Gravitational waves from a spinning particle orbiting a Kerr black - PowerPoint PPT Presentation

Gravitational waves from a spinning particle orbiting a Kerr black hole Ryuichi Fujita ( 藤 田 龍 一 ) Yukawa Institute for Theoretical Physics, Kyoto University [with Norichika Sago (Kyushu University, Japan), in preparation] Gravity and Cosmology 2018, YITP, Feb. 6, 2018 Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 1 / 28

Motivation • To study extreme-mass ratio inspirals (EMRIs) as GW sources using black hole perturbation theory � One of the main targets for LISA l Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 2 / 28

Motion of a spinning particle in Kerr spacetime M µ <<M µ GW • Zeroth order in the mass ratio O [( µ/ M ) 0 ]: � Geodesic orbits with ( E , L z , C ) • First order in the mass ratio O [( µ/ M ) 1 ]: Deviation from the geodesic orbits because of � Radiation reaction � Spin of the particle, ... How important is the spin of the partcle for orbits and GWs? Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 3 / 28

Equations of motion of the spinning particle � Mathisson-Papapetrou-Pirani (MPP) equation: ∗ Neglect higher multipoles than quadrupole, accurate up to the linear order in the spin − 1 D d τ p µ ( τ ) 2 R µ νρσ ( z ( τ )) v ν ( τ ) S ρσ ( τ ) , = D d τ S µ ν ( τ ) 2 p [ µ ( τ ) v ν ] ( τ )(= 0) , = ∗ v µ ( τ ) = dz µ ( τ ) / d τ : four-velocity ∗ p µ ( τ ): four-momentum ∗ S µ ν ( τ ): spin tensor ⇒ 14 degrees of freedom for 10 equations � Spin supplementary condition (4 equations): S µ ν ( τ ) p ν ( τ ) = 0 (determines COM of the particle) Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 4 / 28

Energy fl ux in the adiabatic approximation • T orbit � T radiation [ T orbit = O ( M ), T radiation = O ( M 2 /µ )] � Energy balance argument � dE � GW � � µ 2 � � � � 2 � m ω | 2 � Z H | Z ∞ = + α � m ω , � m ω 4 πω 2 dt � �� � � �� � t � m In fi nity part Horizon part where ω = m Ω φ , α � m ω ∝ ω − mq / (2 r + ) and � d τ R in / up Z ∞ , H � m ω ( r ) T � m ω ( r ) , � m ω ∼ R in / up � m ω ( r ) : Homogeneous solutions of the radial Teukolsky equation T � m ω ( r ) : Source term constructed from energy-momuntum tensor � Energy-momuntum tensor : � � �� � p ( α v β ) δ (4) ( x − z ( τ )) S µ ( α v β ) δ (4) ( x − z ( τ )) T αβ = d τ √− g − ∇ µ √− g . Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 5 / 28

How to calculate the energy fl ux • Solve the EOM for a spinning particle: v µ ( τ ), p µ ( τ ), S µ ν ( τ ) • Construct energy-momentum tensor � The source term of the Teukolsky equation T � m ω ( r ) • Solve the Teukolsky equation R in / up � m ω ( r ) � The analytic method by Mano et al. (1995) � m ω ( r ) ∼ � a ν ∗ R in / up n F n + ν ( r ) ∗ a ν n +1 α ν n + a ν n β ν n + a ν n − 1 γ ν n = 0 • Calculate the amplitude of each mode Z ∞ / H � m ω � dE � GW � � 2 � � Z ∞ / H � � • Sum over all modes ∼ � � m ω dt � m Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 6 / 28

spin-aligned binary in circular orbit • As a fi rst step: � Circular and equatorial orbits � Particle’s spin is parallel to the BH spin M µ <<M µ GW Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 7 / 28

PN fl ux for spin-aligned binary in circular orbit (spin) 1 (spin) 2 (spin) 3 3.5PN (NNLO) 3PN (NLO) 3.5PN (LO) PN [Boh´ e+(2013)] [Boh´ e+(2015)] [Marsat (2015)] 2.5PN for µ ’s spin BHP ∗ ∗ [Tanaka+(1996)] This 6PN for µ ’s spin ∗ ∗ work [+ BH absorption] [ ∗ : In BHP, PN fl uxes are derived without expanding in M ’s spin] � nPN means ( M Ω φ ) 2 n / 3 correction to leading order � 4PN means ( M Ω φ ) 8 / 3 correction to leading order where M Ω φ is the orbital frequency Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 7 / 28

Phase shift due to the particle’s spin � − 1 d Ω φ � dE � � dE • Orbital phase Φ ≡ Ω φ ( t ) dt = Ω φ d Ω φ dt • Phase shift due to µ ’s spin: δ Φ = Φ (ˆ s � = 0) − Φ (ˆ s = 0) � Early inspiral for one-year observation of LISA φ ) 1 / 3 ∼ 0 . 2, ( M Ω ( f ) φ ) 1 / 3 ∼ 0 . 25 and ( M , µ ) = (10 5 , 10) M � ∗ ( M Ω ( i ) � Late inspiral for one-year observation of LISA φ ) 1 / 3 ∼ 0 . 3, ( M Ω ( f ) φ ) 1 / 3 ∼ 0 . 4 and ( M , µ ) = (10 6 , 10) M � ∗ ( M Ω ( i ) ( a , s ) = (0 . 9 M , 0 . 9 µ ) Early inspiral Late inspiral δ Φ at 1.5PN 3.96 2.14 δ Φ at 2PN -1.25 -1.04 δ Φ at 2.5PN 1.17 1.51 δ Φ at 3PN -0.64 -1.28 δ Φ at 3.5PN 0.34 1.06 δ Φ at 4PN -0.14 -0.66 δ Φ at 4.5PN 0.05 0.41 δ Φ at 5PN -0.01 -0.19 δ Φ at 5.5PN -0.0003 0.02 δ Φ at 6PN 0.003 0.07 Φ up through 6PN 700472.95 1051632.90 Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 9 / 28

Summary • Gravitational waves from a spinning particle around a Kerr BH � 6PN energy fl ux for circular and spin-aligned orbits ∗ New terms beyond 3.5PN ∗ BH absorption is also derived � Phase shift due to the particle’s spin δ Φ ∗ δ Φ � 1 at 4PN and beyond for typical binaries in the LISA band [ δ Φ = Φ (ˆ s � = 0) − Φ (ˆ s = 0)] • Future � Circular and slightly inclined orbits ∗ Spin-spin precessions � Eccentric and spin-aligned orbits in the equatorial plane ∗ Periastron shift � More generic orbits? Coupling between orbit and spins ∗ Ω φ , Ω r , Ω θ and Ω spin − prec ? Ryuichi Fujita (YITP) GWs from a spinning particle in Kerr 10 / 28