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How to test the Einstein gravity using gravitational waves. Gravitational waves. Takahiro Tanaka (YITP, Kyoto university ). Based on the work in collaboration with R. Fujita, S. Isoyama, H. Nakano, N. Sago. Binary coalescence. Clean system. (Cutler et al, PRL 70 2984(1993)). - PowerPoint PPT Presentation
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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
7
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)
8
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.
9
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.
10
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
11
§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.
12
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
13
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))
15
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
16
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.
17
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)
18
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.
19
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
20
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
21
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
23
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
24
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
25
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.
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