1 重力波放出の反作用 Takahiro Tanaka ( Kyoto university ) Gravitational waves 素粒子・天文合同研究会 「初期宇宙の解明と新たな自然像」

2 Gravitation wave detectors LISA ⇒DECIGO/BBO TAMA300 ⇒LCGT LIGO⇒adv LIGO

3 Various sources of gravitational waves Earth-based Interferometer Binary coalescence –  -ray bursts Spinning NS –LMXB –SN remnant GW background SN formation –high kick velocity taken from Cutler & Thorne (2002)

4 Various sources of gravitational waves Space Interferometer taken from Cutler & Thorne (2002) Guaranteed binary sources –WD-WD –AM CVn –LMXB Supermassive BH –merger –formation from a super massive star Stochastic BG –WD binary noise –primordial

5 Rough estimate of GW frequencies and amplitude Frequency Amplitude of continuous sources Amplitude of inspiraling binary sources

6 ＤＥＣＩＧＯ space interferometer with shorter arm length high frequencies ⇒ more cycles acceleration of the universe better angular resolution 1deg ⇒ 1min 川村さんの 学会トラペより

7 Binary coalescence Inspiral phase (large separation) Merging phase - numerical relativity,  ray burst Ringing tail - quasi-normal oscillation of BH for precision test of general relativity Clean system Negligible effect of internal structure (Cutler et al, PRL (1993)) for detection (Berti et al, gr-qc/ ) for parameter extraction Accurate prediction of the wave form is requested

8 Do we need to predict accurate wave form ？ We know how higher expansion goes. Template (prediction of waveform) 1PN 1.5PN for circular orbit though we need to know template to exclude unphysical parameter region from search to some extent. ⇒ Only for detection, higher order templates may not be necessary?

9 We need higher order accurate template for precise measurement of parameters (or test of GR). For large  or small , higher order coefficients can be important. For TAMA best sensitivity, error due to noise errors coming from ignorance of higher order coefficients ~10 -2 / ~10 -4 /  Wide band observation is favored to determine parameters ⇒ Multi-band observation will require more accurate template

10 相対論のテストという観点では しかし、 Spin があると・・・ bound from solar system  current bound: Cassini  > 4×10 4  Future LATOR mission  > 4×10 8 － 1 PN の振動数依存 性 Scalar-tensor 理論 LISA で 1.4M ◎ +400M ◎ の場合：  > 4×10 5 DECIGO はもっとすごいはず (Berti et al, gr-qc/ )  > 4×10 4 ふたつは見ている効果が違う スカラー波の放出 vs PN correction また、コンパクト星は大きな scalar charge を持つ可能性もある。

11 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 § ２ Methods to predict waveform Red ○ means determination based on balance argument

12 Black hole perturbation  v/c can be O(1)  M>>  : master equation Linear perturbation Gravitational waves Regge-Wheeler formalism (Schwarzschild) Teukolsky formalism (Kerr) Mano-Takasugi-Suzuki’s method (systematic PN expansion)

13 Teukolsky formalism Teukolsky equation First we solve homogeneous equation Newman-Penrose quantities Angular harmonic function projection of Weyl curvature 2 nd order differential operator

14 Green function method at r →∞ up downinout Boundary condi. for homogeneous modes Construct solution with source by using Green function. Wronskian

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

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17 We need to directly evaluate the self-force acting on the particle, but… Radiation reaction for General orbits in Kerr black hole background Radiation reaction to the Carter constant Schwarzschild “ constants of motion ” E, L i ⇔ Killing vector Conserved current for GW corresponding to Killing vector exists. Kerr conserved quantities E, L z ⇔ Killing vector Q ⇔ Killing vector × In total, conservation law holds.

18 §3 Adiabatic approximation for Q We evaluate the radiative field instead of the retarded field. Self-force is computed from the radiative field, and it determines long time average of the change rate of E,L z,Q. differs from energy balance argument. For E and L z the results are consistent with the balance argument. (Gal’tsov ’82) For Q, it has been proven that the estimate by using the radiative field gives the correct long time average. (Mino ’03) Key point: Under the transformation every geodesic is transformed back into itself. Radiative field does not have divergence at the location of the particle.

19 A remarkable property of the Kerr geodesic equations is with Only discrete Fourier components arise Skipping many interesting details, we just present the final formula: By using, r- and  - oscillations can be solved separately. Periodic functions of periods Simplified dQ/dt formula (Sago, Tanaka, Hikida & Nakano PTPL(’05)) amplitude of partial wave evaluated at infinity

20 Summary up to here Basically this part is Z  simplified

21 Second order waveform second order leading order 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. the leading order self-force

22 Higher order in  Post-Teukolsky formalism Perturbed Einstein equation expansion 2 nd order perturbation : post-Teukolsky equation (1)  construct metric perturbation h  from  (1) (2) derive T  (2)  taking into account the self-force : Teukolsky equation linear perturbation

23 Electro-magnetism (DeWitt & Brehme (1960)) cap1 cap2 tube §４ Self-force in curved space Abraham-Lorentz-Dirac

24 tail-term Tail part of the self-force Retarded Green function in Lorenz gauge curvature scattering tail direct direct part ( S- part)tail part ( R- part)

25 Matched asymptotic expansion Extension to the gravitational case (Mino et al. PRD 55(1997)3457, see also Quinn and Wald PRD 60 (1999) ) Extension is formally non-trivial. １）equivalence principle e=m ２）non-linearity mass renormalization near the particle） small BH(  )+perturbations |x|/(GM)<< 1 far from the particle） background BH( M )＋perturbation G  /|x| << 1 matching region

26 Gravitational self-force Tail part of the metric perturbations E.O.M. with self-force = geodesic motion on Retarded Green function in harmonic gauge direct part ( S- part)tail part ( R- part) curvature scattering tail direct (MiSaTaQuWa equation) Extension of its derivation is non-trivial, but the result is a trivial extension.

27 Since we don’t know the way of direct computation of the tail ( R - part), we compute Both terms on the r.h.s. diverge ⇒ regularization is needed Mode sum regularization Coincidence limit can be taken before summation over { Decomposition into spherical harmonics Y { m modes finite value in the limit r→r 0

28 can be expanded in terms of ・ S- part is determined by local expansion near the particle. S- part (Skip) : spatial distance between x and z ・ Mode decomposition formulae (Barack and Ori (’02), Mino Nakano & Sasaki (’02)) { where

29 We usually evaluate full- and S- parts in different gauges. But it is just a matter of gauge, so we can neglect this term as long as the gauge transformation  R stays small. Gauge problem cannot be evaluted directly in harmonic gauge (H) gauge transformation connecting two gauges is divergent in general. can be computed in a convenient gauge (G). We do not know how to evaluate decompose it as Force in the hybrid-gauge

30 What is the remaining problem? Basically, we understand how to compute the self-force in the hybrid-gauge. But actual computation is still limited to simple 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. 2 nd order perturbation : post-Teukolsky equation Both terms on the right hand side are gauge dependent. but T (2) in total must be gauge independent. regularization ？ What we want to know is the second order wave form We need the self-force and the second order source term simultaneously.

31 Conclusion Adiabatic radiation reaction for the Carter constant is now almost ready to compute. We need to evaluate the second order source term and the self-force as a pair. second order leading order The leading order self-force is also almost ready to compute. However, due to gauge degrees of freedom, it might be the case that only the secular change of constants of motion in the self-force has physical meaning. If so, there is no merit in computing the self-force directly before the second order perturbation is solved.