Gravitational-Wave Echoes from Spinning Exotic Compact Objects: Numerical Waveforms from the Teukolsky Equation

Gravitational-wave echoes from spinning exotic compact objects: numerical waveforms from the Teukolsky equation

Shuo Xin,1 Baoyi Chen,2 Rico K. L. Lo,3 Ling Sun,3, 4 Wen-Biao Han,5 Xingyu Zhong,5 Manu Srivastava,6 Sizheng Ma,2 Qingwen Wang,7 and Yanbei Chen8 1Tongji University, Shanghai 200092, China 2Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA 3LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA 4OzGrav-ANU, Centre for Gravitational Astrophysics, College of Science, The Australian National University, ACT 2601, Australia 5Shanghai Astronomical Observatory, Shanghai, 200030, China 6Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India 7Perimeter Institute and the University of Waterloo, Canada 8TAPIR, California Institute of Technology, Pasadena, CA 91125, USA (Dated: May 27, 2021) We present numerical waveforms of gravitational-wave echoes from spinning exotic compact objects (ECOs) that result from binary black hole coalescence. We obtain these echoes by solving the Teukolsky equation for the ψ4 associated with gravitational waves that propagate toward the horizon of a Kerr spacetime, and process the subsequent reflections of the horizon-going wave by the surface of the ECO, which lies right above the Kerr horizon. The trajectories of the infalling objects are modified from Kerr geodesics, such that the gravitational waves propagating toward future null infinity match those from merging black holes with comparable masses. In this way, the corresponding echoes approximate to those from comparable-mass mergers. For boundary conditions at the ECO surface, we adopt recent work using the membrane paradigm, which relates ψ0 associated with the horizon-going wave and ψ4 of the wave that leaves the ECO surface. We obtain ψ0 of the horizon-going wave from ψ4 using the Teukolsky-Starobinsky relation. The echoes we obtain turn out to be significantly weaker than those from previous studies that generate echo waveforms by modeling the ringdown part of binary black hole coalescence waveforms as originating from the past horizon.

I. INTRODUCTION Since the ECOs have nearly the same external spacetime geometry as BHs, geodesic motions around them are identi- The detection of binary black hole (BBH) mergers, e.g., cal to those around BHs, therefore one way to probe ECOs is GW150914 [1], and binary neutron star (BNS) collisions, through tidal interactions during a binary inspiral process [21– e.g., GW170817 [2], has opened the era of gravitational wave 24]. An additional approach is via GW echoes [25–27], (GW) astronomy. In the third observing run of Advanced namely the GWs reflected from the near-horizon region of the LIGO [3] and Virgo [4], completed in March 2020, the up- final ECO formed during the merger process. One prominent graded detectors reached further in space and observed a sig- feature of these echoes is that the lag of echoes behind the nificantly increased number of events. The LIGO-Virgo Col- main wave corresponds to the exponentially small distance be- laboration has confirmed 50 GW events in total during the tween the ECO surface and the location of the horizon. Thus, first, second, and the first half of the third observing runs (as of GW echoes can be used to probe even Planck-scale structures October 1st, 2019) [5, 6]. With this new messenger to observe near the horizon [25]. Even though there is still no statisti- signals from strong gravity regime, we are now able to test cally significant evidence for echoes in existing GW data [28– theories of gravity in ways that was inaccessible before [7– 37], echo signals would be a promising candidate for probing 10]. physics beyond GR. arXiv:2105.12313v1 [gr-qc] 26 May 2021 In General Relativity (GR), black holes (BHs) are the stan- Many studies about GW echoes have been carried out for dard models for compact stellar remnants of the gravitational spherically symmetric, non-spinning ECOs, whose external collapse of massive stars at the end of their lives and for spacetime is Schwarzschild. For instance, Mark et al. studied massive objects at centers of galaxies. However, exotic mat- echo modes of scalar waves in some ECO models by solving ter equations of states [11], phase transitions [12, 13], or ef- the scalar perturbation equations with reflecting boundaries fects of quantum gravity [14–17] allow the existence of Exotic [38]. Du et al. solved GW echo modes based on the Sasaki- Compact Objects (ECOs), whose external spacetime has the Nakamura formalism and studied its contribution to stochas- same geometry as a BH — except in a small region near the tic background [39]. Huang et al. developed the Fredholm horizon (i.e., with size M, where M is the BH mass) [18– method and a diagrammatic representation of echo solution 20]. Searching for and detecting ECOs would push the fron- for general wave equations [40]. Maggio et al. studied the tier of fundamental physics. Without detection, upper limits ringdown of non-spinning ECOs [41]. on ECO properties derived from the search can be used to Astrophysically, we expect that merging BHs are spinning, quantitatively test “how black are BHs”, thereby confirming at least to some extent. Indeed, the LIGO-Virgo Collabora- the existence of the event horizon, the boundary of a region tion has detected mergers of spinning binary BHs [6]. In- within which signals cannot be sent to distant observers. ferring these spins has led to better understandings of these 2

BHs’ formation history and their stellar environments [42]. (fiducial observers used in the membrane paradigm). Since More importantly, even non-spinning merging BHs will gen- the current focus of GW astronomy is on binaries with nearly erally result in significantly spinning remnants. For example, comparable masses, we tune our point-particle trajectory in an equal-mass, non-spinning BH binary in a quasi-circular such a way that the waveform at infinity matches the Numer- merger leads to a final BH with J/M2 ∼ 0.7 (with units ical Relativity (NR) surrogate waveform from non-precessing G = c = 1), where J is the remnant angular momentum. This binaries with comparable masses; this is similar to the ap- motivates the study of GW echoes from spinning ECOs. proach taken by Ref. [51]. In this paper, we assume that the external spacetime geom- The paper is organized as follows. In Sec. II, we briefly re- etry of a spinning ECO is Kerr, except in a small region with view the Teukolsky formalism and obtain trajectories of parti- distance M near the horizon, and pose a boundary condi- cles in Kerr spacetime whose gravitational waves (in terms of tion at the boundary of that region. Here we need to point the Newman-Penrose ψ4 scalar) at infinity match those from out that, while in the spherically symmetric case, one can use comparable-mass binaries that merge from quasi-circular in- the Birkhoff’s theorem to argue that the exterior spacetime spirals. In Sec. III, we discuss how to obtain echoes at infinity of a spherical ECO should be Schwarzschild in GR, a sim- from ψ4 waveforms that go down the horizon, in particular ilar argument does not exist for spinning objects. For spin- deducing a conversion factor from the echoes obtained in the ning ECOs, one eventually needs to consider the compos- SN formalism to those obtained by imposing more physical ite effects of spacetime deviation and near-horizon boundary boundary conditions on curvature perturbations. In Sec. IV, condition. Note that mapping exterior spacetime geometry of we discuss features of the GW echoes, demonstrating the ef- compact objects (or “bumpy BHs”) has been a subject of ex- fect of the conversion factor obtained in the previous section, tensive studies [43–45]. Recent studies have further obtained and highlight a subtlety that leads to discrepancy between ψ4 non-spherically symmetric fuzzball geometries that arise from directly obtained using the Teukolsky formalism and ψ4 ob- spacetime microstates [15, 46]. tained using approximations imposed by Refs. [49–51]. The Let us now get to effects of the boundary condition at the main conclusions are summarized in Sec. VI. ECO surface. Nakano et al. [47] have constructed a model for echoes from spinning ECOs, where the asymptotic behav- II. PARTICLE FALLING INTO A BH ior of solutions to the Teukolsky equation is used to analyze the reflectivity and echo modes, but the incident wave toward the horizon is phenomenological. Micchi and Chirenti stud- In this section, we briefly review the computation of wave- ied effects of the rotation using a scalar charge falling into forms of a particle falling into a Kerr BH, both at infinity and a Kerr spacetime [48]. Wang et al. [49], Maggio et al. [50] near the horizon. We follow the prescription of the effective and Micchi et al. [51] obtained echo waveforms from spin- one-body (EOB) and Teukolsky formalism [58] and construct ning ECOs by first deducing waves that propagate toward the particle trajectories in the Kerr spacetime that decay due to ra- horizon from waves that propagate toward infinity, and then diation reaction, in such a way that the waveforms at infinity imposing a reflectivity for Sasaki-Nakamura (SN) functions. match those from comparable-mass binaries obtained from Sago et al. [52] studied echoes from a particle radially falling NR surrogate models [59]. Here the study is restricted to non- into a spinning ECO. spinning binaries. The mass M and the angular momentum One caveat when studying echoes from ECOs comes from per unit mass a of the Kerr spacetime correspond to those of the instability of ECOs, either from the structure of the ECO the remnant formed by the binary merger, which can be ob- itself, including superradiance for spinning ECOs [53, 54] and tained from the initial total mass and mass ratio of the binary the existence of stable photon orbits [55], or from the energy via NR surrogate models [60, 61]. content of GWs that propagate toward the ECO, which may induce the ECO to collapse [56, 57]. Despite all such insta- A. GWs emitted by a particle falling into a Kerr BH bilities, we advocate keeping an open mind about echoes. The superradiant instability can either be quenched or lead to non- spinning ECOs. We might also argue that instabilities that Let us first consider GWs emitted by a particle falling into cause gravitational collapse can simply cause the event hori- a Kerr BH. We use the Newman-Penrose scalar curvature ψ4, zon to grow, while non-local effects of quantum gravity would which can be decomposed into frequency and angular compo- keep appearing right outside the new location of the growing nents in the Boyer-Lindquist coordinates event horizon. ψ4(t, r, θ, φ) In this article, we construct echoes from spinning ECOs Z +∞ 4 X aω imφ −iωt using the Teukolsky formalism. We evolve a point particle = ρ dω R`mω(r) −2S `m(θ)e e , (1) in quasi-circular orbits, until it finally plunges into a Kerr −∞ `m BH. Our approach improves from previous work, notably − with ρ = (r − ia cos θ) 1. Here S aω is the spin-weighted Refs. [49–51], by: (i) using gravitational-wave waveform to- −2 `m spheroidal harmonic with eigenvalue E [62], while R is wards the horizon directly computed in the Teukolsky formal- `m `mω the solution to the radial Teukolsky equation, ism, and (ii) using a boundary condition on the ECO surface ! that is connected to tidal tensors measured locally by zero- 2 d 1 dR`mω ∆ − V(r)R`mω = −T`mω(r), (2) angular-momentum observers floating right above the horizon dr ∆ dr 3 with the potential and

2 hole Z ∞ ∞ 0 0 K + 4i(r − M)K B R (r )T`mω(r ) V(r) = − + 8iωr + λ, (3) ZH BH = `mω dr0 `mω . (11) ∆ `mω in ∞ ∆(r0)2 2iωB`mωD`mω r+ 2 2 2 2 where K = (r + a )ω − ma, λ = E`m + a ω − 2amω − 2, and 2 2 In particular, at r → +∞, ψ4 is related to GW polarizations h+ ∆ = r − 2Mr + a . The source term T`mω(r) is determined by the mass and trajectory of the particle (discussed in Sec. II B). and h× by Homogeneous solutions for the radial Teukolsky equation 1 have two types of asymptotic behaviors each, near the horizon ψ4(r → ∞) = (h¨ + − ih¨ ×). (12) and at inﬁnity, respectively. We are particularly interested in 2 two combinations of such solutions, namely the one that is The GWs we observe at a distance r, latitude angle Θ and purely in-going near the horizon, azimuthal angle Φ, is given by:  hole 2 −ipr∗ →  B`mω∆ e r r+ BH − BH| H  h+ ih× (r,Θ,Φ,t) R`mω(r) =  (4) Z +∞ ∞  out 3 iωr∗ −1 in −iωr∗ 2 X Z  → ∞ `mω aω −iω(t−r∗) B`mωr e + r B`mωe r , − dω S , e . = 2 −2 `m(Θ Φ) (13) r −∞ ω and the one that is purely out-going at inﬁnity `m

 out ipr∗ 2 in −ipr∗ Here we note that the spin-weighted spheroidal harmonics  D e + ∆ D e r → r+  `mω `mω differ from the spin-weighted spherical harmonics, and are R∞ (r) =  (5) `mω  frequency dependent. In NR waveform catalogs, as well as  D∞ r3eiωr∗ r → ∞. `mω in surrogate models, GW strains at infinity are decomposed Here we define into spin-weighted spherical harmonics h i X p = ω − mΩ , Ω = a/(r2 + a2), (6) NR NR NR + + + r h+ − ih× = h (t) −2Y`m(Θ, Φ) . (14) (Θ,Φ,t) `m `m with Ω+ being the horizon’s rotation angular frequency, and the tortoise coordinate r∗ as In this paper, we treat the mode-mixing in the spheroidal harmonics as negligible, writing S aω ≈ Y . This allows us 2Mr r − r 2Mr r − r −2 `m −2 `m + + − − ∞ r∗ = r + log − log . (7) NR to make a simple connections between Z`mω and h : r+ − r− 2M r+ − r− 2M `m

in, out, hole, ∞ in, out, hole, ∞ ∞ The quantities B and D are related to the Z`mω − 2 ↔ hNR . (15) transmissivity and reflectivity of the compact object; their val- ω2 `mω ues here are subject to a choice of conventions, for which we follow the convention of Hughes [63]. We compute these ho- We also focus on the ` = 2 contributions, with m = ±2. Note mogeneous solutions numerically with the help of the SN for- that −2Y22(Θ, Φ) and −2Y2−2(Θ, Φ) predominantly emit toward malism based on the codes developed in [64, 65] (see review the northern hemisphere (Θ < π/2) and southern hemisphere in the Appendix). (Θ > π/2), respectively. Both m = +2 and −2 contributions When the central object is a BH, we look for solutions that are equally important to describe the “(2,2)” waveform. Fur- are only in-going at the horizon and only out-going at infinity, thermore, for non-precessing binaries with angular momen- ∗ − by imposing tum along the z axis (Θ = 0), we have h`m( f ) = h`−m( f ). The studies in this paper are based on this scenario.  Z∞ BHr3eiωr∗ r → ∞  `mω RBH =  (8) `mω   H BH 2 −ipr∗ Z`mω ∆ e r → r+. B. Trajectory of particles in Kerr and the Teukolsky source terms This uniquely determines a solution that can be obtained from the Green’s function approach, We aim to use Teukolsky waveforms to approximate those ∞ Z r H 0 0 from coalescence of BHs with comparable masses. First of R (r) R (r )T`mω(r ) RBH (r) = `mω dr0 `mω all, in order for the ringdown frequencies to match up, we set `mω 2iωBin D∞ ∆(r0)2 `mω `mω r+ the mass and spin of the Kerr background spacetime equal to H Z ∞ ∞ 0 0 R (r) R (r )T`mω(r ) those of the remnant BH of a comparable-mass binary merger. + `mω dr0 `mω , (9) in ∞ 0 2 We then evolve particle trajectories by modifying the Kerr 2iωB D r ∆(r ) `mω `mω geodesic equation, adding generalized forces that implement from which we can read off the effect of radiation reaction, in such a way that the late inspiral, merger, and ringdown parts of the waveforms match Z ∞ H 0 0 1 R (r )T`mω(r ) Z∞ BH = dr0 `mω , (10) those from comparable-mass binaries, obtained from surro- `mω in ∆(r0)2 2iωB`mω r+ gate models. 4

For a trajectory in the Boyer-Lindquist system, In the above integrals, the integration variable t parametrizes parametrized as xµ(τ) = (t(τ), r(τ), θ(τ), φ(τ)), our modi- locations on the trajectory of the in-falling particle, with fied equations are written as t → −∞ corresponding to the beginning of the infall, and dxµ t → +∞ corresponding to when the particle approaches r+. = uµ, (16) The integrand approaches zero for t → +∞. For t → −∞, we dτ duµ apply a window which selects part of the trajectory within a = −Γµ uρuσ + F µ . (17) finite distance from the BH/ECO. We note that as t → +∞, al- dτ ρσ though the individual terms, e.g., Am¯ m¯ 0 in Eq. (24) can diverge, µ The radiation reaction force F can be obtained from the GW the sum of those terms approaches zero due to the cancellation ˙ ˙ energy flux E, angular momentum flux Lz and rate of change between the terms. of Carter constant Q˙, by solving the following equations: t t φ Eu˙ = −gttF − gtφF , (18) t t φ C. Waveform at infinity: calibration with surrogate models L˙zu = gtφF + gφφF , (19) L L˙ Qu˙ t = 2g2 uθF θ + 2 cos2 θa2EE˙ + 2 cos2 θ z z , (20) In order to obtain the trajectory that leads to a Teukolsky θθ sin2 θ waveform matching the surrogate waveform, we take the (2, 2) µ ν gµνu F = 0. (21) component of the surrogate waveform, evaluate its instantaneous angular frequency ω , as well as its rate of change In this paper, we focus on non-precessing binaries. Hence we (2,2) ω˙ , and make sure that, before reaching the Innermost Sta- consider equatorial circular orbits and impose Q˙ = 0; E˙ and L˙ (2,2) z ble Circular Orbit (ISCO), the orbital frequency of the particle are determined phenomenologically such that the Teukolsky follows the a corresponding evolution with ω = 1/2ω by waveforms (in the BH case) match those obtained from NR. orb (2,2) adjusting the radiation reaction forces. We turn off the radia- We also assume that the ECO does not modify these forces. tion reaction forces after the particle reaches ISCO. The am- From the numerical trajectory xµ(τ), or alternatively the 3- plitude of the final waveform from this orbit is rescaled so that dimensional trajectory as a function of time r(t), θ(t), φ(t), the the normalization is the same as that of the surrogate model. source term in the Teukolsky equation is given by (see Ap- Note that the time axis of the surrogate model waveform is pendix A for details) also rescaled to units of remnant mass. Z ∞ 0 i[ωt−mφ(t)] 2 0 To make comparisons, we take (`, m) = (2, 2), com- T`mω(r ) = dte ∆ (r ) ∞ BH −∞ pute Z22ω , obtain the time-domain waveform via an inverse n Fourier transform [see (13)], and then compare the waveform [A + A + A ]δ(r0 − r(t)) nn0 nm¯ 0 m¯ m¯ 0 with the NR surrogate model. Fig. 1 shows the wavforms 0 + ∂r0 ([Anm¯ 1 + Am¯ m¯ 1]δ(r − r(t))) with mass ratio q = 1 and q = 4 obtained by “NRSur7dq4” 2 0 o model [66], compared with our approximate Teukolsky-based + ∂ 0 [A δ(r − r(t))] . (22) r m¯ m¯ 2 waveforms. For the portion of the waveform shown in the Plugging this source term into Eqs. (10) and (11), we obtain figure, using Advanced LIGO noise spectrum at the design amplitudes of GWs toward infinity, sensitivity, the match [67] between the Teukolsky and the NR ∞ surrogate waveforms are above 0.99 for all M between 10 M 1 Z + ∞ i[ωt−mφ(t)] and 500 M . Z`mω = in dt e 2iωB`mω −∞ ( H R`mω(r(t))[Ann0 + Anm¯ 0 + Am¯ m¯ 0] III. CONSTRUCTING ECHOES dRH `mω − [Anm¯ 1 + Am¯ m¯ 1] In this paper, we assume that the ECO spacetime is iden- dr r(t) tical to a Kerr spacetime except for r∗/M −1. Since GW 2 H ) d R echoes are mainly sourced by the plunge part of the trajectory, `mω A , + 2 m¯ m¯ 2 (23) dr r(t) which is not significantly affected by the radiation reaction, we can neglect the modification to the trajectory due to the and toward the horizon, ECO surface. For these reasons, as a particle falls towards the hole Z +∞ ECO, we keep the same Teukolsky equation, with the source H B`mω i[ωt−mφ(t)] Z`mω = ∞ in dt e term in Eq. (2) given by Eq. (22). However, we need to change 2iωD`mωB`mω −∞ −4 ( the boundary condition for ρ ψ4 to a more general form ∞ R (r(t))[Ann0 + Anm¯ 0 + Am¯ m¯ 0]  `mω Z∞ECOr3eiωr∗ r → ∞  `mω ∞ RECO =  (25) dR`mω `mω  − [A + A ]  in 2 −ipr∗ out ipr∗ nm¯ 1 m¯ m¯ 1 Z`mω∆ e + Z`mωe r → r+, dr r(t) 2 ∞ ) d R where Zout appears due to the “reflection” from the ECO sur- + `mω A . (24) `mω 2 m¯ m¯ 2 in dr r(t) face, while Z`mω is modified since additional waves propagate 5

FIG. 1. Comparison between the NR surrogate model NRSur7dq4 and waveforms generated from the Teukolsky code with phenomenological trajectory described by Eq. (21) (see text for details), for q = 1 (top two panels) and q = 4 (bottom two panels). The time t is shifted so that the waveform starts at a certain frequency during inspiral, which is consistent with the starting time of the phenomenological trajectory.

toward the ECO upon reflection from the inner side of the Kerr X`mω that satisfies the SN equation potential barrier near the light ring. 2 d X`mω dX`mω In Sec. III A, we first prescribe reflectivity within the SN − F(r) − U(r)X = 0. (26) 2 dr `mω framework, following previous literature. However, it turns dr∗ ∗ out that for Kerr spacetime, it has a more direct physical The transformation between the SN function and Teukolsky meaning to impose boundary conditions in terms of ψ0 and radial function is given by ψ , which are tied to curvature perturbations experienced 4 " ! # by observers near the future and past horizons, respectively. 1 β,r ∆X`mω β d ∆X`mω R`mω = α + √ − √ . (27) We obtain echo formulas from such boundary conditions in η ∆ r2 + a2 ∆ dr r2 + a2 Sec. III B. In Sec. III C, we review the reflectivity models used in subsequent sections. The potentials F(r) and U(r) in Eq. (26) and functions α, β, and η in Eq. (27) can be found in Eqs. (3.4)–(3.9) of Ref. [68] (also see Appendix B). To derive RH and R∞, we use two homogeneous solutions of the SN equation, which have purely sinusoidal dependence on r∗ due to the short-ranged potential: A. Boundary condition imposed on SN functions  Ahole e−ipr∗ r → r  `mω + XH (r) =  (28) `mω   out iωr∗ in −iωr∗ 1. SN formalism and boundary condition A`mωe + A`mωe r → ∞,  Cout eipr∗ + Cin e−ipr∗ r → r  `mω `mω + X∞ (r) =  (29) The simplest way to describe the ECO’s reflection of GWs `mω   ∞ iωr∗ → ∞ is to use the SN formalism (see Appendix B). We use fields C`mωe r . 6

Because these X’s are directly used to compute the corre- We can further write sponding R’s, there are relations between the amplitudes A, RECOT BH C here and B, D in Eqs. (4) and (5), given in Appendix B. For K = `mω `mω , (37) ∞ hole `mω ECO BH convenience, we can set C`mω = A`mω = 1. Using linearity, 1 − R`mω R`mω we can also write BH BH where R`mω and T`mω are amplitude reflecitity and transmissivity of the BH potential barrier — or the SN potential in ECO in −ipr∗ out ipr∗ → −∞ ∞ X`mω = ξ`mωe + ξ`mωe , r∗ (30) Eq. (26); in terms of asymptotic expressions of X`mω, they can be written as with 1 Cin T BH = , RBH = `mω . (38) ξout = RECOξin , (31) `mω out `mω out `mω `mω `mω C`mω C`mω ECO where R`mω is the ECO reflectivity. This is the baseline ap- In Eq. (36), the out-going wave at infinity, in the case of an proach taken by most previous literature [39, 49–51], except ECO, is the out-going wave in the case of a BH plus echoes, for [47], where using energy reflectivity is proposed. As dis- which are determined by the horizon-going wave in the BH ECO 2 cussed in more detail later in this paper, although |R`mω | cor- case. responds to energy reflectivity in the Schwarzschild case, it is not generally true for Kerr BHs, or spinning ECOs. We in- troduce more physically motivated reflectivites in Sec. III C 3. Echoes in Teukolsky functions and comment on additional subtleties of the SN formalism. At this stage, by comparing Eq. (30) with Eq. (25), as well as Using linearity of the transformation between the SN and Eqs. (4), (5), (28) and (29), we write Teukolsky functions, we have

ξin Cin Ahole 1 ξout Cout ξ∞ Aout C∞ 1 `mω = `mω = `mω = , `mω = `mω . (32) `mω = `mω = `mω = , (39) in in hole hole Zout Dout ∞ out ∞ ∞ Z`mω D`mω B`mω B`mω `mω `mω Z`mω B`mω D`mω D`mω Since the conversion relation (27) between SN and Teukolsky and functions involves derivatives, these formulas are only correct ξBH Cin Ahole 1 when the source terms vanish rapidly enough — which is true `mω = `mω = `mω = . (40) BH in hole hole for sources that are bounded outside the horizon, but not nec- Z`mω D`mω B`mω B`mω essarily true for a particle that plunges into the horizon. See ∞ hole Sec. III B 1 for more discussions. Here we use the conventions C`mω = A`mω = 1. We can rewrite Eq. (36) in terms of the Teukolsky functions

RECORBH 2. Echoes in the SN Formalism Z∞ ECO = Z∞ BH + `mω `mω Z∞ eff, (41) `mω `mω ECO BH `mω 1 − R`mω R`mω In the BH case, we obtain with  ∞ iωr BH ∗ → ∞ ∞  ξ`mω e r D BH  ∞ eff `mω H BH X`mω(r) = (33) Z`mω ≡ Z`mω . (42)  − in  H BH ipr∗ D`mω ξ`mω e r → r+, in the SN formalism. Then in the ECO case, we need to su- For very compact ECOs, substantial phase shift exists in RECO, causing substantial time delays between neighboring perimpose an additional homogeneous solution to form a new `mω solution, echoes, which allows +∞ ECO BH ∞ X n X (r) = X (r) + cX (r). (34) ∞ ECO ∞ BH ECO BH ∞ eff `mω `mω `mω Z`mω = Z`mω + R`mω R`mω Z`mω . (43) n=1 We assume that the source term does not change for ECOs. After imposing boundary condition (30) at the ECO surface, Here the nth item is effectively the nth echo in the full wave- we obtain form. The physical understanding of the expansion above is that the nth echo is reflected n times by the ECO surface and ECO R`mω − c = ξH BH ≡ K ξH BH. (35) n 1 times by the potential barrier (of SN or Teukolsky equa- out ECO in `mω `mω `mω n C`mω − R`mω C`mω tions), propagates for an additional 2 times the distance between the potential barrier and the ECO surface (in terms of ∞ Using the asymptotic form of X`mω, we obtain r∗), and finally transmits through the potential barrier. We can obtain GW strain h from Z via Eq. (13). Eqs. (41) ∞ ECO H BH ∞ BH ξ`mω = K`mωξ`mω + ξ`mω . (36) and (43) reduce to those in Maggio et al. [50] and Wang 7

ℛ to to T future null infinity future null infinity ℛ BH T Z ∞ eﬀ BH BH H H Z Z transmitted v ~ first echo v Z ∞ future horizon= future horizon= e v Z ∞ v wave ﬀ to 0 BH to 0

~ ring down wave reflected ~ ring down wave wave

incoming

wave v= v= v v 0 0 from

infalling past horizon past null infinity particle from BH Barrier ECO Surface BH Barrier trajectory

FIG. 2. Spacetime diagrams illustrating generation of echoes when a particle plunges into a massive ECO (left panel), and the construction ∞ eff H BH ∞ eff of Z from Z (right panel), with Z being the transmitted wave at future null infinity. In the left panel, we label v = v0 at which the particle plunges into the future horizon; at this point ZH BH has a local feature. In the upper-right region of the panel, we also label the approximate locations of ringdown and the first echo in Z∞ ECO. In the right panel, we indicate that one needs to supply the incoming wave from the past horizon, in order for the BH barrier to reflect it and generate ZH BH. et al. [49] (in their “inside” prescription), if we make the sub- Therefore, we write stitution of RECOT BH Z∞ ECO = Z∞ BH + `mω `mω Z˜H BH, “inside” prescription: Z∞ eff ← Z∞ BH RD. (44) `mω `mω ECO BH `mω `mω 1 − R`mω R`mω ∞ BH RD (45) Here Z`mω is the ringdown part of the waveform at infinity in the BH case. where we define The quantity Z∞ eff, as defined in Eq. (42), has the follow- `mω D∞ ing physical meaning. If we replace the spacetime of a parti- Z˜H BH ≡ `mω ZH BH, (46) hole `mω cle plunging into Kerr with a linear perturbation of Kerr, and B`mω replicate the waveform at infinity by sending in a wave from which remains finite as ω → +∞. By absorbing the the past horizon, in such a way that the waveform toward the D∞ /Bhole factor, we have future horizon agrees with ZH BH, then the waveform at infin- `mω `mω ∞ ity is given by Z eff. 1 Z +∞ R∞(r0)T (r0) `mω Z˜H BH = dr0 `mω The situation is illustrated in Fig. 2. As we can see 2iωBin ∆2(r0) from the figure, in the region “above” the trajectory of the `mω r+ Z +∞ H BH 1 − particle, we have a vacuum spacetime, and [Z ]v>v0 = i[ωt mφ(t)] in ∞ ∞ BH ∞ eff = in dte [(D /D )Z ]v>v0 . It is then plausible that Z`mω can be 2iωB`mω −∞ approximated by the ringdown part of Z∞ BH, or Z∞ BH RD, as ( `mω `mω ∞ Maggio et al. [50] and Wang et al. [49] have done in the “from R`mω(r(t))[Ann0 + Anm¯ 0 + Am¯ m¯ 0] inside” prescription [see Eq. (44)]. dR∞ However, our results turn out to differ rather significantly `mω − [Anm¯ 1 + Am¯ m¯ 1] from these prescriptions. More details are given in Sec. IV C. dr r(t) ∞ D /Din ∼ 2 ∞ ) Here we point out two features, namely `mω `mω 0 when d R ∞ `mω ∼ in → ∞ + Am¯ m¯ 2 . (47) ω mΩ+, and D`mω/D`mω diverges quickly when ω + . 2 ∞ eff dr r(t) This means that Z`mω vanishes when ω ∼ mΩ+ (i.e., for ra- diations that are co-rotating with the horizon), and tends to This is related to Z˜∞ BH by replacing the Green function RH ∞BH ∞ infinity as ω → +∞ [69]. This is clearly different from Z`mω . with R [cf. Eqs. (10)–(11)]. 8

a/M = 0 a/M = 0.7 2 2

1 1 T T R R / 0 / 0 real part SN SN

R R imaginary part 1 1 magnitude − − mΩ+ 2 2 − 2 1 0 1 2 − 2 1 0 1 2 − − Mω − − Mω

(a) a/M = 0 (non-spinning) (b) a/M = 0.7 (spinning)

FIG. 3. The real (blue) and imaginary (green) parts and absolute values (black) of the reﬂectivity conversion factor [for (`, m) = (2, 2)] as a function of ω for (a) a/M = 0 and (b) a/M = 0.7. The vertical line at non-zero ω in the right panel indicates ω = mΩ+. For a non-spining ECO, the SN and Teukolsky reﬂectivities are related to each other with a pure phase shift; while for a spinning ECO, the factors have substantial frequency dependence, but remain at the order of unity at frequencies interested for compact binary mergers.

B. Boundary conditions directly imposed on Teukolsky A particular example of such subtleties is as follows. As functions one tries to evaluate the SN source terms, there are two de- grees of freedom in the form of the integration constants. In this section, we discuss some subtleties in the SN formal- They lead to different leading-order field amplitudes at infinity ism, and directly impose boundary conditions on Teukolsky (in terms of 1/r) and at the horizon (in terms of r − r−). Only functions. The SN formalism remains as a computational tool after accounting for the correction terms up to the second or- in obtaining homogeneous solutions to the Teukolsky equa- der, one can obtain the correct Teukolsky amplitudes. This tion. presents an ambiguity for how to impose boundary conditions for X.

1. Subtleties in the SN formalism 2. Physical boundary conditions from ψ0 and ψ4 As argued by Nakano et al., from the Wronskian of the radial equations, the energy reflectivity, in terms of the SN re- In a companion paper [70], we have determined the relation ECO out in flectivity R , is given by [47] between Z`mω and Z`mω by considering tidal tensor fields of fiducial observers near the horizon. To summarize the results, E˙ |C d |2 out = `mω `mω |RECO|2, we connect ψ0 and ψ4 to the tidal tensors of ficucial observers 5 2 2 2 2 2 `mω E˙in 16(2Mr+) (p + 4 )(p + 16 )|η (r+)| near the horizon: (48) ∆ Σ ∗ 2 2 2 with E ∼ − ψ0 − ψ , Σ = r + a cos θ . (50) √ 4Σ ∆ 4 2 2 = M − a /(4Mr+) (49) −2 −ipr∗ Note that it is the in-going piece of ψ0 (∼ ∆ e ) and the +ipr∗ equal to half the surface gravity. More specifically, this is the out-going piece of ψ4 (∼ e ) that dominate this expres- energy flux that emerges from the past horizon divided by the sion, with both contributing to E at the order of 1/∆ since energy flux that goes into the future horizon [70]. Note that the effect of GWs is heavily blue-shifted for near-horizon ob- the modulus of this conversion factor is not unity for a/M , 0 servers. Since the in-going piece of ψ0 is externally applied to (see Fig. 3). the ECO, while the out-going piece is generated by the ECO, As we use the SN formalism to derive echoes, we need to the ratio of these two terms can then be viewed as a local tidal note two important subtleties: (i) as we convert between X Love number of the ECO. and R, it is important to keep both the first two leading orders We can find the the in-going ψ0 components via the Teukolsky-Starobinsky identity in r − r+ near the horizon, and keep those in 1/r near infinity; (ii) the particle plunges toward the horizon, and therefore the in in Y`mω = σ`mωZ`mω, (51) source term for Teukolsky and SN equations does not vanish with as r∗ → −∞. Such subtleties affect the evaluation of in-going 4 2 2 energies down the horizon, as well as the boundary conditions 64(2Mr+) ip(p + 4 )(−ip + 4) σ`mω = . (52) near the ECO surface. C`mω 9

Here C`mω is the Starobinsky constant, given by In Fig. 3, we plot the real, imaginary parts and modulus of RECO/RECO T. 2 2 2 2 2 2 2 |C`mω| = (Q + 4aωm − 4a ω )[(Q − 2) + 36aωm − 36a ω ] At this stage, aside from subtleties of the Teukolsky- + (2Q − 1)(96a2ω2 − 48aωm) Starobinsky relation, Eq. (58) provides the reflectivity of the RECO 2 2 2 ECO in the SN frame work, `mω , in terms of the physically + 144ω (M − a ), ECO T defined ECO reflectivity, R`mω . We can insert Eq. (58) into Im C`mω = 12Mω, Eqs. (41) and (45) to obtain echoes that arise from these phys- p 2 2 ical boundary conditions. Re C`mω = + |C`mω| − (ImC`mω) , (53) with Q = E + a2ω2 − 2aωm, where E is the spheroidal `m `m 3. Echoes in terms of Teukolsky reflectivity eigenvalue [see discussions below Eq. (1)]. This has previ- ously been applied to computing energy and angular momentum carried by GWs into the horizon. However, we need to be We can now write echo waveforms in terms of the ampli- careful here because the Starobinsky-Teukolsky identity may tudes of Teukolsky functions. In terms of reflectivity, we have not work in the presence of source terms — while here we RECORBH = RECO TRBH T, (59) do have a particle plunging into the horizon. This remains an `mω `mω `mω `mω unaddressed issue in this paper. where By considering tidal distortions of zero-angular-momentum fiducial observers very close to the horizon, we obtain m+1 in BH T (−1) D`mω R`mω = σ`mω out . (60) m+1 4 D`mω out (−1) ECO T in ∗ Z = R Y − − . (54) `mω 4 `mω ` m ω We can also write Here the Teukolsky reflectivity RECO T can be related to the re- +∞ `mω ∞ ECO ∞ BH X ECO T BH Tn ∞ eff ECO T Z`mω = Z`mω + R`mω R`mω Z`mω . (61) sponse of the ECO to external driving. Its modulus, |R`mω |, corresponds to the energy reflectivity of the ECO surface. n=1 Here we have ignored the mixing between different `-modes, Here |RBH T|2 directly gives the energy reflectivity of the BH which is a general feature due to the distortion of spacetime `mω potential barrier, including superradiance at frequencies ω < geometry by the spin of the ECO. As it turns out, we also need mΩ . The condition |RBH TRECO T| < 1 needs to be satisfied to consider (Zout , Zout ∗ ) and (Zin , Zin ∗ ), since generi- + `mω `mω `mω `−m−ω `mω `m−ω such that the instability does not happen in the ECO. cally an ECO couples between these modes. Nevertheless, We can also express the echo waveform in terms of the in- in the most commonly considered situation of an equatorial, going ψ component toward the horizon in the BH case, YH BH, quasi-circular orbit, we have 0 `mω as ∗ Z = Z , (55) ECO T `mω `−m−ω R J`mω Z∞ ECO = Z∞ BH + `mω YH BH, (62) `mω `mω ECO T BH T `mω as described by Maggio et al. [50]. This indicates that only 1 − R`mω R`mω one reflectivity needs to be considered. However, when the particle has an inclined orbit, relation (55) no longer holds, where we define and thus the echoes have a more complex form. m+1 ∞ (−1) D`mω Assuming Eq. (55) to hold, we have J`mω = out . (63) 4 D`mω (−1)m+1 out ECO T in Here, the in-going ψ0 takes the form of curvature perturba- Z`mω = σ`mωR`mω Z`mω. (56) 4 tions of fiducial observers near the horizon. Response of struc- Assuming a linear relation between the SN and Teukolsky tures in the observers’ frame gives rise to a local reflectiv- RECO ψ functions, we can write ity `mω , leading to the out-going 4 with a conversion fac- m+1 ECO tor of (−1) R`mω /4. This ψ4 is then transmitted to infinity Xout Cout Din Zout with a factor of D∞ /Dout applied. Since Eq. (62) does not RECO = `mω = `mω `mω `mω . (57) `mω `mω `mω Xin Dout Cin Zin use the Teukolsky-Starobinsky transformation (which is only `mω `mω `mω `mω valid for homogeneous solutions), it provides a more straight- This leads to forward way to compute echoes.

(−1)m+1 Cout Din RECO = σ `mω `mω RECO T `mω 4 `mω out in `mω C. Models for ECO reflectivity D`mω C`mω 5/2 − m 4(2Mr+) η(r+)(p 2i)(p + 4i) ECO T We consider two types of reflectivity, namely a parame- = (−1) R`mω . C`mωd`mω terized Lorentzian reflectivity and a Boltzman-type reflectiv- (58) ity [49]. 10

1. Lorentzian reﬂectivity 2. Boltzmann reﬂectivity

In the Lorentzian case, we assume the reflection takes place Considering wave reflection by a thermal atmosphere, at a fixed position of r = b, or r∗ = b∗ [Eq. (7)]. At position Wang et al. [49] and Oshita et al. [71] proposed the follow- r = b, the proper distance δ along the radial direction toward ing Boltzmann reflectivity, given by the horizon is given by ! " # B |p| p s R`mω = exp − exp −i log(γ|p|) , (68) Z b 2 2 2 2TH πTH √ r+ + a cos θ p δ = grrdr ≈ b − r+, (64) r+ Mr+κ with Hawking temperature √ for δ M. For BHs with a/M not too close to unity, this κ r − r 1 − a2 T = = + − = √ . (69) leads to H 2 2 2π 4π(r+ + a ) 4πM(1 + 1 − a2)

1 b − r+ r− r+ − r− b∗ ≈ r+ + log − log We may replace TH with a free parameter TQH to generalize 2κ 2M 2κr+ 2M the Boltzmann reflectivity, 1 δ ≈ log p . (65) | | ! " # κ r2 + a2 cos2 θ B p p + R`mω = exp − exp −i log(γ|p|) . (70) 2TQH πTQH Another way of measuring the closeness to the horizon is via the red-shift of zero-angular-momentum observers at a con- We fix the temperature to the Hawking temperature, i.e., stant r = b, with TQH = TH, until in Sec. V, where we relax this condition to explore the detectability of echoes that arise from a broader s class of reflectivity models. Similar to the Lorentzian re- 4Mr+κ p B α = b − r . (66) flectivity, R`mω depends on ω via p = ω − mΩ+, leading to 2 2 2 + r+ + a cos θ the peak reflectivity (equal to unity) for modes with zero frequency viewed by observers co-rotating with the horizon, or For a , 0, both δ and α depend on θ. This can be understood ω ∼ mΩ+, and vanishing reflectivity for |ω − mΩ+| TH. as the deformation of spherical symmetry due to the spin. In Boltzmann reflection does not take place at a fixed point, Ref. [70], we choose to set the reflection surface at a constant but for waves oscillating near the quasi-normal mode (QNM) red shift α (i.e., the reflectivity has the same phase for all val- frequency, ω ≈ <[ωQNM]. The effective distance traveled by ues of θ when α is a constant), which leads to mixing between waves at this frequency, in terms of r∗, due to the phase factor the modes with different `. In this paper, for simplicity, we in Eq. (68), is given by assume that the Lorentzian reflectivity is a constant at r = b eff B (or r∗ = b∗) and can be written as 2r∗ = log(γ|<[ωQNM] − mΩ+|)/πTH. (71) ! eff B iΓ Similar to the Lorentzian case, here −2r∗ corresponds to the RL = ε e−2ib∗ p. (67) `mω p + iΓ time lag between echoes. As shown in Ref. [49], the Boltzmann reflectivity leads to a Here the quantity ε ∈ (0, 1) parametrizes the amplitude re- stable ECO. Basically, the BH potential barrier has a reflectiv- flectivity of the ECO surface. Note that R depends on ω ity higher than unity for ω < mΩ+, and the ECO simply needs to have a reflectivity that decreases fast enough as p increases only via p = ω − mΩ+, the frequency of oscillations measured by observers co-rotating with the horizon of the Kerr from zero. spacetime (even though it is covered by the ECO surface at r = b). The quantity Γ characterizes a relaxation rate of the ECO surface, which corresponds to an impulse response func- IV. HORIZON WAVEFORMS AND ECHOES tion ∼ e−Γt in the time domain and imposes a low-pass filtering of waves upon reflection in the frequency domain. For distant In this section, we discuss numerical features of GWs that observers, GWs with frequencies |ω−mΩ+| . Γ have the high- go down the horizon and echoes generated using several mod- est reflectivity. Note in particular, that peak reflectivity takes els of the ECO reflectivity. place at ω ∼ 2Ω+ for m = 2 and ω ∼ −2Ω+ for m = −2. As argued by Refs. [49, 53], as long as ε is not too close to unity, the ECO is stable under the Lorentzian reflecitivity. A. Prescriptions for computing echoes −2ib∗ p L The phase factor e in R`mω corresponds to a time delay of −2b∗. If we consider that the ringdown wave is generated We first review existing prescriptions and show how our roughly at r∗ ≈ 0, the term −2b∗ provides an estimate of the prescription is connected to and differs from these previous time delay between the main wave and the first echo, as well studies. In existing literature, in particular Wang et al. [49] as time delays between neighbouring echoes. and Maggio et al. [50], echoes are obtained from the out-going 11

FIG. 4. Advanced-time trajectory and horizon waveform ZH BH, with and without self field, corresponding to a trajectory fitted to the NR waveform shown in Fig. 1 (Top row: q = 1, bottom row: q = 4). The left panels show the advanced time verses coordinate time. In the late stage of the particle plunge, advanced time converges to a constant. All the late-time pieces in the integration for ZH BH accumulate near this epoch, resulting in a dip near t/M = 1047, as shown in the middle panels. The right panels show the frequency spectrum |ZH BH(ω)|, which peaks at the QNM frequency of the final BH.

GWs at infinity and the SN transmissivities and reflectivities. In the “inside prescription” of [49] and [50], one has RECORBH h∞ECO = h∞BH + [h∞BH] , (72) 1 − RECORBH RD where “RD” represents the ringdown part of the binary black hole coalescence waveform h∞BH. In this work, following Ref. [70], we propose that it is in fact RECO T that follows the reflectivity models described in Sec. III C, since RECO T is directly connected to the tidal fields measured by fiducial observers near the horizon. One must take a reflectivity from Sec. III C, and convert it into a SN reflectivity RECO using Eq. (58). The second difference between this study and previous FIG. 5. The in-going energy spectrum of the merger waveform with work is that we obtain the in-going ψ4 wave toward the hori- q = 1. The dashed curve shows the energy spectrum directly com- zon directly from Teukolsky formulation. This is equivalent puted. The solid curve shows the energy spectrum of the interpolated to obtaining it directly from the SN formalism for ψ4, e.g., waveform after removing the self-field part. done in Ref. [52] for radially in-falling particles. We insert the in-going Teukolsky amplitude ZH BH, obtained from Eq. (24), into Eq. (41) to compute echoes. This is equivalent to using B. Features of horizon waveforms Eqs. (45)–(47). We would like to mention that instead of computing in- H BH going wave via ψ4 (or the SN formalism for ψ4), one can In Fig. 4, we plot the horizon waveform Z22 for q = 1 and also compute the in-going ψ0 directly, and then use Eq. (62) q = 4. There is a feature in the time-domain horizon wave- to compute echoes. As discussed, since the reflection on the form, right at the advanced time when the particle plunges into ECO surface is really a relation between the in-going ψ0 and the horizon. This differs from the discussion for scalar fields the out-going ψ4, this approach is more direct and not subject by Mark et al. [38], and has to do with curvature perturbations to uncertainties of whether the in-going ψ4 can be converted due to a point particle. As shown in the figure, the feature oc- into the in-going ψ0 reliably using the Teukolsky-Starobinsky curs at a rather early time in the horizon waveform. Therefore relation when the point particle plunges into the horizon. We most of the echo does arise from GWs that hit the ECO after leave this for future work. the point particle plunges into the horizon. 12

FIG. 6. Echoes for an equal-mass binary merger (q = 1) with Lorentzian (top two rows) and Boltzmann (bottom two rows) reflectivities imposed on SN or Teukolsky radial functions. Top row: Impose Lorentzian reflectivity on SN functions. The left panel shows how and b∗/M in Lorentzian reflectivity change the magnitude and separation of echoes (MΓ = 0.5). The right panel shows how Γ impacts the shape of the first echo (with ε = 0.2, b∗/M = −75). Second row: (similar to the top row) Impose Lorentzian reflectivity on Teukolsky radial functions. Third row: Impose Boltzmann reflectivity on SN functions. The left panel shows how γ in the Boltzmann reflectivity changes the echoes. The eff B right panel shows how γ impacts the shape of the first echo [time is shifted by 2r∗ , as defined in Eq. (71), to align with the first echo]. Bottom row: (similar to the third row) Impose Boltzmann reflectivity on Teukolsky radial functions. Imposing reflectivities on Teukolsky functions (the more physical approach) tends to generate echoes with lower amplitudes than imposing the same reflectivities on the SN functions; this can be understood from the frequency content of ZH BH and the conversion factors shown in Fig. 3.

To compute the energy of the GWs going down the horizon, which leads to Eq. (4.44) in Ref. [72]: we need to convert ψ4 to ψ0, using the Teukolsky-Starobinsy dEhole X 64ωp(p2 + 42)(p2 + 162)(2Mr )5 identity and then the Hartle formula of BH area increase, = + |ZH BH|2 . dω π|C |2 `mω `m `mω (73) 13

This leads to diverging energy going down the horizon, even 3. Removal of “self field” for each individual `, as indicated in Fig. 5. However, at least in the Schwarzschild case, this energy should not diverge for In Fig. 7, we investigate the impact of removing the feature each `, as shown in Ref. [73]. We suspect that this is due to of ZH BH near the location where the particle plunges into the the fact that the Teukolsky-Starobinsky identity does not ap- future horizon. We compare the original first echo, the one ply to our case, where the particle plunges into the horizon, resulted from a smoothed version of ZH BH, as well as the one but leave the detailed study for future work. In our current obtained by completely removing the part before the plunge formulation, the divergence of the energy flux is a direct con- H BH H BH from Z . The differences caused by the removal of these sequence of the “self-field” feature of Z near the location features are small. where the particle plunges. Smoothing out this feature in the time domain (see middle panels of Fig. 4) can fix the energy divergence (blue curve in Fig. 5) without significantly affect- ing echo waveforms, as discussed later in Sec. IV C 3. 4. Polarization of the echoes

For the waveforms and reflectivity models considered here, ∗ we have R`−m−ω = R`mω for both ECO and BH reflecitivites, C. Features of echoes ∗ and h`−m(−ω) = h`m(ω) for all waveforms, including the main ECO BH wave and the echoes. For example, the fact that R`mω R`mω = ∗ In Fig. 6, we plot the GR and echo waveforms for binaries ECO BH R`−m−ωR`−m−ω is shown explicitly in Fig. 8. This means with q = 1, using Lorentzian (the first and second rows) and that, for the models considered here, we do not see the polar- Boltzman reflectivities (the third and fourth rows), either im- ization mixing pointed out by Maggio et al. [50]. Note that the posing them on the SN functions (the first and third rows), or effect of polarization mixing is absent as long as we consider on Teukolsky functions (the second and fourth rows) using the pairs of ±m simultaneously, while for an individual m, such prescription described in Ref. [70]. We zoom in and show the polarization mixing can still exist. details of the first echo in each row on the right panels. In Fig. 9, we plot both the + and × polarizations, for the GR wave and echoes, and for inclination angle Θ = 0 (face on) and Θ = π/2 (edge on) — indeed, the polarization state 1. Lorentzian reflectivity of the echos traces that of the main wave. The echoes are approximately circularly polarized for the face-on case, and linearly polarized for the edge-on case. RL For Lorentzian reflectivity , ε simply scales the magni- We also see that even by accounting for the `-`0 mixing n | | tude of the n-th echo by ε , while 2b∗ shifts the time-domain effects due to the ECO rotation, as considered in Ref. [70], separation between echoes (effects of ε and b on the first echo polarization mixing is still absent. This is because the equato- are shown in the left panels of the first two rows in Fig. 6). rial symmetry is not broken by the deformation of the space- The bandwidth Γ of the reflectivity acts as a low-pass filter in time geometry due to the ECO rotation. To observe polar- the reference frame of the ECO surface, therefore it filters out ization mixing proposed by Maggio et al. [50], one can con- | − | frequency components with ω mΩ+ . Γ. By comparing the sider precessing binaries, whose source modes can be ex- first and second rows of Fig. 6, we find that the new bound- ∗ plicitly decomposed into those with h`−m(−ω) = h`m(ω) and ary condition, imposed on curvature perturbations (the sec- h (−ω) = −h∗ (ω) [70]. ond row), gives rise to slightly weaker echoes than those with `−m `m the condition imposed on the SN functions (the first row), but does not modify the qualitative features of the echoes. This is consistent with the conversion factor in Eq. (58) with an ab- D. Comparison between prescriptions solute value smaller than unity in the frequency band of the echoes, i.e., around ω ∼ mΩ+ and higher toward ωQNM (as We finally compare our numerical waveforms with the re- shown in Fig. 3). sults from the “inside” formulation [49, 50], as outlined in Eq. (72). The comparison is shown in Fig. 10 using (a) a Lorentzian reflectivity and (b) a Boltzmann reflectivity. We can see that the magnitude of echoes are substantially dif- 2. Boltzmann reflectivity ferent. The differences between these models in frequency domain are shown on the right panels. The echoes obtained The Boltzmann reflectivity RB only has one free parameter using our model are weaker than those from the “inside” pre- H BH γ, which simply shifts the separation between echoes (as well scription. This is because Z does not peak at ω ∼ mΩ+ in as between the first echo and the GR wave) in the time do- our method (it peaks at ωQNM, as indicated in Fig. 4); while for eff B main by 2r∗ , as given by Eq. (71). Similar to the Lorentzian the “inside” model, it naturally peaks near ω ∼ mΩ+ by using in ∞ ∞ BH ∞ BH case, the new boundary condition on curvature perturbations (D /D )Z (or Z ) to estimate the in-going ψ4 toward leads to slightly weaker echoes, for the same reason discussed the horizon. Thus, the echoes from our method are suppressed above. by ω − mΩ+ when ω ∼ mΩ+, compared to the inside model. 14

FIG. 7. Impact of the particle self-field on echoes using the Lorentzian reflectivity for SN functions with = 0.2, MΓ = 1, b∗ = −75M (left) and the Boltzman reflectivity for SN functions with γ/M = 0.2 (right). Both panels display the first echo directly computed using the particle plunging orbit (red dashed curve), after removing self-field region (blue solid curve), and after removing all early-time waves (yellow solid curve).

1.0 1.0 0.5 0.5 BH BH R R 0.0 0.0 ECO ECO R R Re 0.5 Im 0.5 − m = 2 − m = 2 m = 2, ﬂipped m = 2, ﬂipped 1.0 − 1.0 − − − 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 − − Mω − − Mω

(a) (b)

FIG. 8. The real part (left panel) and the imaginary part (right panel) of RECORBH when ` = |m| = 2 using the Lorentzian reflectivity `mω h `mω i with = 1 and MΓ = 0.21. For m = −2, the real part is flipped such that Re RECO(−ω)RBH(−ω) is shown instead, and the imaginary h i part is also flipped such that −Im RECO(−ω)RBH(−ω) is plotted. This shows explicitly that the equatorial symmetry is preserved such that ∗ ECO BH ECO BH R`mω R`mω = R`−m−ωR`−m−ω .

It seems peculiar that the results from our model and the the fiducial observers, and hence a better approximation for inside model have a large discrepancy. As clarified above, the echoes. This is because ψ0 associated with the in-going one can see that the spacetime region above the plunge trajec- wave is directly responsible for tidal perturbations for the fidu- tory in the spacetime diagram (left panel of Fig. 2) is free of cial observers, and we obtain in-going ψ0 by applying the sources, and therefore ψ4 in this region should be character- Teukolsky-Starobinsky identity to the in-going ψ4. However, ized by a homogeneous solution with no incoming wave from strictly speaking, the Teukolsky-Starobinsky identity only ap- the past null infinity, or R∞ in Eq. (5). However, since the con- plies to homogeneous solutions. One hint that this approach version from ψ4 at infinity to ψ4 on the future horizon is done may not be sound is the fact that the amount of energy going in the Fourier domain, and the homogeneous solution is only down the horizon computed this way diverges for the (2, 2) valid for part of the future horizon, additional transient waves mode alone; the same behavior is also seen in Ref. [52] for may need to be added to this conversion, e.g., corresponding radially plunging particles. As discussed before [73], for a to the poles of the frequency-domain conversion factor. Thus, plunging point particle, even though the total in-going GW the ZH BH computed using our method is a more faithful rep- energy summed over all `’s are divergent, the energy corre- resentation of ψ4 going down the horizon. sponding to each individual ` does not diverge. We believe it is necessary to directly compute the in-going ψ , in order to However, a more faithful horizon-going ψ does not guar- 0 4 completely resolve the above issue. antee a better approximation for curvature perturbations for 15

Θ = 0 Θ = π/2 0.010 0.0050 plus plus 0.005 cross 0.0025 cross

0.000 0.0000 rh/M rh/M

0.005 0.0025 − −

0.010 0.0050 − 1200 1400 1600 1800 − 1200 1400 1600 1800 t/M t/M

(a) (b)

FIG. 9. The plus and cross polarizations of echoes when viewing the ECO at two diﬀerent angles, Θ = 0 (left panel) where the echoes are circularly polarized and Θ = π/2 (right panel) where the echoes are linearly polarized (no cross polarization) using our prescription and the Lorentzian reﬂectivity model with = 1 and MΓ = 0.21.

(a) Using Lorentzian reﬂectivity with parameters MΓ = 0.15, = 0.25, and b∗ = −75M

(b) Using Boltzmann reﬂectivity with parameters γ/M = 1 and TQH = TH

FIG. 10. Comparison between the echoes generated using the “inside” prescription and the method in this paper. Left panel: the ﬁrst echoes in time domain; middle panel: the second echoes in time domain; right panel: the ﬁrst and second echoes in frequency domain. Note that Refs. [50] and [51] both correspond to the “inside” prescription. The echoes in this paper are weaker than those obtained using the “inside” formulation.

V. DETECTABILITY which is deﬁned as [74]

Z ∞ ˜ 2 2 |h( f )| ρopt = 4 d f , (74) 0 S n( f ) In this section, we discuss the detectability of echoes with current and future detectors. To quantify the detectability, where S n( f ) is the one-sided noise power spectral density of a one can compute the optimal signal-to-noise ratio (SNR) ρopt, detector, and h˜( f ) is the strain measured by the detector which 16 is given by Figure 13 and 14 show the SNR and detectability of echoes in the TQH–γ parameter space for the Boltzmann reflectiv- h˜( f ) = F+h˜ +( f ) + F×h˜ ×( f ), (75) ity model assuming the Advanced LIGO and Cosmic Ex- plorer at their design sensitivities, respectively. Similar to with F+,× being the detector response to the plus and the cross the Lorentzian reflectivity model, echoes obtained from the polarization respectively. inside prescription are only detectable in a small part of the Following Ref. [74], we define a new quantity H+,×( f ) that parameter space explored (−1 ≤ log10 TQH/TH ≤ 1, −20 ≤ factors out the 1/dL dependence on the luminosity distance dL log10 γ ≤ 0) with Advanced LIGO, while we would not see [75] for each polarization, where any echoes from our model with second-generation detectors. Detecting echoes would become more promising with next- 1 h˜ +,×( f ) = H+,×( f ). (76) generation detectors. Interesting, from the plots we see that dL the detectability of echoes is generally independent of γ. The direction and orientation averaged optimal SNR hρ2i is then given by [74] VI. CONCLUSIONS 4 1 Z dΩ Z ∞ |H (Θ, Φ, f )|2 + |H (Θ, Φ, f )|2 hρ2i = d f + × , 2 π S f In this paper, we compute GW echoes from merging ECOs 5 dL 4 0 n( ) (77) that arise from the waves reflected by the surface of spinning where the angle bracket h...i denotes average over the sky lo- ECOs. The exterior spacetime of a spinning ECO is mod- cation angles of the source with respect to the detector, the eled as Kerr spacetime except in a small region above the polarization angle and the polar angles of the detector with horizon where a reflecting boundary exists. We obtain the respect to the source (with dΩ = sin ΘdΘdΦ). If we only echo waveforms by first computing the ψ4 of the GWs that consider the ` = |m| = 2 modes, the averaging over the ori- travel toward the horizon of the final BH in the case of BBH entation can also be done analytically. In fact, it is given by mergers, and then computing the subsequent reflection from [76] the ECO surface and the Kerr potential barrier, in the case of ECO. More specifically, we solve the Teukolsky equation for 16 1 Z ∞ |H (Θ = 0, Φ, f )|2 ψ sourced by an inspiraling particle that eventually plunges hρ2i d f + . 4 = 2 (78) 25 d 0 S n( f ) into the horizon of a Kerr BH. In order to model binaries with L comparable masses, which is the most interesting case with Similarly, we can also compute the maximal ρopt by setting existing GW events, we have adopted an EOB-like approach: the source to be face-on (Θ = 0) and directly above a detector by modifying the trajectory of the infalling particle and cali- (F+,× = 1), i.e. both optimally oriented and optimally located. brating the GWs at infinity to match NR surrogate waveforms, We compute both the direction-and-orientation averaged, we can obtain the horizon-going GWs that approximate those as well as the maximal optimal SNR of the first five echoes of comparable-mass mergers. For comparable-mass binaries, in Advanced LIGO at the design sensitivity [77] and Cosmic our approach is only designed for echoes reflected by the sur- Explorer [78] with both the Lorentzian and Boltzmann reflec- face of the final ECO. Nevertheless, from Fig. 7 (and the as- tivities, using the inside prescription and the prescription in sociated discussions), we show that most of the echoes arise this paper. However, we only use the Teukolsky formulation from the reflection of waves that propagate toward the final for reflectivity; the corresponding transformed SN reflectivity horizon after the particle plunges. This justifies our approxi- becomes weaker (see Fig. 3 for ` = m = 2 where most of the mation. wave contents are concentrated in the positive frequencies). In the comparison with previous work, we use prescriptions These differ from the treatment in Ref. [51]. of the ECO reflectivity that are better connected to spacetime Figure 11 and 12 show the SNR and detectability of echoes geometry near the ECO surface (obtained from a companion in the –Γ parameter space for the Lorentzian reflectivity paper [70]). More specifically, the reflectivity RECO T in our model assuming the Advanced LIGO and Cosmic Explorer at method is directly related to the tidal response of the ECO their design sensitivities, respectively. For Advanced LIGO, surface to the external curvature perturbations due to incom- we see that the echoes computed using the inside prescrip- ing GWs. This reflectivity can also be converted into an effec- tion are only detectable in a small part of the parameter space, tive reflectivity for SN functions using Eq. (58). Numerically, while the echoes obtained using the prescription in this pa- this conversion factor leads to discrepancies in the resulting per are too weak to be detected in the parameter space that waveforms, although it does not qualitatively modify the main we explore here (0 ≤ ≤ 1, 0 ≤ Γ/κ ≤ 1). This implies features of the echo waveform. that if our prescription is correct, we would not be able to de- The echo waveforms we obtain here are significantly tect echoes with second-generation terrestrial detectors, and weaker than those obtained by Maggio et al. and from the “in- would require next-generation detectors in order to test the side” formulation of Wang et al., because the ψ4 we obtain by existance of ECOs via GW echoes. Indeed, Fig. 12 indicates directly solving the Teukolsky equation turns out to be much that with Cosmic Explorer, a much larger fraction of the – smaller in magnitude than those obtained in previous studies Γ parameter space allows detection for echoes from both the (see Sec. IV D for details). As shown in Sec. V, the echoes inside model and our prescription. obtained in this paper are not expected to be detectable us- 17

M = 60M , q = 1, dL = 100 Mpc M = 60M , q = 1, dL = 100 Mpc 1.0 80 1.0

ρopt 80 0.8 = 8 0.8 60 L 60 L 0.6 /d 0.6 /d 1 1

40 ∝ ∝ 40 0.4 0.4 opt opt ρ ρ 20 0.2 0.2 20 Advanced LIGO at design sensitivity Advanced LIGO at design sensitivity using the inside prescription using our prescription 0.0 0 0.0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Γ/κ Γ/κ

(a) Inside prescription (b) Our prescription

FIG. 11. SNR and detectability of echoes in the –Γ parameter space for the Lorentzian reflectivity model assuming the Advanced LIGO design sensitivity [77] with Necho = 5 at a luminosity distance of dL = 100 Mpc. The solid contour corresponds to the maximal SNR ρopt = 8 as the detection threshold. Here we set b∗ = −100M. This choice of b∗ is not expected to affect the detectability as it mostly affects the time delay between echoes. We see that echo signals from the inside prescription are only detectable in a small part of the parameter space, while the echoes obtained using the method in this paper are generally too weak to be detected.

M = 60M , q = 1, dL = 100 Mpc M = 60M , q = 1, dL = 100 Mpc 1.0 > 80 1.0 > 80

0.8 0.8 60 60 L L /d /d 0.6 0.6 2

1 ρ 1 h opt q = 8 i

40 ∝ 40 ∝ ρ 2 h opt ρ 0.4 q = 8 0.4 opt opt = 8 opt ρ i

opt ρ ρ = 8 20 20 0.2 0.2 Cosmic Explorer Cosmic Explorer using the inside prescription using our prescription 0.0 0 0.0 0 0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00 Γ/κ Γ/κ

(a) Inside prescription (b) Our prescription

FIG. 12. Same as Fig. 11 but with Cosmic Explorer at its design sensitivity [78]. The solid contours correspond to the maximal SNR ρopt = 8 as the detection threshold, while the dash-dotted contours correspond to the location-and-orientation averaged SNR of 8. We see that the detectable –Γ parameter space using the inside prescription is much larger, with the threshold of being detectable reaching as low as ≈ 0.2. With Cosmic Explorer, the echoes computed using our prescription are now strong enough to be detected, but with a smaller detectable parameter space compared with that using the inside prescription. ing the second-generation detectors, and we would need the ACKNOWLEDGMENTS next-generation detectors to test ECOs via GW echoes. One subtlety in our calculation is about obtaining the reﬂec- ECO T tivity for ψ4 on the ECO surface. When applying the R Research of B. C. and Y. C. are supported by the Simons between the in-going and out-going ψ4, the Starobinsky- Foundation (Award Number 568762), the Brinson Founda- Teukolsky relation between the in-going ψ0 and the in-going tion, and the National Science Foundation (Grants PHY– ψ4 has been assumed. Strictly speaking, this relation only ap- 2011968, PHY–2011961 and PHY–1836809). Research of plies to homogeneous solutions of the Teukolsky equation and S. X. and Q. W. was done in part during their visits to the may not apply to the situation here. A more direct approach, California Institute of Technology, which was supported by to be studied in future work, is to compute the in-going ψ0, funds from the Simons Foundation. R. K. L. L. and L. S. ac- and then obtain the ψ4 of the echoes by applying reﬂectivities knowledge support of the National Science Foundation and to the ECO surface. the LIGO Laboratory. LIGO was constructed by the Cali- 18

M = 60M , q = 1, dL = 100 Mpc M = 60M , q = 1, dL = 100 Mpc ρ opt 1.0 = 8 60 1.0 60

ρopt = 8 ) 0.5 ) 0.5 H H 40 L 40 L /T /T /d /d 1 1 QH QH

0.0 ∝ 0.0 ∝ T T ( ( 10 10 opt opt

20 ρ 20 ρ log 0.5 log 0.5 − − Advanced LIGO at design sensitivity Advanced LIGO at design sensitivity using the inside prescription using our prescription 1.0 0 1.0 0 − 20 15 10 5 0 − 20 15 10 5 0 − − − − − − − − log10(γ/M) log10(γ/M)

(a) Inside prescription (b) Our prescription

FIG. 13. SNR and detectability of echoes in the TQH–γ parameter space for the Boltzmann reﬂectivity model assuming the Advanced LIGO design sensitivity [77] with Necho = 5 at a luminosity distance of dL = 100 Mpc. The solid contour corresponds to the maximal optimal SNR ρopt = 8 as the detection threshold. We see that echoes obtained using the inside prescription are louder and detectable in part of the TQH–γ parameter space, while echoes obtained using our method are too weak to be detected.

M = 60M , q = 1, dL = 100 Mpc M = 60M , q = 1, dL = 100 Mpc 1.0 > 60 1.0 > 60

) 0.5 ) 0.5 H H 40 L 40 L /T /T

/d 2 /d ρopt = 8 1 h i 1

QH QH q

0.0 2 ∝ 0.0 ρopt = 8 ∝ T ρopt = 8 T ( h i ( q 10 10 opt opt

ρopt = 8 20 ρ 20 ρ log 0.5 log 0.5 − − Cosmic Explorer Cosmic Explorer using the inside prescription using our prescription 1.0 0 1.0 0 − 20 15 10 5 0 − 20 15 10 5 0 − − − − − − − − log10(γ/M) log10(γ/M)

(a) Inside prescription (b) Our prescription

FIG. 14. Same as Fig. 13 but with Cosmic Explorer at its design sensitivity [78]. The solid contours correspond to the maximal optimal SNR ρopt = 8 as the detection threshold, while the dash-dotted contours correspond to the location-and-orientation averaged SNR of 8. Again, we see that the detectable TQH–γ parameter space with the inside prescription is much larger for Cosmic Explorer, compared with Fig. 13 for Advanced LIGO. Echoes computed using our prescription are also loud enough to be detected, but again the detectable parameter space is smaller compared with the inside prescription. The plots also indicate that the detectability of echoes is generally independent of the value of γ. fornia Institute of Technology and Massachusetts Institute of from the Gordon and Betty Moore Foundation. Technology with funding from the United States National Sci- ence Foundation, and operates under cooperative agreement PHY–1764464. Advanced LIGO was built under award PHY– 0823459. R. K. L. L. would also like to acknowledge support from the Croucher Foundation. L. S. also acknowledges Appendix A: Source term in Teukolsky equation the support of the Australian Research Council Centre of Ex- cellence for Gravitational Wave Discovery (OzGrav), project Consider a massive point particle with the trajectory xµ = number CE170100004. W.H. is supported by by NSFC (Na- (t, r(t), θ(t), φ(t))(expressed in Boyer-Lindquist coordinates) tional Natural Science Foundation of China) no. 11773059 in Kerr spacetime with instantaneous energy E and angular The computations presented here were conducted on the Cal- momentum in z direction Lz. Following [79], the source term tech High Performance Cluster partially supported by a grant T`mω induced by this trajectory is (see Eq. (2.24) of [79]): 19

Taking Eq. (B1) into Teukolsky equation, one can find the Z ∞ equation for SN function X`mω: 0 i[ωt−mφ(t)] 2 0 T`mω(r ) = dte ∆ (r ) d2X dX −∞ `mω − F(r) `mω − U(r)X = 0. (B2) n 0 ∗ 2 ∗ `mω [Ann0 + Anm¯ 0 + Am¯ m¯ 0]δ(r − r(t)) (dr ) dr 0 + ∂r0 ([Anm¯ 1 + Am¯ m¯ 1]δ(r − r(t))) A detailed discussion of the SN formalism can be found in 2 0 o Ref. [63]. Expressions for functions α, β, η and the potentials 0 − + ∂r [Am¯ m¯ 2δ(r r(t))] , (A1) F(r), U(r) can be found in Eqs. (3.4)–(3.9) of Ref. [68]. For completeness, we list η here: where the coefficients are given by c c c c η = c + 1 + 2 + 3 + 4 , (B3) −2 0 r r2 r3 r4 A C ρ−2ρ−1L†{ρ−4L† ρ3S }, nn0 = 2 nn ¯ 1 2( ) ∆√ with 2 2 iK K A = C ρ−3 + ρ + ρ¯ (L†S ) − a sin θS (¯ρ − ρ) , mn¯ 0 ∆ mn¯ ∆ 2 ∆ c0 = −12iωM + λ(λ + 2) − 12aω(aω − m), (B4) " 2 # c = 8ia[3aω − λ(aω − m)], (B5) −3 K K K 1 Am¯ m¯ 0 = − ρ ρ¯Cm¯ m¯ S −i∂r − + 2iρ , ∆ ∆2 ∆ c = −24iaM(aω − m) + 12a2[1 − 2(aω − m)2], (B6) √ 2 3 2 2 2 −3 † c3 = 24ia (aω − m) − 24Ma , (B7) Amn¯ 1 = ρ Cmn¯ [L2S + ia sin θ(¯ρ − ρ)S ], ∆ c = 12a4. (B8) K 4 A = − 2ρ−3ρ¯C S (i( + ρ), m¯ m¯ 1 m¯ m¯ ∆ The SN equation admits two homogeneous solutions having −3 Am¯ m¯ 2 = − ρ ρ¯Cm¯ m¯ S, purely sinusoidal asymptotic behavior since the potential U(r) (A2) is short-ranged: with H hole −ipr∗ X`mω = A`mωe , r → r+, " #2 (B9) 1 dr H out iωr∗ in −iωr∗ → ∞ C = E(r2 + a2) − aL + Σ , X`mω = A`mωe + A`mωe , r , nn 4Σ3dt/dτ z dτ " # and −ρ 2 2 dr Cmn¯ = √ E(r + a ) − aLz + Σ × ∞ out ipr∗ in −ipr∗ 2 dτ X = C e + C e , r → r+, 2 2Σ dt/dτ `mω `mω `mω (B10) " !# (A3) ∞ ∞ iωr∗ L X = C e , r → ∞. i sin θ aE − z , `mω `mω sin2 θ The homogeneous equations XH,∞ are related to RH,∞ by 2 " !#2 ρ Lz Eq. (B1) and thus the asymptotic amplitudes Ahole, Ain, Aout, Cm¯ m¯ = i sin θ aE − . ∞ ∞ 2Σdt/dτ sin2 θ Cin, Cout, C , Bhole, Bin, Bout, and Din, Dout, D have following relations [79]: The operators appearing in the coefficients are defined by 1 Bin = − Ain , † m `mω 4ω2 `mω Ls = ∂θ − + aω sin θ + scotθ. (A4) sin θ 4ω2 Bout = − Aout , √ `mω −12iωM + λ(λ + 2) − 12aω(aω − m) `mω The coefficients are different from [79] by a factor of 1/ 2π hole −1 hole due to normalization. The spin-weighted spheroidal harmon- B`mω =d`mωA`mω, ics functions we use are normalized as (while in [79] this ex- (B11) pression is normalized to 1): in −1 in D`mω =d`mωC`mω, √ √ Z π 2 2 aω 2 1 4p 2Mr+(2Mr+ p + i M − a ) [− S (θ)] sin θdθ = . (A5) out out 2 `mω D`mω = − C`mω, 0 2π η(r+) 2 ∞ 4ω ∞ D`mω = − C`mω. Appendix B: Sasaki-Nakamura formalism −12iωM + λ(λ + 2) − 12aω(aω − m) (B12) where Transformation between Teukolsky function R`mω and SN p h − − 2 2 2 function X`mω is given by[63]: d`mω = 2Mr+ (8 24iMω 16M ω )r+ " ! # + (12iam − 16M + 16amMω + 24iM2ω)r 1 β ∆X β d ∆X + ,r `mω − `mω i R`mω = α + √ √ . (B1) − 2 2 − 2 η ∆ r2 + a2 ∆ dr r2 + a2 4a m 12iamM + 8M . (B13) 20

We note the subtlety that since the transformation between X not vanish, one has to expand the coeﬃcients of X and R up and R contains derivatives, in regions where the source does to the second order in 1/r at inﬁnity and ∆2 near horizon in order to obtain the correct transformations.

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