Gravitational wave echoes induced by a point mass plunging to a black hole
aa r X i v : . [ g r- q c ] O c t Prog. Theor. Exp. Phys. , 00000 (22 pages)DOI: 10.1093 / ptep/0000000000 Gravitational wave echoes induced by a pointmass plunging to a black hole
Norichika Sago , , ∗ and Takahiro Tanaka , , Department of Physics, Kyoto University, Kyoto 606-8502, Japan Advanced Mathematical Institute, Osaka City University, Osaka 558-8585, Japan Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japan ∗ E-mail: [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Recently, the possibility of detecting gravitational wave echoes in the data stream sub-sequent to the binary black hole mergers observed by LIGO was suggested. Motivatedby this suggestion, we presented templates of echoes based on black hole perturbationsin our previous work. There, we assumed that the incident waves resulting in echoes aresimilar to the ones that directly escape to the asymptotic infinity. In this work, to extractmore reliable information on the waveform of echoes without using the naive assump-tion on the incident waves, we investigate gravitational waves induced by a point massplunging into a Kerr black hole. We solve the linear perturbation equation sourced bythe plunging mass under the purely outgoing boundary condition at infinity and a hypo-thetical reflection boundary condition near the horizon. We find that the low frequencycomponent below the threshold of the super-radiant instability is highly suppressed,which is consistent with the incident waveform assumed in the previous analysis. Wealso find that the high frequency mode excitation is significantly larger than the oneused in the previous analysis, if we adopt the perfectly reflective boundary conditionindependently of the frequency. When we use a simple template in which the same wave-form as the direct emissions to infinity is repeated with the decreasing amplitude, thecorrelation between the expected signal and the template turns out to decrease veryrapidly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index E01, E02, E31, E38
1. Introduction
Since the direct detection of gravitational waves (GWs) was reported by LIGO and Virgocollaboration [1, 2], the reported number of black hole merger events is rapidly increasing[3, 4]. These data allow us to carefully examine the nature of black holes.The possible presence of gravitational wave echoes is one of the intriguing topics stimulatedby GW observations. Abedi et al. [5] analyzed the data succeeding to the BBH merger eventsobserved by LIGO during O1 observation run to search for signals of GW echoes, and claimedthat they found a tentative evidence for the echoes at false detection probability of 0.011.Motivated by this work, several groups have done the follow-up analyses [6–12].GW echoes after a compact binary coalescence (CBC) can be a probe of the exotic natureof black holes, since no GW echo is expected if the resulting object after coalescence is justan ordinary classical black hole. There is a possibility that GW echoes are induced, if theresulting object after merger is an exotic compact object (ECO) without horizon [13, 14], c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. .g. gravastar [15], wormhole [16, 17], firewall [18], and so on (Also see [19], which gives acomprehensive review on ECOs and their tests). Once GW echoes are observed, it would bepossible to extract the information about the ECO from the waveforms of the echoes.Several works on the construction of the waveform of the GW echoes have been donebased on the black hole perturbation theory [20–26]. In the case of an ordinary black holespacetime, one can calculate GWs by solving the perturbation equation ( e.g. the Teukolskyequation in Kerr case) under the pure-outgoing condition at infinity and the pure-ingoingone on the horizon. Even in the case of ECOs, we can consider the possibility that only asmall neighborhood of the horizon might be modified, i.e. , the same spacetime as a black holemight be realized outside a hypothetical near-horizon boundary located slightly outside thehorizon. Then, the perturbations in ECO spacetime can be expressed by the same equationas in general relativity, except for the modified boundary condition. The reflective boundarycondition at the near-horizon boundary, instead of the pure-ingoing one, leads a series ofGW echoes. As a result of this modified boundary condition, extra GWs due to the reflectionby the inner boundary is added to GWs observed at infinity, which can be calculated bymultiplying the transfer function to the GWs that should have fallen into the horizon in theordinary setup.The transfer function consists of the reflectance and the transmittance determined by theeffective potential of the perturbation equation, the position and reflectance of the innerboundary. The behavior of the transfer function has been studied well in previous research[20–26] by solving the scattering problem in one-demension. By contrast, the waveform ofthe incident GWs, which is nothing but the ingoing waves absorbed by the black hole, hasnot been investigated extensively, especially in the Kerr case. In this paper we study thisissue.In Sec. 2 after recapitulating the basic equations for the black hole perturbation theorybased on the Sasaki-Nakamura equation, we make it clear how to compute the ingoingwaveform. The variable that we use in the Sasaki-Nakamura equation is the one obtainedby a transformation from the Teukolsky variable ψ . For this Teukolsky variable ψ , theamplitude of the ingoing waves are suppressed in the asymptotic regions at infinity andnear the horizon, compared with the outgoing ones. Because of this, we cannot calculate theenergy of the waves falling into the horizon directly from the asymptotic waveform of theSasaki-Nakamura variable computed by using the standard Green function method. Herewe give the explicit formula for the energy spectrum ingoing to the horizon. Furthermore,based on this flux formula, we develop a method to impose a reflective boundary near thehorizon, which applies even in the presence of the source term. In Sec. 3 we present theresults of numerical calculation of echo waveform with the reflective boundary condition.We also discuss the detectability of the resulting waveform by using the previously proposedways to generate the echo templates. Section 4 is dedicated for conclusion. . Basic equations The Kerr metric in the Boyer-Lindquist coordinates, ( t, r, θ, ϕ ), is given by g µν dx µ dx ν = − (cid:18) − M r Σ (cid:19) dt − M ar sin θ Σ dt dϕ + Σ∆ dr +Σ dθ + (cid:18) r + a + 2 M a r Σ sin θ (cid:19) sin θ dϕ , (1)where Σ = r + a cos θ , ∆ = r − M r + a , M and aM are the mass and the angularmomentum of the black hole, respectively.In this work, we consider a point mass which is initially at rest at infinity and falls into ablack hole on the equatorial plane ( θ = π/ r dtdτ = − a ( a − L ) + r + a ∆( r ) P ( r ) ,r drdτ = − p R ( r ) ,r dϕdτ = − ( a − L ) + a ∆( r ) P ( r ) , (2)where L is the angular momentum of the point mass, P ( r ) = r + a − aL and R ( r ) =2 M r − L r + 2 M r ( a − L ) . (These equations correspond to the geodesic equations with E = 1 and C = 0, where E and C are the specific energy and Carter parameter of the pointmass.) The Sasaki-Nakamura equation is given by [30] (cid:18) d dr ∗ − F ddr ∗ − U (cid:19) X lmω ( r ∗ ) = S lmω , (3)where r ∗ is defined by dr ∗ dr = r + a ∆ . The functions of F and U in Eq. (3) are given by F = ∆ r + a γ ′ γ , (4) U = ∆ U ( r + a ) + G + dGdr ∗ − ∆ Gr + a γ ′ γ , (5) here a prime means a differentiation with respect to r , and α = 3 iK ′ + λ + 6∆ r − iK ∆ β,β = ∆ (cid:18) − iK + ∆ ′ − r (cid:19) ,γ = α (cid:18) α + β ′ ∆ (cid:19) − β ∆ (cid:18) α ′ + β ∆ V (cid:19) ,G = − ∆ ′ r + a + r ∆( r + a ) ,U = V + ∆ β (cid:18)(cid:18) α + β ′ ∆ (cid:19) ′ − γ ′ γ (cid:18) α + β ′ ∆ (cid:19)(cid:19) ,V = − K + 4 i ( r − M ) K ∆ + 8 iωr + λ,K = ( r + a ) ω − am, with the eigenvalue of the spheroidal harmonics, λ . (Here, we choose the free functions inthe equations given in Ref. [31] as f = h = 1 and g = ( r + a ) /r .)The source term in the right hand side of Eq.(3) for a plunge orbit in Eq.(2) is expressedby S lmω = γ ∆ W T ( r + a ) / r exp (cid:18) − i Z r K ∆ dr (cid:19) , (6)with W T = W nn + W n ¯ m + W ¯ m ¯ m , (7)1 µ W nn = f exp( iχ ) + Z ∞ r dr f exp( iχ ) + Z ∞ r dr Z ∞ r dr f exp( iχ ) , (8)1 µ W n ¯ m = g exp( iχ ) + Z ∞ r dr g exp( iχ ) , (9)1 µ W ¯ m ¯ m = h exp( iχ ) + Z ∞ r dr h exp( iχ ) + Z ∞ r dr Z ∞ r dr h exp( iχ ) , (10) here f = − ω r √R ( r + a ) S c , (11) f = f S c (cid:20) (S + ( aω − m )S ) iar + S c (cid:26) a − r ) r ( r + a ) + R ′ R + iη (cid:27)(cid:21) , (12) f = iω r √R ( r + a )∆ (cid:18) − P √R (cid:19) " { S + ( aω − m )S } iar +S c ( a r ( r + a ) + 2 rr + ( L − a ) − ( P + √R ) ′ P + √R + iη ) , (13) η = ( aω − m )( a − L ) √R − am ∆ (cid:18) − P √R (cid:19) , (14) g = − a − Lω (S + ( aω − m )S ) r r + a , (15) g = g (cid:20) a r ( r + a ) + iη (cid:21) , (16) h = − r h , h = − rh , h = S ( a − L ) √R , (17)S = − S aωlm ( π/ , S = ddθ − S aωlm ( π/ , (18)S c = (cid:18) aω − m − iar (cid:19) [S + ( aω − m )S ] − λ , (19) χ = ωt − mϕ + Z r K ∆ dr . (20)(These source terms are given in Ref. [33] or Ref. [34].) − S aωlm ( θ ) is the spin-weightedspheroidal harmonics with the spin weight −
2. Noticing that γ = O ( r ) for r → ∞ and γ = O (( r − r + ) ) for r → r + , and R = O ( r ) for r → ∞ in the present setup, we find theasymptotic fall-off behaviors of the source term are given by S lmω = O (cid:0) r − / (cid:1) for r → ∞ ,O (( r − r + )) for r → r + . (21)which correspond to W T = O ( r / ) and W T = O (1), respectively. The Sasaki-Nakamuravariable X lmω ( r ∗ ) can be converted to the Teukolsky variable R lmω ( r ∗ ) by the formula R lmω ( r ∗ ) = Λ[ X lmω ( r ∗ )] + ( r + a ) / γ S lmω , (22)where the differential operator Λ is defined byΛ[ X ( r ∗ )] ≡ γ (cid:20) ( α ∆ + β ′ ) √ r + a X ( r ∗ ) − β ∆ ddr (cid:18) ∆ √ r + a X ( r ∗ ) (cid:19)(cid:21) . (23)Here, we should stress that the source term of the Sasaki-Nakamura equation is obtainedby radially integrating the source term of the Teukolsky equation twice. Therefore, it shouldcontain two arbitrary integration constants. The different choice of the source term S lmω eads to a different solution of X lmω , but the resulting Teukolsky variable should be invariant.In fact, R lmω derived by using Eq. (22) is invariant under the simultaneous transformations: X lmω → ˜ X lmω = X lmω + δX lmω , S lmω → ˜ S lmω = S lmω + δS lmω , (24)where δX lmω = − √ r + a r exp (cid:18) − i Z K ∆ dr (cid:19) × (cid:2) (2 r − a + λr − irK + 3 ir K ′ ) c +(6 M r − a + λr − irK + 3 ir K ′ ) c r (cid:3) , (25) δS lmω = γ ∆( c + c r )( r + a ) / r exp (cid:18) − i Z r K ∆ dr (cid:19) . (26)In other words, δX lmω and δS lmω satisfy the following relation:0 = Λ[ δX lmω ] + ( r + a ) / γ δS lmω . (27)One can also verify that the solution of the Sasaki-Nakamura equation sourced by δS lmω isgiven by δX lmω .If we specify c and c as c = − W T ( r + ) + r + W ′ T ( r + ) , c = − W ′ T ( r + ) , , (28)we find that ˜ S lmω = ( O ( r − ) for r → ∞ ,O (( r − r + ) ) for r → r + , (29)which correspond to W T = O ( r ) for r → ∞ and W T = O (( r − r + ) ) for r → r + , respectively.Compared with the asymptotic behavior of the original source term (21), the fall-off of thenew source term ˜ S lmω is much faster near the horizon, while, as an expence to pay, thefall-off at infinity is slower. Let X ∞ , + and X ∞ , − stand for the homogeneous solutions of the Sasaki-Nakamuraequation (3) that satisfy the purely outgoing (+) and purely ingoing ( − ) conditions at infin-ity, respectively. We omit the index lmω for simplicity, unless it is necessary. In the samemanner, let X H, ± denote the purely outgoing and purely ingoing homogeneous solutionsnear the horizon. The asymptotic forms are given by X ∞ , ± = e ± iωr ∗ , for r → ∞ X H, ± = e ± ikr ∗ for r → r + . (30)In a similar manner, we define the homogeneous solutions of the radial Teukolsky equation, R ∞ , ± and R H, ± , which satisfy R ∞ , ± = r ± e ± iωr ∗ for r → ∞ ,R H, ± = ∆ ∓ e ± ikr ∗ for r → r + . (31)Substituting X ∞ , + into Eq. (22), we obtain the relationΛ[ X ∞ , + ] = Γ ∞ , + R ∞ , + , Γ ∞ , + ≡ ω iωM − λ ( λ + 2) + 12 aω ( aω − m ) . (32) hen we apply this formula to an inhomogeneous solution, the source term in Eq. (22)does not contribute independently of whether we use the original source term S lmω or themodified one ˜ S lmω , In the same way, the relation between X H, + and R H, + can be obtainedas Λ[ X H, + ] = Γ H, + R H, + , Γ H, + ≡ − i √ M r + k ( r + − M − ikM r + ) γ ( r + ) , (33)where γ ( r + ) = − ω r + 8(9 kM + iλ ) ωr + − k M − ikM ( λ + 3)+12 iωM + λ ( λ + 2) + 48 ikM r + . (34)To obtain the relation between X H, − and R H, − through Eq. (22), the higher order correc-tions of X H, − with respect to ( r − r + ) are required, because the leading and the sub-leadingorder terms vanish in the expression for R H, − . The asymptotic form of the purely ingoingsolution is given up to O (( r − r + ) ) by X H, − = (cid:2) a ( r − r + ) + a ( r − r + ) + O (( r − r + ) ) (cid:3) e − ikr ∗ , (35)where a and a are determined so that Eq. (3) without the source term is satisfied at eachorder. Thus determined values of a and a are a = ( r + − M )( r + ( λ − M r + + 8 M ) − kM mar M r + ( r + − M − ikM r + )( r + − M ) , (36) a = 116 M r ( r + − M − ikM r + )( r + − M − ikM r + )( r + − M ) × n m a r (cid:2) (4 M k + 2 ikM − r − iM ( M k + 1) r + − M (cid:3) +2 maM r + (cid:2) − kr − ir (2 − ikM λ + 13 ikM )+ iM r (12 − M k − ikM λ + 35 ikM ) − i (4 + 5 ikM ) M r + + 4 iM (cid:3) − r (1 − ikM ) − M r (6 − ikM λ + 47 ikM − λ )+ M r (156 − ikM λ + λ + 170 ikM − λ ) − M r (244 − ikM λ − λ + 25 ikM − λ )+ M r (678 + λ − ikM − λ ) − M r + (111 − ikM − λ ) + 112 M o . (37)Substituting Eq. (35) into Eq. (22) with Eqs. (36) and (37), we obtainΛ[ X H, − ] = Γ H, − R H, − , Γ H, − ≡ √ M r + ( r + − M − ikM r + )( r + − M − ikM r + ) . (38)Here, one remark is in order when we apply this formula to an inhomogeneous solution. Ingeneral we cannot neglect the source term contribution in Eq. (22). Neglecting the sourceterm contribution can be justified only when we use the scheme in which the source term issuppressed near the horizon. Therefore, this relation can directly apply only when we considerthe asymptotic behaviors of ˜ X lmω , the solution obtained by considering the modified sourceterm ˜ S lmω . .4. Inhomogeneous solution with the ordinary boundary conditions Introducing a new variable ξ , defined by dξ = γdr ∗ , the Sasaki-Nakamura equation can berewritten as (cid:18) d dξ − Uγ (cid:19) X = Sγ . (39)We construct the retarded Green function for this equation by using the homogeneoussolutions that satisfy appropriate boundary conditions, X H, − and X ∞ , + , as G ( ξ, ξ ′ ) = 1 W G (cid:0) X H, − ( ξ ) X ∞ , + ( ξ ′ ) θ ( ξ ′ − ξ ) + X ∞ , + ( ξ ) X H, − ( ξ ′ ) θ ( ξ − ξ ′ ) (cid:1) , (40)where W G is the Wronskian of X ∞ , + and X H, − , defined by W G = W [ X ∞ , + , X H, − ] = X H, − ( ξ ) ddξ X ∞ , + ( ξ ) − X ∞ , + ( ξ ) ddξ X H, − ( ξ ) , which is guaranteed to be constant in ξ . Using the Green function, we obtain the retardedsolution X S ( r ∗ ) = X ∞ , + ( r ∗ ) W G Z r ∗ −∞ dr ′∗ X H, − ( r ′∗ ) S ( r ′∗ ) γ + X H, − ( r ∗ ) W G Z ∞ r ∗ dr ′∗ X ∞ , + ( r ′∗ ) S ( r ′∗ ) γ . (41)This solution X S satisfies the boundary condition that there is no incoming wave from theinfinity, because X S does not contain such a component owing to the retarded nature thatthe above Green function possesses by construction, and the source term is suppressed inthe limit r → ∞ . The boundary condition at r → r + is also verified because of the retardednature of the Green function and the fact that the source term does not contribute to therelation between X and R for the outgoing waves.The asymptotic form of Eq. (41) at infinity is given by X S = A ∞ X ∞ , + + O ( r − / ) , A ∞ ≡ W G Z ∞−∞ dr ′∗ X H, − ( r ′∗ ) S ( r ′∗ ) γ , for r → + ∞ . (42)Using the relation (32), we translate Eq. (42) into the Teukolsky variable at infinity as R = Λ[ X S ] + ( r + a ) / γ S = Z ∞ R ∞ , + + O ( r / ) , Z ∞ = Γ ∞ , + A ∞ . (43)To convert Eq. (41) into the Teukolsky variable R near the horizon, we should make useof the invariance of R under the transformation of Eq. (24) with (28). Consider the solutionfor the transformed source, ˜ S , through the Green’s function as: X ˜ S ( r ∗ ) = X ∞ , + ( r ∗ ) W G Z r ∗ −∞ dr ′∗ X H, − ( r ′∗ ) ˜ S ( r ′∗ ) γ + X H, − ( r ∗ ) W G Z ∞ r ∗ dr ′∗ X ∞ , + ( r ′∗ ) ˜ S ( r ′∗ ) γ . (44)Now we consider the difference between X S and X ˜ S : X ˜ S − X S = X H, − W G Z ∞ ξ dξ ′ X ∞ , + ( ξ ′ ) δS ( ξ ′ ) + X ∞ , + W G Z ξ −∞ dξ ′ X H, − ( ξ ′ ) δS ( ξ ′ )= X H, − W G W [ X ∞ , + , δX ] ∞ ξ + X ∞ , + W G W [ X H, − , δX ] ξ −∞ = δX + W [ X ∞ , + , δX ]( ∞ ) W G X H, − − W [ X H, − , δX ]( −∞ ) W G X ∞ , + , (45) here we use the following relation for an arbitrary homogeneous solution, X : Z ξ ξ dξX ( ξ ) δS ( ξ ) = Z ξ ξ dξX ( ξ ) (cid:18) d dξ − Uγ (cid:19) δX ( ξ )= (cid:20) X ( ξ ) ddξ δX − (cid:18) ddξ X ( ξ ) (cid:19) δX (cid:21) ξ ξ + Z ξ ξ dξ (cid:20)(cid:18) d dξ − Uγ (cid:19) X ( ξ ) (cid:21) δX ( ξ )= [ W [ X ( ξ ) , δX ( ξ )]] ξ ξ . (46)From Eqs. (22) and (45), we obtain R = Λ[ X S ] + ( r + a ) / γ S = Λ[ X ˜ S − δX − w ∞ , X H, − + w H, X ∞ , + ] + ( r + a ) / γ S = Λ[ X ˜ S − w ∞ , X H, − + w H, X ∞ , + ] + ( r + a ) / γ ˜ S , (47)where w ∞ , = W [ X ∞ , + , δX ]( ∞ ) W G , w H, = W [ X H, − , δX ]( −∞ ) W G . (48)Since both X H, − and δX satisfy the purely ingoing boundary condition at r → r + , we find w H, = 0. From this fact and ˜ S = O (( r − r + ) ), we find the asymptotic form of R near thehorizon is given by R = Z H R H, − + O (( r − r + ) ) , (49)with Z H = Γ H, − ( ˜ A H − w ∞ , ) , ˜ A H = 1 W G Z ∞−∞ dr ′∗ X ∞ , + ( r ′∗ ) ˜ S ( r ′∗ ) γ . (50) We consider the case in which the ingoing wave to the black hole is reflected by a hypotheticalboundary near the horizon. To describe such a situation, we first consider the solutionobtained by replacing the pure-ingoing solution in the Green function, Eq. (40) with the onethat contains the reflection waves, X H, − → X H, ref ≡ X H, − + R b Φ X H, + , = (1 − R b Φˆ r ) X H, − + R b Φˆ tX ∞ , + , (51)where R b is the reflectance on the boundary defined by the square root of the ratio betweenthe energies of the ingoing and outgoing waves, and Φ is the factor that takes care of thenon-trivial relation between the amplitude of the Sasaki-Nakamura variable and the energyspectrum. The explicit formulae for Φ will be provided later. In the second equality, we ntroduce ˆ r and ˆ t that satisfy the relation X H, + + ˆ rX H, − = ˆ tX ∞ , + , which are the apparentreflection and transmission coefficients of the scattering problem for an incident outgoingwave from the horizon side reflected by the angular momentum barrier. Using the Greenfunction with the above replacement (51), we obtain a solutionˆ X S ( r ∗ ) = X ∞ , + ( r ∗ )ˆ W G Z r ∗ r ∗ b dr ′∗ X H, ref ( r ′∗ ) S ( r ′∗ ) γ + X H, ref ( r ∗ )ˆ W G Z ∞ r ∗ dr ′∗ X ∞ , + ( r ′∗ ) S ( r ′∗ ) γ , (52)where r ∗ b is the value of r ∗ on the reflective boundary surface, and ˆ W G is the Wronskianbetween X ∞ , + and X H, ref ,ˆ W G = W [ X ∞ , + ( r ∗ ) , X H, ref ( r ′∗ )] = (1 − R b Φˆ r ) W G . (53)The coefficients w ∞ , and w H, = 0 in the right hand side of Eq.(47) are now replaced withˆ w ∞ , = W [ X ∞ , + , δX ]( ∞ )ˆ W G = w ∞ , − R b Φˆ r , ˆ w ref , = W [ X H, ref , δX ]( r ∗ b )ˆ W G = R b Φ1 − R b Φˆ r W [ X H, + , δX ]( r ∗ b ) W G . (54)In contrast to the original case, ˆ w ref , does not vanish because of the presence of the outgoingcomponent in X H, ref . As a result, we obtain R = Λ[ ˆ X ˜ S − ˆ w ∞ , X H, ref + ˆ w ref , X ∞ , + ] , (55)as an expression for the Teukolsky variable valid near the horizon. The existence of theterm with X ∞ , + , which contains the outgoing component near the horizon, in Eq.(55) isinconsistent with the boundary condition of R that we impose near the horizon. To makethe requested retarded boundary conditions satisfied, we need to add a homogeneous solutionto the solution (52) as ˆ X mod ( r ∗ ) = ˆ X S ( r ∗ ) − ˆ w ref , X ∞ , + ( r ∗ ) . (56)The additional term proportional to X ∞ , + ( r ∗ ) does not disturb the purely outgoingboundary condition at infinity. This modified solution behaves asymptotically at infinitylike ˆ X mod = ( ˆ A ∞ − ˆ w ref , ) X ∞ , + + O ( r − / ) , for r → + ∞ , (57)with ˆ A ∞ ≡ W G Z ∞ r ∗ b dr ′∗ X H, ref ( r ′∗ ) S ( r ′∗ ) γ ≈ W G Z ∞−∞ dr ′∗ X H, ref ( r ′∗ ) S ( r ′∗ ) γ , (58)where in the last approximate equality we neglect the error due to the change of the integra-tion range. The asymptotic amplitude ˆ A ∞ in the current problem can be expressed in termsof the amplitudes A ∞ and ˜ A H evaluated in the original case with the ordinary boundary onditions asˆ A ∞ = 1(1 − R b Φˆ r ) W G Z ∞−∞ dr ′∗ (1 − R b Φˆ r ) X H, − ( r ′∗ ) + R b Φˆ tX ∞ , + ( r ′∗ ) γ S ( r ′∗ )= A ∞ + R b Φˆ t (1 − R b Φˆ r ) W G Z ∞−∞ dr ′∗ X ∞ , + ( r ′∗ ) n ˜ S ( r ′∗ ) − δS ( r ′∗ ) o γ = A ∞ + R b Φˆ t − R b Φˆ r (cid:20) ˜ A H − W [ X ∞ , + , δX ]( ∞ ) W G + W [ X ∞ , + , δX ]( −∞ ) W G (cid:21) = A ∞ + R b Φˆ t − R b Φˆ r (cid:20) ˜ A H − w ∞ , + ˆ r ˆ t w H, + W [ X H, + , δX ]( −∞ )ˆ tW G (cid:21) = A ∞ + R b Φˆ t − R b Φˆ r (cid:16) ˜ A H − w ∞ , (cid:17) + ˆ w ref , . (59)Using the relation of Eq. (32), we translate Eq. (57) into the Teukolsky variable in the limit r → ∞ as ˆ R = Λ[( ˆ A ∞ − ˆ w ref , ) X ∞ , + + O ( r − / )] + ( r + a ) / γ S X = ˆ Z ∞ R ∞ , + + O ( r / ) , (60)with ˆ Z ∞ = Z ∞ + K Z H , K = R b Φˆ t − R b Φˆ r Γ ∞ , + Γ H, − . (61)The relation (61) is the gravitational wave counterpart of Eq. (2.25) in Ref. [21], and K shown here is the transfer function. Here, we determine Φ in the expression for X H, ref given by (51). The corresponding Teukolskyradial function is derived as R H, ref = Λ h X H, ref i = Γ H, − R H, − + Γ H, + R b Φ R H, + . (62)Here, we quote the formulae for the energy spectra of the ingoing and outgoing waves acrossthe boundary surface given by [20, 29] (cid:18) dEdω (cid:19) H, − = µ ǫ H, − πω (cid:12)(cid:12) Γ H, − (cid:12)(cid:12) , (cid:18) dEdω (cid:19) H, + = µ ǫ H, + πω (cid:12)(cid:12) Γ H, + R b Φ (cid:12)(cid:12) , (63)with ǫ H, − = 256(2 M r + ) ( k + 4 ǫ )( k + 16 ǫ ) kω | C SC | , (64) ǫ H, + = ω k (2 M r + ) ( k + 4 ǫ ) , (65) ǫ = √ M − a M r + , (66) | C SC | = (cid:2) ( λ + 2) + 4 aωm − a ω (cid:3) (cid:2) λ + 36 aωm − a ω (cid:3) +48 aω (2 λ + 3)(2 aω − m ) + 144 ω ( M − a ) . (67) ince the reflectance on the boundary R b should be defined to satisfy | R b | = (cid:18) dEdω (cid:19) H, + (cid:30) (cid:18) dEdω (cid:19) H, − = (cid:12)(cid:12)(cid:12)(cid:12) ǫ H, + Γ H, + R b Φ ǫ H, − Γ H, − (cid:12)(cid:12)(cid:12)(cid:12) , (68)we obtain Φ = ǫ H , − Γ H , − ǫ H , + Γ H , + e − ikr ∗ b . (69)For later use, we also introduce the phase shift on the boundary given by∆ φ ≡ arg(R b ) = arg (cid:18) Γ H, + A ref e ikr ∗ b Γ H, − e − ikr ∗ b (cid:19) . (70)Since R b depends on the unknown property of the inner boundary, it cannot be specifieduniquely. Later, we discuss a few simple models of R b . In Sec. 2.5, we introduce ˆ r and ˆ t that satisfy X H, + + ˆ rX H, − = ˆ tX ∞ , + . These coefficients aredetermined by the property of the scattering due to the angular momentum barrier. By usingthe current notation, the reflectance and the phase shift given in Ref. [20] are representedby R = (cid:12)(cid:12)(cid:12)(cid:12) ǫ H, − Γ H, − ǫ H, + Γ H, + (cid:12)(cid:12)(cid:12)(cid:12) | ˆ r | , φ − ( f ) = arg (cid:18) ˆ r Γ H, − Γ H, + (cid:19) . (71)ˆ r and ˆ t can be calculated from the Wronskians between the homogeneous solutions asˆ r = − W [ X ∞ , + , X H, + ] W G , ˆ t = W [ X H, + , X H, − ] W G . (72)
3. Results
Based on the formulation shown in the previous section, we compute the gravitational wavesproduced by a point particle which is initially at rest at infinity and falls on the equatorialplane to a black hole. We take M = 1 for all computation in this paper. To check our numerical code, we first compute the energy of GWs radiated to infinity byconsidering an infalling particle with the ordinary boundary conditions. In Fig.1, we showthe energy spectra of ( l, m ) = (2 , , (2 , , (2 , , (3 , , (4 ,
4) in the case of a = 0 . M and L = 2 . M . This plot is consistent with that in Fig.3(a) of [33], except for the difference of thefactor 2, which comes from the difference in the definition of the energy spectrum ( dE/dω ):the spectrum in [33] corresponds to the one-side spectrum density, while ours is the two-sideone.Next we compute the energies both radiated to the infinity and absorbed by the blackhole with ordinary boundary conditions for several sets of the parameters ( a, L ). In Fig.2,we show both the energy spectra of the ( l, m ) = (2 ,
2) mode for a = { . , . , . , . } M and L = { . , . , . } L + c , where L ± c is the critical value of the angular momentum L ± c = ± M (1 + p ∓ a/M ) . In the above equation, the upper sign corresponds to corotating orbits and the lower tocounterrotating ones. Our focus is on plunge orbits, and hence we only discuss L that satisfies L − c < L < L + c . -4 -3 -2 -1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 a =0.85, L =2.6 a =0.85, L =2.6 d E / d ω M ω ( l , m )=(2, 2)( l , m )=(2, 1)( l , m )=(2, 0)( l , m )=(3, 3)( l , m )=(4, 4) Fig. 1
Energy spectrum radiated to infinity in the case of a corotating orbit with a =0 . M and L = 2 . M . -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 a =0.1 a =0.1 d E / d ω M ω L =0.9 L c+ (infinity) L =0.5 L c+ (infinity) L =0.1 L c+ (infinity) L =0.9 L c+ (horizon) L =0.5 L c+ (horizon) L =0.1 L c+ (horizon) -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 a =0.3 a =0.3 d E / d ω M ω L =0.9 L c+ (infinity) L =0.5 L c+ (infinity) L =0.1 L c+ (infinity) L =0.9 L c+ (horizon) L =0.5 L c+ (horizon) L =0.1 L c+ (horizon) -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 a =0.5 a =0.5 d E / d ω M ω L =0.9 L c+ (infinity) L =0.5 L c+ (infinity) L =0.1 L c+ (infinity) L =0.9 L c+ (horizon) L =0.5 L c+ (horizon) L =0.1 L c+ (horizon) -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 a =0.7 a =0.7 a =0.7 a =0.7 a =0.7 a =0.7 a =0.7 a =0.7 d E / d ω M ω L =0.9 L c+ (infinity) L =0.5 L c+ (infinity) L =0.1 L c+ (infinity) L =0.9 L c+ (horizon) L =0.5 L c+ (horizon) L =0.1 L c+ (horizon) Fig. 2
Energy spectra of GWs radiated to infinity and to the horizon of ( l, m ) = (2 , a, L ). We choose four values of the Kerr parameter, a = 0 . a = 0 . a = 0 . a = 0 . L = 0 . L + c (red), L = 0 . L + c (green), and L = 0 . L + c (blue) from top to bottom.The energy spectra of GWs absorbed by the horizon is largely suppressed at low frequen-cies. This result turns out to be consistent with the feature of the waveform model proposedin our previous work [11, 20]. However, the spectra emitted to infinity and to the horizon are ot similar at all. The suppression at low frequencies suggested in Ref.[20] is the one causedby the small transmission coefficient due to the angular momentum barrier, while in thepresent model the original amplitude of GWs to be reflected by the hypothetical boundarynear the horizon is already suppressed at low frequencies. This lack of similarity means thatit cannot be an optimal choice to use the waveform of GWs directly emitted to infinity asthe seed to generate the echo template.We can also find that the energy spectrum to the horizon is dominated by the higherfrequency band than the quasi-normal mode (QNM) frequency. Here, we consider GWsexcited by an in-falling point mass. If we replace the source with a finite size body, theremight arise some suppression at high frequencies for ingoing waves falling into the blackhole. To suppress the influence of the high frequency modes, which might be an artifact ofusing the point particle model, it might be appropriate to reduce the amplitude of the highfrequency component contained in the waves reflected by the hypothetical boundary nearthe horizon.There is also another motivation to introduce a cutoff to the high frequency modes, relatedto the expected property of the hypothetical reflective boundary near the horizon. There aresome proposals suggesting that only low frequency gravitational waves are reflected back bythe boundary [35–37]. In Ref. [35] actually proposed was the discretization of the horizonarea, from which we naively expect the avoidance of the absorption of low frequency GWssince the state after the absorption is absent. References [36, 37] proposed very differentindependent arguments that suggest the reflective boundary selective to low frequency GWs.To take into account such a possibly expected property of the boundary, here we consider afew simple models that give a frequency dependence to the reflectance of the boundary nearthe horizon. The simplest one is the sharp cut-off model, given byR b = ( | ω | < ω c )0 (otherwise) , (73)where ω c > b = exp " − (cid:18) ω − ω QNM σ ω (cid:19) , (74)where ω QNM is the frequency of the least damped quasi-normal mode for the m = 2 modein Kerr spacetime (here we use the fitting function given in[38]), and σ ω is a free parameter.We refer to this model as the Gauusian model. In addition, we consider a reflectance modelproposed in Ref. [36, 37], R b = e −| k | /T H , (75)where T H := ( r − a ) / (4 πr + ( r + a )) is the Hawking temperature. We refer to this modelas the quantum black hole (QBH) model.In Fig.3, we show the transfer function | ( ǫ ∞ , + /ǫ H, − ) K| with the various choices of thereflectance on the inner boundary, for ( l, m ) = (2 ,
2) and a = 0 .
7. The factor | ( ǫ ∞ , + /ǫ H, − ) | is multiplied so that the plot shows the square root of the ratio of the energy flux that reachesthe infinity compared with that falls into the black hole in the case without the reflectiveboundary. We fix the position of the boundary surface at r ∗ b = − M for all cases. -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 | ( ǫ ∞ , + / ǫ H , − ) K | M ω cutoff ( ω c =0.6)cutoff ( ω c =0.8)cutoff ( ω c =1)cutoff ( ω c =2) -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 | ( ǫ ∞ , + / ǫ H , − ) K | M ω QBH modelGaussian ( σ ω =0.1)Gaussian ( σ ω =0.2)Gaussian ( σ ω =0.3) Fig. 3
Transfer functions with the reflective surface near the horizon, ( ǫ ∞ , + /ǫ H, − ) |K| . Herewe plot the cases with the reflectance of Eq. (73) with the cutoff parameter ω c = 0 . , . , , σ ω = (0 . , . , .
3) and (75)on the right panel. We fix ( l, m ) = (2 , a = 0 . r ∗ b = − M for all cases.Once we obtain Z H under the ordinary boundary conditions and transfer function K , wecan construct the echo amplitude and its waveform measured at infinity under the presenceof the reflective boundary near the horizon, through Eq. (61). -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4-200 0 200 400 600 800 1000 1200 cutoff ω c =1cutoff ω c =1 r h ( u ) u / M Fig. 4
Waveform of the ( l, m ) = (2 ,
2) mode for the cutoff model with ω c = 1. Here weset ( a, L ) = (0 . , . L + c ) and r ∗ b = − M . The time domain waveform observed at infinity can be constructed from the asymptoticform of the Teukolsky variable, R = Zr e iωr ∗ by h ( u ) = Z dωe − iωu H ( ω ; Z ) , H ( ω ; Z ) = − r X lm Zω e imϕ √ π S( θ ) (76) ith u = t − r ∗ . From Eq. (61), the waveform in the reflective boundary case, ˆ h ∞ ( u ) is givenby ˆ h ∞ ( u ) = h ∞ ( u ) + h echo ( u ) , (77)where h ∞ ( u ) = Z dωe − iωu H ( ω ; Z ∞ ) , (78)correspond to the waveform observed at infinity in the ordinary BH boundary case, and h echo ( u ) = Z dωe − iωu H ( ω ; K Z H ) , (79)is that of the series of echoes in the reflective boundary case. As a demonstration, we givethe time-domain waveform of the ( l, m ) = (2 ,
2) mode for the cutoff model with ω c = 1 inFig. 4. Even with the high-frequency cutoff, one can recognize that the first echo is veryloud.In our previous work, we constructed the waveform of echoes using the merger-ringdownwaveform observed at infinity as the seed, instead of the waveform of GWs falling into theblack hole. Namely, in the present context our previous waveform would correspond to theone obtained by replacing Z H in Eq. (79) with Z ∞ ,¯ h echo ( u ) = Z dωe − iωu H ( ω ; K Z ∞ ) . (80)To compare h echo and ¯ h echo , we introduce the overlap between two waveforms by ρ = max ∆ t, ∆ φ, ¯ r ∗ ( h echo | ¯ h echo ) p ( h echo | h echo ) p (¯ h echo | ¯ h echo ) , (81)with ( f | g ) ≡ π Z ∞−∞ [ F ( ω ) G ∗ ( ω ) + F ∗ ( ω ) G ( ω )] dω, where ∆ t , ∆ φ are the shifts of time and phase between the two waveforms, and ¯ r ∗ b , which isthe location of the reflective boundary surface for ¯ h echo , controls the time interval betweenneighboring echoes. Here, we assume white noise spectrum, which would be a good approx-imation as long as we are interested in the waveform whose power is localized in a narrowfrequency band. In the above equation, we marginalize ∆ t , ∆ φ and ¯ r ∗ b in ¯ h echo to maxi-mize ρ , while we fix the other parameters to the same values as h echo . We also define a newestimator for the detectability of the signal when we use a specified echo template, effectivefiltered amplitude (EFA), byEFA = max ∆ t, ∆ φ, ¯ r ∗ b ( h echo | ¯ h echo ) p ( h ∞ | h ∞ ) p (¯ h echo | ¯ h echo ) . (82)This value roughly estimates the amplitude of the echo signal when we project the data byusing the template ¯ h echo relative to the amplitude of h ∞ . In Table 1 we show the values of ρ and EFA for each model with a = 0 . L = 0 . L + c and r ∗ b = − M . When ω c in the cutoffmodel gets large, the overlap decreases because the fraction of the high frequency modes in h echo , which is not included in ¯ h echo , increases. On the other hand, the EFA gets smallerwith the decrease of ω c because setting ω c smaller simply reduce the signal contained in odel ρ EFA ρ ′ EFA ′ γ (cutoff) ω c = 0 . .
903 0 .
347 0 .
291 0 .
112 0 . . . . . . ω c = 0 . .
673 0 .
588 0 .
162 0 .
142 0 . . . . . . ω c = 1 0 .
443 0 .
623 0 .
101 0 .
142 0 . . . . . . ω c = 2 0 .
144 0 .
627 0 . .
142 0 . . . . . . σ ω = 0 . .
945 0 .
329 0 .
303 0 .
105 0 . . . . . . σ ω = 0 . .
839 0 .
445 0 .
245 0 .
130 0 . . . . . . σ ω = 0 . .
724 0 .
508 0 .
195 0 .
137 0 . . . . . . .
962 0 . .
537 0 . . . . . . . Table 1
Overlap and EFA between two waveforms of echoes for ( l, m ) = (2 , a, L ) = (0 . , . L + c ) and r ∗ b = − M . The corresponding values for L =0 . L + c are shown in parentheses for comparison.the data. A similar behavior of ρ and EFA can be found in the Gaussian models. The EFAfor the QBH model is very small compared to the other cases because the amplitude of theechoes is largely suppressed by the transfer function as shown in the right panel of Fig. 3.Next we evaluate the decline rates of the echoes. The echo term K Z H in Eq. (61) iscomposed of the sum of the contributions from individual echoes, which correspond to therespective terms in the series expansion of the transfer function, K Z H = ∞ X n =1 K ( n ) Z H , K ( n ) ≡ (R b Φˆ r ) n − (R b Φˆ t ) Γ ∞ , + Γ H, − . (83)Namely, K ( n ) Z H corresponds to the amplitude of the n -th echo. From this, the whole echowaveform is expressed by a simple additive sum of the waveforms of the individual echoesas h echo = ∞ X n =1 h ( n )echo , h ( n )echo = Z dωe − iωu H ( ω ; K ( n ) Z H ) (84)In Fig. 5 we show | H ( ω ; K ( n ) Z H ) | , the absolute values of the waveform of the n -th echo fortwo representative models. The left panel is the plot for the cutoff model given in Eq. (73)with ω c = 1, while the right panel for the QBH model given in Eq. (75). The cutoff model isidentical to the case of the perfectly reflective boundary for M ω <
1. The first echo containsa large amplitude of high frequency modes, but they disappear in the second and laterechoes since the reflectance due to the angular momentum barrier, ˆ r , is almost zero. We also resent the case of QBH, because the relative amplitude below the threshold frequency ofthe super-radiance instability is significantly larger in the late-time echoes than the othercases. Of course, this is not because the lower frequency modes are enhanced but becausethe higher frequency modes are largely suppressed in the QBH model. ω c =1cutoff ω c =1 A b s o l u t e v a l u e o f r H ( u ) M ω n =1 n =2 n =3 n =4 n =5 A b s o l u t e v a l u e o f r H ( u ) M ω n =1 n =2 n =3 n =4 n =5 Fig. 5
Plots of the absolute values of H ( ω ; K ( n ) Z H ) for ( l, m ) = (2 , ω c = 1, while the right panel for the QBH model (75).In Fig. 6 we also plot | H ( ω ; K ( n ) Z ∞ ) | , the absolute values of the waveform of the n -thecho generated by using Z ∞ as the seed, instead of Z H . The left panel is the plot for thecutoff model with ω c = 1, while the right panel for the QBH model. cutoff ω c =1cutoff ω c =1 A b s o l u t e v a l u e o f r H ( u ) M ω n =1 n =2 n =3 n =4 n =5 QBH modelQBH model A b s o l u t e v a l u e o f r H ( u ) M ω n =1 n =2 n =3 n =4 n =5 Fig. 6
Plots of the absolute values of H ( ω ; K ( n ) Z ∞ ) for ( l, m ) = (2 , ω c = 1, the right panel for the QBH model in Eq. (75).With the identification of the waveform of each echo mentioned above, we define thefollowing quantities A n ≡ ( h ( n )echo | h ( n )echo )( h echo | h echo ) , B n ≡ ( h ( n )echo | ¯ h ( n )echo ) p ( h echo | h echo ) p (¯ h echo | ¯ h echo ) , (85)where B n is evaluated with the same common values of ∆ t , ∆ φ and ¯ r ∗ b as those used in themaximization in Eq. (81). A n is the relative amplitude of each echo to the total echoes, and B n is the relative overlap between the n -th echo component of h echo and that of ¯ h echo . -5 -4 -3 -2 -1
0 5 10 15 20 25 30 cutoff ω c =1cutoff ω c =1 n A n B n C n -5 -4 -3 -2 -1
0 5 10 15 20 25 30
QBH modelQBH model n A n B n C n Fig. 7
Plots of A n , B n and C n for ( l, m ) = (2 , ω c = 1, and the right panel for the QNM model (75). Here we fix( a, L ) = (0 . , . L + c ) and r ∗ b = − M .For comparison, we also consider an even simpler waveform model of echoes:¯ h ′ echo ( u ) = ∞ X n =1 ¯ h ′ ( n )echo ( u ) , ¯ h ′ ( n )echo ( u ) = ( − n γ n − h ∞ ( u − ( n − t echo ) , (86)where γ and ∆ t echo , independent of the frequency, are corresponding to the damping fac-tor and the time interval between successive echoes used in the analysis in Ref. [5]. Thismodel just repeats exactly the same waveform with a minus sign, periodically with decayingamplitude. We define the overlap and EFA between h echo and ¯ h ′ echo as ρ ′ = max ∆ t ′ , ∆ φ ′ ,γ, ∆ t echo ( h echo | ¯ h ′ echo ) p ( h echo | h echo ) q (¯ h ′ echo | ¯ h ′ echo ) , (87)EFA ′ = max ∆ t ′ , ∆ φ ′ ,γ, ∆ t echo ( h echo | ¯ h ′ echo ) p ( h ∞ | h ∞ ) q (¯ h ′ echo | ¯ h ′ echo ) , (88)where we marginalize ∆ t ′ , ∆ φ ′ , γ and ∆ t echo to maximize ρ ′ and EFA ′ . In Table 1, we showthe values of ρ ′ , EFA ′ and γ for the maximization. Both ρ ′ and EFA ′ are significantly smallerthan ρ and EFA. This means that the template proposed in Ref. [20] better captures thefeature of the echo signal expected by the model with a reflective boundary near the horizon.At the same time we find that the best fit value for the decay rate γ is not so large. This isnot consistent with the results of the data analysis presented in Ref. [5].In a similar manner to B n , we also define C n ≡ ( h ( n )echo | ¯ h ′ ( n )echo ) p ( h echo | h echo ) q (¯ h ′ echo | ¯ h ′ echo ) . (89)In Figs. 7 and 8, we show the plots of A n , B n and C n as functions of n for a representativecase with a = 0 . L = 0 . L + c . B n is much smaller than A n , which means that the echowaveform by using GWs emitted to infinity as the seed is not really a good approximation.Nevertheless, B n decreases much less rapidly than C n .In Fig. 8, we give a comparison of A n , B n and C n for several models. Here, we adopt( a, L ) = (0 . , . L + c ) as the representative values. The left panels are the plots for the cutoff -5 -4 -3 -2 -1
0 5 10 15 20 25 30 cutoffcutoff | A n | n ω c =0.6 ω c =0.8 ω c =1.0 ω c =2.0 10 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 | A n | n QBH modelGaussian ( σ ω =0.1)Gaussian ( σ ω =0.2)Gaussian ( σ ω =0.3)10 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 cutoffcutoff | B n | n ω c =0.6 ω c =0.8 ω c =1.0 ω c =2.0 10 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 | B n | n QBH modelGaussian ( σ ω =0.1)Gaussian ( σ ω =0.2)Gaussian ( σ ω =0.3)10 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 cutoffcutoff | C n | n ω c =0.6 ω c =0.8 ω c =1.0 ω c =2.0 10 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 | C n | n QBH modelGaussian ( σ ω =0.1)Gaussian ( σ ω =0.2)Gaussian ( σ ω =0.3) Fig. 8
Comparison of A n , B n and C n for several models. The left panels are the plots forthe cutoff models while the right panels for the Gaussian models and the QBH model. Herewe fix ( a, L ) = (0 . , . L + c ) and r ∗ b = − M .models. All the cases behave very similarly, i.e. , the higher cutoff frequency leads to thesmaller magnitude for the second and later echoes because the higher frequency modes,which are contained only in the first echo waveform, are emphasized. In the right panelswe give the plots for the Gaussian models and the QBH model. These plots show that theGaussian model with broader frequency band decays more rapidly at the beginning butmore slowly at a late time. This is because the model contains both the rapidly escapinghigh frequency modes and the long-lasting low frequency modes.
4. Conclusion
We have investigated the expected feature of the waveform of GW echoes in the model witha hypothetical reflective boundary near the horizon. As a model which is easy to handle, e consider perturbations induced by a particle falling into a Kerr black hole, instead of abinary coalescence. In the latter case, it is not so clear how to impose the modified reflectiveboundary condition near the horizon of the black hole newly formed after the merger. Bycontrast, imposing the reflective boundary condition at the boundary is mathematicallywell-posed in the case of black hole perturbation.We used the Sasaki-Nakamura equation to calculate GWs induced by a point mass fallinginto a Kerr black hole. We clarified the method how to compute GWs absorbed by theblack hole by using the Sasaki-Nakamura equation, and developed a prescription to imposethe reflective boundary condition near the horizon. For simplicity, our computation wasrestricted to the case in which the point mass is in an equatorial orbit and initially at restat r = ∞ , but its angular momentum was varied. Independently of whether the angularmomentum is small or large, the obtained spectrum of GWs absorbed by the black hole isdominated by modes with a higher frequency than that of the fundamental quasi-normalmode. As a result, the echo signal obtained by introducing a reflective boundary near thehorizon is also dominated by high frequency modes.If we assume that the echo waveform is given by a simple repetition of the waveformof GWs emitted to infinity, a significantly large fraction of echo signal is contained in thefrequency band lower than the QNM frequency. However, our analysis suggests that a simplereflective boundary model will not predict large power in echoes at such low frequencies.In Ref. [11] we reanalyzed LIGO data searching for the echo signal after binary blackhole merger. However, the use of our templates that take into account the reflection rateof the angular momentum barrier did not improve the significance of the signal suggestedin Ref. [5]. The main difference in the templates used in these two analyses is in the lowfrequency bands. If there exists echo signal dominated by lower frequency modes, our analysispresented in this paper suggests that we need to consider more complicated model than themodel with a simple reflective boundary near the horizon. Acknowledgment
This work was supported by JSPS KAKENHI Grant Number JP17H06358 (and alsoJP17H06357), A01:
Testing gravity theories using GWs , as a part of the innovative researcharea, “GW physics and astronomy: Genesis”. T.T. also acknowledges the support from JSPSKAKENHI Grant No. JP20K03928. The work of N.S. is partly supported by JSPS Grant-in-Aid for Scientific Research (C), No. JP16K05356, and Osaka City University AdvancedMathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theo-retical Physics JPMXP0619217849). Some numerical computations were carried out at theYukawa Institute Computer Facility.
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