Gravitational waves from
Extreme mass ratio inspirals
Gravitational Radiation Reaction Problem
BH
重力波
Gravitational
waves
Takahiro Tanaka
(Kyoto university)
1
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
– γ- ray burst
• Stochastic background
– GWs from the early universe
– Unresolved foreground
2
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
3
• Inspiral phase (large separation)
(Cutler et al, PRL 70 2984(1993))
Clean system
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 71:084025,2005)


Merging phase - numerical relativity
recent progress in handling BHs
Ringing tail - quasi-normal oscillation of BH
4
Extreme mass ratio inspirals (EMRI)
• LISA sources 0.003-0.03Hz
→ merger to M ~ 105 M ◎  5 106 M ◎
white dwarfs (m=0.6M◎),
neutron stars (m=1.4M◎)
BHs (m=10M◎,~100M◎)
• Formation scenario
m
X
BH M
GW
– 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
although still very
(Gair et al, CGQ 21 S1595 (2004))
5
uncertain.(Amaro-Seoane et al, astro-ph/0703495)
•
m≪M
Radiation reaction is weak
Large number of cycles N before
plunge in the strong field region
m
M BH
重力波
Roughly speaking,
difference in the
number of cycle
DN>1 is detectable.
• High-precision determination of orbital parameters
• maps of strong field region of spacetime
– Central BH will be rotating: a~0.9M
6
Probably clean system
•Interaction with accretion disk
(Narayan, ApJ, 536, 663 (2000))
,assuming almost spherical accretion (ADAF)
3
vrel
t df 
4 log G 2 msatellite
1
1





M
m
m

  2
 yr
 4.5 1012 6




10 M ◎  10M ◎   10 M Edd 
Frequency shift
due to interaction
obs. period
Df Tobs
~1yr

f
t df
Change in number of cycles
DN  Df Tobs
Tobs
 fTobs
t df
7
Theoretical prediction of Wave form
Template in Fourier space
5 / 6
m
A
,   m 3 5M 2 5 ,  
M
20 3 DL
h  f   A f 7 / 6ei   f 
  2 f tc  c 
1
3
  f 5 / 3 1  20  743  11  u 2 / 3   16  u  
128
9  331 4 


 
u  M f O v
3
1.5PN
1PN
for quasi-circular orbit
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
8
parameter estimate ∝ signal to noise ratio
Test of GR
Effect of modified gravity theory
Scalar-tensor type
Mass of graviton
3
  f 5 / 3 u 2 / 3  1   3715  55   128  g  u 2 / 3   16  u  
128
3
 756 9



3
u  M f  Ov 
 2M 2
1
 g  2  a d

Dipole radiation ＝ －1 PN
g
w BD
  
Current constraint on dipole radiation:
wBD＞140, (600)
4U 1820-30（NS-WD in NGC6624)
(Will & Zaglauer, ApJ 346 366 (1989))
Constraint from future observation:
LISA 107M◎BH+107M◎BH:
graviton compton wavelength
g > 1kpc
(Berti & Will, PRD71 084025(2005))
Constraint from future observation:
LISA 1.4M◎NS+400M◎BH: wBD > 2×104
(Berti & Will, PRD71 084025(2005))
Decigo1.4M◎NS+10M◎BH： wBD >5×109 ?
9
Black hole perturbation
G m g  8GT m
1
2 
g m  g m  hm  hm  
BH
 M≫m
 v/c can be O(1)
BH
重力波
Gravitational
waves
Linear perturbation
 G m h 1   8 GT 1m
L 1  4
 g T 1 :master equation
Regge-Wheeler formalism (Schwarzschild)
Teukolsky formalism
(Kerr)
Mano-Takasugi-Suzuki’s method (systematic PN expansion) 10
Teukolsky formalism
T  m T m
Teukolsky equation
2nd order differential operator
projected Weyl curvature
L   4  g T
First we solve homogeneous equation
L  0
   R r Y  ,   e iw t

2
r

  , m, w

  Rr   0
Angular harmonic function
Construct solution using Green fn. method.
1
up
4
in
iw t











   up
x
Z


x

g
d
x
'
R
r
Y

,

e
T  x   










s W
up
Wronskian s W  s R  r s Rin
at r →∞
1 
 ~ h  i h
2


dE
2
  Z 
: energy loss rate
dt

dLz
m
2
  Z  : angular momentum
dt
 w
loss rate
11
Leading order wave form
Energy balance argument is sufficient.
dEGW
dEorbit

dt
dt
df
Wave form 
for quasi-circular orbits, for example.
dt
df dEorbit dEorbit

dt
df
dt
leading order
 
dEorbit

0
 Om   O m 2
dt
dEorbit
 geodesic   Om   O m 2
df
 
self-force
effect
12
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.
m GW  
EGW   d t m 
E
  E In total, conservation law holds.
orbit
gw
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.
13
Adiabatic approximation for Q,
which differs from energy balance argument.
• 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.
rad 

ret 
adv
hm  hm  hm
2
(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.
2T 
1
1

Q  lim
m
T 
T
T
d
Radiative field is not
divergent at the
location of the particle.

Q

 rad 
F
h
m
u 

Regularization of the
self-force is
unnecessary!
14
Simplified dQ/dt formula
(Sago, Tanaka, Hikida, Nakano, Prog. Theor. Phys. 114 509(’05))
•
Self-force f  is explicitly expressed in terms of hm as
1 

f   g  u  u  h ;  h ;  h ; u  u 
2
dQ
Killing tensor associated with Q
 2 K m u m f
d
Q  K u m u



m
*
hm  m
s
Complicated operation is necessary
for metric reconstruction from the
master variable.
after several non-trivial manipulations
• We arrived at an extremely simple formula:
2
dQ
r 2  a 2 Pr  dE
aPr  dL
nr  r
2
2
2 
Z l ,m ,w
dt
D
dt
D
dt
w
l , m ,w wln,rm,n
Only discrete Fourier components exist
Pr   E r 2  a 2  aL


w w
nr ,n
m
 m  nr r  n  

D  r 2  2Mr  a 2

15
Use of systematic PN expansion of BH perturbation.
Small eccentricity expansion
General inclination
(Ganz, Hikida, Nakano, Sago, Tanaka, Prog. Theor. Phys. (’07))
16
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
 Om   O m 2 
dEorbit

0
dt
dEorbit
 geodesic   Om   O m 2
df
 
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.
17
Summary up to here
s  

1
up
4
out


g
d
x
'

s  
s  sT
s W
Basically this part is Z
*
hm   ss m
s
f



1 
  g  u u  h ;  h ;  h ; u  u 
2
dQ
 2K m u m f 
d
simplified


dQ
r 2  a2 P r 
2
dt
D
dE
aP r 
2
dt
D
dL
nr r
2 
Z l ,m ,w
n
,
n
dt
w
r 
l ,m ,w wl ,m
18
Second order wave form
df dEorbit

dt
dt
dEorbit
df
leading order
 
dE orbit

0
 Om   O m 2
dt
dEorbit
 geodesic   Om   O m 2
df
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.
19
Higher order in m
Post-Teukolsky formalism
 G m h  8 GT m  2 m h, h
Perturbed Einstein equation
h  h1  h  2  
expansion

1

 2
linear perturbation
 G m h 1  8 GT 1m
L 1  4  g T 1 :Teukolsky equation
 
2nd order perturbation
 G m h 2   2 m h 1 , h 1  8 GT 2 m
 


(1) construct metric perturbation hm from  (1)
(2) derive T (2)m taking into account the self-force
L 2   4  g T 2  : post-Teukolsky equation
20
§４ Self-force in curved space
Abraham-Lorentz-Dirac
Electro-magnetism (DeWitt & Brehme (1960))
z ( )

cap2
cap1
tube
e2
m 
e
2c 2
21
tail-term
Retarded Green function in Lorenz gauge
G ret m x, z   u m  x, z   v m  x, z 
direct part (S-part) tail part (R-part)
1
 x, z   distance along geodesic 
2
x
direct
z ( )
tail
Tail part of the self-force

Ftail x   e d 'v m  x, z 'u z '
m

curvature scattering
22
Extension to the gravitational case
Extension is formally non-trivial. mass renormalization 2
e
m

m

m

m

１）equivalence principle e=m
0
0
e
2c 2
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)<< 1
far from the particle）
background BH(M)＋perturbation
Gm /|x| << 1
23
Gravitational self-force
Extension of its derivation is non-trivial, but the result is a trivial extension.
Retarded Green function in harmonic gauge
x
direct
z ( )
tail
G ret m x, z   u m  x, z   v m  x, z 
direct part (S-part)
tail part (R-part)
curvature scattering Tail part of the metric perturbations

hR  x    d 'v m  x , z 'T  z '
m

E.O.M. with self-force = geodesic motion on gm  hm
R
(MiSaTaQuWa equation)
24
Since we don’t know the way of direct computation of the
tail (R-part), we compute
F  [ R ]   lim (F  [ f ull (x )]  F  [ S (x )])
x z ( )
Both terms on the r.h.s. diverge ⇒ regularization is needed
Mode sum regularization
Decomposition into spherical harmonics Y{m modes
F  [h full ]( x)   F [ full ]( x) , F  [ S ]( x)   F [ S ]( x)


Coincidence limit can be taken before summation over {
F  [h R ]( )   lim F [h full ( x)]  F [h S ( x)]

x  z  
l
1
1 r 
     Pl cos  
r  r0
l r  r 
finite value in the limit r→r0
25
S-part
・S-part is determined by local expansion near the particle.
z ( )
can be expanded in terms of
 : spatial distance between x and z
{
( x  z )  (T , R, , )
R b c  d

a


f abcd z  eq ( x) , u  eq ( x) 

 eq ( x )
x
 ret ( x)
・Mode decomposition formulae
(Barack and Ori (’02), Mino Nakano & Sasaki (’02))
F(S),  A L  B  C / L  D
where
C   D  0
L    12
26
Gauge problem
We usually evaluate full- and S- parts in different gauges.

S
lim F  [h R (x )]  lim (F  [hfull
(
x
)]

F
[
h
H
H  (x )])
x z ( )
cannot be evaluted
directly in harmonic
gauge (H)
x z ( )

S

full
 lim (F  [hfull
(
x
)]

F
[
h
(
x
)]

F
[

h
G
H 
HG  (x )])
x z ( )
can be computed in a
convenient gauge (G).
gauge transformation
connecting two gauges
lim F  [hfull
H G  ( x)] is divergent in general.
x  z ( )

S
lim (F  [hfull
(
x
)]

F
[
h
G
H  (x )]) also diverges.
x z ( )
lim F  [hfull
H G  ( x)] cannot be evaluated without error.
x  z ( )
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 m (xm→xm+m) to be small.
27
28
Hybrid gauge method (Mino-Barack-Ori?)
gauge transformation
hG  (x )  hH  (x )  HG  hH  
S
hRH  ( x)  hfull
(
x
)

h
H
 H  ( x)
 
 h   h  (x )  
 h  
 h   also automatically stays finite
full
S
 hfull
(
x
)



h

h
G
HG  H 
H  (x )
 hfull
G  (x )  HG
hRH  stays finite ⇒ HG
S
H
S
H
HG
R
H
R
H
R
if it is determined by local value of hH  . (T.T.)
A similar but slightly different idea was proposed by Ori.
 
S
S
hRHyb ( x)  hfull
(
x
)



h

h
RW 
HRW  H 
H  ( x)
We can compute the self-force by using hRHyb
29
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
 G m h 2   2 m h 1 , h 1  8 GT 2 m
 


Both terms on the right hand side are gauge dependent.
L 2   4  g T 2  : post-Teukolsky equation
but T (2) in total must be gauge independent.
regularization
 G m h 2 R   2 m 2h 1S  h 1R , h 1R  8 GT 2 m h 1R ？
We need the regularized self-force and
the regularized second order source term simultaneously.
30
§２ Methods to predict wave form
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
○
○ ○○ ○ ○○ ○ ○ ○ ○
○
○ ○ ○○
○
○○
○
BH perturbation
Post
Teukolsky
μ4
post-Newton
○ : done
Red ○ means determination based on balance argument
31
Standard post-Newtonian approximation
G m g  8GT m
 g g m   m  h m
h m ,  0
□flat h m  16GT m  m h
B
A
source
C

Post-Minkowski expansion (B+C)
n 
vacuum solution
hm   G n hm
n
Post-Newtonian expansion (A+B)
△ flat h m  16GT m   t2 h m  m h
slow motion  t  v i
GM
v / c  1 
 r
2
c
32
Green function method
Boundary condi. for homogeneous modes
up
down
in
out
Construct solution with source by using Green function.
s
 

1
up
4
in
iw t










x

g
d
x
'
R
r
Z

,

e
sT  x   
s

s 
s 

s W
Wronskian
at r →∞

1 


~
h

i
h
2


2

s W  s R
up


 r s Rin
d 2 E ( out)
r2


2 2
dt d
4w
2
33
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  a
t , r , ,    t , r , , 
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.
□Gadv / ret    4 x  x 
□G rad   0
34
Metric re-construction in Kerr case
 
x, x' T  z  ' Chrzanowski
hm x     g d x' Gm
(‘75)
Mode function
4
ret
s
Assume factorized form of Green function.
1
 ret 
up
out




G
x
,
x
'


x


s m
s
m
  x ' r  r '  
s
  s sW

Compute ψ following the definition.
comparison
m
s  s D hm  x   

s  



for metric
perturbation
m
D
s  m  s 
s
1
up
4
out
m
s    x   g d x ' s  m  x 'T
 s sW
1
up
4
out
s     g d x ' s   sT
s W
Calculation using Green
function for 
 g d 4 x' s  outm  x'T m   s   g d 4 x' s out s m T m
 
since the relation holds for arbitrary T
x '   ss m s
s
*
by integration by

s mis obtained from s m
parts.
Further, using the Starobinsky identity, one can also determine s . 35
out
m
*
out

Constants of motion for geodesics in Kerr
← definition of Killing tensor
36
Hint: similarity between expressions
for dE/dt and dQ/dt
• Energy loss can be also evaluated from the self-force.
dE    
dQ   ~ m  
   t  
  K m u  
d 
  x  z  
d 
  x  z  
just –iw after mode decomposition
• Formula obtained by the energy balance argument:
2
dE
2
dL
m
   Z l ,m ,w
   Z l ,m,w
d
d
l , m ,w
l , m ,w w

Z l ,m,w    mTm

x  z  
d
← amplitude of the partial wave
• dQ/dt formula is expected to be given by
dQ
   Zˆ l ,m ,w Z l ,m ,w
d
l , m ,w
d
ˆ
with Z l ,m,w  
K u    m Tm
 iw
x  z  
 

37
Further reduction
• A remarkable property of the Kerr geodesic equations is
 dr 
 d 


  Rr

   
with d  dr / 
 d 
 d 
By using , r- and  -oscillations can be solved independently.
dt
r 2  a2
2
 a aE sin   L 
E r 2  a 2  aL
d
D
d
L  a

  aE 
  E r  a   aL 
d
sin 
D
2
2





2
2


dt
t    t  t 

d
Periodic functions of periods 2  r 1 ,2 1
• Only discrete Fourier components arise
1
nr , n
w  wm  dt / d m d / d  nr  r  n  
• In general for a double-periodic function
r 
1
lim
T  2T
 
 r 






d

f
g

,
g


T
2 2
T
r 
2
 

2 r 1
0
dr 
21
0
d f g r  r , g    
38
Final expression for dQ/dt in
adiabatic approximation
After integration by parts using the relation in the previous slide,


dQ
r 2  a 2 Pr 
2
dt
D
dE
aPr 
2
dt
D
dL
nr  r
2 
Z l ,m ,w
dt
w
l , m ,w wln,rm,n

2

Pr   E r 2  a 2  aL
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}.



  
 
j

dd

I



I
r r r Ir jIj ,,r r I Ij ,j ,  Def.  r,r ,
   r ,r ,
t  t r  I j  ,  r   t   I j  ,    
dt
d
d
j
  ...
39
40
41
42
leading order
second order
 Om   Om 2 
dE orbit

0
dt
dEorbit
 geodesic   Om   O m 2
df
 
43
Probably clean system
•Interaction with accretion disk
(Narayan, ApJ, 536, 663 (2000))
3
vrel
10 M 6
t df 
 4.5 10
yr
2

4 log G msatellite
mm1
典型的な値としては
  10 2
m
M
：almost spherical

2
4 r vr
accretion (ADAF)
M
 T c

M Edd 4 Gmp
vrel  vK
vr   vK
  0.1
ts 
相互作用による
frequencyの変化
Df DT

f
t df
~ 4.5 107 yr (  0.1)
M  10 6 M 6 M sol
M  m M Edd
msatellite  10 m1M sol
cycle数の変化に焼きなおすと
観測期間
f DT 2 N DT
DN  Df DT 

t df
t df
44
Test of GR
(Berti & Will, PRD71 084025(2005))
Scalar-tensor type の重力理論の変更
]
  
3
 M f 5 / 3 u 2 / 3  1   3715  55   128  g  u 2 / 3   16  u  
128
3
 756 9



双極子放射＝－1 PNの振動数依存性
5s  s 
  1 2
64wBD
2
 
u  M f  O v 3
NS同士では同じscalar chargeをもっているので４重極
2
放射がleadingになってしまう。その場合、s1  s2  ≪1
双極子放射からのwBDに対する制限は4U 1820-30（NS-WD in
globular cluster NGC6624) からwBD＞140, (600)が得られている。
45
(Will & Zaglauer, ApJ 346 366 (1989))
number of cycles in LISA band for BH-NS systems
-1
wBD
-1
wBD
Parameter estimateにおける error  =10
-1
wBD
-1
wBD
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
46
LISAで 1.4M◎+400M◎の場合： wBD > 4×105
DECIGOはもっとすごいはず
Spinを考慮するとがあると・・・
wBD > 2×104
bound from Solar system
 current bound:
Cassini wBD > 2×104
 Future LATOR mission
wBD > 4×108
(Plowman & Hellings, CQG 23 309(’06) )
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。
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
47
重力の伝播速度の変更
(Berti & Will, PRD71 084025(2005)より)
massive gravitonのphase velocity
m2
1
cphase f    1 
 1 2
w
2w 2
2gf
D   d a 2
k
2
D


D  2f Dt  2fD Dcphase f   2
gf
3
  f
  
128
 2DM
g 
2g

5 / 3
振動数に依存し
た位相のずれ
 2 / 3

128
 3715 55
 2/3
1 
 
 g  u   16   u  
u
3
 756 9



gravitonがmassを持っている効果
number of cycles in LISA band for BH-BH systems
48
We need higher order accurate template
for precise measurement of parameters (or test of GR).
error due to noise
Di  1 
ortho-normalized parameters
For TAMA best sensitivity,
errors coming from ignorance of higher order coefficients
are @3PN ~10-2/ , @4.5PN ~10-4/
For large  or small   m/M ,
higher order coefficients can be important.
Wide band observation is favored to determine parameters
⇒ Multi band observation will require more
accurate template
49
Gravitation
wave detectors
LISA
⇒DECIGO/BBO
TAMA300
CLIO
⇒LCGT
LIGO⇒adv LIGO
VIRGO, GEO
50