Gravitational tuning forks and hierarchical triple systems
GGravitational tuning forks and hierarchical triple systems
Vitor Cardoso and Francisco Duque
CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico – IST,Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
Gaurav Khanna
Department of Physics and Center for Scientific Computing and Visualization Research,University of Massachusetts, Darthmouth, MA 02747Department of Physics, The University of Rhode Island, Kingston, RI 02881
We study gravitational wave (GW) emission in the strong-field regime by a hierarchical triplesystem composed of a binary system placed in the vicinity of a supermassive black hole (SMBH). TheLIGO-Virgo collaboration recently reported evidence for coalescences with this dynamical origin.These systems are common in galactic centers and thus are a target for the space-based LISAmission as well as other advanced detectors. Doppler shifts, aberration, lensing and strong amplitudemodulations are features present in the GW signal from these systems, built into our frameworkand with no need for phenomenological patches. We find that the binary can resonantly excite thequasinormal modes of the SMBH, as in the resonant excitation of two tuning forks with matchingfrequencies. The flux of energy crossing the SMBH horizon can be significant, when compared withthat from standard extreme-mass-ratio inspirals. Therefore, these triple systems are an outstandingprobe of strong-field physics and of the BH nature of compact objects.
Introduction.
Since the birth of the gravitational-wave(GW) era in 2015 [1], dozens of GW events have beendetected [2]. Other detectors will soon join the ground-based network and further improve our ability to mea-sure GWs in the 1 − Hz frequency range [3, 4]. Thespace-based LISA mission will extend detection to the ∼ − − − Hz window. GWs with these frequen-cies are emitted in galactic centers by supermassive blackholes (SMBHs) and extreme-mass-ratio inspirals (EM-RIs), but also by cosmological sources [5, 6]. The cover ofsuch a broad spectrum will allow us to test General Rela-tivity with unprecedented precision over a wide range ofscales, and to answer questions regarding the nature ofcompact objects, of dark matter and dark energy [5, 6].However, recent results question the validity of the“standard” binary system. During its third observationrun, the LIGO-Virgo collaboration detected three BH bi-nary coalescences [7–10], unlikely to be composed by twofirst-generation BHs [11, 12]. Instead, their componentsare thought to be remnants of previous coalescences,forming what is called a “hierarchical merger” [9, 11–14]. Generally, these require the presence of a thirdbody to induce coalescence. The Zwicky Transient Fa-cility [15, 16] reported an electromagnetic counterpartto one of these events, GW195021 [17], consistent withthe presence of the BH binary in an active galactic nuclei(AGN) [18–22], reinforcing the claim that its componentswere part of a hierarchical triple system. “Hierarchical”here refers to the distinct length scales between the orbitof the BH binary and the one of its center-of-mass (CM)around the third body. Hierarchical triple systems arecommon in a variety of astrophysical scenarios, such as,globular clusters [13, 23], AGNs [18, 24, 25], and otherdense stellar environments [26–28]. Around 90% of lowmass binaries with periods shorter than 3 days are ex-pected to belong to some hierarchical structure [29–31]. The above motivated recent studies on the dynamicsand GW emission in hierarchical triple systems Kozai-Lidov resonances, in particular, have attracted some at-tention [32–34]. These describe secular changes in the bi-nary eccentricity and inclination with respect to the orbitdescribed by its CM around the third object. This mech-anism triggers periods of high eccentricity ( e ∼
1) whereGW emission increases significantly, potentially inducingcoalescence in eccentric orbits detectable by LISA [35–38], which may enter the LIGO-Virgo band still at higheccentricities [23, 39–41]. Moreover, it can lead to GWbursts at periapsis [42, 43]. A direct integration of theequations of motion confirms that GWs from these sys-tems have unique features [43], which may be detectedindirectly via radio observations of binary pulsars [44].There are also attempts at modeling the effects of a thirdbody directly into the waveform. These include Dopplershifts [45–49], relativistic beaming effects [50, 51], grav-itational lensing [52, 53] and other dynamical effects intriple systems caused by the third-body [54–56].Studies so far are restricted to the (post-)Newtonianregime and cannot capture strong-field effects. Here, wetake a first step towards this direction, and investigateGWs from binaries around SMBHs. Our methods canprobe resonant excitation of quasinormal modes (QNMs)in triple systems, and capture for free all of the relativisticeffects which have so far been included at a phenomeno-logical level only. We adopt units where c = G = 1. Setup: Hierarchical triple systems.
We are inter-ested in a setup where a small binary (SB) of compactobjects is in the vicinity of a “large” BH (larger than allthe lengthscales of the SB), as illustrated in Fig. 1. TheSB is taken to be a small perturbation in a backgrounddescribed by the geometry of the massive BH, which invacuum must belong to the Kerr family. We use Boyer-Lindquist coordinates { t, r, θ, ϕ } [57] in our study and a r X i v : . [ g r- q c ] J a n define Σ := r + a cos θ and ∆ := r − M r + a . Thereis an event horizon at r + = M + √ M − a .The SB is modeled as composed of two point particles ± . The SB components also carry a scalar charge α in oursetup. If τ denotes the proper time of each point particlealong the world line z µ ( τ ) = ( t ( τ ) , r ( τ ) , θ ( τ ) , ϕ ( τ )),the corresponding stress-energy tensor is T µν ( x ) ± = m ± (cid:90) + ∞−∞ δ (4) ( x − z ( τ )) dz µ dτ dz ν dτ dτ , (1)with (cid:82) (cid:82) (cid:82) (cid:82) δ (4) ( x ) √− gd x ≡ m ± is the rest massof each component of the compact binary.First-order perturbations on the Kerr spacetime aredescribed by Teukolsky’s master equation [58] L s Ψ =Σ T , where L is a second-order differential operator, s refers to the “spin weight” of the perturbation field (e.g., s = 0 , ± T isa spin-dependent source term [58].To compute the source T , we need to prescribe the mo-tion of the SB. We take the CM at r = R ( τ ) to either bestatic at some fixed radius, to describe a timelike equa-torial circular orbit around a Kerr BH, or then a simpleplunge. For the SB inner motion, we take elliptic orbitsaround the CM, such that ϕ ± = Ω CM t ± (cid:15) ϕ sin ω t , θ ± = π/ ± (cid:15) θ cos ω t , (2)where (cid:15) θ , (cid:15) ϕ (cid:28) δR θ ≡ (cid:15) θ R , δR ϕ ≡ (cid:15) ϕ R of the SB and Ω CM is the angularvelocity of the CM. Note that Ω CM and ω are coordinatefrequencies, while the proper oscillation frequency of theSB, ω (cid:48) , is obtained by a rescaling with the time compo-nent of the 4-velocity of the CM, i.e. ω (cid:48) = U t CM ω . Forconcreteness, we focus exclusively on equal-mass binaries, m ± = m and a highly eccentric orbit with (cid:15) θ = 0 (we donot see any qualitatively new phenomena in the generalcase; this particular choice could mimic high-eccentricitybinaries driven by Kozai-Lidov resonances).A physical relation between (cid:15) ϕ and ω must be im-posed. In the SB’s rest frame, δR (cid:48) ϕ ∝ / ( ω (cid:48) ) / , wherethe prime refers to proper quantities. For SBs on circulargeodesics, for example, doing the appropriate rescaling ω (cid:48) = U t CM ω and δR ϕ = ∆ / Σ · δR (cid:48) ϕ , we find (cid:15) ϕ ∝ ∆Σ 1 R ( U t CM ω ) / . (3)We are looking for possible resonances in this triplesystem, which may happen when the forcing frequencyequals natural frequencies of the system. There are threeimportant frequencies in the problem: that of the CM,that of null geodesics on the light ring (LR), and theangular velocity of the BH horizon Ω H = a/ (2 M r + ) [59].Close to the BH all are of order O (1 /M ), which in fact arealso of the order of the QNM frequencies of the centralBH [60]. To have M ω ∼
1, we need to ensure δR/m ∼ ( M/m ) / . For a SMBH with M ∼ − M (cid:12) , likeSagittarius A*, and a SB composed by stellar-mass BHs,this would correspond to δR/m ∼ − . Therefore, FIG. 1. Equatorial slice of a spacetime with a hierarchicaltriple system, where one component is a central SMBH. Weplace a small binary (SB) of frequency ω orbiting the SMBH.At the innermost stable circular orbit (ISCO), timelike cir-cular motion is marginally stable. High-frequency GWs are(semi-) trapped at the light ring (LR). Such motion is unsta-ble, and can be associated with the “ringdown” excited duringmergers. Among other effects, here we show that the LR canbe excited by tuning ω . (cid:96) s a/M Mω QNM / Mω LR Mω ISCO s R LR s R ISCO
TABLE I. Frequency Mω X which maximizes the energy out-put of a SB standing at location X close to a SMBH, in a given( (cid:96), (cid:96) ) mode, as measured by the ratio s R ( s = 0 , − the SB can probe the central BH while still well withinthe inspiral phase of its evolution. Numerical implementation.
We used two differentnumerical schemes to solve Teukolsky’s equation. Oneworks in the time domain, and it smooths the pointlikecharacter of the SB constituents [61–64]. The other tech-nique is based on separation of angular variables usingspheroidal harmonics [65] in the frequency domain, whereone can apply standard Green function techniques [66–70]. Both approaches are well documented and have beenwidely tested in the past. Both codes were comparedwith analytical estimates in the low-frequency regime,obtained using matched asymptotic techniques [70–72].Results from these independent codes are consistent witheach other and with analytical estimates.
Resonant excitation of QNMs.
We now use the SBas a tuning fork, placing it at some fixed radius, with itsCM fixed with respect to distant observers, and letting
FIG. 2. Energy output when a SB stands at the ISCO ofa SMBH of spin a = 0 . M , as a function of the orbital fre-quency of the SB components, ω . The modal energy output,as measured by − R , peaks at a finite ω extremely well de-scribed by the lowest QNM (cf. Table I). Also shown is theflux integrated over all modes: it has a substantial componentgoing down the SMBH horizon, and the total flux at infinity ismodulated by QNM contributions. Here, ˆ ω (cid:96)m ≡ Mω QNM / its frequency ω vary. In flat space, this system radiatesa (time-averaged) scalar flux in the (cid:96) , m mode ( J ν ( z ) isa Bessel function of first kind [73]) ˙ E N (cid:96) m = m α (cid:15) ϕ Γ ( (cid:96) + 3 / √ π (cid:96) ! R m ω J (cid:96) +1 / ( R ω ) , (4)and a similar but more cumbersome expression for theNewtonian gravitational-wave flux − ˙ E N (cid:96) m . Define anestimate of the SMBH impact through the ratio s R (cid:96) m = s ˙ E (cid:96) m / s ˙ E N (cid:96) m . (5)Our results indicate that at large distances R this ratiotends to unity, as it should on physical grounds.Figure 2 shows the behavior of − R as the SB fre-quency ω changes, for an SB sitting at the ISCO ofa SMBH. The behavior is similar for other modes andfields. We observe a peak which we identify as a reso-nant excitation of the (cid:96) = m = 3 QNM. As shown inTable I, the location of the peak is well described by thelowest QNM frequency [60], for general binary locations.When the SB is placed at the LR, the agreement is ex-cellent (better than 1% for scalars, and 4% for GWs forthe lowest modes (cid:96) m modes). Recall that QNMs can beinterpreted as waves marginally trapped in unstable or-bits on the photon-sphere [74]. We therefore arrive at thefirst major result of this paper: a hierarchical triple sys-tem behaves as a driven harmonic oscillator [75], wherethe SB is the external harmonic force and the central BHthe (damped) oscillator.This behavior is analogous to the Purcell effect inquantum electrodynamics [76], describing the enhance-ment in the spontaneous decay of a quantum emitter in-side a cavity, when its frequency matches those of themodes of the field inside the cavity. Our results are con-sistent with recent findings [77], namely that the spatially independent (i.e. independent of R ) contribution to thepower spectrum in Fig. 2 is described by a Lorentziancurve R ∝ ω / ( ω + 4 Q ( ω − ω QNM ) ), where Q is the quality factor of the central BH. Our results areconsistent with and extend those of Ref. [78], where res-onant excitation of QNMs was observed for EMRIs ineccentric orbits, during passage on the periapsis. The ef-fect is stronger the closer the particle can get to the LR,as also conclude in Ref. [79].As a rule of thumb, the flux peaks at lower frequenciesthe further the SB is placed from the BH, in agreementwith blueshift/redshift corrections. Note that R smallerthan unity does not imply that the system is emittingless energy than expected, since a portion of the radia-tion falls into the BH. Also, a possible CM orbital mo-tion contributes to a shift in the resonant frequencies by ± m Ω CM , fully consistent with our results. The maximumvalue of R in the entire ( R, ω ) parameter space does notoccur precisely at the LR, but close to it. The maximumis attained at locations R closer to the horizon for large (cid:96) . Finally, the magnitude of the resonance grows with (cid:96) .For a fixed CM location R and multipole (cid:96) we searchedfor ω for which s R is a maximum s R peak . We find anexponential dependence on (cid:96) , s R peak ∼ a + b exp( c · (cid:96) ), atlarge (cid:96) with a, b, c constants. Total integrated flux.
Ours is a mode decompositionin terms of harmonics of the central BH, thus radiationhas support in higher modes as the binary is placed fur-ther away from it [65, 80]. In general, therefore, thelowest modes will not be dominant and one needs to suma sufficient amount of modes to understand total fluxes.Already for a SB at the ISCO of a non-rotating BH wefind that the GW flux at infinity is comparable to thatat the horizon of the SMBH. As seen in Fig. 2, the effectis more dramatic when spin is included, the flux crossingthe horizon can be orders of magnitude larger than thatat infinity, even including superradiant modes [81]. Thispeculiar aspect is due to the similar length scales of thecentral BH horizon and the radiation wavelength. GWsare then efficiently absorbed by the BH, in clear contrastwith the inspiral phase of an EMRI, whose wavelength ismuch larger than the BH radius. This is our second ma-jor result: hierarchical triple systems where the SMBHoccupies a large fraction of the SB’s sky will naturallyprobe strong field physics, since the fraction of radiationthat falls into the SMBH is non-negligible. This will beessential for dynamical evolutions of these systems, par-ticularly when accounting for radiation reaction effects.For a fixed radius R , the field has support on higher (cid:96) modes as the SB is vibrating at higher frequencies ω . Ifthe SB is close enough to the BH, it can resonantly excitethe QNMs, leading to characteristic peaks in the fluxat infinity/horizon, as seen in Fig. 2. These structurescorrespond to the single multipolar excitations studied inthe previous section. Waveforms: Doppler, aberration & lensing.
As aby-product of our methods, we can calculate waveformsfrom SBs close to SMBHs, which feature interesting rel-
220 240 260 280 300 320 340 - - FIG. 3. Teukolsky function Ψ measured by an (anti)alignedstationary observer at r = 75 M , for a SB with constant properfrequency Mω (cid:48) = 1 . r = 30 M withzero initial velocity. The dotted lines correspond to the CMcontribution to the signal. The SB crosses r = 10 M at t ∼ M , the ISCO at t ∼ M and the LR at t ∼ M . ativistic effects. Figure 3 shows the GW signal producedwhen a SB, of constant proper frequency ω (cid:48) falls radi-ally from rest into a non-rotating SMBH. The signal isshown for observers sitting along the merger direction,podal and anti-podal. The observer aligned with the SBsees it moving away, and a GW signal that is progres-sively redshifted both kinematically and gravitationally(the shifts – barely visible to the naked eye, are presentand agree with expectations). An anti-aligned observersees a blueshifted signal. As the SB crosses the LR, theradiation it emits is semi-trapped and the signal ringsdown: the large frequency of the signal is still dictatedby the SB, but is now modulated by a low frequency( ∼ . /M ) decay ( ∼ e − . t ). The parameters of such de-cay and low-frequency modulation agree remarkably wellwith the frequency and damping time of null geodesicsat the LR. Imprints of the binary nature of the SB areclearly left on the ringdown stage, that differs visiblyfrom that generated by a point-mass.Finally, Fig. 4 shows the GW measured by stationaryobservers at large distances, for a SB on circular motionat the ISCO of a non-rotating BH. These are signals cal-culated from first-principles. We removed the (linear)CM contribution, which only induces a low-frequencymodulation. Observers on the equatorial plane see gravi-tational and Doppler-induced frequency shifts, consistentwith analytical predictions [45, 82] when the CM is mov-ing towards the observer. The amplitude of the wavecan vary by orders of magnitude because of relativisticbeaming [43, 50, 51] and gravitational lensing [53, 83].The former focuses the radiation along the direction ofmotion, and is significant for fast CM motion. The max-imum amplitude does not occur precisely when the SBis moving towards the observer ( t ∼ M in Fig. 4) butslightly before, when the SB is still behind the BH withrespect to the observer. This is due to lensing by the cen-tral BH, which distorts the path taken by GWs and con- - - FIG. 4. Teukolsky function Ψ measured by a stationary ob-server at large distances (either edge- or face-on, θ = π/ , T CM ≈ M and at t = 0the observer is aligned with the SB. Doppler effect inducesfrequency shifts, relativistic beaming and gravitational lens-ing modulations in the amplitude. The maximum blue-shiftis well described by ω max = ω (cid:48) Υ (Υ + v CM ) / (Υ − v CM ), withΥ = (cid:112) − M/R , Mω (cid:48) = 1 the proper SB frequency and v CM is the CM velocity [45, 82]. centrates radiation on certain directions, amplifying thesignal [84, 85]. This effect is more relevant for larger fre-quencies, when the radiation wavelength is much smallerthan the BH radius. On the other hand, observers fac-ing the plane of motion “face-on” ( θ = 0) do not mea-sure such modulations, since the motion of the CM isnow transverse. The only feature is a modulation in am-plitude coming from the CM motion (at second order),which has also been reported in Post-Newtonian studiesof triple systems [43]. Discussion.
We show that a stellar-mass binary sys-tem (or any other radiator) in the vicinity of a SMBH isan exquisite probe of strong gravity. Under special cir-cumstances, the binary can resonantly excite the modesof the SMBH, a unique opportunity to probe the Kerrgeometry and the presence of horizons in the cosmos.Such classes of hierarchical triple systems are abun-dant in AGNs, and thus our results have implicationsfor GW astronomy, in particular for LISA which is spe-cially designed to detect GWs originated in galactic cen-ters [6]. We can estimate if a SB can get close enoughbefore being tidally disrupted due to the Hills mecha-nism [86–88]. This occurs if the tidal forces induced bythe BH overcome the binary’s self gravity, which hap-pens at a radius R t ∼ δR ( M/ m ) / . The SB fre-quency will be related to its separation by the Kepler’slaw ω ∼ (cid:112) m /δR . We thus find R t (cid:46) / ( M ω ) / M .Already for M ω = 0 .
2, we find that tidal disruptionhappens at R t ∼ . M , smaller than the ISCO of aSchwarzschild BH. Thus, SBs very close to a central BHand oscillating at relevant frequencies of the system haveastrophysical interest. This is supported by more so-phisticated numerical works [89]. We neglected spin-spineffects in the motion of the SB. The corrections are pro-portional to σ = qJ/m , with J the angular momentumof the SB [90]. Again using Kepler’s law, one finds thatcorrections to the motion scale like σ ∝ q / , which areextremely small for the systems we consider.A follow-up to our work is to study the capacity of GWdetectors to distinguish between these systems and iso-lated binaries. In particular, it is important to quantifythe systematic errors incurred in parameter estimationsfrom a signal originated in a hierarchical triple, using GWtemplates for isolated binaries. Moreover, it is importantto extend our study to other motions. An interestingcase is a SB describing a high-eccentricity orbit arounda spinning SMBH. Such eccentric orbits can be formednaturally in non-trivial environments [91]. In these or-bits, the SB gets closer to the LR, which enhances theresonant excitation of the SMBH [78] and may lead tomanifestations of superradiance [81]. Another interestingtriple system is a pair same-sized BHs and a third lightercompact object orbiting around them. These spacetimeshave been shown to have global properties not present inisolated BHs (e.g. global QNMs) [92, 93] and our resultssuggest that the lighter object can excite these globalmodes. Acknowledgments.
We thank Ana Carvalho for pro- ducing some of the figures in this work. We are grate-ful to B´eatrice Bonga, Emanuele Berti, Hirotada Okawaand Paolo Pani for useful comments and suggestions.We thank UMass Darthmouth and Waseda Universityfor warm hospitality while this work was being finalized.F.D. is indebted to Nur Rifat and Asia Haque for helpprovided during his stay in UMass Darthmouth. V.C.acknowledges financial support provided under the Eu-ropean Union’s H2020 ERC Consolidator Grant “Matterand strong-field gravity: New frontiers in Einstein’s the-ory” grant agreement no. MaGRaTh–646597. F.D. ac-knowledges financial support provided by FCT/Portugalthrough grant No. SFRH/BD/143657/2019. G.K.would like to acknowledge support from the Na-tional Science Foundation (NSF) under awards PHY-2106755 and DMS-1912716. This project has re-ceived funding from the European Union’s Hori-zon 2020 research and innovation programme un-der the Marie Sklodowska-Curie grant agreement No101007855. We thank FCT for financial support throughProject No. UIDB/00099/2020. We acknowledge fi-nancial support provided by FCT/Portugal throughgrants PTDC/MAT-APL/30043/2017 and PTDC/FIS-AST/7002/2020. The authors would like to acknowl-edge networking support by the GWverse COST ActionCA16104, “Black holes, gravitational waves and funda-mental physics.” [1] LIGO Scientific Collaboration and Virgo Collaboration,B. P. Abbott et al. , Phys. Rev. Lett. , 061102 (2016).[2] LIGO Scientific, Virgo, R. Abbott et al. , 2010.14527.[3] KAGRA, T. Akutsu et al. , Nature Astron. , 35 (2019),[1811.08079].[4] M. Punturo et al. , Classical and Quantum Gravity (2010).[5] L. Barack et al. , Class. Quant. Grav. , 143001 (2019),[1806.05195].[6] E. Barausse et al. , 2001.09793.[7] LIGO Scientific, Virgo, R. Abbott et al. , Phys. Rev. D , 043015 (2020), [2004.08342].[8] LIGO Scientific, Virgo, R. Abbott et al. , Astrophys. J.Lett. , L44 (2020), [2006.12611].[9] LIGO Scientific, Virgo, R. Abbott et al. , Phys. Rev. Lett. , 101102 (2020), [2009.01075].[10] LIGO Scientific, Virgo, B. Abbott et al. , Astrophys. J.Lett. , L3 (2020), [2001.01761].[11] B. Liu and D. Lai, 2009.10068.[12] G. Fragione, A. Loeb and F. A. Rasio, Astrophys. J. ,L26 (2020), [2009.05065].[13] M. A. Martinez et al. , Astrophys. J. , 67 (2020),[2009.08468].[14] W. Lu, P. Beniamini and C. Bonnerot, 2009.10082.[15] M. J. Graham et al. , Publ. Astron. Soc. Pac. , 078001(2019), [1902.01945].[16] E. C. Bellm et al. , PASP , 018002 (2019),[1902.01932].[17] M. Graham et al. , Phys. Rev. Lett. , 251102 (2020),[2006.14122]. [18] I. Bartos, B. Kocsis, Z. Haiman and S. M´arka, Astrophys.J. , 165 (2017), [1602.03831].[19] N. C. Stone, B. D. Metzger and Z. Haiman, Mon. Not.Roy. Astron. Soc. , 946 (2017), [1602.04226].[20] B. McKernan et al. , ApJ , L50 (2019), [1907.03746].[21] R. Genzel, F. Eisenhauer and S. Gillessen, Rev. Mod.Phys. , 3121 (2010).[22] A. M. Ghez et al. , The Astrophysical Journal , 1044(2008).[23] M. Zevin, J. Samsing, C. Rodriguez, C.-J. Hasterand E. Ramirez-Ruiz, Astrophys. J. , 91 (2019),[1810.00901].[24] N. C. Stone, B. D. Metzger and Z. Haiman, MonthlyNotices of the Royal Astronomical Society ,946 (2016), [https://academic.oup.com/mnras/article-pdf/464/1/946/18512767/stw2260.pdf].[25] X. Chen and W.-B. Han, Communications Physics , 53(2018), [1801.05780].[26] R. M. O’Leary, Y. Meiron and B. Kocsis, Astrophys. J.Lett. , L12 (2016), [1602.02809].[27] C. L. Rodriguez et al. , MNRAS , 2109 (2016),[1601.04227].[28] S. F. P. Zwart and S. L. W. McMillan, (2000).[29] A. Tokovinin, S. Thomas, M. Sterzik and S. Udry,A&A , 681 (2006), [astro-ph/0601518].[30] T. Pribulla and S. M. Rucinski, Astron. J. , 2986(2006), [astro-ph/0601610].[31] T. Robson, N. J. Cornish, N. Tamanini and S. Toonen,Phys. Rev. D , 064012 (2018), [1806.00500].[32] Y. Kozai, AJ , 591 (1962). [33] S. Naoz, Annual Review of Astronomy and Astrophysics , 441 (2016), [https://doi.org/10.1146/annurev-astro-081915-023315].[34] E. Poisson and C. M. Will, Gravity: Newtonian, Post-Newtonian, Relativistic (Cambridge University Press,2014).[35] B.-M. Hoang, S. Naoz, B. Kocsis, W. Farr and J. McIver,Astrophys. J. Lett. , L31 (2019), [1903.00134].[36] L. Randall and Z.-Z. Xianyu, 1902.08604.[37] L. Randall and Z.-Z. Xianyu, 1907.02283.[38] B. Deme, B.-M. Hoang, S. Naoz and B. Kocsis, Astro-phys. J. , 125 (2020), [2005.03677].[39] F. Antonini and H. B. Perets, Astrophys. J. , 27(2012), [1203.2938].[40] F. Antonini et al. , The Astrophysical Journal , 65(2016).[41] B.-M. Hoang, S. Naoz, B. Kocsis, F. A. Rasio andF. Dosopoulou, The Astrophysical Journal , 140(2018).[42] B. Kocsis and J. Levin, Phys. Rev. D , 123005 (2012).[43] P. Gupta, H. Suzuki, H. Okawa and K.-i. Maeda, Phys.Rev. D , 104053 (2020), [1911.11318].[44] H. Suzuki, P. Gupta, H. Okawa and K.-i. Maeda,2006.11545.[45] S. Cisneros, G. Goedecke, C. Beetle and M. Engelhardt,Monthly Notices of the Royal Astronomical Society ,2733 (2015), [https://academic.oup.com/mnras/article-pdf/448/3/2733/6007665/stv172.pdf].[46] Y. Meiron, B. Kocsis and A. Loeb, Astrophys. J. ,200 (2017), [1604.02148].[47] L. Randall and Z.-Z. Xianyu, Astrophys. J. , 75(2019), [1805.05335].[48] K. W. Wong, V. Baibhav and E. Berti, Mon. Not. Roy.Astron. Soc. , 5665 (2019), [1902.01402].[49] W.-B. Han and X. Chen, Mon. Not. Roy. Astron. Soc. , L29 (2019), [1801.07060].[50] A. Torres-Orjuela, X. Chen, Z. Cao, P. Amaro-Seoaneand P. Peng, Phys. Rev. D , 063012 (2019),[1806.09857].[51] A. Torres-Orjuela, X. Chen and P. Amaro-Seoane, Phys.Rev. D , 083028 (2020), [2001.00721].[52] J. M. Ezquiaga and M. Zumalac´arregui, 2009.12187.[53] J. M. Ezquiaga, D. E. Holz, W. Hu, M. Lagos and R. M.Wald, 2008.12814.[54] H. Yu and Y. Chen, 2009.02579.[55] B. Bonga, H. Yang and S. A. Hughes, Phys. Rev. Lett. , 101103 (2019), [1905.00030].[56] H. Yang, B. Bonga, Z. Peng and G. Li, Phys. Rev. D , 124056 (2019), [1910.07337].[57] R. P. Kerr, Phys. Rev. Lett. , 237 (1963).[58] S. A. Teukolsky, Astrophys. J. , 635 (1973).[59] J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astro-phys. J. , 347 (1972).[60] E. Berti, V. Cardoso and A. O. Starinets, Class. Quant.Grav. , 163001 (2009), [0905.2975].[61] W. Krivan, P. Laguna, P. Papadopoulos and N. Anders-son, Physical Review D , 3395–3404 (1997).[62] R. Lopez-Aleman, G. Khanna and J. Pullin, Class.Quant. Grav. , 3259 (2003), [gr-qc/0303054]. [63] E. Pazos- ´Avalos and C. O. Lousto, Physical Review D (2005).[64] P. A. Sundararajan, G. Khanna and S. A. Hughes, Phys.Rev. D76 , 104005 (2007), [gr-qc/0703028].[65] E. Berti, V. Cardoso and M. Casals, Phys. Rev. D ,024013 (2006), [gr-qc/0511111], [Erratum: Phys.Rev.D73, 109902 (2006)].[66] M. Davis, R. Ruffini, W. Press and R. Price, Phys. Rev.Lett. , 1466 (1971).[67] Y. Mino, M. Sasaki, M. Shibata, H. Tagoshi andT. Tanaka, Prog. Theor. Phys. Suppl. , 1 (1997),[gr-qc/9712057].[68] V. Cardoso and J. P. Lemos, Phys. Lett. B , 1 (2002),[gr-qc/0202019].[69] E. Berti et al. , Phys. Rev. D , 104048 (2010),[1003.0812].[70] V. Cardoso, A. del Rio and M. Kimura, Phys. Rev. D100 , 084046 (2019), [1907.01561].[71] A. Starobinsky, Sov. Phys. JETP , 28 (1973).[72] E. Poisson, Phys. Rev. D , 1497 (1993).[73] NIST Digital Library of Mathematical Functions ,http://dlmf.nist.gov/, Release 1.0.26 of 2020-03-15,F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I.Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller,B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.[74] V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T.Zanchin, Phys. Rev. D , 064016 (2009), [0812.1806].[75] H. Georgi, The Physics of Waves (Prentice Hall, 1993).[76] E. M. Purcell, H. C. Torrey and R. V. Pound, Phys. Rev. , 37 (1946).[77] C. Sauvan, J. P. Hugonin, I. S. Maksymov andP. Lalanne, Phys. Rev. Lett. , 237401 (2013).[78] J. Thornburg, B. Wardell and M. van de Meent, Phys.Rev. Res. , 013365 (2020), [1906.06791].[79] R. H. Price, S. Nampalliwar and G. Khanna, Phys. Rev.D , 044060 (2016), [1508.04797].[80] L. Gualtieri, E. Berti, V. Cardoso and U. Sperhake, Phys.Rev. D , 044024 (2008), [0805.1017].[81] R. Brito, V. Cardoso and P. Pani, Lect. Notes Phys. ,pp.1 (2015), [1501.06570].[82] C. T. Cunningham and J. M. Bardeen, ApJ , L137(1972).[83] J. M. Ezquiaga, W. Hu and M. Lagos, 2005.10702.[84] Y. Nambu and S. Noda, Class. Quant. Grav. , 075011(2016), [1502.05468].[85] Y. Nambu, S. Noda and Y. Sakai, Phys. Rev. D ,064037 (2019), [1905.01793].[86] J. G. Hills, Nature , 687 (1988).[87] E. Addison, P. Laguna and S. Larson, 1501.07856.[88] H. Suzuki, Y. Nakamura and S. Yamada, 2009.06999.[89] H. Brown, S. Kobayashi, E. M. Rossi and R. Sari, Mon.Not. Roy. Astron. Soc. , 5682 (2018), [1804.02911].[90] P. I. Jefremov, O. Y. Tsupko and G. S. Bisnovatyi-Kogan,Phys. Rev. D , 124030 (2015), [1503.07060].[91] V. Cardoso, C. F. Macedo and R. Vicente, 2010.15151.[92] L. Bernard, V. Cardoso, T. Ikeda and M. Zilh˜ao, Phys.Rev. D100