Gravitational wave templates from Extreme Mass Ratio Inspirals
GGravitational wave templates from Extreme Mass RatioInspirals
V. Skoup´y
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University,CZ-180 00 Prague, Czech RepublicAstronomical Institute of the Czech Academy of Sciences, Boˇcn´ı II 1401/1a, CZ-141 00Prague, Czech Republic
G. Lukes-Gerakopoulos
Astronomical Institute of the Czech Academy of Sciences, Boˇcn´ı II 1401/1a, CZ-141 00Prague, Czech Republic
Abstract.
An extreme mass ratio inspiral takes place when a compact stellarobject is inspiraling into a supermassive black hole due to gravitational radiationreaction. Gravitational waves (GWs) from this system can be calculated using theTeukolsky equation (TE). In our case, we compute the asymptotic GW fluxes of aspinning body orbiting a Kerr black hole by solving numerically the TE both intime and frequency domain. Our ultimate goal is to produce GW templates forspace-based detectors such as LISA.
Introduction
The Laser Interferometer Space Antenna (LISA) is a future space based gravitational wave(GW) detector planned to launch in 2030s [Amaro-Seoane et al., 2017]. LISA will consist ofthree spacecrafts forming an equilateral triangle with sides 2.5 million km long on heliocentricorbit. Changes in the distance between the spacecrafts will be monitored by Michelson-like in-terferometers with high precision. When a GW passes this constellation, the spacetime betweenthe spacecrafts will be quasi-periodically stretched and contracted in the direction perpendicu-lar to the propagation of the wave. Hence, LISA will be able to detect the wave, including itsamplitude and phase, by measuring the changes in the distance between the three spacecrafts.This GW observatory will be sensitive in frequencies around 10 − Hz.One type of events LISA will be able to detect are extreme mass ratio inspirals (EMRIs). Itis expected that the centre of galaxies host supermassive black holes with masses in the range of10 –10 M (cid:12) . An EMRI takes place when a stellar mass black hole or a neutron star is inspiralinginto a supermassive black hole while losing energy and angular momentum due to gravitationalradiation reaction. Such a system is emitting GWs that should be detectable far away from thesource in the mHz bandwidth.A GW signal from an EMRI can provide important information about the parametersof the system such as the masses of the objects, their spins etc. Actually EMRIs’ detectionwill give us the opportunity to map the spacetime around a supermassive black hole to highaccuracy. Since the parameter analysis of the detected GW signal depends on the accuracy ofthe waveform templates, it is important to model an EMRI with adequate precision and findtheoretical waveforms from EMRI systems for various parameters.The radiation reaction that acts on a moving particle can be split in two parts: a dissipativeand a conservative one. In this work, we focus on the dissipative part, which can be calculatedfrom the energy and angular momentum fluxes at the horizon of the black hole and at infinity.To obtain the leading term in the evolution of the GW phase, it is sufficient to consider onlytime averages of the dissipative part [Barack and Pound, 2019]. This approximation is calledadiabatic. In this case, the particle is slowly shifted from one orbit to another on time scalemuch larger than the orbital period.In this paper, we first review the properties of a spinning test particle moving in the Kerr a r X i v : . [ g r- q c ] J a n KOUPY AND LUKES-GERAKOPOULOS: GW TEMPLATES FROM EMRI spacetime. Then we summarize the Teukolsky formalism, which allows us to calculate the energyand angular momentum fluxes along with the waveforms. Using this formalism we calculatethe energy fluxes from circular equatorial orbits of spinning particles around Schwarzschild andKerr black holes. Subsequently, we use these fluxes to adiabatically evolve circular equatorialorbits and to find the effects of spin of the secondary object on the GW phase. Throughout thepaper, we use geometrical units G = c = 1, where G is the gravitational constant and c is thespeed of light, and the metric signature ( − +++). Spinning test particles in the Kerr geometry
The Kerr geometry, which describes a rotating black hole in vacuum, is represented inBoyer-Lindquist coordinates ( t, r, θ, ϕ ) by the metric [Misner et al., 2017]d s = − ∆Σ (cid:0) d t − a sin θ d ϕ (cid:1) + Σ∆ d r + Σd θ + sin θ Σ (cid:0)(cid:0) r + a (cid:1) d ϕ − a d t (cid:1) (1)where ∆ = r − M r + a , Σ = r + a cos θ . (2)This metric depends on two parameters: the mass of the black hole M and the Kerr parameter a . The internal angular momentum (spin) of a Kerr black hole is a M . At the radius r + = M + √ M − a , where ∆ = 0, the outer event horizon is located. In this paper, we are dealingonly with the region r > r + . For a = 0 the Kerr spacetime reduces to the Schwarzschild one.A compact test object in general relativity can be characterized just by its multipoles[Dixon, 1964]. For example, a rotating black hole or a neutron star moving in a Kerr backgroundcan be modeled using a pole-dipole approximation, where only the mass and the spin of thesecompact objects are taken into account, reducing them to a spinning test particle. The pole-dipole approximation holds as long as the size of the test body is smaller than the scale of thebackground curvature. The stress-energy tensor of a spinning test particle reads [Ehlers, 1979] T µν = 1 √− g (cid:32) v ( µ p ν ) v δ − ∇ α (cid:32) S α ( µ v ν ) v δ (cid:33)(cid:33) (3)where g is the determinant of the metric, v µ = d x µ d τ is the four-velocity, p µ is the four-momentum, S µν is the spin tensor, δ = δ ( r − r p ( t )) δ ( θ − θ p ( t )) δ ( ϕ − ϕ p ( t )) where r t ( t ), θ p ( t ) and ϕ p ( t )are the coordinates of the particle depending at the coordinate time t . The conservation of thestress-energy tensor (3) leads to the Mathisson-Papapetrou-Dixon (MPD) equations [Mathisson,1937; Papapetrou, 1951; Dixon, 1964]D p µ d τ = − R µνκλ v ν S κλ , (4)D S µν d τ = p µ v ν − p ν v µ , (5)where τ is the proper time and R µνκλ is the Riemann tensor.The centre of mass for an extended body in general relativity is not uniquely defined. Tofix the centre of mass for a spinning body, one has to specify the so called spin-supplementarycondition (SSC). In this work we use the Tulczyjew-Dixon SSC [Dixon, 1964], which reads S µν p µ = 0 . (6)and closes the MPD system. Actually, this SSC allows an explicit relation of the dependence ofthe four-velocity v µ on the four-momentum p µ [Ehlers and Rudolph, 1977]. KOUPY AND LUKES-GERAKOPOULOS: GW TEMPLATES FROM EMRI
The magnitude of the spin is defined as S = S µν S µν /
2, while the mass of the particle is µ = − p µ p µ . Both of these quantities are conserved under Tulczyjew-Dixon SSC. Instead ofthe measure of the spin S , its dimensionless counterpart σ = S/ ( µM ) is often used in EMRIstudies. For this dimensionless spin holds σ (cid:39) µ/M ≡ q , i.e. it is of the order of an EMRI massratio.There are two Killing vectors in the Kerr geometry ξ µ ( t ) = ∂x µ ∂t , ξ µ ( ϕ ) = ∂x µ ∂ϕ , (7)providing respectively two conserved quantities for the spinning particle E = − ξ µ ( t ) p µ + 12 ξ ( t ) µ ; ν S µν , (8) J z = ξ µ ( ϕ ) p µ − ξ ( ϕ ) µ ; ν S µν . (9)These quantities can be interpreted at infinity as the energy and the component of the totalangular momentum parallel to the rotational axis of the central black hole ( z -axis). Teukolsky equation
The mass ratio q of an EMRI lies between 10 − and 10 − . Thanks to this, the GWs fromsuch systems have relatively low amplitudes and perturbation theory can be employed. UsingNewman-Penrose formalism it is possible to find equations governing the perturbation of theWeyl tensor projected on some tetrad.Teukolsky [1973] found the master equation (Teukolsky equation, TE) (cid:32) (cid:0) r + a (cid:1) ∆ − a sin θ (cid:33) ∂ ψ∂t + 4 M ar ∆ ∂ ψ∂t∂ϕ + (cid:18) a ∆ − θ (cid:19) ∂ ψ∂ϕ − ∆ − s ∂∂r (cid:18) ∆ s +1 ∂ψ∂r (cid:19) − θ ∂∂θ (cid:18) sin θ ∂ψ∂θ (cid:19) − s (cid:18) a ( r − M )∆ + i cos θ sin θ (cid:19) ∂ψ∂ϕ − s (cid:32) M (cid:0) r − a (cid:1) ∆ − r − ia cos θ (cid:33) ∂ψ∂t + (cid:0) s cot θ − s (cid:1) ψ = 4 π Σ T , (10)which governs scalar, neutrino, electromagnetic and gravitational perturbations of the Kerrspacetime. s denotes the spin weight of the field and ψ is a projection of the field quantity on atetrad depended on s , while T is the source term. For GWs at infinity it is useful to calculatethe quantity Ψ = ρ ψ for s = −
2, where ρ = − / ( r − ia cos θ ). Then the source term consistsof derivatives of the stress-energy tensor projected on the tetrad.This equation is usually decomposed into azimuthal m -modes ψ ( t, r, θ, ϕ ) = ∞ (cid:88) m = −∞ ψ m ( t, r, θ ) e imϕ , ψ m ( t, r, θ ) = 12 π (cid:90) π ψ ( t, r, θ, ϕ ) e − imϕ d ϕ . (11)This replaces the derivatives d/d ϕ with im . Time domain
It is possible to numerically solve the (2+1)dimensional TE in the time domain. Wesolve TE equation in the so called horizon-penetrating hyperboloidal (HH) coordinates ( τ, ρ, θ ).Hypersurfaces of constant time in these coordinates are light-like at the horizon and null infinityand compactified in the radial direction, which allows us to deal with the boundary conditions.By defining the quantity φ = ∂ τ ψ m one can split the system into two first order in time and KOUPY AND LUKES-GERAKOPOULOS: GW TEMPLATES FROM EMRI second order in space differential equations, which then can be solved by using the methods oflines [Harms et al., 2014].The energy flux d E ∞ /d t and the angular momentum fluxes at infinity can be calculatedfrom the strain h = h + − ih × , where + and × are the polarizations of the GW. The secondderivative with respect to time of the strain is asymptotically proportional to the field quantity¨ h ∝ ψ at infinity. The energy flux d E H (cid:14) d t and the angular momentum fluxes can be calculatedat the horizon as well. Frequency domain
It is also possible to solve the TE (10) in the frequency domain by Fourier transformationin time, i.e. ψ ( t, r, θ, ϕ ) = (cid:88) l l (cid:88) m = − l (cid:90) ∞−∞ d ω s S aωlm ( θ ) s R lmω ( r ) e − iωt + imϕ . (12)By employing this transform separated ordinary differential equations can be derived for thespin weighted spheroidal harmonic function s S aωlm ( θ ) and the radial function s R lmω ( r ). Thisseparation is an advantage of the frequency domain method, since it highly reduces the compu-tational cost. On the other hand, only GWs from bound multiperiodic orbits without dissipationcan be calculated by a summation over discrete frequencies. Thus, different methods have tobe employed to calculate the GW fluxes from inspiralling orbits.The energy flux at infinity is given by [Drasco and Hughes, 2006]d E ∞ d t = (cid:88) l,m (cid:12)(cid:12) Z H lm (cid:12)(cid:12) πω m , (13)where the amplitudes Z H lm can be calculated as a convolution of the radial function R Up lmω , whichis regular at the horizon, and the source term derived from the stress-energy tensor (3). Forcircular equatorial orbits of spinning particles with spin parallel to the z -axis at radius r theseamplitudes read Z H lm = A R Up lmω ( r ) + A d R Up lmω d r ( r ) + A d R Up lmω d r ( r ) + A d R Up lmω d r ( r ) . (14)We have derived independently the exact expressions of these coefficients A i and cross checkedthem with those of Piovano et al. [2020]. This derivation is possible thanks to the fact that theorbital frequency Ω = d ϕ /d t is constant for circular equatorial orbits, and, hence, each m -modeconsists of only one frequency ω m = m Ω. Similar expressions can be derived for the flux at thehorizon and the angular momentum fluxes. We have checked that the frequency domain resultsagree for various orbits with the time domain approach results.We have calculated the total energy fluxes of spinning particles moving on circular equa-torial trajectories around a black hole with their spin parallel to the z -axis for several values ofthe frequency parameters y ≡ ( M Ω) / . These calculations have taken place on a black holebackground for a = 0, a = 0 . M and a = 0 . M , while the spin of the secondary σ rangedbetween − . . .
1. The dependence of the flux on the spin was fitted with apolynomial and the linear part was extracted to obtain F = d E ∞ d t + d E H d t = F + σ F + O (cid:0) σ (cid:1) , (15)where F denotes the total energy flux, F is a constant term corresponding to the flux froma non-spinning particle and F is the term linear in spin. Both terms are plotted in Fig. 1.The radial and angular functions and their derivatives were calculated using the Black HolePerturbation Toolkit (BHPT) [BHPT contributors, 2020]. KOUPY AND LUKES-GERAKOPOULOS: GW TEMPLATES FROM EMRI a a Ma M y ℱ ℱ N a a Ma M - - - y ℱ ℱ N Figure 1.
Dependence of total energy flux (15) normalized by the quadrupole formula F N =32 q y / y = ( M Ω) / . Dots indicate datacalculated using the equations (13) and (15), while solid lines show post-Newtonian (PN) resultscalculated using the BHPT [Nagar et al., 2019] (5.5PN for a = 0 and 2.5PN for a (cid:54) = 0). ThePN approximation fails for large y especially for cases a (cid:54) = 0 because the PN order is lower. Adiabatic inspiral
A geodesic orbit in Kerr can be characterized by its constants of motion, i.e. the energy E , the z -component of the angular momentum J z and the Carter constant Q [Schmidt, 2002].For a spinning particle, the Carter constant is in general missing, it can be only retrieved whenthe MPD system is linearized in spin [Witzany, 2019]. Hence, when the particle is orbiting onthe equatorial plane and its spin is parallel to the orbital angular momentum and z -axis, onecan use energy E and angular momentum J z to characterize the orbit. Actually, for circularequatorial orbits only one parameter such as energy, radius or orbital frequency is needed.Due to the conservation of the energy and the angular momentum, the change of theseparameters must be opposite to the energy and angular momentum fluxes at the horizon andat infinity. The rate of change of the orbital frequency isdΩd t = − F (Ω) d E dΩ . (16)We have derived the dependence of energy E on frequency parameter y linear in spin σ for Kerrblack hole: E ( y ) = 1 − xyx / (cid:112) − x − xy − σ y / (cid:0) x − xy (cid:1) x / (cid:112) − x − xy , (17)where x = √ − a Ω. This result agrees with the equation (82) of Harms et al. [2016] for a = 0and to the first order in spin with the equation (39a) of Hinderer et al. [2013].Because the spin σ scales as the mass ratio q , the phase of the GW can be written as thefollowing expansion [Piovano et al., 2020]Φ( t ) = 1 q Φ ( t ) + σq Φ ( t ) + O (cid:18) σ q (cid:19) , (18)where the first term is of adiabatic order and the second term is correction caused by the spinof the secondary object. The frequency of the m -modes is m Ω and the dominant mode is m = 2. This implies that the GW phase is Φ( t ) = 2 ϕ ( t ). Suppose that the azimuth angle is φ ( t ) = ϕ ( t ) + σϕ ( t ) + O (cid:0) σ (cid:1) , then we can solve the system of equations d ϕ /d t = Ω( t ) and(16) perturbatively to find that the correction to the phase is Φ ( t ) = 2 qϕ ( t ). To obtain anadiabatic inspiral the fluxes F and F from Fig. 1 were interpolated with a 3rd order Lagrange KOUPY AND LUKES-GERAKOPOULOS: GW TEMPLATES FROM EMRI interpolation. The obtained results for Φ are shown in Fig. 2 for different a ; they are inagreement with those provided by Piovano et al. [2020], that have followed a different approachto obtain them. a a Ma M q tM - Φ Figure 2.
Corrections to the GW phase caused by the spin of the secondary object. The initialfrequency Ω is the same as the Ω for r = 10 . a . This plot is identical to the plot inFigure 3 of Piovano et al. [2020]. Conclusions
Our main results are the following: • We have numerically calculated the energy fluxes from circular equatorial orbits of spin-ning particles with spin parallel to the z -axis. • We have used the above results to independently verify the results provided by Piovanoet al. [2020] by perturbatively solving the equations for the azimuthal angle ϕ and theorbital frequency Ω to find correction to the GW phase caused by the spin of the secondaryobject. These results are shown in Fig. 2 and agree with those of Piovano et al. [2020].In a future work, these fluxes will be compared with fluxes computed using the time do-main code (Teukode) for an inspiralling orbit to check whether the adiabatic approximation isjustified. Acknowledgments.
Computational resources were supplied by the project ”e-InfrastrukturaCZ” (e-INFRA LM2018140) provided within the program Projects of Large Research, Developmentand Innovations Infrastructures and this work makes use of the Black Hole Perturbation Toolkit. Theauthors would also like to acknowledge networking support by the COST Action CA16104.
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