Gravitational waves from Extreme mass ratio inspirals Gravitational - - PowerPoint PPT Presentation

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 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 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)

◎ ◎

M M M

6 5

10 5 10 ~

• 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

yr 10 10 10 10 5 . 4

1 2 1 6 12

• Edd

M m M M M & &

◎ ◎

• Interaction with accretion disk

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

df

• bs

t T f f

• df
• bs
• bs
• bs

t T fT T f N

• Frequency shift

due to interaction Change in number of cycles ,assuming almost spherical accretion (ADAF)

• satellite

rel df

m G v t

2 3

log 4

• bs. period

~ 1yr

8

Theoretical prediction of Wave form

We know how higher expansion proceeds.

f i

e f A f h

• 6

/ 7

Template in Fourier space

• L

u u f t f

c c

• 16

4 11 331 743 9 20 1 128 3 2

3 / 2 3 / 5

• M

M D A

L

• ,

, 20 1

5 2 5 3 6 / 5 3

• 3

v O f M u

• 1.5PN

for 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).

• bservational error in

parameter estimate ∝ signal to noise ratio 1PN c.f.

9

Test of GR Scalar-tensor type

• L

L u u u f

g

• 16

3 128 9 55 756 3715 1 128 3

3 / 2 3 / 2 3 / 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))

3

v O f M u

• 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))

• d

a M

g g

• 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

• GT

G 8

• g

L

• 2

1

• h

h g g

BH

v/c can be O(1)

BH

重力波

M≫

• 1

1

8 GT G

• h

:master equation

Linear perturbation

1 1

4 T g L

• Gravitational

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

Mano-Takasugi-Suzuki’s method (systematic PN expansion)

• L
• x

T e Y r R x d g W x Z x

t i in s up up

• ,

' 1

4

11

Teukolsky formalism

T g L

• 4
• T

T

Teukolsky equation First we solve homogeneous equation

• t

i

e Y r R

• ,

Angular harmonic function

• 2
• r

R

r

L

2nd order differential operator

• h

i h & & & & 2 1 ~

• 2

Z dt dE

at r →∞

in s r up s s

R R W

• Construct solution using Green fn. method.

Wronskian

• L

projected Weyl curvature

• 2

Z m dt dLz

• : energy loss rate

: angular momentum loss rate

• ,

,m l

12

Leading order wave form

• 2
• O

O dt dEorbit

• dt

df

• Energy balance argument is sufficient.

dt dE dt dE

• rbit

GW

• Wave form for quasi-circular orbits, for example.

df dE dt dE dt df

• rbit
• rbit
• 2

geodesic

• O

O df dEorbit

• leading order

self-force effect

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 constant

Schwarzschild “constants of motion” E, Li ⇔ Killing vector Conserved current for GW corresponding to Killing vector exists.

gw

• rbit

E E & &

• Kerr

conserved quantities E, Lz ⇔ Killing vector Q ⇔ Killing vector

×

In total, conservation law holds.

• GW

GW

t d E

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.

• 2

adv ret rad

h h h

• rad

h F d Q

u Q T

T T T

• 2

1 1 lim

&

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
• u

u h h h u u g f

; ; ;

2 1

• f

u K d dQ 2

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

Killing tensor associated with Q

• s

h

*

• Complicated operation is necessary

for metric reconstruction from the master variable.

• n

r n m l

m l m l r r

Z n dt dL r aP dt dE r P a r dt dQ

, ,

, , 2 , , 2 2

2 2 2

• aL

a r E r P

• 2

2

• n

n m

r r n n m

r ,

Only discrete Fourier components exist after several non-trivial manipulations

• We arrived at an extremely simple formula:
• u

u K Q

2 2

2 a Mr r

16

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

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

17

Summary

Adiabatic radiation reaction for the Carter constant has been computed.

• 2
• O

O dt dE orbit

• 2

geodesic

• O

O df dE orbit

• 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

• f the wave form.