103 6

hhh iRL

2

1,

zRLGR

RL

dz

dz

dz

dzdzHf

0 2

22/5

02,, 1

2

7164exp

hh 3, 10exp OfRLGR h

Propagation of GWs as an example of GR test

RRgxdS *4

4

Ghost free bi-gravity

Both massive and massless gravitons exist. → oscillation-like phenomena

,16

~~

16gL

G

Rg

G

RgL matter

NN

(in preparation)

5

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

Methods to predict wave form

Red ○ means determination based on balance argument

6

Extrime Mass Ratio Inspiral(EMRI)

• Inspiral of 1 ～ 100Msol BH of NS into the super massive BH at galactic center (typically 106Msol)

Many cycles before the coalescence ～O(M/) allow us to determine the orbit precisely.

The best place to test GR.

Very relativistic wave form can be calculated using BH perturbation

Clean system

BH

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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:simple master equation

Linear perturbation

11 4 TgL

Gravitationalwaves

Regge-Wheeler-Zerilli formalism (Schwarzschild)Teukolsky formalism (Kerr)

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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

Killing tensor

Killing vector for rotational sym.

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Evolution of general orbits

uu 1

If we know four velocity u at each time accurately, we can solve the orbital evolution in principle. On Kerr background there are four “constants of motion”

Normalization of four velocity:

Energy:

Angular momentum:

tuE

uLz

Killing vector for time translation sym.

Carter constant: KuuQ

Quadratic and un-related to Killing vector(simple symmetry of the spacetime)

uQLE z ,,One to one correspondence

constant in case of no radiation reaction

2 0 OOdt

dEorbit

2 geodesic OOdf

dEorbit We need to know the secular evolution of E,Lz,Q.

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We need to directly evaluate the self-force acting on the particle, but it has never been done for general orbits in Kerr because of its complexity.

The issue of radiation reaction to Carter constant

E, Lz ⇔ Killing vector

Conserved current for the field corresponding to Killing vector exists. 　　　　　

GWEE

However, Q ⇔ Killing vector×

As a sum conservation law holds.

GWGW tdE

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§3 Adiabatic approximation for Q

• T << RR

　　 T: orbital period

RR : timescale of radiation reaction

As the lowest order approximation, we assume that the trajectory of a particle is given by a geodesic specified by E,Lz,Q.We evaluate the radiative field

instead of the retarded field.Self-force is computed from the radiative field, and it determines the change rates of E,Lz,Q.

2advretrad hhh

radhFdddQ

u

Q

T

T

TT

2

11lim

which is different from energy balance argument.

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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

a geodesic is transformed back into itself.

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

Divergent part is common for both retarded and advanced fields.

,,,,,, rtrt

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Outstanding property of Kerr geodesic

• Only discrete Fourier components arise in an orbit

partdependent -partdependent -

rd

dt

nnddmddt rr

nnm

r //1,

rRd

dr

2

2

d

d

222 cosar

dd

r- and -oscillations can be solved independently.

similar

d

d

d

dtttt r

Periodic functions with periods 11 2,2 r

Introducing a new time parameter by

Final expression for dQ/dt in adiabatic approximation

2

,,,,

,,

222

nrn

mlmlml

rrz An

dt

dLrg

dt

dErf

dt

dQ

After a little complicated calculation, miraculous simplification occurs

This expression is similar to and as easy to evaluate as dE/dt and dL/dt.

xdA mlml4

,,,, termsourcefunction mode

amplitude of the partial wave

2

,,,,

mlmlA

dt

dE 2

,,,,

ml

mlAm

dt

dL

(Sago, Tanaka, Hikida, Nakano PTPL(’05))

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Key point: Under Mino’s transformation

a geodesic is transformed back into the same geodesic.

,,,,,, rtrt

(Mino’s time)

r

rrr jj However, for resonant case:

(separation from max to rmax) has physical meaning.

Under Mino’s transformation, a resonant geodesic with transforms into a resonant geodesics with .

with integer jr & j

Resonant orbit

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xxGxxGxxG symradret ,,,For the radiative part (retarded-advaneced)/2, a formula similar to the non-resonant case can be obtained: 2

,,,,,,

22

222

nnmlnnml

rr

r

rA

n

dt

dLraP

dt

dErPar

dt

dQ

NmlNmlNml

r BA,,

,,,,2

jjr

r

r

Nnn rr

nnmlNml rAA ,,,,, nnmlrNml r

AnB ,,,,, Nnn rr

(Flanagan, Hughes, Ruangsri, 1208.3906)

dQ/dt at resonance

This is rather trivial extension. The true difficulty is in evaluating the contribution from the symmetric part.

We recently developed a method to evaluate the symmetric part contribution for a scalar charged particle and there will not be any obstacle in the extension to the gravity case.

Sum for the same frequency is to be taken first.

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dt

d 2

Mdt

d Impact of the resonance on the phase

evolution

: duration staying around resonance

1 MOtres

If for c,

MOres

MOres : overall phase error due to resonance

≠O((0)

0dt

d

: frequency shift caused by passing resonance

cM

Odt

d

Odt

d c cc

Mdt

d

22

2

If stays negative, resonance may persist for a long time.

Oscillation period is much shorter than the radiation

reaction time

(gravitational radiation reaction)

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ConclusionAdiabatic radiation reaction for the Carter constant is as easy to compute as those for energy and angular momentum.

We derived a formula for the change rate of the Carter constant due to scalar self-force valid also in the resonance case.

2 0 OOdt

dEorbit

2 geodesic OOdf

dEorbit

second order leading order

Hence the leading order waveform whose phase is correct at O(M/) is also ready to compute.

The orbital evolution may cross resonance, which induces O((M/)1/2) correction to the phase.

Extension to the gravitational radiation reaction is a little messy, but it also goes almost in parallel.

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zx

symrt

T

TT d

draar

r

rPd

T

q

d

dQ

22lim

The symmetric part (retarded+advanced)/2 also becomes simple.

dzxGqx symsym ,

,zzr-oscillationoscillation

,

lim

rzx

symr

rr

T

TrT d

drdd

T

q

,

22lim

rzx

sym

rtr

T

TrT d

d

r

raPar

r

rPdd

T

q

,,,,,, rrGedxxG symttisym

xxGxxG symsym ,, ,,,,,,,,,, rrGrrG symsym

zxzx

symt xxGdd

,,

zxzx

symGiddd,

, xx

0,,

zxzx

symGiddd xx↔ ’ →

d

dq

sym2

zzGdd

TsymT

TT

sym ,2

1lim:

=0

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To compute ,

Regularized field

Tail part gives the regularized self-field.Direct part must be subtracted.

zxvzxuzxG ret ,,,

geodesic along distance2

1, zx

Hadamard expansion of retarded Green function

x

)(z

curvature scattering

tail

direct direct part tail part

31, zxOzxu

regularization is necessary.

Easy to say but difficult to calculate especially for the Kerr background.

(DeWitt & Brehme (1960))

We just need

But what we have to evaluate here looks a little simpler than self-force.

zzGdd

TsymT

TT

sym ,2

1lim:

zzGzzGdd

TdirsymT

TT,,

2

1lim:

Regularization is necessary

zx

radrad

uqKd

dqQ 22

drops after long time average

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Simplified dQ/dt formula

• Self-force is expressed as ,uugqf

fuK

d

dQ2

Sago, Tanaka, Hikida, Nakano PTPL(’05)

zx

radrt

T

TT d

draar

r

rPd

T

q

d

dQ

22lim

/ ddMino time:

Substituting the explicit form of K,

222 cosar aLarErP 22

Instead of directly computing the tail, we compute

Both terms on the r.h.s. diverge in the limit z±()→ z().

zzGzzGddT

dirsymT

TT,,

2

1limlim:

0

22

: 211

tzzwith

ml

mlmlsym zzdzzG

,,,,,,

Nml

mlml zzmNmN

,,,,,, periodic source

Nmlmlml

mNi zzemNmN

,,,,,,

21

Nm

symmN

mNi zzGe,

,: 21

zzGdd

Tsym

mN

T

TT

symmN ,

2

1lim:

Fourier coefficient of with respect to {1,2}.

sym

is just the

22

21

21

dir dirmN is finite and calculable.

dirsym

0

lim

Novel regularization method

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mN

dirmN

symmN

mNidirsym e 21

00lim:lim:

(sym)-(dir) is regular

We can take →0 limit before summation over m & N

mN

dirmN

symmN

Difference from the ordinary mode sum regularization:Compute the force and leaves l-summation to the end.

Divergence of the force behaves like 1/2. l-mode decomposition is obtained by two dimensional integral.

marginally convergent

However, l-mode decomposition of the direct-part is done (not for spheroidal) but for spherical harmonic decomposition.

m & N-sum regularization seems to require the high symmetry of the resonant geodesics. is periodic in

{1,2}.

does not fit well with Teukolsky formalism

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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

Parallel to the case of a scalar charged particle.