Gravitational waves from Extreme mass ratio inspirals Gravitational Radiation Reaction Problem

1. Gravitational waves from Extreme mass ratio inspirals Gravitational Radiation Reaction Problem Gravitational waves Takahiro Tanaka 　　(Kyoto university)

2. 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

3. 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

4. Clean system (Cutler et al, PRL 70 2984(1993)) Negligible effect of internal structure • Inspiral phase (large separation) • Merging phase -　numerical relativity recent progress in handling BHs • Ringing tail -　quasi-normal oscillation of BH 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)

5. Extreme mass ratio inspirals (EMRI) m X • LISA sources 0.003-0.03Hz 　　　 → merger to white dwarfs (m=0.6M◎), neutron stars (m=1.4M◎) BHs (m=10M◎,~100M◎) • Formationscenario • star cluster is formed • large angle scattering encounter putthe body into a highly eccentric orbit • Capture and circularization due to gravitational radiation reaction ~last three years: eccentricity reduces1-e →O(1) • Event rate: a few ×102events for 3 year observation by LISA BH M GW (Gair et al, CGQ 21 S1595 (2004)) although still very uncertain. (Amaro-Seoane et al, astro-ph/0703495)

6. m≪M Radiation reaction is weak Large number of cycles N before plunge in the strong field region m M 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

7. Probably clean system (Narayan, ApJ, 536, 663 (2000)) • Interaction with accretion disk ,assuming almost spherical accretion (ADAF) Frequency shift due to interaction Change in number of cycles obs. period ~1yr

8. Theoretical prediction of Wave form Template in Fourier space 1.5PN for quasi-circular orbit 1PN • 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 parameter estimate ∝signal to noise ratio

9. Test of GR Effect of modified gravity theory Scalar-tensor type Mass of graviton Dipole radiation ＝ －1 PN Current constraint on dipole radiation: wBD＞140, (600) 4U 1820-30（NS-WD in NGC6624) Constraint from future observation: LISA-107M◎BH+107M◎BH: graviton compton wavelength lg > 1kpc (Will & Zaglauer, ApJ 346 366 (1989)) (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 ?

10. Black hole perturbation • M≫m • v/c can be O(1) Gravitational waves Linear perturbation :master equation Regge-Wheeler formalism (Schwarzschild) Teukolsky formalism (Kerr) Mano-Takasugi-Suzuki’s method (systematic PN expansion)

11. Teukolsky formalism Teukolsky equation 2nd order differential operator projectedWeyl curvature First we solve homogeneous equation Angular harmonic function Construct solution using Green fn. method. Wronskian atr →∞ : energy loss rate : angular momentum loss rate

12. Leading order wave form Energy balance argument is sufficient. Wave form for quasi-circular orbits, for example. leading order self-force effect

13. Radiation reaction for General orbits in Kerr black hole background • Radiation reaction tothe Carter constant Schwarzschild “constants of motion”E, Li⇔ Killing vector Conserved current for GW corresponding to Killing vector exists. In total, conservation law holds. Kerrconserved quantitiesE, 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.

14. 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. . . . (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 byE,Lz,Q. Radiative field is not divergent at the location of the particle. Regularization of the self-force is unnecessary!

15. SimplifieddQ/dt formula (Sago, Tanaka, Hikida, Nakano, Prog. Theor. Phys. 114 509(’05)) • Self-forcef ais explicitly expressed in terms of hmn as Killing tensor associated with Q Complicated operation is necessary for metric reconstruction from the master variable. after several non-trivial manipulations • We arrived at an extremely simple formula: Only discrete Fourier components exist

16. Use of systematic PN expansion of BH perturbation. • Small eccentricity expansion • General inclination (Ganz, Hikida, Nakano, Sago, Tanaka, Prog. Theor. Phys. (’07))

17. 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 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.

18. Summary up to here Basically this part is ZL simplified

19. Second order wave form leading order 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.

20. linear perturbation :Teukolsky equation 2nd order perturbation (1)construct metric perturbationhmnfromz (1) (2) deriveT(2)mntaking into accountthe self-force : post-Teukolsky equation Higher order in m Post-Teukolsky formalism Perturbed Einstein equation expansion

21. §４ Self-force in curved space Abraham-Lorentz-Dirac Electro-magnetism (DeWitt & Brehme (1960)) cap1 tube cap2

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

23. Extension to the gravitational case mass renormalization • Extension is formally non-trivial. • １）equivalence principlee=m • 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

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

25. Since we don’t know the way of direct computation of thetail (R-part), we compute Both terms on the r.h.s. diverge ⇒ regularization is needed • Mode sum regularization Decomposition into spherical harmonicsY{mmodes Coincidence limit can be taken before summation over{ finite value in the limit r→r0

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

27. We usually evaluate full- and S- partsin different gauges. Gauge problem cannot be evaluted directly in harmonic gauge (H) can be computed in a convenient gauge (G). gauge transformation connecting two gauges is divergent in general. also diverges. cannot be evaluated without error. 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 parameterxm(xm→xm+xm) to be small.

29. Hybrid gauge method (Mino-Barack-Ori?) gauge transformation stays finite ⇒ also automatically stays finite if it is determined by local value of . (T.T.) A similar but slightly different idea was proposed by Ori. We can compute the self-force by using

30. 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 Both terms on the right hand side are gauge dependent. : post-Teukolsky equation butT (2) in total must be gauge independent. regularization ？ We need the regularizedself-force and the regularized second order source term simultaneously.

31. §２　Methods to predict wave form Post-Newton approx. ⇔ BH perturbation • Post-Newton approx. • v < c • Black hole perturbation • m1 >>m2 BH pertur- bation Post Teukolsky post-Newton ○ : done Red ○means determination based on balance argument

32. Standard post-Newtonian approximation C B A source l Post-Minkowski expansion (B+C) vacuum solution Post-Newtonian expansion (A+B) slow motion

33. Construct solution with source by using Green function. Wronskian at r →∞ Green function method Boundary condi. for homogeneous modes up down in out

34. 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 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.

35. Metric re-construction in Kerr case Mode function for metric perturbation Assume factorized form of Green function. Chrzanowski (‘75) Compute ψ following the definition. comparison Calculation using Green function for y since the relation holds for arbitraryT by integration by parts. is obtained from Further, using the Starobinsky identity, one can also determine zs .

36. Constants of motion for geodesics in Kerr ← definition of Killing tensor

37. Hint: similarity between expressions for dE/dt and dQ/dt • Energy loss can be also evaluated from the self-force. • Formula obtained by the energy balance argument: • dQ/dt formula is expected to be given by just –iw after mode decomposition ← amplitude of the partial wave with

38. Further reduction • A remarkable property of the Kerr geodesic equations is with • Only discrete Fourier components arise • In general for a double-periodic function By usingl, r- and q -oscillations can be solved independently. Periodic functions of periods

39. Def. Final expression for dQ/dt in adiabatic approximation After integration by parts using the relation in the previous slide, 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 anddQ/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}.

43. leading order second order

44. Probably clean system (Narayan, ApJ, 536, 663 (2000)) • Interaction with accretion disk 典型的な値としては ：almost spherical accretion (ADAF) 相互作用によるfrequencyの変化 cycle数の変化に焼きなおすと 観測期間

45. ] • Test of GR (Berti & Will, PRD71 084025(2005)) Scalar-tensor type の重力理論の変更 双極子放射＝－1 PNの振動数依存性 NS同士では同じscalar chargeをもっているので４重極放射がleadingになってしまう。その場合、 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。 (Will & Zaglauer, ApJ 346 366 (1989))

46. number of cycles in LISA band for BH-NS systems 他の全てのparameterが与えられている場合 Parameter estimateにおける error r=10 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

47. 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

48. 重力の伝播速度の変更 (Berti & Will, PRD71 084025(2005)より) massive gravitonのphase velocity 振動数に依存した位相のずれ gravitonがmassを持っている効果 number of cycles in LISA band for BH-BH systems

49. We need higher order accurate template • for precise measurement of parameters (or test of GR). error due to noise ortho-normalized parameters For TAMA best sensitivity, errors coming from ignorance of higher order coefficients are@3PN~10-2/h , @4.5PN ~10-4/h For larger or smallh = m/M , higher order coefficients can be important. Wide band observation is favored to determine parameters ⇒　Multi band observation will require more accurate template

50. Gravitation wave detectors TAMA300 　CLIO 　⇒LCGT LISA ⇒DECIGO/BBO LIGO⇒adv LIGO VIRGO, GEO