Gravitational waves from Extreme mass ratio inspirals Gravitational - PowerPoint PPT Presentation

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

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 2

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 3

• Inspiral phase (large separation) 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, PRD 71 :084025,2005 ) � Merging phase - numerical relativity recent progress in handling BHs � Ringing tail - quasi-normal oscillation of BH 4

Extreme mass ratio inspirals (EMRI) • LISA sources 0.003-0.03Hz → merger to � � 5 6 M ~ 10 M 5 10 M ◎ ◎ � white dwarfs ( � =0.6 M ◎ ), X neutron stars ( � =1.4 M ◎ ) BH BHs ( � =10 M ◎ , ~100 M ◎ ) M • Formation scenario GW – star cluster is formed – large angle scattering encounter put the body into a highly eccentric orbit – Capture and circularization due to gravitational radiation reaction ~last three years: eccentricity reduces 1 -e → O(1) • Event rate: a few ×10 2 events for 3 year observation by LISA (Gair et al, CGQ 21 S1595 (2004)) 5 although still very uncertain. (Amaro-Seoane et al, astro-ph/0703495)

• � ≪ M Radiation reaction is weak Large number of cycles N before plunge in the strong field region � BH Roughly speaking, M 重力波 difference in the number of cycle � N >1 is detectable. • High-precision determination of orbital parameters • maps of strong field region of spacetime – Central BH will be rotating: a ~0.9 M 6

Probably clean system •Interaction with accretion disk (Narayan, ApJ, 536 , 663 (2000)) ,assuming almost spherical accretion (ADAF) 3 v � rel t � � � df 2 4 log G m satellite � � 1 1 � � � � � & M m � � � � � � 12 4 . 5 10 yr � � � � � & 6 2 � � � � 10 M 10 M 10 M ◎ ◎ Edd Frequency shift Change in number of cycles due to interaction obs. period � f � T T � � � � obs N f T fT obs ~ 1 yr obs obs t f t df df 7

Theoretical prediction of Wave form Template in Fourier space � � � � 5 / 6 � � 1 � � � � � � � � 7 / 6 i f 3 5 2 5 � h f A f e A , M , � D M 3 20 L � � � � 3 � � 20 743 11 � � � � � � � � � � � � � � � � � � 5 / 3 � � 2 / 3 � L 2 f t f � 1 u 16 u � c c � � � � 128 9 331 4 � � 1.5PN 1PN � � � 3 u M f O v for quasi-circular orbit 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). observational error in c.f. ∝ signal to noise ratio parameter estimate 8

Test of GR Effect of modified gravity theory Scalar-tensor type Mass of graviton � � � � 3 � � 3715 55 128 � � � � � � � � � � � � � � �� � � � � � 5 / 3 � � 2 / 3 2 / 3 L � L f � u 1 u 16 u � g � � � � 128 756 9 3 � � � � � 3 � u M f O v 1 2 M � � � � � � Dipole radiation ＝ － 1 PN 2 a d � � g 2 g BD Current constraint on dipole radiation: Constraint from future observation: � BD ＞ 140, (600) LISA � 10 7 M ◎ BH+10 7 M ◎ BH: 4U 1820-30 （NS-WD in NGC6624) graviton compton wavelength � g > 1kpc (Will & Zaglauer, ApJ 346 366 (1989)) Constraint from future observation: (Berti & Will, PRD 71 084025(2005)) LISA � 1.4 M ◎ NS+400 M ◎ BH: � BD > 2 × 10 4 (Berti & Will, PRD 71 084025(2005)) Decigo � 1.4 M ◎ NS+10 M ◎ BH ： � BD >5 × 10 9 ? 9

Black hole perturbation � � �� �� � � GT G g 8 � � � � � � � � BH 1 2 L g g h h �� �� �� �� � M ≫ � BH � v / c can be O (1) 重力波 Gravitational waves Linear perturbation � � � � � � �� �� � � � 1 1 G h 8 GT : master equation � � � � � � � � 1 1 L 4 g T Regge-Wheeler formalism (Schwarzschild) Teukolsky formalism (Kerr) Mano-Takasugi-Suzuki’s method (systematic PN expansion) 10

Teukolsky formalism Teukolsky equation �� T � � T �� 2 nd order differential operator � � � � L 4 g T projected Weyl curvature First we solve homogeneous equation � � � l , m , � � � � � � � � � � � � � i t L 0 , R r Y e � � � � � � � Angular harmonic function � � � 2 L R r 0 r Construct solution using Green fn. method. � � � � 1 � � � � � � L � � � � � � � � � � � � � � � � � up up 4 in i t x Z x g d x ' R r Y , e T x � � � � � W � � � � s Wronskian � � up in W R R � � � s s r at r →∞ s dE � � � 2 : energy loss rate � � Z 1 � & & & & � � � h dt ~ h i � � 2 dL z m � � � 2 : angular momentum Z � 11 � dt loss rate �

Leading order wave form Energy balance argument is sufficient. dE dE � � GW orbit dt dt df � Wave form for quasi-circular orbits, for example. dt df dE dE � orbit orbit leading order dt dt df � � � � dE orbit � � � � � 2 0 O O dt � � � � � � dE orbit � � � � � 2 geodesic O O df self-force effect 12

Radiation reaction for General orbits in Kerr black hole background Radiation reaction to the Carter constant “ constants of motion ” E , L i ⇔ Killing vector Schwarzschild Conserved current for GW corresponding to Killing vector exists. � � � � � � � � GW E d t �� GW & & � � In total, conservation law holds. E E orbit gw conserved quantities E , L z ⇔ Killing vector Kerr × Q ⇔ Killing vector We need to directly evaluate the self-force acting on the particle, but it is divergent in a naïve sense. 13

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,L z ,Q. � � 2 (Mino Phys. Rev. D67 084027 (’03)) � � � � � � � � :radiative field rad ret adv h h h �� �� �� At the lowest order, we assume that the trajectory of a particle is given by a geodesic specified by E,L z ,Q. � � � 1 lim � � 1 � � T Q & � � � rad Q d F h �� � � � � � 2 T u T T Radiative field is not Regularization of the divergent at the self-force is location of the particle. unnecessary! 14

Simplified dQ/dt formula (Sago, Tanaka, Hikida, Nakano, Prog. Theor. Phys. 114 509(’05)) Self-force f � is explicitly expressed in terms of h �� as • � � � � 1 � �� � � � � � � � � � f g u u h h h u u �� � �� � �� � ; ; ; 2 dQ Killing tensor associated with Q � � � 2 u f K � � � d Q � � � K u u � � Complicated operation is necessary � � �� � * h �� for metric reconstruction from the s master variable. after several non-trivial manipulations • We arrived at an extremely simple formula: � � � � � � � � 2 2 dQ r a P r dE aP r dL n � 2 � � � r r 2 2 2 Z � � � � l , m , dt dt dt � � � n , n � r l , m , l , m � � aL Only discrete Fourier components exist � � � � � 2 2 P r E r a � � � � � � � � � � � n r , n � m n n � � � � � � � 2 2 r 2 Mr a m r r 15

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