PHYSICAL REVIEW D 103, 104014 (2021)

Adiabatic waveforms for extreme mass-ratio inspirals via multivoice decomposition in time and frequency

Scott A. Hughes ,1 Niels Warburton ,2 Gaurav Khanna ,3,4 Alvin J. K. Chua ,5 and Michael L. Katz 6 1Department of Physics and MIT Kavli Institute, Cambridge, Massachusetts 02139, United States 2School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland 3Physics Department, University of Massachusetts, Dartmouth, Massachusetts 02747, United States 4Department of Physics, University of Rhode Island, Kingston, Rhode Island 02881, United States 5Theoretical Astrophysics Group, California Institute of Technology, Pasadena, California 91125, United States 6Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam-Golm, Germany

(Received 4 February 2021; accepted 7 April 2021; published 11 May 2021)

We compute adiabatic waveforms for extreme mass-ratio inspirals (EMRIs) by “stitching” together a long inspiral waveform from a sequence of waveform snapshots, each of which corresponds to a particular geodesic orbit. We show that the complicated total waveform can be regarded as a sum of “voices.” Each voice evolves in a simple way on long timescales, a property which can be exploited to efficiently produce waveform models that faithfully encode the properties of EMRI systems. We look at examples for a range of different orbital geometries: spherical orbits, equatorial eccentric orbits, and one example of generic (inclined and eccentric) orbits. To our knowledge, this is the first calculation of a generic EMRI waveform that uses strong-field radiation reaction. We examine waveforms in both the time and frequency domains. Although EMRIs evolve slowly enough that the stationary phase approximation (SPA) to the Fourier transform is valid, the SPA calculation must be done to higher order for some voices, since their instantaneous frequency can change from chirping forward (f>_ 0) to chirping backward (f<_ 0). The approach we develop can eventually be extended to more complete EMRI waveform models—for example, to include effects neglected by the adiabatic approximation, such as the conservative self-force and spin-curvature coupling.

DOI: 10.1103/PhysRevD.103.104014

I. INTRODUCTION detectors like LISA. A typical EMRI will execute ∼104 − 105 ’ A. Extreme mass-ratio inspirals and self-forces orbits in LISA s band as gravitational wave backreaction shrinks its binary separation. Because the The large mass-ratio limit of the two-body problem is a small body orbits in the larger black hole’s strong field, laboratory for studying strong-field motion in general EMRI waves are highly sensitive to the nature of the large relativity. By treating the spacetime of such a binary as black hole’s spacetime. EMRI events will make it possible an exact black hole solution plus a perturbation due to the to precisely map black hole spacetimes, weighing black less massive orbiting body, it is possible to analyze the holes’ masses and spins with exquisite accuracy, and ’ binary s dynamics and the gravitational waves it generates testing the hypothesis that astrophysical black holes are using the tools of black hole perturbation theory. Because described by the Kerr spacetime [2]. the equations of perturbation theory can, in many circum- To achieve these ambitious science goals for EMRI stances, be solved very precisely, the large mass-ratio limit measurements, we will need waveform models, or tem- serves as a high-precision limit for understanding the two- plates, that accurately match signals in data over their body problem in general (see, e.g., Ref. [1]). full duration. Such templates will provide guidance to This limit is also significant thanks to the importance algorithms for finding EMRI signals in detector noise, and of extreme mass-ratio inspirals (EMRIs) as gravitational will be necessary for characterizing astrophysical sources. wave sources. Binaries formed by the capture of stellar- Developing such models is one of the goals of the self-force mass (μ ∼ 1–100 M⊙) compact bodies onto relativistic program, which seeks to develop equations describing the 6 orbits of black holes with M ∼ 10 M⊙ in the cores of motion of objects in specified background spacetimes, galaxies will generate low-frequency (f ∼ 0.01 Hz) gravi- including the interaction of that object with its own tational waves, right in the sensitive band of space-based perturbation to that spacetime—i.e., including the small

2470-0010=2021=103(10)=104014(32) 104014-1 © 2021 American Physical Society HUGHES, WARBURTON, KHANNA, CHUA, and KATZ PHYS. REV. D 103, 104014 (2021) body’s “self-interaction” [3]. Taking the background space- and we define the oscillations about this average as ð1Þ ð1Þ ð1Þ time to be that of a black hole, self-forces can be developed δGα ðqr;qθ;JÞ≡Gα ðqr;qθ;JÞ−hGα ðJÞi. The oscillations ∼ using tools from black hole perturbation theory, with the vary about zero on a rapid orbital timescale To M; the mass ratio of the system ε ≡ μ=M (where μ is the mass of average evolves on a much slower dissipative inspiral ∼ 2 μ ∼ ε the orbiting body, and M the mass of the black hole) timescale Ti M = ,orTi To= . Because of the large serving as a perturbative order counting parameter. separation of these two timescales for EMRIs, the oscil- To make this more quantitative, we sketch the general lations nearly average away during an inspiral. For most form of such equations of motion in the action-angle orbits, neglecting the impact of the oscillations introduces ð Þ¼ ðεÞ formulation used by Hinderer and Flanagan [4]. Let qα ≐ errors of O To=Ti O . ð Þ qt;qr;qθ;qϕ be a set of angle variables which describe The simplest model for inspiral which captures the the motion of the smaller body in a convenient coordinate strong-field dynamics of black hole orbits is known as the adiabatic approximation. It amounts to solving the system, let Jβ ≐ ðJt;Jr;Jθ;JϕÞ be a set of actions asso- ciated with those motions, and let λ be a convenient time following variants of Eqs. (1.1) and (1.2): variable. The motion of the smaller body is then governed dqα dJα ð1Þ by a set of equations with the form ¼ ωαðJÞ; ¼ εhGα ðJÞi: ð1:5Þ dλ dλ dqα ð1Þ 2 ð2Þ ¼ ω ðJÞþε α ð ;JÞþε α ð ;JÞþ In words, we treat the short-timescale orbital dynamics as λ α g qr;qθ g qr;qθ ; d geodesic, but we use the orbit-averaged impact of the self- ð Þ 1:1 force on the actions Jα. This amounts to including the “dissipative” part of the orbit-averaged self-force, since to dJα ð1Þ 2 ð2Þ ð1Þ ¼ ε α ð ; JÞþε α ð ; JÞþ ð Þ OðεÞ, the action of hGα i is equivalent to computing the λ G qr;qθ G qr;qθ : 1:2 d rates at which an orbit’s energy E, axial angular momentum L , and Carter constant Q change due to the backreaction of The frequency ωα describes the rate at which the angles z accumulate per unit λ, neglecting the self-interaction. The gravitational-wave emission. The adiabatic approximation treats inspiral as “flow” through a sequence of geodesics, term ωαðJÞ thus characterizes the geodesic motion of the ðnÞ with the rate of flow determined by the rates of change small body in the black hole background. The terms gα ’ of E, Lz, and Q [7]. describe how the small body s trajectory is modified by the It is worth noting that the picture we have sketched breaks ðnÞ self-interaction at OðεnÞ; the terms Gα describe how the down near the so-called resonant orbits [8–11]. These are actions (which are constant along geodesics) are modified. orbits for which the frequency ratio ωx=ωy ¼ nx=ny, where Note that the self-interaction terms only depend on the nx and ny are small integers. Resonances have been shown to — angles qr and qθ because black hole spacetimes are arise from the gravitational self-force itself [8] (for which axisymmetric and stationary, these terms are independent x ¼ r and y ¼ θ), as well as from tidal perturbations from of qt and qϕ. Many aspects of this problem are now under stars or black holes that are near an astrophysical EMRI control at OðεÞ (see, e.g., Ref. [5]), and work is rapidly system [10,11] (for which x ¼ r or θ,andy ¼ ϕ). Near proceeding on the problem at Oðε2Þ [6]. resonances, some terms change from oscillatory to nearly As the issue of the smaller body’s motion is brought constant; neglecting their impact introduces errors of under control, more attention is now being paid to the Oðε1=2Þ. As such, they will be quite important, contributing gravitational waveforms that arise from this motion. That is perhaps the leading postadiabatic contribution to waveform the focus of this paper. phase evolution. We neglect the impact of resonances in this analysis, though where appropriate we comment on B. The adiabatic approximation and its use their importance and how they may be incorporated into future work. The forcing terms in Eqs. (1.1) and (1.2) can be further The computational cost associated with even the rather decomposed by splitting them into averages and oscilla- simplified adiabatic waveform model is quite high. As we tions about their average. Consider the first-order forcing ð1Þ outline in Sec. III, computing adiabatic backreaction term Gα . We set involves solving the Teukolsky equation for many multi- poles of the radiation field and harmonics of the orbital ð1Þð ; JÞ¼h ð1ÞðJÞi þ δ ð1Þð ; JÞ ð Þ Gα qr;qθ Gα Gα qr;qθ ; 1:3 motion; tens of thousands of multipoles and harmonics may need to be calculated for tens of thousands of orbits. To where the averaged contribution is produce EMRI models which capture the most important Z Z elements of the mapping between source physics and 1 2π 2π h ð1ÞðJÞi ¼ ð1Þð ; JÞ ð Þ Gα 2 dqr dqθGα qr;qθ ; 1:4 waveform properties, the community has developed several ð2πÞ 0 0 kludges as a stopgap for data analysis and science return

104014-2 ADIABATIC WAVEFORMS FOR EXTREME MASS-RATIO … PHYS. REV. D 103, 104014 (2021) studies. The analytic kludge of Barack and Cutler [12] multipole of the radiation and harmonic of the fundamental essentially pushes post-Newtonian models beyond their orbital frequencies. The voice-by-voice decomposition was domain of validity. Their match with fully relativistic suggested long ago to one of us by L. S. Finn, and was first models is not good, but they capture the key qualitative presented in Ref. [15] for the special case of spherical Kerr features of EMRI physics. Almost as importantly, the orbits (i.e., orbits of constant Boyer-Lindquist radius, analytic kludge is fast and is easy to implement; as such, including those inclined from the equatorial plane). This it has been heavily used for many LISA measurement paper corrects an important error in Ref. [15] (which left studies. out a phase that is set by initial conditions) and extends this The numerical kludge of Babak et al. [13] is closer to the analysis to fully generic configurations. spirit of the adiabatic approximation we describe here, in The approach that we present has several important that it treats the small body’s motion as a Kerr geodesic, but features. First, we find that the data which must be stored to it uses semianalytic fits to strong-field radiation emission describe the waveform voice by voice evolve smoothly to describe inspiral. Wave emission is treated with a crude over an inspiral. This suggests that these data can be multipolar approximation based on the small body’s sampled at a relatively low cadence, and we can then build coordinate motion. Despite the crudeness of some of the a high-quality waveform using interpolation. Such behavior underlying approximations, the numerical kludge fits had been seen in earlier work [15]; this analysis demon- relativistic waveform models fairly well. It is, however, strates that this behavior is not unique to spherical orbits, slower and harder to use than the analytic kludge, and as but is generic. We describe the methods we have used for such has not been used very much. More recently, Chua and our initial exploration, and that were used in a related study Gair [14] showed that one can significantly improve [16] focusing on the rapid evaluation of waveforms for data matches to relativistic models by using an analytic kludge analysis. Reference [16] showed that the chasm between augmented with knowledge of the exact frequency spec- the computational demands of accurate modeling and trum of Kerr black hole orbits. efficient analysis can be bridged, but much work remains The shortcomings of the kludges illustrate that, ulti- in order to “optimize” these methods—for example, in mately, one needs waveform models that capture the determining the best way for the numerical data to be strong-field dynamics of Kerr orbits and that accurately sampled and interpolated. describe strong-field radiation generation and propagation Second, we show that this framework works well in both through black hole spacetimes. Waveforms based on the the time and frequency domains. The frequency-domain adiabatic approximation are the simplest ones that accu- construction is particularly interesting. Thanks to their rately include both of these effects. Though the adiabatic extreme mass ratios, EMRIs evolve slowly enough that approximation misses important aspects from neglected the stationary-phase approximation (SPA) to the Fourier pieces of the self-force, they get enough of the strong-field transform should provide an excellent representation of the physics correct that they will be effective and useful tools frequency-domain form. However, for some EMRI voices, for understanding the scope of EMRI data analysis chal- the instantaneous frequency grows to a maximum and then lenges, and for accurately assessing the science return that decreases. For such voices, the first time derivative of the EMRI measurements will enable. frequency vanishes at some point along the inspiral. The standard SPA is singular at these points. We show that C. This paper including information about the second derivative fixes Although the computational cost of making adiabatic this behavior, allowing us to compute frequency-domain inspiral waveforms is high, the most expensive step of this waveforms for all EMRI voices. calculation—computing a set of complex numbers which Third, we show that several of the waveform’s param- encode rates of change of (E, Lz, Q), as well as the eters can be included in a simple way which effectively gravitational waveform’s amplitude—need only be done reduces the dimensionality of the waveform parameter once. These numbers can be computed in advance for a space. These parameters are angles which control the initial range of astrophysically relevant EMRI orbits, and then position of the orbit in its eccentric radial motion, and its stored and used to assemble the waveform as needed. The initial polar angle in the range of motion allowed by its goal of this paper is to lay out what quantities must be orbital inclination. They are initial conditions on the computed, and to describe how to use such precomputed relativistic analog of the “true anomaly angles” used in data to build adiabatic waveforms. Newtonian celestial mechanics. Associated with these Our particular goal is to show how to organize and store initial anomaly angles are phases, originally introduced the most important and useful data needed to assemble in Ref. [17], which correct the complex amplitudes asso- the waveforms in a computationally effective way. A key ciated with the waveform’s voices. By comparing with element of our approach is to view the complicated EMRI output from a time-domain Teukolsky equation solver waveform as a sum of simple “voices.” Each voice [18,19], we show that these phases allow one to match corresponds to a mode ðl; m; k; nÞ representing a particular any allowed initial condition with very little computational

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∼ 2 μ ¼ ε cost. This means that we can generate a suite of data using timescale associated with the inspiral, Ti M = To= . only a single initial choice of the anomaly angles, and then The adiabatic waveform is only accurate up to corrections transform to initial conditions corresponding to any other of order the system’s mass ratio, essentially because it choice. This significantly reduces the computational cost assumes that all time derivatives only include information associated with covering the full EMRI parameter space. about the system’s “fast” time variation. One way in which Finally, it should not be difficult to extend this framework a postadiabatic analysis will improve on these results will to include at least some important effects beyond the adiabatic be by including information about time derivatives with approximation, many of which are discussed in detail in respect to the slow variation along the inspiral. Refs. [20,21]. For example, both conservative self-forces and Section V describes how to compute multivoice signals spin-curvature coupling have orbit-averaged effects that are in the frequency domain. Because EMRI systems are small, but that secularly accumulate over many orbits slowly evolving, the stationary phase approximation [22–25]. Such effects can be included in this framework (SPA) should accurately describe the Fourier transform by allowing the relativistic anomaly angles discussed above of EMRI signals. However, because the evolution of certain (which in the adiabatic limit are constant) to evolve over the voices is not monotonic, the “standard” SPA calculation inspiral; the phases associated with these angles will evolve as can fail, introducing singular artifacts at moments when a well. We also expect that the impact of small oscillations (such voice’s frequency evolution switches sign. We review the as arise from both self-forces and spin-curvature coupling) standard SPA Fourier transform and show how by includ- can likewise be incorporated, perhaps very efficiently using a ing an additional derivative of the frequency it is straight- near-identity transformation [26]. forward to correct this artifact. We conclude this section by As we were completing the bulk of the calculations showing how to combine multiple voices to construct the which appear here, a similar analysis of adiabatic EMRI full frequency-domain EMRI waveform. waveforms was presented by Fujita and Shibata [27]. Their In Sec. VI, we present various important technical details analysis focuses to a large extent on the measurability of describing how we implement this framework for the EMRI waves by LISA, confining their discussion to results we present in this paper. We strongly emphasize eccentric and equatorial sources. Like us, they also take that there is a great deal of room for improvement on the advantage of the fact that one can precompute the most techniques described here. We have not, for example, expensive data on a grid of orbits, and then assemble the carefully assessed the most effective method for laying waveform by interpolation. We are encouraged by the fact out the grid of data on which we store information about that their independently developed framework is substan- adiabatic backreaction and waveform amplitudes, nor have tially similar to what we present here. we thoroughly investigated efficient methods for interpo- lating these data to off-grid points (e.g., Ref. [16]). These D. Organization of this paper important points will be studied in future work, as we begin The remainder of this paper is organized as follows. Since assessing how to take this framework and use it to develop inspiral in the adiabatic approximation is treated as a sequence EMRI waveforms in support of LISA data analysis and of geodesic orbits, we begin by reviewing the properties of science studies. Kerr geodesics in Sec. II. Nothing in this section is new; it is In Secs. VII and VIII, we present examples of adiabatic included primarily to keep the manuscript self-contained, and EMRI waveforms. In both of these sections, we show to allow us to carefully define our notation and the meaning of examples of the complete time-domain waveform produced important quantities which are used elsewhere. Section III by summing over many voices, as well as the (much simpler) reviews how we solve the Teukolsky equation and use its structure of representative voices which contribute to these solutions in order to calculate adiabatic backreaction on an waveforms. Section VII shows results for inspiral into orbit. This allows us to compute how a system evolves from Schwarzschild black holes, presenting the details of an ¼ 0 2 orbit to orbit, as well as the gravitational-wave amplitude inspiral with small initial eccentricity (einit . ) and ¼ 0 7 produced by that orbit. This material is likewise not new, another with high initial eccentricity (einit . ). but is included for completeness, as well as to introduce and Section VIII examines several examples for inspiral into explain all relevant notation. Kerr, including a case that is spherical, a case that is In Sec. IV, we lay out how one “stitches” together data equatorial with high eccentricity, and one example that is describing radiation from geodesics to construct an adia- generic, both eccentric and inclined. Although generic EMRI batic waveform. This construction essentially amounts to waveforms based on various “kludge” approximations have taking the solution to the Teukolsky equation for a geodesic been presented before, to our knowledge the generic example orbit and promoting various factors which are constants on shown in Sec. VIII is the first calculation that uses strong- geodesics into factors which vary slowly along an inspiral. field backreaction and strong-field wave generation for the This construction introduces a two-timescale expansion: entire computation. We find that there is a great deal of some quantities vary on the “fast” timescale associated similarity between qualitative features of the Kerr and ∼ “ ” with orbital motions, To M; others vary on the slow Schwarzschild waveforms. As such, our presentation of

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Kerr results is somewhat brief, concentrating on the most dλ is related to an interval of proper time dτ by dλ ¼ dτ=Σ, important highlights and differences as compared to the where Σ ¼ r2 þ cos2θ, and where r and θ are the Boyer- Schwarzschild cases. Lindquist radial and polar coordinates. With this para- For all the cases we examine, we demonstrate how one metrization, motion in Boyer-Lindquist coordinates is can account for a system’s initial conditions by adjusting a governed by the equations set of phase variables which depend on the initial values of the anomaly angles that parametrize the system’s radial and dr 2 ¼½Eðr2 þ a2Þ − aL 2 − Δ½r2 þðL − aEÞ2 þ Q polar motions. We calibrate our calculations in one case dλ z z (presented in Sec. VII) by comparing with an EMRI ≡ ð Þ ð Þ waveform computed using a time-domain Teukolsky equa- R r ; 2:1 tion solver [18,19]. Interestingly, in this case we find a θ 2 small phase offset that, for most initial conditions, secularly d 2 2 2 2 2 ¼ Q − cot θLz − a cos θð1 − E Þ accumulates, causing the waveforms computed with this dλ ’ paper s techniques to dephase from those computed with ≡ ΘðθÞ; ð2:2Þ the time-domain code by up to several radians over the course of an inspiral. We argue that this is an artifact of the 2 dϕ 2MraE a L adiabatic approximation’s neglect of terms which vary on ¼ 2θ þ − z λ csc Lz Δ Δ the slow timescale, and is not unexpected. We show in this d ≡ Φ ð ÞþΦ ðθÞ ð Þ comparison case that we can empirically compensate for r r θ ; 2:3 much of the dephasing using an ad hoc correction that corrects some of the neglected “slow-time” evolution. dt ðr2 þ a2Þ2 2MraL ¼ E − a2sin2θ − z Though this correction is not rigorously justified, its form dλ Δ Δ suggests that the dephasing may arise from a slow-time ≡ T ðrÞþTθðθÞ: ð2:4Þ evolution of the phase variables which are set by the orbit’s r initial conditions. We have introduced Δ ¼ r2 − 2Mr þ a2. The quantities E, Our conclusions are presented in Sec. IX. Along with ’ μ summarizing the main results of this analysis, we describe Lz, and Q are the orbit s energy (per unit ), axial angular μ μ2 plans and directions for future work. Chief among these momentum (per unit ), and Carter constant (per unit ). plans is to investigate how to optimize the implementation These quantities are conserved along any geodesic; in order to make EMRI waveforms as rapidly as possible, choosing them specifies an orbit, up to initial conditions. λ ¼ Σ τ which will be very important in order for these waveforms Writing d=d d=d puts these equations into more to be usable for LISA data and science analysis studies, as familiar forms typically found in textbooks, such as – well as investigating how to include postadiabatic effects Eqs. (33) and (32a) (32d) of Ref. [28]. with small modifications of the framework we present here. Equations (2.1) and (2.2) tell us that bound Kerr orbits θ λ We also plan to publicly release the code and data used in are periodic in r and when parametrized using : this study, and we describe the status of our plans as this ðλÞ¼ ðλ þ Λ Þ θðλÞ¼θðλ þ Λ Þ ð Þ analysis is being completed. r r n r ; k θ ; 2:5

where n and k are each integers. Simple formulas exist II. KERR GEODESICS for the Mino-time periods Λr and Λθ [31]; we define the We begin by discussing bound Kerr geodesics. The most associated frequencies by ϒr;θ ¼ 2π=Λr;θ. important aspects of this content are discussed in depth The motions in t and ϕ are the sums of secularly elsewhere [28–32]; we briefly review this material for this accumulating pieces and oscillatory functions: paper to be self-contained, as well as to introduce notation and conventions that we use. Certain lengthy but important tðλÞ¼t0 þ Γλ þ Δtr½rðλÞ þ Δtθ½θðλÞ; ð2:6Þ formulas are given in Appendix A.

ϕðλÞ¼ϕ0 þ ϒϕλ þ Δϕr½rðλÞ þ Δϕθ½θðλÞ: ð2:7Þ A. Mino-time formulation of orbital motion ϕ Consider a pointlike body of mass μ orbiting a Kerr black In these equations, t0 and 0 describe initial conditions, hole with mass M and spin parameter a ¼jSj=M (where S Γ ¼h ð Þiþh ðθÞi ð Þ is the black hole’s spin angular momentum in units with Tr r Tθ ; 2:8 G ¼ 1 ¼ c), and use Boyer-Lindquist coordinates (with the angle θ measured from the black hole’s spin axis) ϒϕ ¼hΦrðrÞi þ hΦθðθÞi; ð2:9Þ to describe its motion. We use Mino time as our time parameter describing these orbits. An interval of Mino time and

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Δ ½ ðλÞ ¼ ½ ðλÞ − h ð Þi ≡ Δ ðλÞ θ ¼ ð Þ tr r Tr r Tr r tr ; cos min sin I: 2:17

Δtθ½θðλÞ ¼ Tθ½θðλÞ − hTθðθÞi ≡ ΔtθðλÞ; ð2:10Þ We have found that xI ≡ cos I is a particularly good parameter to describe inclination: x varies smoothly from Δϕ ½ ðλÞ ¼ Φ ½ ðλÞ − hΦ ð Þi ≡ Δϕ ðλÞ I r r r r r r r ; 1to−1 as orbits vary from prograde equatorial to Δϕθ½θðλÞ ¼ Φθ½θðλÞ − hΦθðθÞi ≡ ΔϕθðλÞ: ð2:11Þ retrograde equatorial, with Lz having the same sign as xI. Schmidt [29] first showed how to compute ðE; Lz;QÞ The quantity Γ describes the mean rate at which observer for generic Kerr orbits; a particularly clean representation time t accumulates per unit λ; the Mino-time frequency ϒϕ is provided by van de Meent [32]. We summarize his describes the mean rate at which ϕ accumulates per unit λ. formulas in Appendix A, tweaking them slightly to use our ð Þ The associated period is Λϕ ¼ 2π=ϒϕ. Simple formulas preferred parameter set p; e; xI . t ϕ t0 likewise exist for Γ and ϒϕ [31]. The averages used in Initial conditions for and are set by the parameters ϕ Eqs. (2.8)–(2.11) are given by and 0 given in Eqs. (2.6) and (2.7). To set initial conditions on r and θ, we introduce anomaly angles χ and χθ to Z r 1 Λr reparametrize those coordinate motions: hfrðrÞi ¼ fr½rðλÞdλ; ð2:12Þ Λ 0 r ¼ p ð Þ Z r ; 2:18 1 þ e cosðχr þ χr0Þ 1 Λθ h ðθÞi ¼ ½θðλÞ λ ð Þ fθ Λ fθ d : 2:13 qffiffiffiffiffiffiffiffiffiffiffiffiffi θ 0 θ ¼ 1 − 2 ðχ þ χ Þ ð Þ cos xI cos θ θ0 : 2:19 The ratio of the Mino-time frequencies to Γ gives the χ ¼ 0 χ ¼ 0 ¼ ϕ ¼ ϕ λ ¼ 0 observer-time frequencies: We set θ , r , t t0, and 0 when . The phase χr0 then determines the value of r at λ ¼ 0, and χθ0 ϒ determines the corresponding value of θ. When χθ0 ¼ 0, Ω ¼ r;θ;ϕ ð Þ r;θ;ϕ : 2:14 θ ¼ θ λ ¼ 0 χ ¼ 0 Γ the orbit has min when ; when r0 , it has ¼ λ ¼ 0 r rmin when . We thus have useful closed-form expressions for all We define the fiducial geodesic to be the geodesic that frequencies, conjugate to both Mino time λ and observer has χθ0 ¼ χr0 ¼ ϕ0 ¼ 0 ¼ t0. We denote with a “check time t, that characterize black hole orbits. mark” accent any quantity which is defined along the fiducial geodesic. For instance, rˇðλÞ is the orbital radius ˇ B. Orbit parametrization and initial conditions along the fiducial geodesic, and θðλÞ is the polar angle θ θ ≤ θ ≤ θ along the fiducial geodesic. For nonfiducial geodesics, we Take the orbit to oscillate over min max, with λ ¼ λ λ θ ¼ π − θ ≤ ≤ define r0 to be the smallest positive value of at which max min, and over rmin r rmax, with ¼ 2 λ ¼ λ r rmin; likewise, θ0 is the smallest positive value of λ at which θ ¼ θ . This means that ¼ p ð Þ min rmin = max : 2:15 1 e ˇ rðλÞ¼rˇðλ − λr0Þ; θðλÞ¼θðλ − λθ0Þ: ð2:20Þ Choosing p, e, and θ is equivalent to choosing the min λ χ integrals of motion E, Lz, and Q. We have found it There is a one-to-one correspondence between θ0 and θ0, θ λ χ particularly convenient to replace min with an inclination and between r0 and r0. A useful corollary is angle1 I, defined by ˇ ˇ ΔtrðλÞ¼Δtrðλ − λr0Þ − Δtrð−λr0Þ; ¼ π 2 − ð Þθ ð Þ I = sgn Lz min: 2:16 ΔtθðλÞ¼Δtˇθðλ − λθ0Þ − Δtˇθð−λθ0Þ; ð2:21Þ

I The angle varies smoothly from 0 for equatorial prograde with analogous formulas for Δϕr and Δϕθ. to π for equatorial retrograde. The definition (2.16) may Some of our definitions differ from those used in 3 seem a bit awkward thanks to the branch associated with Ref. [17]. In that reference, χθ0 and χr0 were not used. the sign of Lz. It is simple to show that 2These definitions account for the fact that these motions 1 θ ðλ þ Λ Þ¼ The angle I has been labeled inc in some previous work, such are periodic: r r0 n r rmin for any integer n,and θðλ þ Λ Þ¼θ as Ref. [33]. We have found this label to be potentially confusing, θ0 k θ min for any integer k. since the Boyer-Lindquist angle θ is measured from the black 3Reference [17] also used slightly different symbols for the hole’s spin axis, whereas the inclination angle is measured from anomaly angles, writing χ for the polar anomaly and ψ for the the plane normal to this axis. We have changed notation to avoid radial. Here, we only use ψ for the Newman-Penrose curvature confusion with the coordinates. scalars.

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Instead, χθ ¼ 0 corresponded to λ ¼ λθ0,andχr ¼ 0 cor- associated gravitational waveform. In this section, we responded to λ ¼ λrθ.AswediscussbrieflyinSecs.IV briefly summarize how these quantities are calculated and IX, the angles χr0 and χθ0 will play an important role using the Teukolsky equation. As with Sec. II, the contents going beyond adiabatic waveforms. In the adiabatic approxi- of this section are discussed at length elsewhere, but are mation, the angles χr0, χθ0,andϕ0 are constant as we move summarized here to introduce critical quantities and con- from geodesic to geodesic. When we include, for example, cepts important for later parts of this analysis. orbit-averaged conservative self-force effects or orbit- averaged spin-curvature forces, we find secularly accumu- A. Solving the Teukolsky equation lating phases associated with each of these motions. The Teukolsky equation [34] describes perturbations to Allowing the angles χr0, χθ0,andϕ0 to evolve during inspiral is a simple and robust way to “upgrade” this the Weyl curvature of Kerr black holes. The version that we framework to include these next-order effects. will use in this analysis focuses on the Newman-Penrose curvature scalar III. ADIABATIC EVOLUTION AND WAVEFORM AMPLITUDES VIA THE ψ ¼ − α ¯ β γ ¯ δ ð Þ TEUKOLSKY EQUATION 4 Cαβγδn m n m ; 3:1 The next critical ingredient to constructing an adiabatic α α inspiral is the backreaction which arises from the orbit- where Cαβγδ is the Weyl curvature tensor, and n and m¯ are averaged self-interaction. The quantities which encode the legs of the Newman-Penrose null tetrad [35]. Teukolsky backreaction also tell us the amplitude of the inspiral’s showed that ψ 4 is governed by the equation

ð 2 þ 2Þ2 ð 2 − 2Þ 4 r a − 2 2θ ∂2Ψ − 4 þ θ − M r a ∂ Ψ þ Mar ∂ ∂ Ψ − Δ2∂ ðΔ−1∂ ΨÞ Δ a sin t r ia cos Δ t Δ ϕ t r r 2 1 a 1 2 aðr − MÞ i cos θ 2 − ∂θðsin θ∂θΨÞþ − ∂ Ψ þ 4 þ ∂ϕΨ þð4cot θ þ 2ÞΨ ¼ 4πΣT : ð3:2Þ sin θ Δ sin2θ ϕ Δ sin2θ

4 The field Ψ ¼ðr − ia cos θÞ ψ 4, and T is a source term The frequency-domain approach we use to solve the whose precise form is not needed here. See Ref. [34] for Teukolsky equation begins by writing ψ 4 in a Fourier and additional details and definitions. multipolar expansion. Writing An important point for our analysis is that 1 ψ 4 ¼ 1 2 ðr − ia cos ϑÞ4 ψ ¼ d ð − Þ → ∞ ð Þ 4 2 hþ ih× as r ; 3:3 Z 2 dt ∞ X∞ Xl i½mφ−ωðt−t0Þ × dω Rlmðr; ωÞSlmðϑ; aωÞe ; −∞ l¼2 m¼−l so ψ 4 far from the source encodes the emitted gravitational waves. These solutions also encode contributions to the ð3:4Þ rates of change of E, Lz, and Q from gravitational-wave backreaction. This is equivalent to the orbit-averaged self- Eq. (3.2) separates [34], with ordinary differential equa- interaction arising from fields which are regular far from tions governing Rlmðr; ωÞ and Slmðϑ;aωÞ. the source (see Ref. [36], as well as additionalp discussionffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The field ψ 4 is measured at the event ðt; r; ϑ; φÞ. (Note 2 2 on this point in Ref. [9]). As r → rþ ≡ M þ M − a the distinction between the orbit’s polar and axial angles, θ (the coordinate radius of the event horizon), ψ 4 encodes and ϕ, and the polar and axial angles at which the field is ’ tidal interactions of the orbiting body with the black hole s measured, ϑ and φ.) The function Slmðϑ; aωÞ is a sphe- event horizon. These solutions encode contributions to the roidal harmonic (of spin-weight −2, left out for brevity); rates of change of E, Lz, and Q from radiation absorbed by this function and methods for computing it are discussed at the horizon, which is equivalent to the orbit-averaged self- length in Appendix A of Ref. [37]. For reasons we will interaction arising from fields which are regular on the explain below, we have also introduced the initial time t0 event horizon [9,36]. Knowledge of ψ 4 in the limits r → ∞ into our expansion (3.4). and r → rþ provides all the data we need to construct The separated radial dependence has simple asymptotic adiabatic inspirals. behavior:

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ð ωÞ → ∞ 3 iωr → ∞ ð Þ ’ θ Rlm r; Zlmωr e ;r ; 3:5 By virtue of the periodicity of the orbit s r and motions ∞;H with respect to Mino time, the function Jlmω can be written − ðω− Ω Þ → H Δ i m H r → ð Þ as a double Fourier series: Zlmω e ;rrþ: 3:6 X∞ X∞ trans ∞;H ∞;H −iðkϒθþnϒ Þλ (We have absorbed a coefficient Clmω into the definition of J ω ¼ J e r ; ð3:12Þ ∞ lm lmkn trans H ¼−∞ ¼−∞ Zlmω, and a coefficient Blmω into the definition of Zlmω; k n see Refs. [38–40] for further discussion of these where quantities.) These asymptotic forms depend on the “tortoise Z coordinate,” Λ ∞ 1 r ϒ λ ;H ¼ λ in r r Jlmkn d re ΛrΛθ 0 Mrþ r Z ð Þ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 1 Λ r r r 2 2 ln θ ϒ λ ∞ − rþ λ ik θ θ ;H½ ðλ Þ θðλ Þ ð Þ M a × d θe Jlmω r r ; θ : 3:13 0 − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMr− r − 1 ð Þ 2 2 ln ; 3:7 M − a r− Combining Eqs. (3.10), (3.12), and (3.13) plus the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relations Ωr;θ;ϕ ¼ ϒr;θ;ϕ=Γ, we find that 2 2 where r ¼ M M − a . The frequency X∞ X∞ ∞ ∞ Z ;H ¼ Z ;H δðω − ω Þ; ð3:14Þ Ω ¼ a ð Þ lmω lmkn mkn H 3:8 k¼−∞ n¼−∞ 2Mrþ where is often called the rotation frequency of the horizon. It describes the frequency at which an observer held infini- ω ¼ mΩϕ þ kΩθ þ nΩ ð3:15Þ tesimally outside the event horizon will orbit the black hole mkn r as seen by distant observers. ∞;H and The quantities Zlmω are computed by integrating homo- ∞ ∞ geneous solutions of the radial Teukolsky equation against Z ;H ¼ e−imϕ0 J ;H =Γ: ð3:16Þ the source term of the separated radial Teukolsky equation. lmkn lmkn Further detailed discussion can be found in Ref. [33], with These coefficients have the symmetry updates to notation and minor corrections in Ref. [41].Of importance for this analysis is that these quantities are ∞;H ðlþkÞ ¯ ∞;H Z − − − ¼ð−1Þ Z ; ð3:17Þ computed by evaluating integrals of the form l; m; k; n lmkn Z ∞ where the overbar denotes complex conjugation. Our code ∞ ∞ ;H ¼ τ iω½tðτÞ−t0 −imϕðτÞ ;H½ ðτÞ θðτÞ ð Þ respects the symmetry in (3.17) to double-precision accu- Zlmω d e e Ilmω r ; : 3:9 −∞ ∞;H racy. We take advantage of this by computing Zlmkn for all l, all m, all k, and n ≥ 0, then using Eq. (3.17) to infer The integration variable τ is proper time along the geodesic, results for n<0. This cuts the amount of computation and we subtract off t0 because it is already accounted for roughly in half. ∞;Hð θÞ in Eq. (3.4). The function Ilmω r; is discussed in As we describe in the following two subsections, the Refs. [33,41]; schematically, it is a Green’s function used ∞ coefficients Zlmkn set the amplitude of the gravitational to solve the radial Teukolsky equation, multiplied by this waves from a specified geodesic orbit, and also encode the ’ equation s source term. Using the properties of Kerr geodesic contribution to the orbit-averaged self-force from fields that – orbits and using the methods developed in Refs. [38 40],itis are regular at null infinity; the coefficients ZH encode the ∞;H½ ðτÞ θðτÞ lmkn well understood how to build Ilmω r ; . contribution to the orbit-averaged self-force from fields that Changing the integration variable from proper time τ to are regular on the event horizon. These sets of coefficients Mino time λ, and using Eqs. (2.6) and (2.7), this becomes are thus of crucial importance for computing adiabatic Z inspiral and its gravitational waveform. ∞ ∞ ∞ ;H ¼ −imϕ0 λ iðωΓ−mϒϕÞλ ;H½ ðλÞ θðλÞ ð Þ Zlmω e d e Jlmω r ; ; 3:10 −∞ B. The gravitational waveform from an orbit ∞;H where we have introduced As shown in Sec. 8 of Ref. [17], the phase of Zlmkn depends on the values of λr0 and λθ0, which in turn depend ∞ ∞ χ χ J ;Hðr; θÞ¼ðr2 þ a2cos2θÞI ;Hðr; θÞ on the initial anomaly angles r0 and θ0. We denote by lmω lmω ˇ ∞;H ∞;H ω½Δ ð ÞþΔ ðθÞ − ½Δϕ ð ÞþΔϕ ðθÞ Zlmkn the value of Zlmkn for the fiducial geodesic. For a i tr r tθ im r r θ ð Þ × e e : 3:11 general geodesic,

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∞ ξ ∞ ∞ ;H ¼ i mkn ˇ ;H ð Þ 2Z Zlmkn e Zlmkn; 3:18 A ¼ − lmkn : ð : Þ lmkn ω2 3 23 mkn where the correcting phase is ∞ ˇ As with Zlmkn, we define Almkn to be the wave amplitude for ξmkn ¼ kϒθλθ0 þ nϒrλr0 the fiducial geodesic, and we have þ ½Δϕˇ ð−λ ÞþΔϕˇ ð−λ Þ − ϕ m r r0 θ θ0 0 ξ ˇ A ¼ ei mkn A : ð3:24Þ ˇ ˇ lmkn lmkn − ωmkn½Δtrð−λr0ÞþΔtθð−λθ0Þ: ð3:19Þ ˇ The data Almkn interpolate very well and should be stored ξ The form of mkn we use is slightly different from that for generating inspiral waveforms. It will be convenient for derived in Ref. [17]. In particular, we have separated out the later discussion to further define dependence on the initial axial phase ϕ0, as shown in Eq. (3.16), and the dependence on t0, which is discussed Hlmkn ¼ AlmknSlmðϑ; aωmknÞ: ð3:25Þ further in Sec. IV. See Ref. [17] for discussion and λ λ derivation of the dependence on θ0 and r0. Because spheroidal harmonics slowly change along an The fact that the initial conditions only influence these inspiral as ωmkn evolves, we find it useful to examine Hlmkn amplitudes via the phase factor ξ means that we only mkn rather than Almkn when computing wave amplitudes during need to compute and store quantities on the fiducial inspiral. geodesic. Using Eq. (3.19), we can then easily convert these results to any initial condition. This vastly cuts down on the amount of computation that must be done to cover C. Adiabatic backreaction ∞ the space of physically important EMRI systems. In Here we summarize how to use the coefficients Z ;H to ξ lmkn addition, notice that mkn can be written compute the adiabatic dissipative evolution, or backreac- tion, on a geodesic. We assume that a body is on a Kerr ξ ¼ ξ þ ξ þ ξ ð Þ mkn m 100 k 010 n 001: 3:20 geodesic orbit, and so is characterized (up to initial conditions) by the orbital integrals E, Lz, and Q. Results ξ ξ ξ For each orbit, one need only compute ( 100, 010, 001)in for dE=dt and dLz=dt have been known for quite a long order to know the phases for all ðl; m; k; nÞ. time [42]; these quantities each split into a contribution As we will see below, the values of these phases are from fields that are regular at infinity, which can be irrelevant for the orbit-averaged backreaction,4 but they are extracted from knowledge of the distant gravitational critical for getting the phase of the gravitational waveform radiation, and a contribution from fields that are regular correct given initial conditions. To write down the gravi- on the black hole’s event horizon. Computing the down- tational waves, we use Eq. (3.3) to relate h to ψ 4 far from horizon contribution is a little more tricky; one must the source. Combining Eqs. (3.4), (3.5), and (3.14),we compute how tidal stresses shear the horizon’s generators, further know that increasing its surface area (or its entropy), and then apply the first law of black hole dynamics to infer the change 1 X ’ ∞ i½mφ−ω ðt−t0Þ in the hole s mass and spin. Results for dQ=dt were first ψ 4 ¼ Z S ðϑ;aω Þe mkn ð3:21Þ lmkn lm mkn derived by Sago et al. [43] and are found by averaging the r lmkn action of the dissipative self-force on a geodesic. It also as r → ∞. Here and in what follows, any sum over the set separates into pieces that arise from fields regular at infinity ðl; m; k; nÞ will be assumed to be from 2 to ∞ for l, from −l and fields regular on the horizon. to l for m, and from −∞ to ∞ for k and n. Let us define The results which we need for our analysis are 1 X ∞ Xj ∞ j2 H Xα j H j2 dE Zlmkn dE lmkn Zlmkn h ≡ hþ − ih ≡ h ¼ ; ¼ ; × lmkn dt 4πω2 dt 4πω2 r lmkn lmkn mkn lmkn mkn 1 X ð Þ i½mφ−ωmknðt−t0ÞÞ 3:26 ¼ AlmknSlmðϑ; aωmknÞe : ð3:22Þ r lmkn dL ∞ X mjZ∞ j2 Combining Eqs. (3.3), (3.21), and (3.22), we see that z ¼ lmkn ; dt 4πω3 lmkn mkn 4The phases are not irrelevant for backreaction as we go dL H X α mjZH j2 z ¼ lmkn lmkn ; ð3:27Þ beyond the adiabatic limit, and indeed play an important role in dt 4πω3 determining the strength of backreaction at orbital resonances [9]. lmkn mkn

104014-9 HUGHES, WARBURTON, KHANNA, CHUA, and KATZ PHYS. REV. D 103, 104014 (2021) ∞ X X dQ ðL þ kϒθÞ dQ H ðL þ kϒθÞ ¼ jZ∞ j2 mkn ; ¼ α jZH j2 mkn : ð : Þ lmkn × 2πω3 lmkn lmkn × 2πω3 3 28 dt lmkn mkn dt lmkn mkn

In these equations,

2 2 2 Lmkn ¼ mhcot θiLz − a ωmknhcos θiE; ð3:29Þ

5 2 2 2 2 3 256ð2MrþÞ ðω − mΩ Þ½ðω − mΩ Þ þ 4ϵ ½ðω − Ω Þ þ 16ϵ ω α ¼ mkn H mkn H mkn H mkn ð Þ lmkn 2 : 3:30 jClmknj

The terms hcot2 θi and hcos2 θi in Eq. (3.29) mean cot2 θ and cos2 θ evaluated at the θ coordinate along the orbit, and then 2 averaged using Eq. (2.13). The terms jClmknj and ϵ in Eq. (3.30) are given by

j j2 ¼½ðλ2 þ 2Þ2 þ 4 ω − 4 2ω2 ðλ2 þ 36 ω − 36 2ω2 Þ Clmkn lmkn a mkn a mkn lmkn am mkn a mkn þð2λ þ 3Þð96 2ω2 − 48 ω Þþ144ω2 ð 2 − 2Þ ð Þ lmkn a mkn am mkn mkn M a ; 3:31

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 − a2 We emphasize that these cutoffs have not been selected ϵ ¼ : ð3:32Þ carefully, but are simply chosen for ease of calculation and 4Mrþ to produce results which are “converged enough” for the exploratory purposes of this paper. A more careful analysis λ The quantity lmkn appearing in Eq. (3.31) is one form of and assessment of how to truncate these sums should be the eigenvalue of the spheroidal harmonic; see Ref. [37] for done before using these ideas to make “production quality” discussion of the algorithm we use to compute it, and waveforms (e.g., for exploring LISA data analysis ques- Appendix C of Ref. [41] for discussion of the various forms tions or science return with EMRI measurements) in order of the eigenvalue in the literature (which are simple to to make sure that systematic errors in waveform modeling convert between). Note that the factor ϵ which appears here are understood and do not adversely affect one’s analysis. (and is not used elsewhere in this paper) is distinct from the As E, Lz, and Q change, we require the system to evolve similar symbol ε ≡ μ=M, the mass ratio. from one geodesic to another. To do this, we let these In evaluating Eqs. (3.26), (3.27), and (3.28), we must orbital integrals change by enforcing a balance law: truncate the infinite sums, with cutoffs determined by the needs of the analysis in question. For the purpose of this dC orbit dC ∞ dC H ¼ − − ; ð3:33Þ paper, we have implemented the following cutoffs: dt dt dt 1. We include all l from 2 to 10. 2. At each l, we include all m from −l to l. for C ∈ ½E; Lz;Q. As these orbital integrals evolve, the 3. We truncate the k sum when the fractional change orbit’s geometry slowly changes in order to keep the system −5 to the accumulated sum is smaller than 10 for on a geodesic trajectory. Appendix B shows how to relate three consecutive values of k. Holding all other rates of change for p, e, and xI to the rates of change of E, indices fixed, contributions to this sum tend to Lz, and Q. fall off monotonically and fairly rapidly with increasing k, so in practice this condition means IV. ADIABATIC INSPIRAL AS DISSIPATIVE that neglected terms change the sum by less than EVOLUTION ALONG A SEQUENCE several ×10−7. OF GEODESICS 4. We truncate the n sum when the fractional change to the accumulated sum is smaller than 10−6 for three We now examine solutions to the Teukolsky equation for consecutive values of n. Especially for e ≳ 0.4, a slowly evolving source. Critical to our analysis is the idea contributions to this sum do not fall off monoton- of a two-timescale expansion: the waveform phase varies “ ” ∼ ically with n until some threshold has been passed on a fast orbital timescale To M, and orbit character- “ ” ∼ 2 μ (see Figs. 5 and 6 of Ref. [44], and Figs. 2 and 3 of istics vary on a slow inspiral timescale Ti M = . Ref. [33]). Once past that threshold, convergence is The two timescales differ by the system’s mass ratio: ¼ μ ≡ ε quick, and we find that neglected terms in this case To=Ti =M . The waveforms we compute in this also change the sum by less than several ×10−7. way are accurate up to corrections of order ε.

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X 1 X Suppose that we have used the rates of change dE=dt, ½ φ−Φ ð iÞ hðtiÞ ≡ h ðtiÞ¼ H ðtiÞei m mkn t ; ð4:3Þ dL =dt, dQ=dt to compute how a system evolves from lmkn lmkn z lmkn r lmkn geodesic to geodesic. We parametrize inspiral by a book- keeper time ti which measures evolution along the inspiral where as seen by a distant observer. We treat the inspiral as a i i i i geodesic at each moment t , and we call the geodesic at this Hlmknðt Þ¼Almknðt ÞSlm½ϑ; aωmknðt Þ ð4:4Þ moment the “osculating” geodesic [45–47]. At each such moment, the osculating geodesic’s energy EðtiÞ, its angular and i i momentum Lzðt Þ, and its Carter constant Qðt Þ are known. 2Z∞ ðtiÞ By our assumption that the inspiral is a geodesic at each A ðtiÞ¼− lmkn : ð4:5Þ i i lmkn i 2 moment, we can reparametrize and determine pðt Þ, eðt Þ, ωmknðt Þ i and xIðt Þ. We can also compute quantities such as the i ∞ i “ ” frequencies Ω θ ϕðt Þ and the amplitudes Z ðt Þ for each Equation (4.3) showcases the multivoice structure of r; ; lmkn ð iÞ ð iÞ geodesic in this sequence. EMRI waveforms: each hlmkn t that contributes to h t constitutes a single “voice” in this waveform. As we will To leading order in ε, the curvature scalar ψ 4 that arises from this sequence of geodesics is given by see in Secs. VII and VIII, even when the waveform is complicated, each voice tends to be quite simple. As discussed in Sec. III, we can compute the amplitudes 1 X ˇ i ∞ i i A for the fiducial geodesic and then correct using the ψ 4ðt Þ¼ Z ðt ÞS ½ϑ; aω ðt Þ lmkn lmkn lm mkn ξ r lmkn phase factor mkn in order to get the wave amplitude for ’ ½ φ−Φ ð iÞ our system s initial conditions. Let us imagine we have × ei m mkn t : ð4:1Þ ˇ computed Almkn on a dense grid of orbits. Knowing i i i ½pðt ;eðt Þ;xIðt Þ, it is then simple to construct the fiducial ∞ ˇ ð iÞ This is Eq. (3.21), but with the amplitude Zlmkn and the amplitudes Almkn t along an inspiral. i ð iÞ frequency ωmkn now functions of t . Notice the dependence To convert from the fiducial amplitudes to Almkn t ,we i on harmonics of the accumulated orbital phase: need the phase factor ξmknðt Þ at each moment along the inspiral. Recall that ξmkn depends on the angles χθ0 and χr0, Z which set the polar and radial initial conditions. An ti i 0 0 important feature of the adiabatic approximation is that Φmknðt Þ¼ ωmknðt Þdt : ð4:2Þ t0 these angles are constant: χθ0 and χr0 do not change as inspiral proceeds. However, the mapping between (χθ0, χr0) ξ i and mkn does change as inspiral proceeds, leading to a This reduces to ω ðt − t0Þ in the limit where the orbit mkn slow evolution in this phase factor. When postadiabatic does not inspiral. Equation (4.2) builds in the dependence physics is included, this story changes: the angles χθ0 and on the initial time t0, which is why we leave this parameter χ 0 slowly evolve under the influence of orbit-averaged out of the factor ξ in Eq. (3.19). r mkn conservative self-forces, and orbit-averaged spin-curvature To justify this inspiraling solution for ψ 4, we substitute interactions. This will change the slow evolution of ξ Eq. (4.1) into the Teukolsky equation, Eq. (3.2). Doing so, mkn and is one way in which conservative effects leave an one finds that it satisfies the Teukolsky equation up to observationally important imprint on EMRI waveforms. errors of the order of the orbital timescale over the inspiral ð Þ ∼ ðεÞ timescale, O To=Ti O . These errors in turn arise from the fact that time derivatives have both fast-time V. FREQUENCY-DOMAIN DESCRIPTION ∂ ∼ 1 ∼ ω contributions, for which t =To mkn, as well as slow- Our description so far has focused on presenting adia- ∂ ∼ 1 ∼ ε time contributions, for which t =Ti =To. In the batic EMRI waveforms in the time domain. We showed that adiabatic approximation, we neglect the slow-time deriv- the time-domain waveform can be regarded as a super- atives, expecting that at any moment their contribution will position of different “voices,” each of which has its own be small as long as the system’s mass ratio is large. Some of slowly evolving amplitude. In this section, we will exploit the errors arising from this neglect can accumulate secu- this multivoice structure to compute the Fourier transform larly, leading to phase errors up to several radians after an of an EMRI waveform, thereby describing these waves in inspiral. Postadiabatic corrections will change the ampli- the frequency domain. tude and phase of Eq. (4.1) in such a way as to correct the Because all quantities in an EMRI evolve slowly (as long adiabatic approximation’s fast-time over slow-time errors. as the two-timescale approximation is valid), our expect- See Ref. [20] for further discussion. ation is that the stationary phase approximation (SPA) will In our applications, we are typically more interested in provide a high-quality approximation to the Fourier trans- i i the waveform hðt Þ than in ψ 4ðt Þ. This is given by form. For some voices, the frequency evolution is not

104014-11 HUGHES, WARBURTON, KHANNA, CHUA, and KATZ PHYS. REV. D 103, 104014 (2021) monotonic: some voices rises to a maximum frequency, and contribution to the integral comes from t0 ≃ 0, or equiv- then their frequency decreases. As we discuss below, it is alently from t ≃ tðfÞ,wehave conceptually straightforward to extend the “standard” SPA Z ∞ calculation in such a circumstance. We begin by reviewing ½2π ð Þ−Φð ð ÞÞ − π _ ½ ð Þð 0Þ2 0 h˜ðfÞ ≃ H½tðfÞei ft f t f e i F t f t dt : ð5:6Þ the standard calculation, then discuss voices whose fre- −∞ quency evolution is not monotonic. We conclude this section by describing how to assemble a frequency-domain This integral can be evaluated with standard methods, and waveform with many voices. we finally obtain

½2π ð Þ−Φð ð ÞÞ−π 4 A. Standard SPA calculation H½tðfÞei ft f t f = h˜ðfÞ ≃ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð5:7Þ We begin by assuming a single voice signal of the form F_ ½tðfÞ

ð Þ − ð Þ ≡ ð Þ¼ ð Þ −iΦðtÞ ð Þ hþ t ih× t h t H t e : 5:1 Note that F_ can be positive or negative, and it appears under the square root. To eliminate ambiguity about the phase of The Fourier transform of this is given by this voice in the frequency domain, we clean this up as Z ∞ follows: h˜ðfÞ ≡ hðtÞe2πiftdt Z−∞ H½tðfÞei½2πftðfÞ−ΦðtðfÞÞ∓π=4 ∞ ˜ð Þ ≃ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ ½2π −Φð Þ h f : 5:8 ¼ HðtÞei ft t dt: ð5:2Þ j _ ½ ð Þj −∞ F t f

To compute the stationary phase approximation to the We choose the minus sign in the exponential if F>_ 0, and ’ Fourier transform, we expand the signal s phase as the plus sign for F<_ 0. This approximation works well when both FðtÞ and HðtÞ change slowly: Φð Þ¼Φð Þþ2π ð − Þþπ _ ð − Þ2 þ ð Þ t tS F t tS F t tS : 5:3

1 dH 1 dF ≪ jHj; ≪ jFj: ð5:9Þ We will define the time tS momentarily. In Eq. (5.3),we F dt F dt have introduced the signal’s instantaneous frequency and ¼ the instantaneous frequency derivative at t tS: It also requires that the signal frequency F evolve mono- — 2 tonically the sign of dF=dt cannot change. 1 dΦ _ dF 1 d Φ F ≡ ; F ¼ ≡ : ð5:4Þ 2π dt dt 2π dt2 tS tS B. Nonmonotonic frequency What if our signal has a frequency which does not evolve We will assume that F_ is small, in a sense to be made monotonically? In particular, what if F rises to a maximum precise below. and then decreases, or falls to a minimum and then Using these definitions, we rewrite the Fourier transform increases? When this occurs, two problems arise with integral: the standard SPA analysis. First, in this circumstance there Z ∞ are multiple solutions to the condition FðtÞ¼f. The signal ˜ − Φð Þ 2π ½ − ð − Þ−ð1 2Þ _ ð − Þ2 hðfÞ¼e i tS HðtÞe i ft F t tS = F t tS dt at frequency f must include contributions from all times −∞ from which the signal frequency F equals the Fourier ½2π −Φð Þ ¼ ei ftS tS _ ¼ 0 Z frequency f. Second, F at the moment that the ∞ 0 2πi½ft0−Ft0−ð1=2ÞF_ ðt0Þ2 0 evolution of F switches sign. The standard SPA Fourier × Hðt þ tSÞe dt : ð5:5Þ −∞ transform is singular at that point. These issues affect EMRI signals, since the frequency associated with many voices On the second line, we changed the integration variable to rises to a maximum and then decreases. In particular, this 0 t ¼ t − tS. The integrand of Eq. (5.5) very rapidly oscil- occurs for EMRI voices which involve harmonics of the lates unless the Fourier frequency f matches the instanta- radial frequency: Ωr reaches a maximum in the strong field, neous frequency F. When this condition is met, the phase is then goes to zero as the inspiral approaches the last stationary: it is approximately constant, varying very stable orbit. slowly due to the contribution from F_ , which is assumed The root cause of the singularity is that the standard SPA _ to be small. assumes that FðtÞ and FðtÞ completely describe the signal’s _ We take the time tS to be the time at which the phase phase. If F vanishes at frequency f, then the calculation is stationary. Let us rewrite it as tðfÞ, the time at which assumes that all times contribute to the Fourier integral at f, FðtÞ¼f. Under the assumption that the largest and the integral (5.6) diverges. (This is consistent with the

104014-12 ADIABATIC WAVEFORMS FOR EXTREME MASS-RATIO … PHYS. REV. D 103, 104014 (2021) fact that the Fourier transform of a constant-frequency where signal is a delta function.) However, for real EMRI signals,

FðtÞ is not constant when F_ ¼ 0; the singularity in the SPA 1 3Φ ̈≡ d ð Þ ð Þ _ ð Þ F 3 : 5:11 analysis is an artifact of our assumption that F t and F t 2π dt completely characterize the signal’s phase. To remove this tS artifact, we need more information about the signal’s phase evolution. Let us therefore include the next term in the Now, we revisit Eq. (5.2) but use Eq. (5.10) to expand the expansion: phase. We again find that tS is the stationary time, for which F ¼ f. However, there may be multiple roots of the equation FðtÞ¼f. Let us define t ðfÞ to be the jth time π j Φð Þ¼Φð Þþ2π ð − Þþπ _ ð − Þ2 þ ̈ð − Þ3 ð Þ¼ _ ≡ _ ½ ð Þ ̈ ≡ ̈½ ð Þ t tS F t tS F t tS 3F t tS ; at which F t f, and write Fj F tj f , Fj F tj f . For EMRI waveforms, N ≤ 2. Each value of j contributes ð Þ 5:10 to the Fourier transform, so we have

Z XN ∞ _ 0 2 ̈ 0 3 ˜ i½2πftjðfÞ−ΦðtjðfÞÞ −iπ½Fjðt Þ þFjðt Þ =3 0 hðfÞ ≃ H½tjðfÞe e dt : ð5:12Þ −∞ j¼1

To perform this integral, we set α ¼ γ þ 2πiF_ , with γ real and positive. We define β ¼ 2πF̈, and use Z ∞ 2 α −αt2=2−iβt3=6 pffiffiffi α3=3β2 3 2 e dt ¼ e K1=3ðα =3β Þ; ð5:13Þ −∞ 3 jβj where KnðzÞ is the modified Bessel function of the second kind. Taking the limit γ → 0, we find

XN _ 2 _ 3 ̈2 ˜ i½2πft ðfÞ−Φðt ðfÞÞ iFj −2πiF =3F _ 3 ̈2 hðfÞ ≃ pffiffiffi H½t ðfÞe j j e j j K1 3ð−2πiF =3F Þ: ð5:14Þ 3 j j̈j = j j j¼1 Fj

This result defines our “extended” SPA. It is useful to examine two limits of Eq. (5.14). To facilitate taking these limits, we define

2π F_ 3 ≡ j ð Þ Xj 3 ̈2 : 5:15 Fj _ ̈ ̈ First, we take Fj to be arbitrary and expand about Fj ¼ 0. To set this up this, we eliminate Fj in Eq. (5.14): rffiffiffi XN _ pffiffiffiffiffi 2 ½2π ð Þ−Φð ð ÞÞ Fj − ˜ð Þ ≃ ½ ð Þ i ftj f tj f qffiffiffiffiffiffi iXj ð− Þ ð Þ h f π H tj f e i Xje K1=3 iXj : 5:16 ¼1 _ 3 j Fj

̈ Expanding about Fj is equivalent to examining Eq. (5.16) for Xj → ∞. Using

F_ qffiffiffiffiffiffij _ −1=2 _ ¼jFjj Fj > 0; _ 3 Fj _ −1=2 _ ¼ ijFjj Fj < 0; ð5:17Þ and, for X real, rffiffiffi pffiffiffiffi π 5 1 −iX −iπ=4 i lim i Xe K1=3ð−iXÞ¼ e 1 − þ ; ð5:18Þ X→∞ 2 72 X we have

104014-13 HUGHES, WARBURTON, KHANNA, CHUA, and KATZ PHYS. REV. D 103, 104014 (2021) XN ½2π ð Þ−Φð ð ÞÞ∓π 4 ̈2 H½t ðfÞei ftj f tj f = 5i F h˜ðfÞ¼ j 1 − j þ : ð5:19Þ j _ j1=2 48π _ 3 j¼1 Fj Fj

The minus sign in the exponential is for F>_ 0; the plus sign is for F<_ 0. Equation (5.19) is an accurate approximation ̈ 2 _ 3 ̈ when jFjj ≪ jFjj . Notice that we recover the standard SPA result, Eq. (5.8), when Fj ¼ 0 and N ¼ 1. ̈ _ _ Next, we allow Fj to be any real value and expand about Fj ¼ 0. To do this, use Eq. (5.15) to replace Fj with powers of ̈ Xj and Fj in Eq. (5.14),

2 3 22=3 XN F̈ = ˜ i½2πft ðfÞ−Φðt ðfÞÞ j 1=3 −iX hðfÞ ≃ H½t ðfÞe j j iX e j K1 3ð−iX Þ; ð5:20Þ 31=6π1=3 j j̈j j = j j¼1 Fj and expand about Xj ¼ 0. Using 2πi=3Γð1Þ 2πi=3Γð− 1Þ 1=3 −iX e 3 2=3 e 3 limiX e K1 3ð−iXÞ¼ 1 − X ; ð5:21Þ →0 = 2=3 2=3 1 X 2 2 Γð3Þ we find

2 3 2πi=3Γð1Þ XN ̈ = π 2=3 Γð− 1Þ _ 2 ˜ e 3 ½2π ð Þ−Φð ð ÞÞ Fj 2π 3 3 Fj hðfÞ ≃ H½t ðfÞei ftj f tj f 1 − e i= þ : ð5:22Þ 31=6π1=3 j jF̈j 3 Γð1Þ ̈4=3 j¼1 j 3 Fj

j _ j2 ≪ j̈j4=3 ¼ 12 ¼ 0 7 ε ¼ 10−3 Equation (5.22) is accurate when Fj Fj . Notice pinit M, einit . , and mass ratio . The full _ that this result is well behaved and finite at Fj ¼ 0, time-domain waveform and additional voices are discussed demonstrating that the extended SPA cures the singularity for this case in more detail in Sec. VIII A. at points where the rate of change of the instantaneous On the left-hand side of Fig. 1, we show this voice’s signal frequency passes through zero. time-domain amplitude and the evolution of its frequency. We note here that the issue of a signal’s frequency and The voice frequency increases until it reaches a maximum, frequency derivative both vanishing was examined by then rapidly decreases, at least until the inspiral ends Klein, Cornish, and Yunes [48] in the context of compa- several hundred M after reaching this maximum. The rable mass binaries with spinning and precessing constitu- Fourier transform, computed using Eq. (5.14) and shown ents. Although superficially similar to the case we discuss in the right-hand panels, has two branches: the branch with here, the root cause of the pathology in their case was rather f>_ 0, shown as the long-dashed curve in the upper right- different. In addition to having an orbital timescale To and hand panel of Fig. 1; and the branch with f<_ 0, shown as an inspiral timescale Ti, the waveforms from precessing the short-dashed curve in this panel. binaries vary on precession timescales Tp which are typically Combining the two branches yields the solid curve intermediate to Ti and To. A given binary typically exhibits shown in the lower right-hand panel of Fig. 1. We overlay ’ precession on multiple timescales, depending on the binary s on this plot a discrete Fourier transform computed using “ spins. Such a waveform may also be considered multi- this time-domain voice; because of the large dynamic range ” voice, due to the evolution of features in different harmon- in the signal’s amplitude, we consider separately a low- ics. One finds in this case that the stationary points of frequency DFT (focusing on data for ti ≲ 5.5 × 105M, for different voices can coalesce, leading to a pathological SPA which f ≲ 0.018M) and a high-frequency DFT (focusing estimate for the waveform’s Fourier transform. Each indi- on data for ti ≳ 5.5 × 105M). We use a Tukey window of vidual voice, however, remains well behaved, in contrast to width 500M to taper these segments of the time-domain the case for EMRIs. As such, their treatment does not need to use additional information about the frequency evolution, as signal. Aside from lobes near the boundaries associated we find is necessary in our analysis. with these windows, we see perfect agreement between the Figure 1 illustrates how these elements come together DFT and the SPA. Notice the interesting structure in the band 0.024 ≲ Mf ≲ 0.037. This arises from beating for a voice that showcases many of the features we have _ discussed here. We show the l ¼ m ¼ 2, k ¼ 0, n ¼ 20 between the contributions along the branches with f>0 voice for an equatorial Kerr inspiral with a ¼ 0.9M, and f<_ 0: at each frequency in this band, the signal

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FIG. 1. An example of the time-domain and frequency-domain structure of one voice: l ¼ m ¼ 2, k ¼ 0, n ¼ 20, for an equatorial −3 Kerr inspiral with a ¼ 0.9M, pinit ¼ 12M, einit ¼ 0.7, and mass ratio ε ¼ 10 . On the left, we show this voice’s amplitude (top) and frequency (bottom) as a function of time over the inspiral. The voice’s frequency increases until it reaches fmax ≃ 0.037=M; it then reverses and falls to ffinal ≃ 0.024=M at the end of inspiral. The spiky features occur when this amplitude passes through zero (note the log scale). On the right, we show the frequency-domain structure computed with the extended SPA. In the top right, we examine contributions to the SPA along the two branches of FðtÞ¼f. The curve with long dashes is the magnitude of the SPA for the branch with f>_ 0; the curve with short dashes is for the branch with f<_ 0. The bottom right shows the final SPA Fourier transform, obtained by summing contributions from the two branches, and compares this to a discrete Fourier transform (DFT). Because of the large dynamic range between the low-frequency and high-frequency behavior of this voice, we separately examine the DFT for the early-time, low-frequency portion (corresponding to ti ≲ 5.5 × 105M, plotted with magenta dashes) and the DFT for the late, high-frequency portion (ti ≳ 5.5 × 105M, plotted with blue dots). Modulo lobes at the boundaries of the two DFTs (associated with the Tukey window used to taper the time-domain signal near these boundaries), the DFTs coincide perfectly with the SPA. Notice the interesting behavior in the band 0.024 ≲ Mf ≲ 0.037. This arises from beating between the contributions along the two branches. contributes at two different times, and with two different The calculation proceeds as before, but now the phases. Additional examples of voices, in both the time and phase is stationary for each voice at some moment. We frequency domains, are shown in Secs. VII and VIII. define

C. Multiple voices 1 dΦ F ¼ V ð5:24Þ Generalization to a multivoice signal is straightforward. V 2π dt Let us write our signal in the time domain X − Φ ð Þ ð Þ¼ ð Þ i V t ð Þ _ ̈ h t HV t e ; 5:23 and likewise define FV, FV. We assume that the condition V FVðtÞ¼f has NV solutions for voice V, and we define where V labels the voice and is shorthand for all the mode tj;VðfÞ to be the jth such solution. (As stated in the previous indices which describe each voice of the waveform. For section, NV will be either 1 or 2 for EMRIs.) The generic EMRIs, V ≡ ðl; m; k; nÞ. stationary-phase Fourier transform is then

3 3 2 X XNV _ 2π _ 2π _ ˜ i½2πft ðfÞ−Φ ðt ðfÞÞ iFj;V i Fj;V i Fj;V hðfÞ¼pffiffiffi H ½t ðfÞe j;V V j;V exp − K1 3 − V j;V j̈ j 3 ̈2 = 3 ̈2 3 ¼1 Fj;V Fj;V Fj;V X V j ˜ ≡ hlmknðfÞ: ð5:25Þ V

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_ _ ̈ We have introduced Fj;V ≡ FV½tj;VðfÞ and Fj;V≡ Using this spacing, we have laid down 40 points between ̈ ¼ þ 0 02 ¼ þ 10 FV½tj;VðfÞ. Depending on the relative values of various pmin pLSO . M and pmax pmin M. Different _ ̈ choices certainly could be used—for example, a grid powers of Fj;V and Fj;V, one can expand the Vth voice as in Eqs. (5.19) or (5.22). reaching to larger p and with a different algorithm for increasing density near pLSO was used in Ref. [16]. The choice of p spacing is an example of an issue that should be VI. IMPLEMENTATION more carefully investigated, and perhaps empirically In this section, we describe various technical details by designed depending on what works best given computing which we implement this formalism for computing EMRI resources and accuracy needs for one’s application. waveforms. To make an adiabatic inspiral and its associated For any X stored on our grid, dX=de → 0 as e → 0. waveform, we lay out a grid of orbits, parametrized by each Empirically, we find it is important to have dense grid orbit’s ðp; e; xIÞ. We store all the data at each grid point coverage for small e in order for this behavior to be needed to construct the inspiral and the waveform. We then accurately captured. For this paper, we have used grids that interpolate to estimate the values of each datum at locations run over 0 ≤ e ≤ 0.8 in steps of Δe ¼ 0.1. Work in away from the grid points. In this section, we describe this progress [52] suggests that this spacing may introduce data grid and the data which are stored on it, and details of small systematic errors in computing the inspiral rate as a how we interpolate data off the grid. We emphasize that function of initial eccentricity; using higher density across there is surely a great deal of room to improve on the at least the small-e part of this range appears to effectively techniques we present here; indeed, we used different address this. More detailed discussion of this point will be algorithms to design our data grid and to perform inter- presented in later work [52]. − polation in a closely related companion analysis (Ref. [16]). For fixed p pLSO and fixed e, all our stored data tends For this initial study, the grids and interpolation techniques to be very smooth and indeed nearly linear as a function we use are chosen for ease of use. In later work, we plan to of xI. Our grid covers the range −1 ≤ xI ≤ 1, with 16 investigate how best to optimize the grids and interpolation points spaced by ΔxI ¼ 2=15 ≃ 0.1333. methods for speed and accuracy of waveform calculation. We store the following data at each grid point: ∞;H ∞;H 1. The rates of change ðdE=dtÞ , ðdLz=dtÞ , and ∞;H A. EMRI data grids ðdQ=dtÞ obtained by summing over many modes until convergence has been reached, using the We store our data on a grid that is rectangular in − convergence criteria described in Sec. III C. p pLSO, e, and xI, where pLSO parametrizes the last 2. The rates of change of the orbital elements stable orbit (LSO). The value of p is easy to calculate as ∞ ∞ ∞ LSO ðdp=dtÞ ;H, ðde=dtÞ ;H, and ðdx =dtÞ ;H obtained e x I a function of and I [49], making it simple to set up a by using the Jacobian described in Appendix B with grid in this space. We set our innermost grid point to ¼ þ 0 02 these fluxes. pmin pLSO . M, slightly outside of the LSO. The ˇ Ω → 0 → 3. The fiducial amplitude Almkn for all modes used to radial frequency r as p pLSO, which means that ð Þ∞ ð Þ∞ ð Þ∞ Ω compute dE=dt , dLz=dt , and dQ=dt . Fourier expansions in r tend to be badly behaved as the So far, we have constructed such datasets for spherical LSO is approached; this can be regarded as a precursor to (e ¼ 0) and equatorial orbits (x ¼1) for a=M ∈ ’ I the small body s plunge into the black hole [50,51] at the ½0; 0.1; 0.2; …; 0.9; 0.95; 0.99, as well as for generic orbits end of inspiral. Very little inspiral remains when the small for spin a ¼ 0.7M covering 0 ≤ e ≤ 0.4. These data are body has reached our choice of p , so we are confident that min produced using the code GREMLIN, a frequency-domain the error incurred by truncating at pmin (rather than closer to Teukolsky solver primarily developed by author Hughes, pLSO) is negligible. That said, it should be emphasized that with significant input from collaborators. The core methods this choice of pmin has not been carefully evaluated. It will be of this code are described in Refs. [33,37] and updated to ’ useful to systematically examine how to select the grid s use methods developed in Refs. [39,40] for solving the “ ” inner edge in future production quality work. homogeneous Teukolsky equation; see Ref. [53] for further Many important quantities vary rapidly near the LSO. ˇ discussion. Of particular importance is the phase of Almkn, which tends Two-dimensional orbits (eccentric and equatorial, or → to rotate rapidly as p pLSO. It is crucial to resolve this spherical and inclined) typically require several hundred behavior in order to compute accurate waveforms. To to several thousand modes in order to converge as described account for this behavior, we use a grid whose density in Sec. III C, reaching ∼104 modes in the strongest fields. increases near pLSO. For this paper, our grid is uniformly The number of modes needed for the convergence of spaced in generic orbits is an order of magnitude or two larger. 1 Each spherical orbit mode requires about 0.1 seconds of ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ CPU time for small values of l, increasing to roughly u − 0 9 : 6:1 p . pLSO 0.25 seconds for modes with l ¼ 10. Computing eccentric

104014-16 ADIABATIC WAVEFORMS FOR EXTREME MASS-RATIO … PHYS. REV. D 103, 104014 (2021) orbit modes is more time consuming, since each mode B. Interpolating and integrating across the grid involves an integral over the radial domain covered by its To find data away from the grid points, in this analysis orbit. This cost also varies significantly by radial mode we use cubic spline interpolation in the three directions. number n. For small l, modes for small eccentricity Because our grid is rectangular in ðp − p ;e;x Þ, this can ≲ 0 2 LSO I (e . ) take on average 1 CPU second or less, be implemented effectively, and is adequate for demonstrat- ≈ 0 5 – medium-eccentricity modes (e . ) average about 5 10 ing how to build adiabatic waveforms and illustrating the ¼ 0 8 CPU seconds, and large-eccentricity modes (e . ) results. Cubic spline interpolation for the individual mode – average 30 40 CPU seconds each. These averages are amplitudes will not scale well to “production-level” code, in skewed significantly by larger values of n, for which the terms of computational efficiency and memory considera- integrand of Eq. (3.13) rapidly oscillates, and the integral tions. In Ref. [16], a set of the present authors used reduced- tends to be small compared to the magnitude of the order methods with machine learning techniques to construct ¼ 10 integrand. At l , these times increase: small- a global fit to the set of mode amplitudes, finding outstanding eccentricity modes take on average up to 20 CPU seconds, efficiency gains in an initial study of Schwarzschild EMRI – medium eccentricity modes average about 40 50 CPU waveforms. Future work will apply these techniques to the seconds, and large eccentricity modes require on average more generic conditions we examine here. 150 seconds. Other data needed to construct the waveform (for The CPU cost per mode is ameliorated by the fact that example, the geodesic frequencies Ωr;θ;ϕ and the phases each orbit ðp; e; x Þ and each mode ðl; m; k; nÞ is indepen- I ξmkn) are calculated at each osculating geodesic as inspiral dent of all others. As such, this problem is embarrassingly proceeds. Substantial computing speed could be gained by parallelizable, and datasets can be effectively generated storing and interpolating such data; indeed, all such data on distributed computing clusters. The datasets described tend to evolve smoothly and fairly slowly over an inspiral, above are publicly available through the Black Hole so it is likely that effective interpolation could be imple- Perturbation Toolkit [54]. Plans to extend these sets, mented. We leave an investigation of what to store and develop further examples, and release the GREMLIN code interpolate versus what to compute for future work and are described in Sec. IX. future optimization.

ð Þ¼ð12 0 2 1Þ FIG. 2. The waveform hþ for inspiral into a Schwarzschild black hole with pinit;einit;xI;init M; . ; (left-hand panels) and ð Þ¼ð12 0 7 1Þ ¼ 0 pinit;einit;xI;init M; . ; (right-hand panels). The waves are observed in the plane of the orbit, so h× , and the mass ratio is ε ¼ 10−3 in all cases. The top panels show the complete inspiral waveform over this domain, with the initial radial anomaly angle χr0 ¼ 2π=3. The bottom panels compare hþ including the phase corrections ξm0n (solid [red] curves) with hþ neglecting these ˇ corrections (i.e., incorrectly using the fiducial amplitudes Alm0n; dashed [blue] curves). In both cases, we compare early times (the first 3000M of inspiral) and late times (an interval of 1000M near the end of inspiral). Neglecting ξm0n leads to significant differences in the waveforms.

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To make an adiabatic inspiral, we construct a sequence of i i i osculating geodesics, parametrized by ½pðt Þ;eðt Þ;xIðt Þ ½ given initial conditions pinit;einit;xI;init . To do this, we interpolate to find dp=dt, de=dt, and dxI=dt at each inspiral time ti, then use a fixed-step-size fourth-order Runge-Kutta integrator to construct the inspiral. This simple method is an obvious point for improvement in future work; indeed, in the related analysis [16], we used a variable-step-size eighth-order Runge-Kutta integrator. Note that both ðdp=dtÞ and ðde=dtÞ are singular as the LSO is approached. This is because the Jacobian (Appendix B) relating these quantities has zero determinant at the LSO. This singularity can adversely affect the accuracy of interpolating these quantities. Multiplying both ðdp=dtÞ ð Þ ð − Þ2 and de=dt by p pLSO , which exactly describes the singularity in the zero-eccentricity limit and approximately describes it in general, yields smooth data which interpolate very well. Another approach is to interpolate the fluxes dE=dt, dLz=dt,anddQ=dt, normalized by post-Newtonian expansions of these quantities in order to “divide out” their most rapid variations across orbits. This would construct FIG. 3. The phase correction ξ100 (bottom panel) and ξ001 (top panel) for inspiral with pinit ¼ 12M, einit ¼ 0.2 at mass ratio a sequence of osculating geodesics parametrized as −3 i i i ε ¼ 10 . In both panels, the solid (black) line corresponds to ½Eðt ;Lzðt Þ;Qðt Þ. Determining the most efficient way to χ 0 ¼ 0, dotted (red) curves show χ 0 ¼ π=6, short-dashed (blue) parametrize these orbits in order to optimize waveform r r curves show χ 0 ¼ π=2, long-dashed (magenta) curves show construction for speed and accuracy are very natural direc- r χr0 ¼ 11π=6, and dot-dashed (green) curves show χr0 ¼ 3π=2. tions for future work. Notice that these curves show only gentle variation until nearly the end, with many changing rapidly as the inspiral approaches VII. RESULTS I: EXAMPLE SCHWARZSCHILD the last stable orbit. EMRI WAVEFORMS AND THEIR VOICES We now present examples of EMRI waveforms and their voices in the time and frequency domains. Our goal is not to exhaustively catalog EMRI waveforms, but just to present examples which showcase the behavior that we find, and how this behavior tends to correlate with source properties. We begin here with results for Schwarzschild; the next section shows Kerr results. Thanks to spherical symmetry, Schwarzschild orbits are confined to a plane, which we define as equatorial. We can thus set Q ¼ 0 and focus on voices with k ¼ 0 (i.e., neglecting harmonics of the θ motion). We examine two ð Þ¼ð12 0 2Þ cases: one starts at pinit;einit M; . and inspirals to ≃ 0 107 ð Þ¼ð12 0 7Þ efinal . ; the other starts at pinit;einit M; . ≃ 0 374 and inspirals to efinal . . In both cases, pfinal follows ¼ð6 þ 2 Þ from the Schwarzschild last stable orbit: pLSO e M. We use mass ratio ε ¼ 10−3 in all the cases we examine, both here and in the following section. Results for other extreme mass ratios can be inferred by scaling durations and accumulated phases with 1=ε.

A. Waveforms in the time domain

The top two panels of Fig. 2 show hþ for the two FIG. 4. The same as Fig. 3, but for an inspiral with pinit ¼ 12M, Schwarzschild inspirals we examine, both with initial einit ¼ 0.7. These phases show more variation in this case than χ ¼ 2π 3 ¼ 0 2 anomaly angle r0 = . The waveform is shown in when einit . , although the curves still show only gentle ’ ¼ 0 the system s equatorial plane, so h× . For the case with variations until the system approaches the end of inspiral.

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¼ 0 2 χ ¼ 0 einit . , we plot contributions from all modes with We show these phases for initial anomaly angles r0 l ∈ ½2; 3; 4, m ∈ ½−l; …;l, n ∈ ½0; …; 10 (as well as (solid [black] curves), π=6 (dotted [red]), π=2 (short-dashed modes simply related by symmetry); for the case with [blue]), 3π=2 (dot-dashed [green]), and 11π=6 (long- ¼ 0 7 einit . , we use the same l and m range, but go over dashed [magenta]). For the small-eccentricity case, both n ∈ ½0; …; 40. As in our discussion of the convergence of ξ001 and ξ100 are nearly flat over the inspiral, though they adiabatic backreaction in Sec. III C, we emphasize that we show significant variation in the very last moments. The have not carefully analyzed how many modes to include. variation is larger in the higher-eccentricity case for ξ001, This range produces visually converged waveforms— changing by almost a radian over the inspiral for χr0 ¼ π=2 additional modes do not change the waveform enough to and 3π=2 even before reaching the large change at the very impact the figures. This is adequate for this paper. end. In all cases, ξ100 and ξ001 are smooth and well behaved. The lower panels of Fig. 2 show the influence of the They are also relatively simple to calculate, only requiring phase ξm0n on the waveform, zooming in on early and late information about the geodesic with parameters p and e. ξ times. The red curves include all m0n corrections, and the Since computing Alm0n is an expensive operation, one ˇ blue curves neglect them, showing inspirals made using should only compute the fiducial amplitudes A 0 and use ˇ lm n the fiducial amplitudes Alm0n. Both early and late in the the phase ξm0n ¼ mξ100 þ nξ001 to convert. inspiral, the phase correction has a noticeable influence. To calibrate how well the phases ξm0n allow us to account This is not surprising, since the impact of ξm0n is to adjust for initial conditions, we compare the waveform assembled the system’s initial conditions—different choices of ξm0n voice by voice with one computed independently using a correspond to physically different inspirals. The influence time-domain Teukolsky equation solver. For the compari- on the large-eccentricity case is particularly strong. son waveform, we compute the worldline followed by an Figures 3 and 4 show how the phases ξ001 (top panels) inspiraling body, use it to build the source for the time- and ξ100 (bottom panels) evolve over these inspirals. domain Teukolsky equation as described in Ref. [18], and then compute the waveform using the techniques developed in Refs. [18,19]. The time-domain solver projects its output

FIG. 5. The waveform for inspiral into a Schwarzschild black −3 hole with pinit ¼ 12M, einit ¼ 0.7, χr0 ¼ 0 at mass ratio ε ¼ 10 computed voice by voice using the methods described here (blue crosses), and computed using a time-domain Teukolsky equation FIG. 6. Same as Fig. 5, but for χr0 ¼ 2π=3; the multivoice data solver for the worldline of an inspiral with these initial conditions. shown here are identical to the l ¼ 2, m ¼ 2 χr0 ¼ 2π=3 data Though only voices with l ¼ 2, m ¼ 2 are included, the multi- shown in Fig. 2. The two waveforms agree very well at early χ ¼ 0 voice data are otherwise identical to the r0 data shown in times, but a secular offset accumulates in this case; as shown in Fig. 2. The top panel shows a stretch Δti ¼ 3000M near the the lower panel, the two waveforms are roughly 2 radians out of beginning of inspiral; the bottom shows a stretch Δti ¼ 1000M phase by ti ≃ 1.5 × 105M. This grows to about 4 or 5 radians by near ti ¼ 1.5 × 105M, roughly the middle of this inspiral. The the end of inspiral. Our hypothesis is that the time-domain two calculations agree perfectly in these two snapshots; we in fact integrator includes slow-time contributions (i.e., contributions find that they hold this agreement all the way to the end from source terms that evolve on the longer inspiral timescale Ti) at ti ≃ 3 × 105M. which are left out of the adiabatic waveform.

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onto spherical ðl; mÞ modes of spin-weight −2; we focus our comparison on voices with l ¼ 2, m ¼ 2. Figures 5–7 summarize the results that we find for ¼ 12 ¼ 0 7 Schwarzschild inspiral with pinit M, einit . . Figure 5 shows what we find when the initial anomaly angle χr0 ¼ 0. In this case, we find that the waveform assembled voice by voice and the time-domain comparison remain in phase for the entire inspiral. In the figure, we compare the two waveforms for a stretch of duration Δti ¼ 3000M at the beginning of inspiral as well as a stretch of duration Δti ¼ 1000M in the middle (near ti ¼ 1.5 × 105M). The two waveforms lie on top of each other in both cases; for this choice of χr0 we find an excellent match all the way to the end at ti ≃ 3 × 105M. Figure 6 shows the inspiral waveforms when we set χr0 ¼ 2π=3. In this case, we find a secular drift which accumulates as inspiral proceeds. The top panel is again a stretch Δti ¼ 3000M from the beginning of inspiral; as in Fig. 5, the two computed waveforms lie on top of one FIG. 7. Same as Fig. 6, but using the ad hoc phase modification another. However, by ti ¼ 1.5 × 105M, the two waveforms introduced in Eq. (7.1). This correction adjusts the orbital phase are about 2 radians out of phase, as can be seen in the lower with a term that depends on the inspiral rate. Though this panel of this figure. This mismatch grows to about 4 or correction has not been rigorously computed, it nonetheless substantially improves the agreement between the two wave- 5 radians by the end of the inspiral. forms, at least over this domain of ti. (As described in the text, As discussed in Sec. IV, our solution neglects the impact the waveforms drift from one another as inspiral continues.) of “slow-time” derivatives on EMRI evolution, leaving This supports the hypothesis that this drift arises due to slow-time out time derivatives of terms which vary on the inspiral variations neglected in the adiabatic approximation. timescale Ti. As such, our solution only solves the Teukolsky

ð Þ¼ð12 0 2Þ FIG. 8. Some of the voices which contribute to hþ for the case pinit;einit M; . , as shown in the left panels of Fig. 2. The i i magnitude of the amplitude jHlm0nðt Þj is on the left; the phase Φm0nðt Þ is on the right. The top panels show voices with l ¼ 4 and m ¼ 4; the bottom panels show l ¼ 2 and m ¼ 2. Solid (red) curves are data with n ¼ 0, short-dashed (blue) curves are n ¼ 1, long- dashed (green) curves are n ¼ 2, dot-dashed (magenta) curves are n ¼ 3, and dotted (black) curves are n ¼ 4. In all cases, both the amplitude and the phase evolve smoothly and simply. As functions of time, these voices can be sampled much less densely than hþ must be sampled in order to accurately track its behavior.

104014-20 ADIABATIC WAVEFORMS FOR EXTREME MASS-RATIO … PHYS. REV. D 103, 104014 (2021) equation (3.2) up to errors of OðεÞ. The time-domain solver, dp=dt connects this phase to the inspiral, and the factors by contrast, finds a solution which, up to numerical dis- 1=p and 1=Ωr;ϕ provide dimensional consistency. The cretization, solves Eq. (3.2) at all orders in To=Ti.Our numerical factor 3=2 was determined empirically. It is hypothesis is that this phase offset is because the time- interesting to note that a slow-time evolution in the domain solver captures at least some of the “slow-time” Newtonian limit yields derivatives which are missed by the adiabatic construction. Interestingly, the magnitude of the offset depends strongly Ω 3 Ω Ω d ¼ − dp − 3 de ð Þ χ χ e 2 : 7:2 on r0. By examining multiple values of r0, we find that dt 2 dt p dt 1 − e the effect varies at least approximately as 1 − cos χr0.This ξ suggests that a slow-time variation in m0n may play a Our empirical phase modification appears to be consistent particularly important role in this secular drift. with a weak-field correction associated with the rate at To test the hypothesis that the offset is due to overlooked which p changes due to inspiral. slow-time terms, we replaced the accumulated phase, We strongly emphasize that Eq. (7.1) is completely Eq. (4.2), with the following ad hoc modification: ad hoc, and has not been justified by any careful calcu- Z lation. However, we find that it does surprisingly well at ti 3 dp=dt Φmodð iÞ¼ Ω 1 þð1 − χ Þ improving the match between the two calculations. m0n t m ϕ cos r0 2 Ω t0 p ϕ Figure 7 is the equivalent of Fig. 6, but with Eq. (7.1) used to compute the phase rather than Eq. (4.2). Notice that 3 dp=dt þ Ω 1 þð1 − χ Þ ð Þ the two waveforms lie on top of one another, at least over n r cos r0 2 Ω dt: 7:1 p r the domain shown here. As inspiral proceeds, our ad hoc fix becomes less accurate: we find a roughly 1-radian i 5 The factor of ð1 − cos χr0Þ in this modification accounts for offset between the two waveforms when t ≃ 2.5 × 10 M, the empirical dependence on χr0 that we found; the factor of

FIG. 10. Some voices with l ¼ 2, m ¼ 2 which contribute to FIG. 9. Some voices with l ¼ 2 and m ¼ −2 which contribute the large-eccentricity waveform shown in Fig. 2. The top panel to hþ for the small-eccentricity case shown in Fig. 2. Colors and shows the voices’ phase; the lower panel shows their amplitudes. definitions are exactly as in Fig. 8. The behavior of modes with Solid (red) curves are data with n ¼ 0, short-dashed (blue) curves n ¼ 3 and n ¼ 4 are interesting: During the inspiral, there are are n ¼ 3, long-dashed (green) curves are n ¼ 6, dot-dashed moments at which the phase evolution reverses direction, (magenta) curves are n ¼ 9, and dotted (black) curves are corresponding to the voice’s instantaneous frequency changing n ¼ 12. In contrast to the small-eccentricity case, the voice with sign. The amplitudes corresponding to these voices pass through n ¼ 0 does not dominate over most of the inspiral. In fact, of the zero at times very close to these moments. The zero passage of voices shown, the one which starts weakest (n ¼ 3) evolves to these amplitudes leads to the sharp appearance that we see here. become the loudest by the end of inspiral. Nonetheless, all The real and imaginary parts of H2−23 and H2−24 are perfectly amplitudes and phases evolve in a smooth and simple way, just as smooth. in the low-eccentricity case.

104014-21 HUGHES, WARBURTON, KHANNA, CHUA, and KATZ PHYS. REV. D 103, 104014 (2021) growing to several radians by the end of inspiral. We find sign. In this low-eccentricity case, the voices with l ¼ 2, nearly identically improved matches examining inspirals m ¼2, n ¼ 0 have the largest amplitudes. with different values of χr0, and for different choices of Figure 10 shows some of the voices that contribute for ε ¼ 0 7 ’ the mass ratio . Interestingly, including terms in de=dt the case with einit . . Again we see that the voices inspired by the weak-field rate of change of Ω does not amplitudes and phases are smooth and well behaved. In help, suggesting that the similarity to the weak-field contrast to the small-eccentricity case, modes with n ¼ 0 formula may be a coincidence. do not dominate here. Of the voices we plot, n ¼ 6 This analysis indicates that the phase offset we find is dominates at early times, though it falls below several consistent with a postadiabatic effect, and therefore is other voices as the inspiral ends. The voice with n ¼ 3 missed by construction when making adiabatic waveforms. starts out weakest, but it becomes strongest roughly half- The surprising effectiveness of our ad hoc fix suggests that way through this inspiral. it may not be too difficult to analytically model this All of the voices we examine have this behavior: both the behavior and improve these waveforms. amplitudes and phases of individual voices evolve smoothly on the inspiral timescale Ti. This property is shared by the B. Waveform voices voices’ frequency-domain behavior. Following Sec. V,we compute the frequency-domain representation of the voices The waveforms shown and discussed in Sec. VII A are fairly complicated, especially for the high-eccentricity case. examined here; Figs. 11 and 12 show our results. Again, By contrast, the individual voices which contribute to these we see that all the voices evolve smoothly over the inspiral. waveforms are very simple, evolving smoothly and simply The apparent spikiness in some cases (for example, the voice ¼ ¼ 4 ¼ 4 ¼ 0 2 on the much longer inspiral timescale. with l m , n for einit . ) is because this ’ Figures 8 and 9 show individual voices that contribute voice s amplitude passes through zero, and we show its ¼ 0 2 magnitude on a log scale. It is also worth noting that the for the case with einit . . A handful of the voices we show look jagged in these figures due to how we have frequency range of different voices varies. This is because presented the data: the amplitude passes through zero in our analysis begins in the time domain, and then transforms some cases, so jHlm0nj appears spiky on a log-linear plot. to the frequency domain using the SPA. Different voices thus These zero passings correspond to moments when the start at different frequencies and reach different frequencies instantaneous frequency Fm0n ¼ð1=2πÞdΦm0n=dt changes at the end of inspiral.

FIG. 11. Frequency-domain representation of the voices shown in Figs. 8 and 9. The top-left panel shows voices with l ¼ m ¼ 2; the bottom-left panel shows voices with l ¼ m ¼ 4; both panels on the right show voices with l ¼ 2, m ¼ −2. In all cases, the solid (red) curves are the frequency-domain amplitudes with n ¼ 0; short-dashed (blue) curves are for n ¼ 1; long-dashed (green) curves are n ¼ 2; dot-dashed (magenta) curves are n ¼ 3; and dotted (black) curves are n ¼ 4. (We use two panels for the voices with m ¼ −2 to showcase the range in amplitude of this case.) Because we generate the waveform in the time domain and use the stationary-phase Fourier transform, each voice spans a slightly different range of frequency.

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The left-hand panels of Fig. 13 show the gravitational waveform we find in this case. For the time-domain waveform, we plot contributions from all modes with l ∈ ½2; 3; 4, m ∈ ½−l; …;l, k ∈ ½0; …; 10, plus modes that are simply related by symmetry. (For inspirals with zero eccentricity, only voices with n ¼ 0 contribute.) The strong influence of spin-orbit modulation can be seen in the lower left-hand panels, which zoom in on early and late times. These lower panels also illustrate the role of the initial polar anomaly angle χθ0, contrasting the waveform with χθ0 ¼ 2π=3 versus the one with χθ0 ¼ 0. The two FIG. 12. Frequency-domain representation of the voices shown waveforms are similar in shape but shifted, consistent with ¼ ¼ 2 in Fig. 10. All curves show voices with l m . The solid the fact that χθ0 controls the system’s initial conditions: (red) curve is the frequency-domain amplitude with n ¼ 0; the χ ¼ 0 θ ¼ θ ¼ 0 χ when θ0 , the orbit is at min when t ; for θ0, short-dashed (blue) curve is for n ¼ 3; the long-dashed (green) θ ¼ 6 ¼ 9 the orbit starts at a value of roughly midway between the curve is for n ; the dot-dashed (magenta) curve is for n ; θ ¼ π − θ and the dotted (black) curve is for n ¼ 12. equator and max min. The right-hand panels of Fig. 13 show several of the voices with l ¼ m ¼ 2 which contribute to this waveform. Both the time-domain and the frequency-domain repre- The trend we have found across all the cases we examine is sentations of these voices can be computed quickly and that voices with jk þ mj¼l are loudest, and they fall off as efficiently. Future work, particularly data-analysis-focused jk þ mj moves away from this peak. This trend can be seen applications which compare EMRI waves to detector noise, in the cases shown here: the strongest voice has k ¼ 0, could benefit substantially by focusing upon the waveform followed by k ¼ −1. In the time domain, the voices with in voice-by-voice fashion, studying which voices are most k ¼ 2 and k ¼ 1 have roughly the same amplitude; the relevant for detection and measurement as a function of voice with k ¼ −2 is the weakest of those shown here. source parameters. Interestingly, the voices with k ¼ 2, k ¼ 1, and k ¼ −2 havepffiffiffiffi similar magnitude in the frequency domain. The factor VIII. RESULTS II: EXAMPLE KERR EMRI 1= F_ which enters the frequency-domain magnitude WAVEFORMS AND THEIR VOICES compensates for the fact that this voice’s time-domain We next consider a few examples of inspiral into Kerr amplitude is smaller by a factor of 2 or 3. Such differences black holes. Although certain important details differ from have important implications for the measurability of these the results discussed in Sec. VII, much of what we find for signals. Kerr inspiral waveforms is qualitatively quite similar to The second case we examine is an equatorial eccentric inspiral, with ðp ;e Þ¼ð12M;0.7Þ. These are the same those we find for the Schwarzschild case. As such, we keep init init this discussion brief, focusing on the most important initial conditions we used for the high-eccentricity inspiral into Schwarzschild we examined in Sec. VII. For Kerr highlights of this analysis. Future work will explore these a ¼ 0 9M waveforms and their properties in depth. with spin . , these initial conditions lead to a We begin in Sec. VIII A with a discussion of two much longer inspiral that goes very deep into the strong constrained cases: one that initially has zero eccentricity, field, becoming nearly circular before the plunge: inspiral Δ i ≃ 6 1 105 but is inclined with respect to the hole’s equatorial plane; lasts for t . × M, roughly twice the duration of and a second case that is equatorial, but starts with large the high-eccentricity Schwarzschild inspiral, and ends ð Þ¼ð2 40 0 057Þ eccentricity. We then discuss one example of fully generic with pfinal;efinal . M; . . (inclined and eccentric) inspiral in Sec. VIII B. The left-hand panels of Fig. 14 show the gravitational waveform for this inspiral. For the time-domain waveform, we include contributions from modes with l ∈ ½2; 3; 4, A. Constrained orbital geometry: m ∈ ½−l; …;l, n ∈ ½0; …; 40, plus modes that are simply Spherical and equatorial inspirals related by symmetry; as in the Schwarzschild cases we We begin our Kerr study with two cases of inspiral into examined, only voices with k ¼ 0 contribute, since there is black holes with spin a ¼ 0.9M. In the first case, we examine no θ motion. The early waveform is qualitatively quite an orbit that is spherical, with large inclination to the black similar to the early large-eccentricity waveform we found ’ ð Þ¼ð10 0 5Þ hole s equatorial plane. We take rinit;xI;init M; . . for Schwarzschild (Fig. 2). The late waveform, by contrast, This inspiral reaches the last stable orbit at r ¼ 3.820M, is quite different, reflecting the fact that the orbit has nearly xI ¼ 0.483. The orbit’s inclination is nearly constant during circularized as it approaches the final plunge. As in the inspiral, with xI decreasing very slightly. A similarly slight Schwarzschild case, we see that the initial radial anomaly decrease of xI is seen in all of the cases we have examined. angle χr0 has a large impact on the waveform.

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FIG. 13. Left panels: the waveform hþ for an example of spherical inspiral into a black hole with a ¼ 0.9M. This example has −3 ðrinit;xI;initÞ¼ð10M; 0.5Þ, is at mass ratio ε ¼ 10 , and is observed in the black hole’s equatorial plane. During inspiral, the orbit’s radius steadily shrinks, while xI decreases very slightly (see text for details). The polar anomaly angle χθ0 ¼ 2π=3. The top-left panel shows the waveform over the entire inspiral; the bottom-left panels zoom in to early and late times. The lower panels also illustrate the influence of the anomaly angle, comparing χθ0 ¼ 2π=3 (solid [red] curves) with inspiral on the fiducial geodesic (dashed [blue] curves). We again see that this angle has a large effect on the waveform, in this case changing the initial orientation so that the frame-dragging induced modulation begins at a different phase. Right panels: some of the l ¼ m ¼ 2 voices which contribute to this waveform, both time domain (top) and frequency domain (bottom). Each line is a different k index: solid (red) is k ¼ 0, short-dashed (blue) is k ¼ 1, long-dashed (green) is k ¼ −1, dot-dashed (magenta) is k ¼ 2, and dotted (black) is k ¼ −2. Across the examples of zero-eccentricity inspiral we have examined, the tendency is that voices with jk þ mj¼l are the strongest, and they fall off as jk þ mj moves away from this peak.

ð Þ¼ The right-hand panels of Fig. 14 show some of the The case we examine begins at pinit;einit;xI;init l ¼ m ¼ 2 voices which contribute to this waveform. An ð12M; 0.25; 0.5Þ. At mass ratio ε ¼ 10−3, inspiral lasts interesting trend we see is that the importance of different for Δti ≃ 3.5 × 105M, at which time the smaller body ð Þ¼ð4 64 0 084 voices changes dramatically during inspiral. The evolution encounters the LSO at pfinal;efinal;xI;final . M; . ; of the n ¼ 0 voice is especially dramatic: it is fairly weak at 0.488Þ. As in the spherical cases, notice that the total early times (and in fact passes through zero during the change in inclination is very small: the change δxI ¼ 0.012 inspiral), but it dominates the waveform at late times. This corresponds to the inclination angle I increasing by about makes sense—at late times, the system’s geometry is nearly 0.79°. Figure 15 shows the trajectory that the smaller body circular, and voices corresponding to radial harmonics play follows in ðp; e; xIÞ. The left-hand panels of Fig. 16 shows a substantially less important role. the time-domain þ-polarization of the waveform that we find in this case, including all modes with l ∈ ½2; 3; 4, B. Generic Kerr inspiral m ∈ ½−l; …;l, k ∈ ½0; …; 10, n ∈ ½0; …; 25 (as well as We conclude our discussion of results by looking at a modes simply related to these modes by symmetry). The Kerr inspiral that is both inclined from the equatorial plane right-hand panel of Fig. 16 and both panels of Fig. 17 show and eccentric. As discussed in Sec. VI, we have not yet some of the voices that contribute to this waveform generated dense datasets covering a wide range of such (l ¼ m ¼ 2, k ¼ 0 voices in Fig. 16; l ¼ m ¼ 2, k ¼2 orbits. The example shown here demonstrates that the voices in Fig. 17). techniques we have developed to build adiabatic EMRI Perhaps not surprisingly, the picture that emerges for waveforms have no difficulty with such cases. Although generic orbits is a blend of features seen in the spherical and generic EMRI waveforms have been developed using equatorial limits. The time-domain waveform has large “kludges,” to our knowledge this is the first generic peaks corresponding to periapsis passage, separated by example that uses strong-field backreaction and strong- lower-amplitude troughs as the orbit moves through apo- field wave generation for the entire calculation. apsis; the relatively small contrast between the peaks and

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FIG. 14. Left panels: the waveform hþ for an example of eccentric equatorial inspiral into a black hole with a ¼ 0.9M. This example ð Þ¼ð12 0 7Þ ε ¼ 10−3 ’ has pinit;einit M; . , is at mass ratio , and is observed in the black hole s equatorial plane. The initial conditions are similar to those used to make the large-eccentricity case shown in Fig. 2, and the two waveforms are similar at early times. However, at this rapid spin, inspiral goes very deep into the strong field, becoming nearly circular before the plunge; we find ðpfinal;efinalÞ¼ð2.40M; 0.057Þ. The late waveform is consistent with low eccentricity late in the inspiral. The top-left panel shows the waveform for χr0 ¼ 2=3 over the inspiral; in addition to zooming in to early and late times, the bottom-left panels illustrate the influence of the anomaly angle, comparing χr0 ¼ 2π=3 (solid [red] curves) with inspiral on the fiducial geodesic (dashed [blue] curves). Right panels: some of the l ¼ m ¼ 2 voices which contribute to this waveform, both time domain (top) and frequency domain (bottom). Solid (red) curves show the voices with n ¼ 0; short-dashed (blue) curves are for n ¼ 3; long-dashed (green) curves are n ¼ 6; dot-dashed (magenta) curves are n ¼ 9; and dotted (black) curves show n ¼ 12.

FIG. 15. The inclined and eccentric trajectory in ðp; e; xIÞ for the generic inspiral we examine. This trajectory, illustrated by the blue curve, begins at ðpinit;einit;xI;initÞ¼ð12M; 0.25; 0.5Þ and proceeds until the smaller body encounters the LSO (a section of which is ð Þ¼ð4 64 0 084 0 488Þ ð Þ illustrated by the orange plane) at pfinal;efinal;xI;final . M; . ; . . A projection of this trajectory into the p; e plane is illustrated by the red curve, along with the projection of the LSO at the final value of xI (black line in lower plane); a projection of this trajectory into the ðp; xIÞ plane is illustrated by the green curve, along with the projection of the LSO at the final value of e (black line on back “wall” of the box).

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FIG. 16. Left panel: the waveform hþ for an example of generic inspiral into a black hole with a ¼ 0.7M. This example corresponds to ð Þ¼ð12 0 25 0 5Þ ε ¼ 10−3 pinit;einit;xI;init M; . ; . , and has a mass ratio ; the small body inspirals until it encounters the LSO at ðpfinal;efinal;xI;finalÞ¼ð4.64M; 0.084; 0.488Þ. We show the waveform as observed in the black hole’s equatorial plane. The upper-left panel shows the waveform over the inspiral; the lower-left panels zoom in to early and late times. The lower-left panels also illustrate the influence of the anomaly angle, comparing χr0 ¼ χθ0 ¼ 2π=3 (solid [red] curves) with the inspiral on the fiducial geodesic (dashed [blue] curves). Right panel: some of the voices that contribute to this waveform. The voices shown here are for l ¼ m ¼ 2, k ¼ 0; the solid (red) curve is n ¼ 0, the short-dashed (blue) curve is n ¼ 1, the long-dashed (green) curve is n ¼ 2, the dot-dashed (magenta) curve is n ¼ 3, and the dotted (black) curve is n ¼ 4. The behavior of these voices is quite simple, with n ¼ 0 being the strongest, and the voices falling away as n increases.

FIG. 17. Additional voices that contribute to the waveform shown in Fig. 16. The left panel shows voices with k ¼ 2; the right panel shows voices with k ¼ −2. As in the right-hand panel of Fig. 16, the solid (red) curves are n ¼ 0, short-dashed (blue) curves are n ¼ 1, long-dashed (green) curves are n ¼ 2, dot-dashed (magenta) curves are n ¼ 3, and dotted (black) curves are n ¼ 4. The behavior is somewhat more complicated for these voices; although larger values of n tend to be weaker, the evolution for small values of n evolves rather differently than in the k ¼ 0 case. For both the cases shown here, the n ¼ 0 voice in particular evolves in importance.

104014-26 ADIABATIC WAVEFORMS FOR EXTREME MASS-RATIO … PHYS. REV. D 103, 104014 (2021) troughs is consistent with this case’s fairly modest initial frequencies and interpolating to compute additional values. eccentricity. Superposed on this structure is a more rapid Knowing tðfÞ is in addition likely to be useful for connecting “whirling” associated with frame dragging. This is espe- the frequency-domain waveforms we describe here to a cially clear late in the waveform when the orbit has moved detector response function. The LISA detector’s response to a small radius, and it resembles behavior seen in the changes with time as the antenna orbits the Sun, introducing waveform for the spherical case at late times. time- and frequency-dependent modulations, and changing For k ¼ 0, the time-domain structure of voices that the sensitivity to the two gravitational-wave polarizations contribute to this waveform is similar to what we saw in as the orientation of the antenna relative to a source varies the low-eccentricity Schwarzschild case shown in Fig. 8: over the orbit. Though both these extensions will complicate n ¼ 0 is the strongest voice, with contributions steadily the development of fast waveforms, they do not change the decreasing as n increases. For k ¼2, we see more fundamental message of Ref. [16]. We are confident that variation: n ¼ 0 often starts out relatively weak, but it adapting those methods with the datasets and framework becomes much stronger late in the inspiral after eccentricity described here will be quite effective. is significantly decreased. The frequency-domain structure Because of the high cost of generating the waveform of the voices is consistent with this picture; given the amplitude data, once computed, the data should be widely already large number of plots in this paper, we do not shared. The datasets we have computed, described in Sec. VI, include figures showing this. have been released through the Black Hole Perturbation Toolkit [54] (hereafter “the Toolkit”). These data were all IX. CONCLUSION computed using a fairly small (roughly 1000-core) in-house In this paper, we have shown how to use precomputed cluster at the MIT Kavli Institute. This scale of cluster is frequency-domain Teukolsky equation solutions to make adequate for making inspiral data for 2D orbits (spherical or EMRI waveforms. In this way, one can compute and store equatorial), but is too small to be useful for fully generic 3D in advance the most computationally expensive aspects of (inclined and eccentric Kerr) datasets. We have ported our EMRI analysis, and then use the methods we describe to frequency-domain Teukolsky equation solver, GREMLIN,to rapidly assemble waveforms using the stored data. We the U.S. National Science Foundation’sXSEDE[55] envi- particularly emphasize the usefulness of the multivoice ronment, and we plan to develop generic datasets there. It waveform structure, which facilitates identifying particu- should be noted, however, that there is a lot to be learned from larly important waveform multipoles and harmonics, both 2D cases about how best to lay out orbits on the grid (for in the time and frequency domains. example, the work in progress mentioned in Sec. VI that The framework and techniques we describe here have shows the importance of dense coverage at small eccentric- not been optimized, so there is much scope for efficiency ity). Datasets released through the Toolkit are likely to be gains. In a companion analysis [16], some of the present revised with some regularity as we learn more about the best authors examined several algorithmic improvements, show- way to lay out the data grids. ing that we can in fact construct analysis-length time- Plans are in place to release the GREMLIN code which was domain EMRI waveforms in the Schwarzschild limit in used to generate these datasets, and which includes tools under a second. There are two ways that the results of the for postprocessing analysis and generating EMRI wave- present work can be incorporated into the framework of forms. A reduced-functionality version of GREMLIN (spe- Ref. [16]. The first is to extend to Kerr inspirals. The main cialized to circular, equatorial orbits) has already been challenges here are due to the higher-dimensional param- released [54]. The generic version will be released after it is eter space, and the need to sum over more modes, since cleaned of proprietary libraries and fully comports with waveforms depend on an additional harmonic frequency. open-source licensing requirements. Inspirals also extend deeper into the strong field, so more In terms of analyses of the source physics, there are modes are likely to make strong contributions to the several natural places to extend what we have done here. waveform. The framework in Ref. [16] is designed to One clear step is to consider how physics beyond the overcome these challenges. Its neural network interpolation adiabatic approximation affects waveforms. As described is a promising technique for dealing with the increased in Sec. VII, we see evidence from comparison with dimensionality, and its use of GPU-based hardware accel- waveforms computed by a time-domain Teukolsky solver eration alleviates the computational burden of summing that neglecting terms which vary on the long-inspiral over thousands of additional ðl; m; k; nÞ modes. timescale leads to an initial-condition-dependent secular The second extension is to incorporate frequency-domain drift in the waveform. Our hypothesis is that this arises due waveforms. For the Schwarzschild limit, Ref. [16] already to a slow evolution in the ξmkn phases described in Sec. III. demonstrates how to efficiently interpolate waveform ampli- In a similar vein, we expect it will not be difficult to tudes; the remaining new challenge is the efficient calculation incorporate the orbit-averaged first-order conservative of the stationary time as a function of frequency. This can self-force. In brief, on average the conservative self-force likely be achieved by sparsely evaluating tðfÞ at a few key changes the rate at which the orbit precesses, and can be

104014-27 HUGHES, WARBURTON, KHANNA, CHUA, and KATZ PHYS. REV. D 103, 104014 (2021) modeled as a slow “anomalous” change to the rate of for raising the issue of the pathology discussed in Ref. [48], periastron advance and the advance of the line of nodes. which was important to consider as a contrast to a similar These anomalous changes can in turn be incorporated into issue we examine. S. A. H.’s work on this problem was an osculating geodesic framework by allowing the param- supported by NASA ATP Grant No. 80NSSC18K1091 and eters χr0, χθ0, and ϕ0 to slowly evolve under the influence NSF Grant No. PHY-1707549. N. W. acknowledges sup- of this force. See, for example, Ref. [46] for further port from a Royal Society–Science Foundation Ireland discussion. A similar effect due to the orbit-averaged Research Fellowship. This publication has emanated from coupling of the smaller body’s spin to the background research conducted with the financial support of Science curvature probably can also be modeled in such a way. It Foundation Ireland under Grant No. 16/RS-URF/3428. should also be possible to include many non-orbit-averaged G. K. acknowledges support from NSF Grants No. PHY- self-forces and spin-curvature couplings [22–25,56] by 2106755 and No. DMS-1912716. A. J. K. C. acknowledges using a near-identify transformation [26]. Such non- support from NASA Grant No. 18-LPS18-0027. M. L. K. orbit-averaged analyses will be needed in order to model acknowledges support from NSF Grant No. DGE-0948017. the impact of resonances [9,11], for example. It is likely that an orbit-averaged-based analysis accurately describes systems away from resonances, but one will need to do a APPENDIX A: FORMULAS FOR E, Lz, AND Q more complicated (and expensive) analysis in the vicinity In this Appendix, we list formulas for the geodesic of each resonance to model how the system evolves constants of the motion E, Lz, and Q as functions of our through each resonance crossing, and practical schemes preferred parameters p, e, and xI ≡ cos I. Formulas for for efficiently combining the two within template models these constants were first worked out by Schmidt [29], and must be devised. Regardless, extensions of this sort may are provided in particularly clean form for generic orbits by make it possible for this framework to incorporate the most van de Meent [32]. We list them here using our preferred important postadiabatic effects quite easily, significantly parametrization, and we also include formulas for spherical improving the ability of these models to serve as templates orbits. for EMRI measurements. The three geodesic constants are given by Finally, we note that recent phenomenological frequency- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi domain gravitational-wave models are being calibrated in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κρ þ 2ϖσ − 2sgnðx Þ σðσϖ2 þ ρϖκ − ηκ2Þ the large-mass-ratio limit with Teukolsky waveforms [57]. E ¼ I ; The directly constructed frequency-domain waveforms ρ2 þ 4ησ presented in this work could be useful in this effort. ðA1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ACKNOWLEDGMENTS ð Þ − ð Þ2 þ ð Þ ð Þ 2 − ð Þ ð Þ ¼ − g ra E g ra h ra f ra E h ra d ra Lz ð Þ ; An early presentation of some of the results and methods h ra described here was given at the 22nd Capra Meeting on ðA2Þ Radiation Reaction in General Relativity, hosted by the Centro Brasiliero de Pequisas Físicas in Rio de Janeiro, L2 Brazil in June 2019; we thank the participants of that ¼ð1 − 2Þ 2ð1 − 2Þþ z ð Þ Q xI a E 2 ; A3 meeting for useful discussions, and particularly Marc xI Casals for organizing the meeting. The time-domain wave- ¼ ð1 − Þ ’ form computations were performed on the MIT/IBM Satori where ra p= e is the coordinate radius of the orbit s GPU supercomputer supported by the Massachusetts Green apoapsis. For generic orbits, the quantities appearing here High Performance Computing Center (MGHPCC) and the are given by CARNiE cluster at UMass Dartmouth. S. A. H. thanks the κ ¼ ð Þ ð Þ − ð Þ ð Þ ð Þ MIT Kavli Institute for providing computing resources and d ra h rp d rp h ra ; A4 support, L. S. Finn for (long ago) suggesting the multivoice ϖ ¼ ð Þ ð Þ − ð Þ ð Þ ð Þ framework for examining EMRI waveforms, Stanislav d ra g rp d rp g ra ; A5 Babak for emphasizing the importance of understanding ρ ¼ ð Þ ð Þ − ð Þ ð Þ ð Þ EMRI waveform structure in the frequency domain, Steve f ra h rp f rp h ra ; A6 Drasco and William T. Throwe for past contributions to η ¼ ð Þ ð Þ − ð Þ ð Þ ð Þ GREMLIN development, Gustavo Velez for insight into the f ra g rp f rp g ra ; A7 Jacobian between ðdE=dt; dLz=dt; dQ=dtÞ and ðdp=dt; σ ¼ ð Þ ð Þ − ð Þ ð Þ ð Þ de=dt; dxI=dtÞ, Talya Klinger for work porting GREMLIN to g ra h rp g rp h ra ; A8 the XSEDE computing environment, and both Talya ¼ ð1 þ Þ ’ Klinger and Sayak Datta for discussions of accuracy and where rp p= e is the coordinate radius of the orbit s systematic error in EMRI datasets. We also thank Leo Stein periapsis, and where

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ð Þ¼Δð Þ½ 2 þ 2ð1 − 2Þ ð Þ d r r r a xI ; A9 system evolves from one geodesic to another in the adiabatic limit. As part of this, we would like to know ð Þ¼ 4 þ 2½ ð þ 2 Þþð1 − 2ÞΔð Þ ð Þ f r r a r r M xI r ; A10 how the geodesic geometry parameters p, e, and xI change due to this backreaction. In this Appendix, we write out the gðrÞ¼2aMr; ðA11Þ details of this procedure. A generic Kerr orbit has turning points in its radial 2 ¼ ð1 − Þ ð1 − x ÞΔðrÞ motion at apoapsis, ra p= e , and at periapsis, hðrÞ¼rðr − 2MÞþ I : ðA12Þ ¼ ð1 þ Þ 2 rp p= e . It also has a turning point in its polar xI motion at θm. (The second polar turning point at π − θm Note that we have switched notation slightly versus yields no new information because of reflection symmetry θ ¼ π 2 θ Refs. [29,32]: we use ϖ in these listspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rather than ϵ to about = .) Recall that m is related to our inclination 2 2 angle I by Eq. (2.16). The radial turning points mean avoid notational collision with ϵ ¼ M − a =2Mrþ,as ð Þ¼0 ð Þ¼0 ð Þ well as with the very similar ε ≡ μ=M. that R ra and R rp , where R r is defined in Θðθ Þ¼0 For spherical orbits with e ¼ 0, r ¼ r ≡ r . In this Eq. (2.1); the polar turning point means that m , a p o ΘðθÞ limit, we use a different form of these quantities: where is defined in Eq. (2.2). We require that these conditions hold as the system evolves due to backreaction, κ ¼ ð Þ 0ð Þ − 0ð Þ ð Þ ð Þ which means that we require d ro h ro d ro h ro ; A13

ϖ ¼ dðr Þg0ðr Þ − d0ðr Þgðr Þ; ðA14Þ d d d o o o o Rðr Þ¼0; Rðr Þ¼0; Θðθ Þ¼0: ðB1Þ dt a dt p dt m ρ ¼ ð Þ 0ð Þ − 0ð Þ ð Þ ð Þ f ro h ro f ro h ro ; A15 Expanding these total time derivatives, we find that the η ¼ ð Þ 0ð Þ − 0ð Þ ð Þ ð Þ f ro g ro f ro g ro ; A16 results can be written as a matrix equation: σ ¼ ð Þ 0ð Þ − 0ð Þ ð Þ ð Þ g ro h ro g ro h ro : A17 0 1 0 1 0 1 JEra JLzra JQra dE=dt dra=dt 0 B C B C B C In these formulas, denotes ∂=∂r. @ A @ A ¼ @ A ð Þ JErp JLzrp JQrp · dLz=dt drp=dt : B2

JExI JLzxI JQxI dQ=dt dxI=dt APPENDIX B: JACOBIAN FROM ðdE=dt; dLz=dt; dQ=dtÞ TO ðdp=dt; de=dt; dxI=dtÞ Computing the various matrix elements Jab, we find simple Once the rates of change, dE=dt, dLz=dt, and dQ=dt, expressions for dra;p=dt and dxI=dt. We examine the have been computed, we use them to compute how a change to the inclination first:

pffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 2 1 − x2 Q − a2ð1 − E2Þð1 − x2ÞðdL =dtÞ − x ðdQ=dtÞ − 2x ð1 − x2Þa2EðdE=dtÞ I ¼ð1 − x2Þ I I z I I I ; ð Þ I 2½ − 2ð1 − 2Þð1 − 2Þ2 B3 dt Q a E xI

2x ð1 − x2ÞL ðdL =dtÞ − x3½ðdQ=dtÞþ2ð1 − x2Þa2EðdE=dtÞ ¼ I I z z I I : ð Þ 2½ 2 þ 4 2ð1 − 2Þ B4 Lz xI a E

Notice that Eq. (B3) is singular when Q → 0 (which Using this, we have coincides with jxIj → 1), and Eq. (B4) is singular when Lz → 0 (jxIj → 0). We use Eq. (B3) when jxIj ≤ 0.5, and 2 2 2 Eq. (B4) when jxIj > 0.5. The two expressions yield 4aMðL − aEÞr − 2Er ða þ r Þ J ≡ z a;p a;p a;p ; identical results except right at their singular points. Era;p Dð Þ ra;p To present our results for dra;p=dt, we first define 4MðaE − L Þr þ 2L r2 J ≡ z a;p z a;p ; Lzra;p Dð Þ ra;p Dð Þ ≡ 2 ½ þð − Þ2 − 2 ½ 2 þ − 2ð1 − 2Þ 2 2 r M Q Lz aE r Lz Q a E r − 2Mr þ a J ≡ a;p a;p : ðB6Þ þ 6 2 − 4 3ð1 − 2Þ ð Þ Qra;p Dð Þ Mr r E : B5 ra;p

104014-29 HUGHES, WARBURTON, KHANNA, CHUA, and KATZ PHYS. REV. D 103, 104014 (2021)

ð Þ We then find [Alternately, one can enforce the conditions dR ro = ¼ 0 0ð Þ ¼ 0 dt , dR ro =dt . Doing so yields solutions for dra;p dE dLz dQ dr =dt and dQ=dt given dE=dt and dL =dt. This is ¼ J þ J þ J : ðB7Þ o z dt Era;p dt Lzra;p dt Qra;p dt how backreaction on spherical orbits was computed [37] before dQ=dt was fully understood [43].] Implementing Once dra;p=dt are known, it is simple to compute dp=dt Eq. (B10), we find and de=dt: dr dE dL dQ dp ð1 − eÞ2 dr ð1 þ eÞ2 dr o ¼ Jc þ Jc z þ Jc ; ðB11Þ ¼ a þ p ; ðB8Þ dt Ero dt Lzro dt Qro dt dt 2 dt 2 dt 2 where the Jacobian elements for spherical orbits are de ð1 − e Þ dr dr ¼ ð1 − eÞ a − ð1 þ eÞ p : ðB9Þ dt 2p dt dt 2aMðL − aEÞ − 2a2Er − 4Er3 Jc ¼ z ; ðB12Þ ¼ 0 ¼ ≡ Er Dcð Þ In the spherical limit, e , ra rp ro, and the r ð Þ¼0 ð Þ¼0 conditions R ra , R rp are redundant. The equa- tions we have derived here do not work in that limit. −2MðL − aEÞþ2L r Jc ¼ z z ; ðB13Þ Spherical orbits are instead governed by the conditions Lzr DcðrÞ ð Þ¼0 0ð Þ¼0 0 ≡∂ ∂ R ro , R ro , where R R= r. Past work [58,59] long ago proved that adiabatic dissipative self- r − M interaction evolves a spherical orbit into a new spherical Jc ¼ ; ðB14Þ Qr Dcð Þ orbit; the first derivation of dQ=dt [43] proved that this r result respected the “spherical goes to spherical” constraint. Given ðdE=dt; dLz=dt; dQ=dtÞ from a spherical orbit, one with can infer dro=dt by enforcing the condition c 2 2 2 2 2 D ðrÞ¼−½a ð1 − E ÞþLz þ Qþ6Mr − 6ð1 − E Þr : d 0ð Þ¼0 ð Þ R ro : B10 ðB15Þ dt

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