1. 1 How to test the Einstein gravity using gravitational waves Takahiro Tanaka (YITP, Kyoto university) Gravitational waves Based on the work in collaboration with R. Fujita, S. Isoyama, H. Nakano, N. Sago

2. (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. 3 Do we need to predict accurate wave form ？ We know how higher expansion goes. Template (prediction of waveform) 1PN 1.5PN for circular orbit ⇒ Only for detection, higher order template may not be necessary? But we need higher order accurate template for the test of GR.

4. 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. Propagation of GWs as an example of GR test Ghost free bi-gravity Both massive and massless gravitons exist. → oscillation-like phenomena (in preparation)

5. 5 Post-Newton approx. ⇔ BH perturbation Post-Newton approx.  v < c Black hole perturbation  m 1 >> m 2 v0v0 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 v 11 00 ○○○○○○○○○○○ μ1μ1 ○○○○○ μ2μ2 ○○○ μ3μ3 ○ μ4μ4 BH pertur- bation post-Newton Post Teukolsky ○ : done Methods to predict wave form Red ○ means determination based on balance argument

6. 6 Extrime Mass Ratio Inspiral(EMRI) Inspiral of 1～100 M sol BH of NS into the super massive BH at galactic center (typically 10 6 M sol ) 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. 7 Black hole perturbation  v/c can be O(1)  M >>  : simple master equation Linear perturbation Gravitational waves Regge-Wheeler-Zerilli formalism (Schwarzschild) Teukolsky formalism (Kerr)

8. 8 Leading order wave form Energy balance argument is sufficient. Wave form for quasi-circular orbits, for example. leading order

9. Killing tensor Killing vector for rotational sym. 9 Evolution of general orbits 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: Killing vector for time translation sym. Carter constant: Quadratic and un-related to Killing vector (simple symmetry of the spacetime) One to one correspondence constant in case of no radiation reaction We need to know the secular evolution of E,L z,Q.

10. 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, L z ⇔ Killing vector Conserved current for the field corresponding to Killing vector exists. However, Q ⇔ Killing vector × As a sum conservation law holds.

11. 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,L z,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,L z,Q. which is different from energy balance argument.

12. 12 For E and L z 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.

13. 13 Outstanding property of Kerr geodesic Only discrete Fourier components arise in an orbit r- and  - oscillations can be solved independently. Periodic functions with periods Introducing a new time parameter  by

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

15. 15 Key point: Under Mino’s transformation a geodesic is transformed back into the same geodesic.  (Mino’s time)  r  However, for resonant case:   (separation from  max to r max ) has physical meaning. Under Mino’s transformation, a resonant geodesic with  transforms into a resonant geodesics with . with integer j r & j  Resonant orbit

16. 16 For the radiative part (retarded-advaneced)/2, a formula similar to the non-resonant case can be obtained: (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. 17 Impact of the resonance on the phase evolution : duration staying around resonance If for  c, : overall phase error due to resonance ≠ O ((  0 ) : frequency shift caused by passing resonance If  stays negative, resonance may persist for a long time. Oscillation period is much shorter than the radiation reaction time (gravitational radiation reaction)

18. 18 Conclusion Adiabatic 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. 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. 19 The symmetric part (retarded+advanced)/2 also becomes simple. r- oscillation  oscillation  ↔  ’  →  =0

20. 20 To compute, Regularized field Tail part gives the regularized self-field. Direct part must be subtracted. Hadamard expansion of retarded Green function curvature scattering tail direct direct part tail part 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. Regularization is necessary

21. drops after long time average 21 Simplified dQ/dt formula Self-force is expressed as Sago, Tanaka, Hikida, Nakano PTPL(’05) Mino time: Substituting the explicit form of K ,

22. Instead of directly computing the tail, we compute Both terms on the r.h.s. diverge in the limit z ± (  ) → z(  ). with periodic source Fourier coefficient of with respect to {  1,  2 }. is just the is finite and calculable. Novel regularization method

23. 23 (sym)-(dir) is regular We can take  →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/  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. 24 Teukolsky formalism Teukolsky equation First we solve homogeneous equation Newman-Penrose quantities Angular harmonic function projection of Weyl curvature 2 nd order differential operator

25. 25 Green function method at r →∞ up downinout Boundary condi. for homogeneous modes Construct solution with source by using Green function. Wronskian Parallel to the case of a scalar charged particle.