How to test the Einstein gravity using gravitational waves

1
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
2
Binary coalescence

• Inspiral phase (large separation)
• Merging phase - numerical relativity
• Ringing tail - quasi-normal oscillation of BH

Clean system
(Cutler et al, PRL 70 2984(1993))
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, gr-qc/0504017 )
3
Do we need to predict accurate wave form?

• We know how higher expansion goes.

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

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

4
Propagation of GWs as an example of GR test

• Chern-Simons Modified Gravity

The evolution of background scalar field q is
hard to detect.
J0737-3039(double pulsar)
(Yunes Spergel, arXiv0810.5541)
Right handed and left handed gravitational waves
are magnified differently during propagation,
depending on the frequencies.

• Ghost free bi-gravity

Both massive and massless gravitons exist. ?n
oscillation-like phenomena
(in preparation)
5
Methods to predict wave form
Post-Newton approx. ? BH perturbation

• Post-Newton approx.
• v lt c

• Black hole perturbation
• m1 gtgtm2

v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11
µ0 ? ? ? ? ? ? ? ? ? ? ?
µ1 ? ? ? ? ?
µ2 ? ? ?
µ3 ?
µ4
BH pertur- bation
Post Teukolsky
post-Newton
? done
Red ? means determination based on balance
argument
6
Extrime Mass Ratio Inspiral(EMRI)

• Inspiral of 1100Msol BH of NS into the super
  massive BH at galactic center (typically 106Msol)

• Very relativistic wave form can be calculated
  using BH perturbation

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

• Clean system

BH
The best place to test GR.
7
Black hole perturbation

• M gtgtm

• v/c can be O(1)

Gravitational waves
Linear perturbation
simple master equation
Regge-Wheeler-Zerilli formalism
(Schwarzschild) Teukolsky formalism
(Kerr)
8
Leading order wave form
Energy balance argument is sufficient.
Wave form for quasi-circular orbits, for
example.
leading order
9
Evolution of general orbits
If we know four velocity um at each time
accurately, we can solve the orbital
evolution in principle.
On Kerr background there are four constants of
motion
constant in case of no radiation reaction
Normalization of four velocity
Energy
Killing vector for time translation sym.
Angular momentum
Killing vector for rotational sym.
Carter constant
Killing tensor
Quadratic and un-related to Killing
vector (simple symmetry of the spacetime)
One to one correspondence
We need to know the secular evolution of E,Lz,Q.
10
The issue of radiation reaction to Carter constant

• E, Lz ? Killing vector
• Conserved current for the field
  corresponding
• to Killing vector exists.

As a sum conservation law holds.

However, Q ? Killing vector

• 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.

11
3 Adiabatic approximation for Q
which is different from energy balance argument.

• T ltlt tRR
• T orbital period
• tRR 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.

12

• For E and Lz the results are consistent with the
  balance argument. (shown by Galtsov 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.

13
Outstanding property of Kerr geodesic
Introducing a new time parameter l by
r- and q -oscillations can be solved
independently.
Periodic functions with periods

• Only discrete Fourier components arise in an orbit

14
Final expression for dQ/dt in adiabatic
approximation
After a little complicated calculation,
miraculous simplification occurs
(Sago, Tanaka, Hikida, Nakano PTPL(05))
amplitude of the partial wave
This expression is similar to and as
easy to evaluate as dE/dt and dL/dt.
15
Resonant orbit

• Key point Under Minos transformation
• a geodesic is transformed back into the
  same geodesic.

However, for resonant case
with integer jr jq
Dl (separation from qmax to rmax) has physical
meaning.
r
l (Minos time)
q
Dl
Dl
Under Minos transformation, a resonant geodesic
with Dl transforms into a resonant geodesics with
-Dl.
16
dQ/dt at resonance
For the radiative part (retarded-advaneced)/2, a
formula similar to the non-resonant case can be
obtained
(Flanagan, Hughes, Ruangsri, 1208.3906)
Sum for the same frequency is to be taken first.
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.
17
Impact of the resonance on the phase evolution
(gravitational radiation reaction)
duration staying around resonance
frequency shift caused by passing resonance
overall phase error due to resonance
?O((m/M )0)
Oscillation period is much shorter than the
radiation reaction time
If for Dl Dlc,
If b stays negative, resonance may persist for a
long time.
18
Conclusion
Adiabatic radiation reaction for the Carter
constant is as easy to compute as those for
energy and angular momentum.
leading order
second order
Hence the leading order waveform whose phase is
correct at O(M/m) is also ready to compute.
The orbital evolution may cross resonance, which
induces O((M/m)1/2) correction to the phase.
We derived a formula for the change rate of the
Carter constant due to scalar self-force valid
also in the resonance case.
Extension to the gravitational radiation reaction
is a little messy, but it also goes almost in
parallel.
19
The symmetric part (retardedadvanced)/2 also
becomes simple.
r-oscillation
q-oscillation
0
(t ?t w ?-w )
20
Regularization is necessary
To compute
,
regularization is necessary.
Regularized field
Hadamard expansion of retarded Green function
tail part
direct part
Tail part gives the regularized
self-field. Direct part must be subtracted.
(DeWitt Brehme (1960))
We just need

Easy to say but difficult to calculate especially
for the Kerr background.
But what we have to evaluate here looks a little
simpler than self-force.
21
Simplified dQ/dt formula
Sago, Tanaka, Hikida, Nakano PTPL(05)

• Self-force is expressed as

drops after long time average
Substituting the explicit form of Kmn,
Mino time
22
Novel regularization method

• Instead of directly computing the tail, we
  compute

with
Both terms on the r.h.s. diverge in the limit
z(t)? z(t).
periodic source
is just the
Fourier coefficient of with respect to
e1,e2.
is finite and calculable.
23
(sym)-(dir) is regular
We can take e ?0 limit before summation over m
N
Difference from the ordinary mode sum
regularization
Compute the force and leaves l-summation to the
end.
Divergence of the force behaves like 1/e 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.
does not fit well with Teukolsky formalism
m N-sum regularization seems to require the
high symmetry of the resonant geodesics.
F is periodic in e1,e2.
24
Teukolsky formalism
projection of Weyl curvature
Teukolsky equation
2nd order differential operator
First we solve homogeneous equation
Angular harmonic function
25
Green function method
Boundary condi. for homogeneous modes
up
down
in
out
Parallel to the case of a scalar charged particle.