PHYSICAL REVIEW D 85, 044035 (2012) Hybrid method for understanding black-hole mergers: Inspiralling case

David A. Nichols* and Yanbei Chen† Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA (Received 31 August 2011; published 13 February 2012) We adapt a method of matching post-Newtonian and black-hole-perturbation theories on a timelike surface (which proved useful for understanding head-on black-hole-binary collisions) to treat equal-mass, inspiralling black-hole binaries. We first introduce a radiation-reaction potential into this method, and we show that it leads to a self-consistent set of equations that describe the simultaneous evolution of the waveform and of the timelike matching surface. This allows us to produce a full inspiral-merger-ringdown waveform of the l ¼ 2, m ¼2 modes of the gravitational waveform of an equal-mass black-hole-binary inspiral. These modes match those of numerical-relativity simulations well in phase, though less well in amplitude for the inspiral. As a second application of this method, we study a merger of black holes with spins antialigned in the orbital plane (the superkick configuration). During the ringdown of the superkick, the phases of the mass- and current-quadrupole radiation become locked together, because they evolve at the same quasinormal-mode frequencies. We argue that this locking begins during the merger, and we show that if the spins of the black holes evolve via geodetic precession in the perturbed black-hole spacetime of our model, then the spins precess at the orbital frequency during the merger. In turn, this gives rise to the correct behavior of the radiation, and produces a kick similar to that observed in numerical simulations.

DOI: 10.1103/PhysRevD.85.044035 PACS numbers: 04.25.Nx, 04.30.w, 04.70.s

complete merger of black-hole binaries, several groups I. INTRODUCTION worked on developing methods that aim to get the most Black-hole-binary mergers are both key sources of out of a given approximation technique. The close-limit gravitational waves [1] and two-body systems in general approximation (see, e.g., [5–8] for early work and [9–12] relativity of considerable theoretical interest. It is common for more recent work) and the Lazarus project (see, e.g., to describe the dynamics and the waveform of a quasicir- [13,14]) both try to push the validity of BHP to early times; cular black-hole binary as passing through three different the effective-one-body (EOB) approach (see, e.g., [15,16] stages: inspiral, merger, and ringdown (see, e.g., [2]). For for the formative work, and [17–20] for further develop- comparable-mass black holes, the three stages correspond ments that allow the method to replicate numerical- to the times one can use different approximation schemes. relativity waveforms) aims to extend the validity of the During the first stage, inspiral, the two black holes can be PN approximation to later times. modeled by the post-Newtonian (PN) approximation as There also have been several methods that do not easily two point particles (see, e.g., [3] for a review of PN theory). fit into the characterization of extensions of PN or BHP As the speeds of the two holes increase while their sepa- theories. For example, the ‘‘particle-membrane’’ approach ration shrinks, the PN expansion becomes less accurate of Anninos et al. [21,22] computes the waveform from (particularly as the two objects begin to merge to form a head-on collisions by extrapolating results from the point- single body). In this stage, merger, gravity becomes particle limit to the comparable-mass case (and taking into strongly nonlinear (and therefore less accessible to ap- account changes to the horizons computed within the proximation techniques). After the merger, there is the membrane paradigm [23]). More recently, white-hole fis- ringdown, during which the spacetime closely resembles sion was used in approximate models of black-hole merg- a stationary black hole with small perturbations [and one ers [24–26], and quite recently, Jaramillo and collaborators can treat the problem using black-hole perturbation (BHP) [27–30] used Robinson-Trautman spacetimes as an ap- theory (see, e.g., [4] for a review of BHP theory)]. proximate analytical model of binary mergers (as part of Because the merger phase of comparable-mass black a larger project correlating geometrical quantities on holes has been so challenging to understand analytically, black-hole horizons with similar quantities at future null there have been many attempts to study it with a variety of infinity). analytical tools. One approach has been to develop PN and Analytical approximations are not limited to BHP theories to high orders in the different approxima- comparable-mass black-hole binaries, and recently there tions. Since neither approximation can yet describe the has been a large body of work on developing techniques to study intermediate- and extreme-mass-ratio inspirals *[email protected] (IMRIs and EMRIs, respectively). Most of these methods †[email protected] aim to produce gravitational waves in ways that are less

1550-7998=2012=85(4)=044035(24) 044035-1 Ó 2012 American Physical Society DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) computationally expensive than computing the exact nu- black holes with transverse, antialigned spins, we found merical solution or computing the leading-order gravita- very good agreement between the two. tional self-force are (see, e.g., [31] for a recent review of There is no reason, a priori, why the same procedure of the self-force). The majority of the approaches rely heavily Paper I (namely, specifying the position of the point par- on BHP techniques combined with some prescription for ticles as a function of time and matching the metrics on a taking radiative effects into account, though not all ap- surface passing through their positions) should not work proximate methods fall into this classification (Barack and for inspiralling black holes as well. The principal difficulty Cutler [32], for example, model EMRIs by instantaneously arises from trying to find a way of specifying the positions Newtonian orbits whose orbital parameters vary slowly of the particles for inspiralling black holes (and thus a over the orbital time scale because of higher-order PN location at which to match the PN and BHP metrics) that effects). A well-known example is that of Hughes [33], does not introduce errors into any of the three stages of the Glampedakis [34], Drasco [35], Sundararajan [36], and inspiral, merger, or ringdown portions of the waveform. their collaborators whose semianalytical approaches are The most important development that we introduce in this often called Teukolsky-based models. These methods de- paper, therefore, is a way of achieving this goal by adding a scribe the small black hole as moving along a sequence of radiation-reaction force to the formalism. In the hybrid geodesics whose energy, angular momentum, and Carter method, we compute a radiation-reaction force by using constant, change from the influence of emitted gravita- the outgoing waves in the exterior BHP spacetime to tional waves. They usually involve some additional pre- modify the PN dynamics in the interior through a scription to treat the transition from the inspiral to the radiation-reaction potential [50]. We show, in this formal- plunge, when the motion is no longer adiabatic. The ism, that introducing a radiation-reaction potential is EOB formalism in the EMRI limit, however, does not equivalent to solving a self-consistent set of coupled equa- require an assumption of adiabatic motion (see, e.g., tions that describe the evolution of the point particles’ [37–42]). By choosing the dynamics of the EMRI to follow reduced-mass motion and the outgoing gravitational radia- the EOB Hamiltonian and a resummed multipolar PN tion, where the particles generate the metric perturbations radiation-reaction force [43], these authors can calculate of the gravitational waves and the waves carry away energy an approximate waveform without any assumption on and angular momentum from the particles (thereby chang- relative time scales of orbital and radiative effects. One ing their motion). can also make an adiabatic approximation with EOB meth- Our principal goal in the paper is to explore this coupled ods, as Yunes et al. [44,45] recently did in their calibration set of evolution equations and show, numerically, that it of the EOB method to a set of Teukolsky-based wave- gives rise to convergent and reasonable results. We will use forms. Lousto and collaborators [46–48] took a different these results to make a refinement of our interpretation of approach to the EMRI problem in their recent work. They the waveform from Paper I, and we will also compare the used trajectories from numerical-relativity simulations of waveform generated by the hybrid method to that from a IMRIs as a way to calibrate PN expressions for the motion numerical-relativity simulation of an equal-mass, nonspin- of the small black hole. They then performed approximate ning inspiral of black holes. The two waveforms agree well calculations of the gravitational waves using the PN tra- during the inspiral phase, but less well during merger and jectories in a black-hole perturbation calculation, and ringdown. The discrepancy at late times is well under- found good agreement with their numerical results. stood: we continue to model the final black hole produced In a previous article [49] (hereafter referred to as from the merger as nonspinning, although, in fact, numeri- Paper I), we showed that for head-on collisions, one can cal simulations have shown the final hole to be spinning match PN and BHP theories on a timelike world tube that relatively rapidly (see, e.g., [51]). Adapting the hybrid passes through the centers of the PN theory’s point parti- approach to treat the final black hole as rotating is beyond cles. The positions of the points particles as a function of the scope of this work, but is something that we will time (and, consequently, the world tube) were chosen investigate in the future. before evolving the waveform. Moreover, they were se- As an application of the hybrid method for inspirals, we lected in such a way that both PN and BHP theories were explore the large kicks produced from black-hole binaries sufficiently accurate descriptions of the spacetime on the with antialigned spins in the orbital plane (the superkick world tube or the errors in the theories did not enter into the configuration [52,53]). As noted by Schnittman et al. [54] waveform. (A plunging geodesic in the Schwarzschild and emphasized to us by Thorne [55], the spins must spacetime worked in Paper I.) This allowed us compute a precess at the orbital frequency during the final stage of complete waveform for all three phases of black-hole- the merger. While Bru¨gmann et al. [56] were able to binary coalescence and gave us a way to interpret the replicate this effect using a combination of PN and different portions of the waveform. Moreover, when we numerical-relativity results, we will need to take a different compared the waveform from the hybrid method with that approach, by using geodetic precession in the exterior of a full numerical simulation of plunging equal-mass Schwarzschild BHP spacetime, to have the spins lock to

044035-2 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) the orbital motion at the merger. When we include the use uppercase variables, and we will use a harmonic gauge. geodetic effect, we are able to recover the correct qualita- For example, we will employ ðT;X;Y;ZÞ when describing tive profile of the kick, although the magnitude does not the Cartesian coordinates of the background Minkowski match precisely. space and ðT;R;; Þ when discussing its spherical-polar We organize the paper as follows: We review the results coordinates. In the perturbed Schwarzschild spacetime, we of Paper I in Sec. II, and we describe the procedure for will use ðt; r; ; ’Þ primarily, though sometimes we will calculating the radiation-reaction force and the resulting also use the light-cone coordinates, ðu; v; ; ’Þ where set of evolution equations in Sec. III. In Sec. IV, we show u ¼ t r ;v¼ t þ r ; (1) the convergence of our waveform, we compare with nu- merical relativity, and we discuss using the hybrid method and to interpret the waveform. Next, we discuss the behavior of r spinning black holes and describe spin precession as a r ¼ r þ M : 2 log M 1 (2) mechanism for generating large black-hole kicks in 2 Sec. V. We conclude in Sec. VI. Throughout this paper, One can match the two coordinate systems, accurate to we set G ¼ c ¼ 1, and we use the Einstein summation linear order in M=R by identifying convention (unless otherwise noted). T ¼ t; ¼ ; ¼ ’; R ¼ r M: (3)

II. A BRIEF REVIEW OF PAPER I For the equal-mass binaries that we study, we will denote the separation by AðtÞ¼2RðtÞ in PN coordinates and In this section, we will review the essentials of the aðtÞ¼2rðtÞ in Schwarzschild coordinates. Moreover, be- formalism from Paper I. In the hybrid method, we divide cause we match the two metrics on a shell passing through the spacetime of an equal-mass, black-hole-binary merger the centers of the point particles, we will indicate the into two regions: a PN region within a spherical shell position of the shell by adding a subscript ‘‘s’’ to the through the centers of the PN theory’s point particles, coordinate radius. For example, we will write RsðtÞ¼ and a perturbed Schwarzschild spacetime outside that AðtÞ=2 or rsðtÞ¼aðtÞ=2 to denote this. For clarity, we shell. Figure 1 shows this at a given moment in time reproduce a table that reviews the essentials of our notation (with one spatial dimension suppressed). For the hybrid in Table I. procedure to work, there must be, at least, a spherical shell Because we are investigating only the lowest-order ef- on which both BHP and PN theories are either simulta- fects in our study of radiation reaction and large black-hole neously valid (to a given level of accuracy) or a way to kicks, we shall only need the lowest-order terms in the PN prevent the errors in the approximations from affecting metric that appeared in Paper I to describe the interior of observables, such as the waveform. By finding good agree- the shell, ment between the hybrid waveform and that of numerical 2 ðl¼2Þ 2 ðl¼2Þ b relativity in Paper I, we found evidence that matching the dS ¼ð1 2M=R 2UN Þdt 8wb dtdx theories on a spherical shell that passes through the PN ðl¼2Þ þð1 þ 2M=R þ 2U ÞðdR2 þ R2d2Þ: (4) theory’s point particles works throughout all three stages of N a head-on black-hole-binary merger: infall, merger, and In the above equation, M is the total mass of the binary, ringdown. d2 is the area element on the unit sphere, dxb ¼ d; d’, To mesh the two descriptions of spacetime, we match the ðl¼2Þ ðl¼2Þ and the additional variables UN and wb are the PN metric to that of the perturbed Schwarzschild black quadrupole parts of the spherical harmonic expansion of hole, which involves relating the two coordinate systems of the binary’s Newtonian potential and gravitomagnetic po- PN and BHP theories. In the PN coordinate system, we will tential, respectively, X2 Uðl¼2Þ U2;mY ; ’ ; A(t) ≡ 2R (t) = 2(r (t)−M) ≡ a(t) − 2M N ¼ N 2;mð Þ (5) s s m¼2 R PN X2 Matching Shell BHP r=R+M wðl¼2Þ w2;mX2;m ; ’ : b ¼ ðoÞ b ð Þ (6) m FIG. 1. (Reproduced from Paper I.) The regions of spacetime ¼2 Y ; ’ and the radial coordinates in the hybrid method (at a given We denote the scalar spherical harmonics by 2;mð Þ, moment in time, with one spatial coordinate suppressed). The 2;m 2;m and the coefficients UN and w are functions of R and t. exterior is a perturbed Schwarzschild and the interior is a PN ðoÞ X2;m ; ’ spacetime. At the position of the shell, the two descriptions of The functions b ð Þ are odd-parity vector spherical spacetime should both be valid, or should be within a region harmonics, whose and ’ coefficients are given by of spacetime that does not heavily influence physical observables Xl;m @ Yl;m ; ’ ; far away. ¼ðcsc Þ ’ ð Þ (7)

044035-3 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) TABLE I. (Reproduced from Paper I.) The notation for the coordinates, the binary separation, and the matching radius. We express these three variables in the PN spacetime, the BHP spacetime, and the matching surface between the two.

PN spacetime Matching shell Perturbed Schwarzschild spacetime

Coordinates (t, R, , ’) ðt; RsðtÞ;;’Þ or ðt; rsðtÞ;;’Þðt; r; ; ’Þ, r ¼ R þ M Binary separation AðtÞ AðtÞ or aðtÞ aðtÞ Matching radius RðtÞ¼AðtÞ=2 RsðtÞ¼aðtÞ=2 M or rsðtÞ¼AðtÞ=2 þ MrðtÞ¼aðtÞ=2

l;m l;m X’ ¼ðsinÞ@Y ð; ’Þ: (8) We, therefore, identify the monopole piece of the PN metric with the unperturbed Schwarzschild metric. A more general description is put forth in Paper I, but here Moreover, at leading order in M=R, we note that the we only take the essential components needed for the perturbations of the two metrics match exactly, calculations in the paper. 2;m 2;m 2;m 2;m Outside of the shell, we write down a perturbed Htt ¼ Hrr ¼ K ¼ 2UN ; (17) Schwarzschild metric, 2;m 2;m h w : ds2 ¼ð1 2M=rÞdt2 þð1 2M=rÞ 1ðdr2 þ r2d2Þ t ¼4 ðoÞ (18) þ hdx dx ; (9) There is then a straightforward procedure that lets one express the metric perturbations in terms of the gauge- h where the nonzero components of the perturbed metric invariant perturbation functions of the Schwarzschild that we shall need in this paper are the quadrupole pieces, spacetime [57] (though in this paper we use the notation ðl¼2Þ h , and they take the form, of [58]), which are typically called the Zerilli function and X2 the Regge-Wheeler function for the even- and odd-parity hðl¼2Þ H2;mY2;m ; ’ ; perturbations, respectively. We reproduce the expressions ð tt ÞðeÞ ¼ tt ð Þ (10) m¼2 below: r r M M 2;m 2 2;m 2 2 2;m 2;m X2 ¼ UN þ 1 UN r@rUN ; hðl¼2Þ H2;mY2;m ; ’ ; ðeÞ 3 2r þ 3M r ð rr ÞðeÞ ¼ rr ð Þ (11) m¼2 (19)

X2 l ð ¼2Þ 2 2;m 2;m 2;m 2;m 2 2;m ðh Þ ¼ r K Y ð; ’Þ; (12) r @ w w : ðeÞ ðoÞ ¼ 2 r ðoÞ ðoÞ (20) m¼2 r In Paper I, we matched the two metrics on a timelike X2 hðl¼2Þ r2 2 K2;mY2;m ; ’ ; tube that we specified before evolving the Regge-Wheeler ð ’’ ÞðeÞ ¼ sin ð Þ (13) m¼2 and Zerilli functions. We assumed that this tube would be spherically symmetric, and we found its radius by first and assuming the reduced-mass motion of the system followed X2 a radial geodesic of a plunging test mass in the background hðl¼2Þ h2;mX2;m ; ’ ; ð t ÞðoÞ ¼ t ð Þ (14) Schwarzschild spacetime and then setting the radius of m¼2 the world tube to be half this distance at each time. This allowed us to use the PN data in the form of the Regge- X2 Wheeler and Zerilli functions, Eqs. (19) and (20), on this hðl¼2Þ h2;mX2;m ; ’ : ð t’ ÞðoÞ ¼ t ’ ð Þ (15) tube to provide a boundary-value problem for the evolution m¼2 of the Regge-Wheeler [59] or Zerilli [60] equations, The subscripts ðeÞ and ðoÞ refer to the parity of the pertur- l l;m 2 l;m V @ ðe;oÞðe;oÞ bations (even and odd, respectively), where we call pertur- ðe;oÞ þ ¼ 0: (21) bations that transform as ð1Þl even and as ð1Þlþ1 odd. @u@v 4 The interior PN metric must match the perturbed The potentials for the Regge-Wheeler (odd-parity) or Schwarzschild metric on a spherical shell between the Zerilli (even-parity) equations are given by two regions. To make this identification, we note that because R ¼ r M, then the term 2M 6M Vl ðrÞ¼ 1 Ul ðrÞ ; (22) ðe;oÞ r r2 r3 ðe;oÞ 2M 1 2M 1 ¼ 1 þ þ O½ðM=RÞ2: (16) r R where ¼ lðl þ 1Þ and

044035-4 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) ð þ 2Þr2 þ 3Mðr MÞ there is a large black-hole potential is confined to a small Ul ðrÞ¼1;Ul ðrÞ¼ ; ðoÞ ðeÞ ðr þ 3MÞ2 space in this diagram. To have Fig. 2 be an effective description of the space- (23) time of a black-hole binary, both PN and BHP theories both where ¼ðl 1Þðl þ 2Þ=2 ¼ =2 1. After numeri- must be sufficiently accurate at the PN point particles (the cally solving the Regge-Wheeler or Zerilli equations black line where we match the metrics), or the PN approxi- above, we computed the gravitational waveforms and the mation could break down if the point particles are well radiated energy and momentum, all of which we found to hidden within the black-hole effective potential. For the be in good agreement with the exact quantities computed errors to stay within the potential, the particles must rap- from numerical-relativity simulations. idly fall to the horizon; thus, one can see in Fig. 2 that the In this paper, while much of the procedure we use for black curve approaches an ingoing null ray asymptotically. matching the metrics is identical to that set forth above, (Recall that u and v are the light-cone coordinates of the there are several important differences that we will discuss BHP spacetime.) Following this trajectory, the perturba- in Sec. III. The most important difference between the first tions induced by the PN spacetime will become strongly paper and the current one arises in how we find the trajec- tory of the system’s reduced mass (and then the timelike tube on which we match the metrics). Before, we chose a region, prior to evolving the Regge-Wheeler and Zerilli ringdown equations, that would not introduce spurious effects into the results; here we determine the position of timelike merger world tube through evolving the position of the reduced homogeneous region H 0: sourcem free mass of the binary subject to a radiation-reaction force. We a 0: source region a will discuss the details of this procedure in the next section. inspiral

III. RADIATION-REACTION POTENTIAL tail dominates v

AND EVOLUTION EQUATIONS 1

In this section, we introduce a radiation-reaction poten- 2 tial into the hybrid method, and we show that it leads to a u set of evolution equations that simultaneously evolve both vdv direct part dominates the outgoing radiation and dynamics of the reduced mass u s v of the system. This, in turn, allows us to produce a full u s inspiral-merger-ringdown waveform. We first qualitatively v v discuss how our method works and how it compares to dv other analytical methods. We then discuss the hybrid method in further detail, and we close this section by FIG. 2 (color online). A spacetime diagram of our method. showing, analytically, that the procedure recovers the cor- The solid black timelike curve depicts the region where we rect Burke-Thorne radiation-reaction potential [50] in the match PN and BHP spacetimes (passing through the centers of weak-field limit. the PN theory’s point particles). Inside this curve, the spacetime can be reasonably approximated by PN theory, whereas outside, A. Qualitative description the spacetime is better described by BHP theory. The yellow (light gray) shade shows the strong-field region, whereas the It is easiest to discuss our method with the aid of the green (gray) shade represents where the black-hole potential is spacetime diagram in Fig. 2. We describe the region within weaker, but the centrifugal barrier of flat space still is important. the solid black timelike curve with the near-zone PN The dark blue (dark gray) shaded region, surrounded by the blue metric, and outside this curve, we use a perturbed (dark gray) dashed lines shows how the value of the perturba- Schwarzschild region. The black line, which passes tions in the exterior (along with the no-ingoing-wave condition) through the PN point particles, is where we match the determines the value of the radiation-reaction potential at the two metrics. We suppress both angular coordinates, so next matching point. The horizontal dashed lines represent the that each point on the curve represents the matching shell region of spacetime where the close-limit approximation or that we discuss in Sec. II. The shaded region represents the Lazarus approach would begin. The red (gray) dashed lines show how one can connect the near-zone behavior to the wave black-hole potential; the yellow (light gray) shade depicts (through lines of constant u), and thereby tie the motion of the the strong-field portion of the black-hole potential (the matching region through the black-hole effective potential to strong-field near zone) and the green (gray) shade shows portions of the waveform. This gives an interpretation of inspiral, the region where centrifugal potential is significant (the merger, and ringdown phases in terms of the direct and scattered weak-field near zone). There is a wave zone near the parts of the waves. Further discussion of this figure is given in horizon (large u), and, consequently, the region where the text of Sec. III.

044035-5 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) redshifted, and they will not escape the black-hole Furthermore, adding a radiation-reaction force to the potential (as Price had found in his description of stellar hybrid method leads to a set of equations that simulta- collapse [61]), because the potential reflects low frequency neously evolve the Zerilli equation (the waveform) and perturbations. the reduced-mass motion of the binary. In Fig. 2 we repre- As in Paper I, we will again be able to interpret different sent schematically how this occurs. We start at a given v [a portions of the waveform by connecting a region of the dark blue (dark gray) dashed line given in Fig. 2] and waveform with the position of the PN binary’s point par- assume that there is a no-ingoing-wave boundary condition ticles in the near zone (via constant values of the light-cone along the line u ¼ 0. In addition, we suppose that we have coordinate, u). In the figure these are the thick red (gray) determined the black-hole-perturbation functions for all dashed lines of constant u. The inspiral part of the wave- smaller values of v, up to the timelike matching surface. form, which propagates directly along the light cone, By evolving the Zerilli equation, Eq. (21), one can find the comes from the part of the trajectory within the weak-field Zerilli function at v þ dv up to the time usðvÞ [within the near zone. Once the trajectory reaches within the strong- dark blue (dark gray) shaded region]. The no-ingoing-wave field near zone, the waves scatter off of the potential and condition combined with the boundary condition on the propagate within the light cone (often referred to as the matching surface, however, fixes how the Zerilli function PN tail part of the wave) in addition to propagating out will evolve to larger values of u. When solved simulta- directly. We view this mixed wave as characteristic of the neously with the Hamiltonian dynamics describing the merger phase. Finally, as the trajectory falls within binary’s motion, this lets one find the position of the re- the effective potential, for a Schwarzschild black hole the duced mass of the binary at v þ dv, denoted by usðv þ dvÞ direct part vanishes and only the scattered waves emerge; in the figure, and the new value of the Zerilli function there. this part is the quasinormal ringing of the final black hole One can evolve the system for all v in such a manner. and should be associated with the ringdown phase. We Including a radiation-reaction force does not greatly distinguish between Schwarzschild and Kerr black holes change the hybrid method as reviewed in Sec. II. The for the ringdown phase, because Mino and Brink [62] and, matching procedure works the same; one modification subsequently, Zimmerman and Chen [63] found that for that comes about is that we must include both the Kerr black holes, frame dragging generates a part of the Newtonian potential and the radiation-reaction potential waveform at the horizon frequency (that decays at a rate in the PN metric (and, therefore, gain an additional term in proportional to the horizon’s surface gravity). This piece of the Zerilli function). The evolution system is now quite the waveform looks like a source, and, thus, only when the different, because it is a coupled system of Hamiltonian final black hole is not spinning do we consider the space- ordinary differential equations and a one-dimensional par- time to appear to be source free. For this reason (and since tial differential equation. We will discuss the system of the matching surface asymptotes to a line of constant v), evolution equations in greater detail after we compare we call the values of v greater than this limiting value the our method with other analytical methods in the next homogeneous region, and the values of v less than this, the subsection. source region. An important development in this paper is that we no B. Descriptive comparison with other analytical longer prescribe the evolution of the reduced mass of the and semi-analytical models system (and thereby a matching region) before evolving In this section, we will compare the similarities and the Regge-Wheeler or Zerilli equations; rather, we specify differences between the hybrid method described above a set of evolution equations for the conservative dynamics and the most closely related methods mentioned in the of the binary, and let the outgoing waves provide back Introduction: the close-limit approximation, the Lazarus reaction onto the dynamics. This, in turn, leads to a self- program, the comparable-mass EOB methods, the consistent system of equations including radiation reac- Teukolsky-based approach, the EOB description of tion. More concretely, we continue to match the PN and EMRIs, and the IMRI calculation calibrated to perturbed Schwarzschild metrics at the centers of the PN numerical-relativity data. The comparison between the theory’s point particles. Moreover, we will again let the hybrid method and the other methods will be descriptive, reduced-mass motion of the binary system follow that of a but we will compare the waveform from the hybrid method point particle in a Schwarzschild background; in this paper, with a numerical-relativity waveform in Sec. IV however, we will use the fact that there are no ingoing To compare with the Lazarus project or the close-limit waves to specify a radiation-reaction potential that acts as a approximation, we again refer to Fig. 2, where we show dissipative force on the Hamiltonian dynamics of the re- two spacelike hypersurfaces (the horizontal dashed lines duced mass. This follows the spirit of the Burke-Thorne labeled by 1 and 2). In the close-limit and Lazarus radiation-reaction potential, but the radiation propagates methods, initial data is posed on these surfaces at a time within a BHP spacetime, and, therefore, also takes the near the merger of the black holes. While these approaches effects of the background curvature into account. have been successful, posing initial data at late times

044035-6 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) makes it more difficult to smoothly connect the initial the near-zone PN solution to an outgoing solution in the inspiral of the binary to the merger and ringdown later. exterior BHP spacetime, whereas the Teukolsky-based Moreover, because the initial data extends inside the black- methods include radiative effects by evolving the orbital hole potential, if it contains high-frequency perturbations, parameters of geodesics from averaged fluxes at infinity. these could escape the potential barrier and enter into the The EOB model of Yunes et al. [45], is a calibration of waveform. The hybrid approach escapes this problem by the EOB method to Teukolsky-based waveforms for setting boundary data on a timelike world tube rather than EMRIs; it, therefore, shares the same similarities and on a spacelike hypersurface. This also lets the method differences as the EOB and the Teukolsky-based methods connect the inspiral, merger, and ringdown portions of discussed above. Han and Cao [42] develop an EOB model the dynamics and waveform more directly. that uses the a Teukolsky-based energy flux (in the fre- The EOB approach, for comparable-mass ratio binaries, quency domain) to treat radiative effects. In comparing only describes times prior to the merger (the hypersurfaces with the hybrid model, therefore, it also falls somewhere between an EOB model and a Teukolsky-based method. 1 and 2 in Fig. 2). To create a full inspiral-merger- ringdown waveform, the EOB method must fit a sequence The recent EOB work of Bernuzzi and collaborators of quasinormal modes to the end of the insprial-plunge [39–41] shares more similarity with the hybrid method, waveform. This procedure makes a very accurate wave- because they evolve the Regge-Wheeler-Zerilli equations form, but it makes connecting the behavior of the space- in the time domain. The most notable specific difference time before and after the merger more difficult. The hybrid (as opposed to the general differences between the hybrid- method, with its interior PN region that falls toward the method and all analytical approaches to EMRIs noted horizon at late times, allows one to make a more clear above) is in the radiation-reaction force. The EOB model connection between the dynamics of the spacetime during uses a high-PN-order, resummed energy flux, whereas (as inspiral and merger to that during ringdown. In its current also noted above) the hybrid method determines radiative implementation, however, it does not produce a waveform effects from directly matching a near-zone PN solution to nearly as accurate as that of the EOB. an outgoing BHP solution. Although the hybrid method is designed for describing We conclude this section by comparing the hybrid method with the recent analytical work of Lousto et al. comparable-mass black-hole binaries, it shares a few sim- [46–48]. They take two approaches to calculating wave- ilarities and has several significant differences from vari- forms for IMRIs perturbatively. In their initial work, they ous approximate techniques that model EMRIs. It is transform the trajectory of the small black hole from their possible to draw a few general comparisons between the numerical-relativity simulations into the Schwarzschild hybrid method and the procedures for studying EMRIs, gauge, and they compute the waveform using this numeri- before moving to more specific comparisons. While the cal trajectory in a BHP calculation. To be able to study a hybrid method evolves perturbations on a black-hole back- wider range of mass ratios, they use PN expressions for the ground (as most EMRI methods do), EMRI methods as- change in frequency and the radial trajectory, but use the sume a source term as the generator of the perturbations in numerical-relativity values of the frequency to calibrate the background. The hybrid approach, however, does not the PN functions. The hybrid method differs from this, have a source term; rather, the perturbations of the back- because it calculates the matching region simultaneously ground come from boundary data that correspond to with the waveform, and it does not use numerical-relativity the multipolar structure of a comparable-mass PN binary. data to calibrate results. Consequently, the hybrid method Because the hybrid approach is a boundary-value problem, does not agree as well with exact results as well as the other the details of the implementation will be different from methods discussed here, but it does present a distinct way those methods that use a point mass as a source term. of calculating the approximate spacetime and gravitational Moving to specific EMRI models, we first compare the waveform. hybrid approach with the Teukolsky-based methods of Sundararajan and collaborators [36] (for example). The C. Radiation reaction and evolution equations hybrid approach is similar to that of [36], in that both use time domain codes and are capable of producing smooth In this section, we will discuss the details of radiation inspiral-merger-ringdown waveforms. An important differ- reaction in the hybrid method. The end result will be the set ence is that the hybrid method calculates the waveform of evolution equations described in Eqs. (48)–(52), and the simultaneously with the evolution of the matching region, majority of this section will be devoted to deriving this whereas the EMRI method of Sundararajan computes system of equations. the trajectory before the evolution (using an adiabatic We begin, as in Paper I, with the PN metric at Newtonian frequency-domain code during insprial, and a prescription order, for the plunge and merger) and then finds the waveform dS2 U dt2 U dR2 R2d2 ; from this trajectory. Moreover, we compute the radiation- ¼ð1 2 NÞ þð1 þ 2 NÞð þ Þ reaction force in the hybrid method by matching (24)

044035-7 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) the same as Eq. (4) of Sec. II, though without the grav- equal-mass binary by symmetry. Throughout this paper, itomagnetic terms. Here, however, we write the Newtonian we focus just on the m ¼2 multipoles, because as one potential (expanded to quadrupole order) as can see from the expressions above, the m ¼ 0 moment l l only evolves due to the radiation-reaction force (for circu- U ¼ Uð ¼0Þ þ Uð ¼2Þ N N N lar orbits), and, therefore, is less significant than the M X2 Q m ¼2 multipoles, which change on the orbital time m 2;m 2 2;m ¼ þ Y þ FmR Y : (25) m r R3 scale. Moreover, the ¼2 quantity is the complex m¼2 conjugate of the corresponding m ¼ 2 quantity, so when The first term is the monopole piece (M is the total mass of we write QðtÞ (or any other variable that might be indexed the binary) and the second term in the sum is the quadru- by m), we refer to the m ¼ 2 variable, and similarly, for Q t m pole parts (and Qm are the quadrupole moments of the ð Þ, we mean the ¼2 element. This way, the notation binary). These two terms above are identical to those of can be simplified by dropping the m label on multipole Paper I, but the second term in the sum (the polynomial in coefficients. Thus, we can write the quadrupole perturba- R with coefficients Fm) is different. One can include the tion as terms proportional to Fm, because like the Newtonian ; ; QðtÞ potential, they are solutions to Poisson’s equation. These U2 2 ¼ U2 2 ¼ þ FðtÞR2; (30) N N R3 terms diverge at infinity (which restricts their use to the near zone), but they cannot be determined from the near- where zone dynamics alone, however. Burke showed [50], using sffiffiffiffiffiffiffi the technique of matched asymptotic expansions, that the MA t 2 3 ð Þ i2ðtÞ terms with coefficients Fm could represent the reaction QðtÞ¼ e ; (31) 10 4 of the binary in the near-zone to radiation losses to infinity. The portion of the potential due to the moments Fm, and FðtÞ, an undetermined function of time, is the therefore, is called the Burke-Thorne radiation-reaction radiation-reaction potential. potential. One can substitute Eq. (30) into Eq. (19) and use the fact In the hybrid method, we will find a similar quantity in that r ¼ R M to find the Zerilli function. Calculating the the interior PN spacetime by matching the PN near-zone Zerilli function introduces many factors of M=R into the solution to a solution in the Schwarzschild exterior with no end result, which, because our calculation is only accurate ingoing waves. Namely, when we assume that there are no to Newtonian order, we will keep only the leading-order ingoing waves from past-null infinity in the exterior BHP terms in R. We find that spacetime, this determines a radiation-reaction potential within the interior PN spacetime. This allows us to incor- 2QðtÞ FðtÞR3 ¼ þ : (32) porate the effects of wave propagation in the background ðeÞ R2 3 black-hole spacetime into the dynamics of the binary. While the Schwarzschild background does not capture We will also shortly need expressions for the derivative of every detail of the curvature of a binary at small separation, the Zerilli function with respect to the tortoise coordinate r we see that it does capture much of the important effects. , Eq. (2), which we compute here as well. Again, we will R Proceeding with the calculation, we assume we have an keep the leading-order expression in , but we will also equal-mass, nonspinning binary in the x-y plane, located at retain the factor of X t X t 1A t t ; t ; ; dr 2M ðR MÞ Að Þ¼ Bð Þ¼2 ð Þðcos ð Þ sin ð Þ 0Þ (26) ¼ 1 ¼ ; (33) dr r ðR þ MÞ where A and B are labels for the two members of the binary. Each black hole has mass M=2, and a straightfor- since, although ðeÞ may be constant on the horizon, @ =@r ward calculation shows that ðeÞ should vanish there [61]. The result of this sffiffiffiffiffiffiffi calculation is that 3 MAðtÞ2 Q ðtÞ¼ e2iðtÞ; @ R M 4QðtÞ 2 (27) ðeÞ F t R2 : 10 4 ¼ 3 þ ð Þ (34) @r R þ M R rffiffiffiffi MAðtÞ2 The Zerilli function satisfies the simple wave equation in Q ðtÞ¼ ; (28) 0 4 5 a potential, Eq. (21). As before, the value of the Zerilli function at the matching surface, RsðtÞ¼AðtÞ=2, provides Q ðtÞ¼Q ðtÞ; (29) a boundary condition for the Zerilli equation on the match- 2 2 ing surface, but now there is an additional boundary con- where the overline stands for complex conjugate, and dition on the Zerilli functions’ derivative with respect to the where the m ¼1 components must be zero for this tortoise coordinate. The two boundary conditions state that

044035-8 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) 8QðtÞ FðtÞAðtÞ3 @ ðtÞ 1 AðtÞþ2M @ ðtÞ t ; _ ðtÞ¼ ðeÞ A_ ðtÞ ðeÞ : ðeÞð Þ¼ 2 þ (35) ðeÞ 2 1 AðtÞ 24 @v 2 AðtÞ2M @r (43) @ ðtÞ AðtÞ2M 32QðtÞ FðtÞAðtÞ2 ðeÞ ¼ þ : (36) A t M 3 @ t =@v @r ð Þþ2 AðtÞ 4 In the above equation, ðeÞð Þ is given by the integral of the Zerilli function up to that time, Eq. (39), By eliminating the unknown function FðtÞ from the above @ t =@r and ðeÞð Þ is given by the boundary condition, equations, one can impose a mixed (Robin) boundary Eq. (37), at that instant. As a result, the only term in condition at the matching surface between the PN and Eq. (43) that is not yet fixed is the expression for A_ ðtÞ. BHP spacetimes, The term A_ ðtÞ specifies the time evolution of the reduced @ t A t M Q t mass of the binary, which, because it is twice the radius of ðeÞð Þ ð Þ2 6 t 80 ð Þ : ¼ ðeÞð Þ (37) the matching surface between the Schwarzschild and PN @r AðtÞþ2M AðtÞ AðtÞ3 metrics, could conceivably evolve via either the PN equa- This specifies a boundary condition at a given moment in tions of motion or those of a particle in the Schwarzschild time, but it does not yet describe how to evolve the match- spacetime. We will choose the latter, for the same reason as ing surface (through evolving the reduced-mass motion of described in Paper I: the Schwarzschild Hamiltonian has the system) and the value of the Zerilli function on this the advantage that a particle falling toward the horizon surface. approaches it exponentially in time, in the limit that the One can determine the value of the Zerilli function at particle is near the horizon. Because we are using this later times through the boundary condition above, and the motion to approximate the region inside of which PN following additional constraint. By integrating the Zerilli theory holds, we want this space to quickly fall toward equation with respect to u, one finds that the horizon as the theory begins to converge slowly. Z Moreover, the motion should move smoothly toward the @ t 1 usðtÞ l ðeÞð Þ Vð ¼2Þ r u0;v du0 horizon (so as not to introduce high-frequency modes that ¼ ðeÞ ð ÞðeÞð Þ @v 4 0 could escape the black-hole effective potential). The PN @ ð0;vÞ equations of motion do not have these desirable features; þ ðeÞ ; @v (38) we consequently favor the point-particle evolution equa- tions in the Schwarzschild spacetime. where we have written r implicitly as a function of u and v, We write the evolution equations for the reduced mass of and usðtÞ denotes the value of u at the matching surface for the system in their Hamiltonian form. As in Paper I, we will a given time t. Having no ingoing waves forces the second describe the dynamics of the reduced mass in PN coordi- term to be zero, so nates, because at late times, this causes the point particles Z in the PN metric to approach the horizon in the external @ t 1 usðtÞ l ðeÞð Þ Vð ¼2Þ r u0;v du0: Schwarzschild spacetime as the reduced mass of the sys- ¼ ðeÞ ð ÞðeÞð Þ (39) @v 4 0 tem does the same. The equations of motion for the re- @ t =@v @ t =@r duced mass, , are Because both ðeÞð Þ and ðeÞð Þ are con- strained at the point of the matching surface, this deter- @H @H ðtÞ A_ ðtÞ¼ ; _ ðtÞ¼ ; mines the evolution of ðeÞ on the matching surface. It is @p ðtÞ @p ðtÞ A (44) easiest to express the Zerilli function on the matching @H surface as a function of time via ðeÞðt; rðtÞÞ. Then, taking p ðtÞ¼ ; p ðtÞ¼F ðtÞ; _ A @A t _ the total derivative, ð Þ d t @ dr @ where the Hamiltonian of a point particle in the ðeÞð Þ _ t ðeÞ ðeÞ ; ðeÞð Þ¼ þ (40) Schwarzschild spacetime is given by dt @t dt @r H A t ;p t ;p t using the fact that ð ð Þ Að Þ ð ÞÞ @ @ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ ðeÞ ðeÞ ðeÞ ¼ 2 ; (41) 2 2 @v @r 2M 2M pAðtÞ pðtÞ @t ¼ 1 1 þ 1 þ : AðtÞ AðtÞ 2 2AðtÞ2 and (45) dr 2M 1 dr ¼ 1 ; (42) dt r dt The radiation-reaction force is given by the derivative of the radiation-reaction potential with respect to ’, and it along with the relationship aðtÞ¼2rðtÞ, one can write should be evaluated at the location of the matching region,

044035-9 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) sffiffiffiffiffiffiffi i!t ðl¼ Þ;F If one assumes a product solution ¼ e c ðRÞ, then F ðtÞ 1 @U 2 15 ðeÞ ¼ N ¼A2ðtÞ =½FðtÞe2iðtÞ: @’ (46) for the radial motion, one must solve the ordinary differ- 2 ential equation ðl¼2Þ;F where UN represents the quadrupole part of the d2 c 6 F t þ !2 c ¼ c : (54) radiation-reaction potential. By solving Eq. (35) for ð Þ dR2 R2 t Q t in terms of ðeÞð Þ [and because ð Þ is proportional to a real amplitude times e2iðtÞ, see Eq. (31)], one can write The solutions for c =R are spherical Hankel functions c =R h !A= j !A= in !A= sffiffiffiffiffiffiffi ¼ 2ð 2Þ¼ 2ð 2Þþ 2ð 2Þ, assuming ! F ðtÞ 15 24 there are no ingoing waves. Here corresponds to the ¼ =½ ðtÞe2iðtÞ: (47) 2 AðtÞ ðeÞ gravitational-wave frequency. We must match this wave zone solution to the PN near-zone expression for the Zerilli With the above relationship between the radiation- function given by Eq. (35); additionally, we must also reaction force and the Zerilli function, there is now a match the derivative of the Hankel function with the radial complete set of evolution equations for the reduced-mass derivative of the PN Zerilli function given in Eq. (36). motion of the system, the Zerilli function on the matching We will write these conditions in the frequency domain, surface, and the Zerilli function in the exterior spacetime. where This system of equations is given by 8Qð!Þ Fð!ÞA3 @H @H Bð!ÞAh ð!A=2Þ¼ þ ; (55) A_ ðtÞ¼ ; _ ðtÞ¼ ; (48) 2 A2 24 @pAðtÞ @pðtÞ 32Qð!Þ Fð!ÞA2 @H B ! Ah0 !A= ; ð Þ 2ð 2Þ¼ þ (56) p_ AðtÞ¼ ; (49) A3 4 @AðtÞ and we must solve for the unknown amplitude Bð!Þ and sffiffiffiffiffiffiffi the radiation-reaction potential Fð!Þ in terms of the quad- 15 24 2iðtÞ Q ! p_ ðtÞ¼ =½ð ÞðtÞe ; (50) rupole moment ð Þ and the spherical Hankel function 2 AðtÞ e h !A= 2ð 2Þ. Since the matching takes place at very large = Z radii, and, by Kepler’s law A! A1 2 for circular orbits, 1 usðtÞ l _ t Vð ¼2Þ r u0;v du0 one can expand the Hankel function in A!=2. This allows ðeÞð Þ¼ ðeÞ ð ÞðeÞð Þ 2 0 one to solve for F as a series in 1=A, whose three lowest AðtÞ2M A_ 6 ðtÞ 80QðtÞ terms are given by ðeÞ ; AðtÞþ2M 2 AðtÞ AðtÞ3 16 8 2 F ¼ !2Q þ !4Q þ i !5Q þ OðA6Þ: (57) (51) 5A3 25A 15

ðl¼2Þ The third term is the familiar Burke-Thorne radiation- 2 V r u; v @ ðeÞ ð ÞðeÞð Þ ðeÞ ¼ ; (52) reaction potential (written in the time domain, this is @u@v 4 proportional to five derivatives of the quadrupole moment). The first two terms resemble 1 PN and 2 PN corrections where the Hamiltonian is given by Eq. (45), the potential to the Newtonian potential in the near zone; however, by Eq. (22), and the quadrupole by Eq. (31). By including a these terms represent the effects of time retardation radiation-reaction force, we arrived at a set of evolution that are needed to match the near-zone solution to an equations that simultaneously evolve the reduced-mass outgoing wave solution in the wave zone. As a result, our motion of the binary and the gravitational waves emitted, method recovers, asymptotically, the expected result. taking into account the back action of the emitted radiation Consequently, the evolution system equations (48)–(52) on the reduced-mass motion. will also give rise to the correct dynamics in the weak-field limit. D. Weak-field analytical solution

First, we will confirm that our procedure recovers the IV. NUMERICAL METHOD AND RESULTS correct Burke-Thorne radiation-reaction potential in the weak-field limit. If we have an equal-mass binary in a We begin this section by describing the numerical circular orbit at a large separation, r r R M, method that we use to solve the system of evolution then the leading-order behavior of the Zerilli equation, equations, Eqs. (48)–(52). We then show that the evolution Eq. (21), is just a wave equation in flat space, equations give rise to reasonable and convergent results. With this established, we compare our waveform with one @2 @2 6 ðeÞ ðeÞ : from a numerical-relativity simulation, and we close this þ ðeÞ ¼ 0 (53) @t2 @R2 R2 section by interpreting the spacetime of the hybrid method.

044035-10 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) A. Numerical methods and consistency checks As in Paper I, the Zerilli function along the matching of the evolution equations surface does not always lie on the uniform grid in the u v Because the set of evolution equations, Eqs. (48)–(52), - plane, and we must be careful when finding the has a somewhat unusual form, we describe our numerical Zerilli function at grid points adjacent to the matching u method in detail, and we present a few basic checks of the surface. For example, at a given value of along the waveform and its convergence. To find the field outside the discretized grid, it is rare that the Zerilli function on the matching surface, we use the same method as that de- matching surface, denoted by scribed in Paper I, a second-order accurate, characteristic t u t ;v t ; W;sð Þ¼ðeÞð sð Þ sð ÞÞ (60) method. If we define the following points on the discretized grid (see the portion on the right, away from the solid line, will actually fall along a grid point (see the left side of in Fig. 3): Fig. 3 near the solid line). Similarly, when evolving the discretized version of Eqs. (48)–(52), it is again unlikely l;m u u; v v ; N ¼ ðeÞ ð þ þ Þ that the Zerilli function along the matching surface at the next value of u (advanced by one unit of u), l;m u u; v ; W ¼ ðeÞ ð þ Þ (58) W;sðt þ tÞ¼ð ÞðusðtÞþu; vsðt þ tÞÞ; (61) l;m u; v v ; e E ¼ ðeÞ ð þ Þ will fall at a grid point or even at the same value of v as the l;m u; v ; S ¼ ðeÞ ð Þ previous earlier value of the Zerilli function, W;sðtÞ. To be able to use Eq. (59) to find the Zerilli function at then discretizing Eq. (52), one can solve for N in terms of u ¼ usðtÞþu for the next grid point in v (which we the other three discretized points and the potential: denote by N;s), we must interpolate the Zerilli function at fixed u ¼ usðtÞ to the same value of v ¼ vsðtÞ as uv l N ¼ E þ W S V ðrcÞðE þ WÞ W;sðt þ tÞ. We will label this point by 8 ðeÞ u t ;v t t : þ Oðu2v; uv2Þ: (59) S;s ¼ ðeÞð sð Þ sð þ ÞÞ (62) As in Paper I, this interpolation does not influence Here rc is the value of r at the center of the discretized grid, (u þ u=2, v þ v=2). the convergence of the algorithm when done with cubic We must evolve this partial differential equation simul- interpolating polynomials. With the value of the Zerilli u ¼ u ðtÞ v taneously with the five ordinary differential equations de- function at s and the nearest grid point in (which scribing the Zerilli function on the matching surface and we will call E;s), one can then find the point N;s using the surface’s position, because all these equations are Eq. (59), where E, W, and S are replaced by E;s, t t coupled together. We solve the ordinary differential equa- W;sð þ Þ, and S;s, respectively. tions using a second-order accurate Runge-Kutta method. As a final note on the numerical methods, we point out that in the evolution equation for the Zerilli function on the @ t =@v matching surface, Eq. (43), the term ðeÞð Þ involves Ψ ∆u N an integral of the Zerilli function times the potential, Eq. (39). Explicitly evaluating this integral adds to the Ψ W ∆v computational expense significantly, so we compared the Ψ @ t =@v Ψ E value of ðeÞð Þ obtained through performing N,s the integral with the value found from evaluating Ψ (t+∆t) Ψ W,s @ t =@v S ðeÞð Þ numerically using a fourth-order finite- Ψ difference approximation of the derivative, calculated E,s Ψ from the Zerilli function in the adjacent exterior BHP S,s spacetime. Since the two agreed to within the numerical Ψ (t ) W,s accuracy of our solution, we used the finite-difference @ t =@v approximation of ðeÞð Þ in our numerical evolutions. FIG. 3. A diagram of how we discretize and evolve the Zerilli We now examine a few consistency checks of the nu- function. The dots represent the Zerilli function evaluated at the merical solutions to the system of evolution equations, grid points (the points of intersection of the dashed lines), and Eqs. (48)–(52). In Fig. 4, we show, in black, the trajectory the solid black line is the matching surface. For all points except those adjacent to the solid line, one can use Eq. (59) directly to of the reduced mass of the binary in the PN coordinates. On numerically evolve the Zerilli function. Near the solid line, one this same figure, we have depicted the Schwarzschild black can use the same procedure as described in Eq. (59), except that hole by a filled black circle, the light ring of this black hole one must interpolate the Zerilli function to the point S;s to use by a red (light) dashed circle, and the innermost stable the same procedure. Further detail is given in the text of this circular orbit (ISCO) by a blue (dark) dashed and dotted section. circle. One can see that the radiation-reaction force causes

044035-11 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012)

|Ψ − Ψ |/M (e),(∆ v)/M (e),1/64

0.01(∆ v)2/M2

−3 10

−4 10

−5 10

−1 10 (∆ v)/M

FIG. 5 (color online). The L2 norm of the Zerilli function at a FIG. 4 (color online). In black, the trajectory of the reduced- given resolution, v=M, minus the Zerilli function at the highest mass motion of the binary, in the PN coordinate system. The blue resolution, ðvÞ=M ¼ 1=64, which we denote by jðeÞ;ðvÞ=M (dark) dotted and dashed circle shows the Schwarzschild ISCO, =M and the red (light) dashed circle depicts the light ring of the ðeÞ;1=64j . We also include a power law proportional to v =M 2 Schwarzschild spacetime. The large filled black circle represents ½ð Þ to indicate the second-order convergence of our result. the horizon. One can see that the binary plunges soon after it reaches the ISCO of the exterior Schwarzschild spacetime. initial few orbits before we include the radiation-reaction force, and we denote the zero of our time to be the moment the matching region to adiabatically inspiral, until it ap- when we let the radiation-reaction force begin acting on proaches the ISCO. Once at the ISCO, it begins plunging the binary. more rapidly toward the light ring, and then falls past the We also calculate the Zerilli function corresponding to light ring and asymptotes to the horizon of the final black these initial conditions, as a function of increasing numeri- hole. cal resolution. In Fig. 5, we show that the Zerilli function at The initial conditions of this evolution correspond to a large constant v, does converge in a way that is consistent binary with a PN separation of Að0Þ¼14 in a circular with the second-order accurate code we are using. We show orbit, with no ingoing gravitational waves from past-null the L2 norm of the difference between the Zerilli function infinity, and with the radiation-reaction force initially set to at a given resolution, which we denote ðeÞ;ðvÞ=M and zero. We do not let the radiation-reaction force enter into the highest resolution, ðvÞ=M ¼ 1=64, which we the dynamics (thereby holding the binary at a fixed sepa- L2 denote by ðeÞ;1=64. The norm, therefore, we write as ration) until we have a stable estimate of the force. At this jðeÞ;ðvÞ=M ðeÞ;1=64j, and we normalize this by the num- point, we include the radiation-reaction force (thereby ber of data points in the evolution, and the mass. We also letting the binary begin its inspiral). To minimize eccen- include a power law, proportional to ½ðvÞ=M2, which tricity, we introduce a small change in the radial momen- indicates the roughly second-order convergence of the tum pAð0Þ that corresponds to the radial velocity of a PN waveform. binary at that separation. Explicitly, we find this value of We then plot the real part of the Zerilli function ex- pAð0Þ by solving tracted at large constant v, for the highest resolution

@H 16 M3 ðvÞ=M ¼ 1=64, in Fig. 6. The top panel depicts the A_ ð Þ¼ ¼ ; 0 3 (63) Zerilli function throughout the full evolution. Because it @pAðtÞ t¼0 5 Að0Þ is difficult to see the slow increase of the amplitude and (see, e.g., [3]), while assuming that pð0Þ continues to have frequency during early times and the smooth transition the value for circular orbits from inspiral to merger and ringdown at late times, we MAð0Þ highlight the early stages of the inspiral in the lower-left pð0Þ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (64) panel, and we depict the merger and ringdown in the lower- A =M pffiffiffi ð0Þ 3 r h ih l right panel. Because 6ðeÞ ¼ ð þ Þ, for the ¼ 2 This is necessary to make the orbit as circular as possible modes at large r [see Eq. (100)], the Zerilli function is once the binary begins to inspiral. We do not show the essentially identical to the gravitational waveform. From

044035-12 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) 0.5 with a numerical-relativity waveform during the inspiral part. We will continue to find the Zerilli function through the procedure describe in Sec. III C using the leading-order expression for the Newtonian potential (and thus also the ]/M

(e) 0 leading-order radiation-reaction potential). We note, how- Ψ [ ever, that when we took the derivative of the Zerilli func- ℜ tion on the matching surface with respect to r, Eq. (36), we kept the factor of ð1 2M=rÞ. This is reasonable, physically, because, although the Zerilli function itself −0.5 may approach a constant on the horizon, its derivative 0 500 1000 1500 2000 2500 3000 with respect to r should vanish. Conversely, if the deriva- t/M tive of the Zerilli function did not vanish, then that could 0.2 0.5 correspond with a perturbation that diverges on the hori- zon. Nevertheless, because the boundary condition only 0 0 takes into account the leading Newtonian expressions, the overall factor of ð1 2M=rÞ is a higher PN correction, −0.5 −0.2 from the point of view of the interior PN spacetime. We, 0 500 1000 1500 2000 2500 2800 2900 3000 t/M t/M therefore, are justified in dropping this term in our leading Newtonian treatment, and we find the agreement between FIG. 6. The top panel shows the real part of the Zerilli function numerical relativity and the hybrid method is helped by throughout the entire evolution, extracted at large constant v. this. It is likely that further adjustments will lead to even The bottom-left panel displays the early part of the same Zerilli better results, though a systematic study of this is beyond function, and the bottom-right zooms in to the merger and the scope of this initial exposition. pringdownffiffiffi portions of the function. Because only a factor of r The modification above results in only a small change 6 differentiates the Zerilli function from times the waveform, to Eq. (36), this can be thought of as the waveform as well. @ ðtÞ 32QðtÞ FðtÞAðtÞ2 ðeÞ ¼ þ ; (65) @r A t 3 this one can see the hybrid method produces a smooth ð Þ 4 inspiral-merger-ringdown waveform. Because the hybrid and it also alters the boundary condition, Eq. (37) of waveform has the correct qualitative features of a full Sec. III C, inspiral-merger-ringdown waveform, it is natural to ask how well it could match a numerical-relativity waveform. @ð ÞðtÞ 6 80QðtÞ e ¼ ðtÞ : (66) A t ðeÞ 3 We, therefore, turn to this question in the next section. @r ð Þ AðtÞ With the exception of these two equations and the fact B. Comparison with numerical relativity that we begin the evolution from a larger initial radius, In this section, we will first discuss how well the wave- Að0Þ¼15:4, we evolve the new system of equations in form compares with a similar waveform from numerical- exactly the same way as that described in detail in relativity simulations. The first part of the section is Sec. IVA. devoted to showing how we can make small modifications For our comparison with a numerical-relativity wave- to the hybrid procedure to make the phase agree well with form, we use the l ¼ 2, m ¼ 2, mode of the waveform that of a numerical-relativity waveform during inspiral from an equal-mass, nonspinning, black-hole binary de- (though the comparison of the amplitudes is less favor- scribed in the paper by Buonanno et al. [18]. In this able). The second part of this section describes why the simulation, the black holes undergo 16 orbits before they hybrid method, in its current implementation, does not merge, and the final black hole rings down. We plot the agree well with numerical-relativity simulations during numerical-relativity waveform in black in Fig. 7, and we the merger and ringdown phases. The reason for the dis- show the equivalent waveform from our approximate crepancy during the late stages of the waveform is well method in red (gray). Recall that the l ¼ 2 modes of the understood (the background spacetime of the hybrid Zerilli function are related to the waveform by method is Schwarzschild, whereas the final spacetime of pffiffiffi r h ih the numerical simulation is Kerr) and could be improved 6ðeÞ ¼ ð þ Þ (67) by modifications to the hybrid method. [see Eq. (100)]. Although the amplitudes of the waveforms do not agree exactly, the fact that the phases match so well 1. Agreement of the waveforms during inspiral throughout the entire inspiral is noteworthy. The approxi- We will briefly describe a small change to the hybrid mate waveform completes one more orbit than the method that leads to a waveform whose phase agrees well numerical-relativity one, and the ringdown portions differ

044035-13 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012)

Hybrid Method ℜ[Mω ] 1.5 NR Numerical Relativity 0.5 ℑ[Mω ] NR ℜ[Mω ] 1 hybrid 0.4 ℑ[Mω ] hybrid 0.5 0.3 /M] ω

2,2 0 M 0.2 [r h ℜ −0.5 0.1

−1 0

−1.5 −0.1 0 500 1000 1500 2000 2500 3000 3500 4000 t/M 3800 3850 3900 3950 4000 t/M FIG. 7 (color online). In black is the real part of the l ¼ 2, m ¼ 2 mode of a numerical-relativity waveform, whereas in red FIG. 8 (color online). The solid curves are the instantaneous (gray) is the equivalent quantity from the approximate method of frequency [see Eq. (68)] of the numerical-relativity waveform; this paper. The agreement of the waveforms’ phases is quite the black curve is the real, oscillatory part and the red (gray) good throughout the entire inspiral, although the amplitudes curve is the imaginary, decaying part. The black, dashed curve differ. The approximate and numerical-relativity waveforms and the red (gray) dashed curve are the real and imaginary parts, differ during ringdown, because the approximate method uses respectively, of the frequency for the hybrid method. The fre- a black-hole with zero spin, whereas the final black hole in the quencies agree quite well during the inspiral, but at late times numerical-relativity simulation has considerable spin. they begin to differ. The qualitative transition from inspiral to merger and ringdown is similar, but the final quasinormal-mode as well. This is not too surprising, however, since the final frequencies that the waveforms approach differ, because the numerical-relativity simulation results in a Kerr black hole of black hole in the numerical-relativity simulation is a Kerr : : dimensionless spin ¼ 0 7, whereas the hybrid waveform is black hole with dimensionless spin 0 7 (see, e.g., generated on a Schwarzschild background. The oscillations in Scheel et al. [51]), whereas our ringdown takes place the hybrid waveform arise from the interference of positive- and around a Schwarzschild (nonspinning) black hole. negative-frequency modes that can arise in a Schwarzschild background, as explained in the text of this section. 2. Differences in the instantaneous frequency during merger and ringdown The discrepancy between the two waveforms at late Solid curves depict the instantaneous frequency of the times in Fig. 7 is most evident in the instantaneous fre- numerical-relativity waveform in Fig. 8; the real (oscilla- quency, often defined as tory) part is the black curve and the imaginary (decaying) part is the red (gray) curve. Similarly, the black dashed _ curve is the real part of the instantaneous frequency of the M! ¼ i ðeÞ ; (68) ðeÞ hybrid method, and the red (gray) dashed curve is its imaginary part. The hybrid and the numerical-relativity r where ðeÞ is the Zerilli function measured at large .We frequencies are in very good agreement for the inspiral calculate this frequency for both the hybrid and the up until the late stages highlighted here. The numerical- numerical-relativity waveforms, and we show the real relativity waveform quickly transitions after the plunge and the imaginary parts (the oscillatory and damping por- and merger to the least-damped l ¼ 2, m ¼ 2 tions, respectively) in Fig. 8. The numerical-relativity quasinormal-mode frequency and decay rate for a Kerr waveform was offset from zero at late times by a small black hole of final dimensionless spin equal to roughly constant of order 104. We subtracted this constant from 0:7 (see, e.g., [64]). The frequency of the hybrid the waveform to find the instantaneous frequency; other- waveform, however, undergoes a similar qualitative tran- wise, when the amplitude of the waveform becomes com- sition, but it approaches the least-damped l ¼ 2, m ¼ 2 parable to this constant, there are spurious oscillations in ringdown frequency of a nonspinning black hole (the the frequency as it becomes dominated by this constant background of the hybrid method). The hybrid method, offset. The hybrid waveform needed no modification. however, oscillates around this value with a frequency that

044035-14 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) is proportional to twice the frequency of this least-damped, ℜ[rh /M] 2,2 l ¼ 2, m ¼ 2 quasinormal mode. 3 ℜ{Ψ [u (v)]} The origin of this oscillation is simple and, in fact, was (e) s explained by Damour and Nagar [38]. For each l and m, there are quasinormal modes with both positive and nega- 2 tive real parts, which both have a negative decay rate. For a

Schwarzschild black hole, the decay rates are the same and 1 the real frequencies are identical, but have the opposite sign. For a Kerr black hole, however, the positive- frequency modes have a lower decay rate than the 0 negative-frequency modes (and the positive frequency is larger in absolute value than the negative frequency is). While a counter-clockwise orbit will tend to excite pre- −1 dominantly the mode with a positive real part, it can also generate the negative real-frequency mode as well. −2 In the hybrid waveform, because the background is Schwarzschild, the positive- and negative-frequency modes decay at the same rate, and they can interfere to 3500 3600 3700 3800 3900 4000 make the oscillations at twice the positive real frequency. u/M In the numerical-relativity waveform, however, the differ- FIG. 9 (color online). The real part of the Zerilli function on ence of the frequencies and decay rates prevents this from the matching surface, the red (gray) dashed curve, and the real happening. part of the gravitational waveform, proportional to the real part of the Zerilli function at large v, (the black solid curve). The two C. Interpreting the hybrid waveform and spacetime functions are nearly out-of-phase for the inspiral, and the wave propagates more or less directly out. During the merger, they Since the phase during inspiral agrees so well, and begin to lose this phase relationship, and during ringdown the because the transition from inspiral to merger and ring- Zerilli function on the matching surface becomes constant. This down is qualitatively similar, this leads one to wonder implies that the ringdown waveform is due just to the waves to what extent the hybrid approach may also be a useful scattered from the potential, as also illustrated in Fig. 10. tool for generating gravitational-wave templates for gravitational-wave searches. To capture the correct ring- This change in phasing between the Zerilli function on down behavior, the hybrid method would need to be ex- the matching surface and that at large v (along a line of tended to a Kerr background; however, it is likely that constant u) allows one to give an interpretation to the calibrated approaches using the effective-one-body method different parts of the waveform. The inspiral occurs when (see, e.g., [18]) or phenomenological frequency-based tem- the waveform propagates out directly, but nearly out-of- plates (see, e.g., [65]) will be more efficient for these phase with the matching surface. The merger is the smooth, purposes. The hybrid approach, as described here, will but brief, transition during which the phase relationship likely be more helpful as a model of how the near-zone between the matching surface and the waveform evolves, motion of the binary connects to different portions of the and the ringdown is the last set of waves that are discon- gravitational waveform. nected from the behavior on the surface (they are the As an example of this, we show the real part of the scattered waves from the potential barrier). gravitational waveform at large v, the black solid curve, We also show in Fig. 10 a contour-density plot of the real and the corresponding value of the Zerilli function on the part of the Zerilli function in the u-v plane during the last matching surface, the red (gray) dashed curve in Fig. 9. few orbits of inspiral, the merger, and the ringdown (for the Interestingly, the Zerilli function on the matching surface evolution discussed in this section). This is a spacetime v and that extracted at large constant are roughly out-of- diagram, where time runs up, and the radial coordinate, r phase with one another during the inspiral; namely, along a increases to the right. The matching surface is the dark ray of constant u, the Zerilli function undergoes nearly one timelike curve running up that turns to a line of constant v half cycle as it propagates out to infinity. This feature is at the end. The region to the left of the surface, the solid also visible in Fig. 10, but it is harder to discern there. This green (gray) is the interior PN region, but we do not show behavior holds through inspiral up to the beginning of the the metric perturbation in this region. On its right is the merger. During the merger, however, the two transition BHP region, where we show the Zerilli function colored so away from the out-of-phase relationship, before the that blue colors (dark gray) are negative and red colors Zerilli function on the matching surface becomes a con- (light gray) are positive. Away from the matching surface, stant during the ringdown (when the reduced-mass of the the Zerilli function oscillates between yellow (light binary falls toward the horizon along a line of constant v). gray) and light blue (darker gray) for several orbits before

044035-15 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) is a low frequency change that occurs within the potential barrier, and is hidden from the region of space outside the potential. In some sense, it is a strong confirmation of Price’s idea that the details of the collapse will be hidden within the potential barrier. At the same time, however, this behavior arises from the fact that the derivative of the Zerilli function with respect to r vanishes on the matching surface. When this condition was neglected in this section, it led to a more regular behavior there. This suggests that it may be worth while to do a more careful analysis of how the Zerilli function and its derivatives near the horizon should scale in the presence of radiation reaction.

V. SPINNING BLACK HOLES, SPIN PRECESSION, AND THE SUPERKICK MERGER In this section, we will incorporate the effects of black- hole spins into our method, with the aim of understanding the large kick that arises from the merger of equal-mass black holes with spins antialigned and in the orbital plane FIG. 10 (color online). A contour-density plot of the real part of the Zerilli function for the evolution discussed in this section. (the superkick configuration). To do this, we will first We only show the last few orbits of the inspiral, followed by the discuss adding odd-parity metric perturbations to the re- merger and ringdown. In this spacetime diagram, time runs up, sults in the previous section. We will then indicate why r increases to the right, and the coordinates u and v run at spin precession is important in producing large kicks and 45 degree angles to the two. The line that starts at nearly constant discuss two ways of implementing spin precession: the PN t and evolves to a line of constant v is the matching surface, and equations of precession and geodetic precession in the to the left of this line, the solid green (gray) region is the interior Schwarzschild spacetime. In our method, we will use the PN region (where we do not show any metric perturbations). The geodetic-precession approach, and we will present numeri- exterior is the BHP region, where we show the Zerilli function. cal results for the kick that uses this equation of spin During inspiral, the Zerilli function propagates out almost di- precession. rectly, and it oscillates between positive, yellow (light gray) colors, and negative, light blue (darker gray). Black dashed contour curves are used to highlight this oscillation. As the A. Odd-parity metric perturbations reduced mass of the binary plunges into the potential during of spinning black holes merger, the amplitude and frequency of the radiation increases, To incorporate the effects of spin into our model, we will but it promptly rings down to emit little radiation, in the upper add the lowest-order metric perturbation arising from using green (gray) diamond of the diagram. There are black dashed spinning bodies in the PN metric, as we did in Paper I. This contours here as well to indicate that there is still oscillation, comes from the metric coefficients even though it is exponentially decaying (and hard to see through j k j k the color scale). 2ijkS n 2ijkS n h A A B B : 0i ¼ 2 2 (69) RA RB inspiral. Each oscillation is bounded between a black, dashed contour curve. As the reduced mass of the binary Here we use the notation of Paper I, where we label the two j plunges toward the horizon, the outgoing waves increase in bodies by A and B. The new variables SA represent the spin frequency and amplitude, which is how we describe the angular momentum of the body, RA is the distance from k transition from the inspiral to the merger phase. The body A, and nA is a unit vector pointing from body A. The merger phase is short, and the black hole rings down variables for body B are labeled equivalently. Since we will (leading to very little gravitational-wave emission in the focus on the extreme kick configuration, we will assume top corner of the diagram). As the reduced mass of the the black holes lie in the x-y plane, at positions XAðtÞ and system approaches the horizon, there is a small wavepacket XBðtÞ [identical to Eq. (26) of Sec. III C], and that the spins of ingoing radiation that accompanies it. are given by We close this section with one last observation. S AðtÞ¼SBðtÞ¼SðcosðtÞ; sinðtÞ; 0Þ; (70) If we were to plot the equivalent quantities to those in Figs. 9 and 10 for the evolution in Sec. III, then one would where S ¼ ðM=2Þ2 is the magnitude of the spin, and is see that the Zerilli function on the matching surface in- the dimensionless spin, ranging from zero to one. creases during ringdown instead of approaching a constant. Under these assumptions, one can show that the This does not have any effect on the waveform, because it Cartesian components of the metric coefficients above are

044035-16 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) 3SAðtÞ As with the even-parity, mass-quadrupole perturbations h X ¼ sin2 sinðtÞ cosððtÞ’Þ; (71) 0 R3 discussed in the previous section, we will only be inter- ested in evolving the m ¼ 2 perturbation (though in this 3SAðtÞ case it is an odd-parity, current-quadrupole moment). The h Y ¼ sin2 cosðtÞ cosððtÞ’Þ; (72) 0 R3 reason for this is subtle, and will be clarified in the next section. Nevertheless, we will mention here that during the 2SAðtÞ merger and ringdown (when the kick is generated), the h Z ¼ sin½ðtÞðtÞ spins precess at the orbital frequency [namely, _ ðtÞ¼ 0 R3 _ ðtÞ]. As a result, the m ¼ 0 part of the perturbations 6SaðtÞ t t þ sin2 cosððtÞÞ sinððtÞ’Þ: (73) which depend on ð Þ ð Þ become constant, and the R3 only changes in the perturbations come from changes in A t One can then convert the Cartesian components into ð Þ. We mentioned in Sec. III C that we would also neglect m spherical-polar coordinates to find that the ¼ 0 part of the even-parity perturbations, because it also evolved from time variations in AðtÞ, which occur on 2SAðtÞ the time scale of the radiation-reaction force (2.5 PN orders h R ¼ sin½ðtÞðtÞ cos; (74) 0 R3 below the leading-order orbital motion). Consequently, because we are interested in the behavior of the binary 2SAðtÞ during merger and ringdown, we can neglect the m ¼ 0 h ¼ sin½ðtÞðtÞ cos 0 R2 parts of the odd-parity metric perturbations for this same 6SAðtÞ reason. In addition, because we are treating just the sin cosððtÞ’Þ sinððtÞ’Þ; (75) m ¼ m ¼ R2 2 perturbations (and the 2 term is the com- plex conjugate of the m ¼ 2 moment), we will again drop m SA t the label on the perturbations. 6 ð Þ 2 h ’ ¼ sin coscosððtÞ’ÞcosððtÞ’Þ: (76) Thus, the relevant piece of the gravitomagnetic potential 0 R2 for our calculation will be As written above, the metric perturbations do not take the sffiffiffiffiffiffiffi SAðtÞ 6 form of an odd-parity vector harmonic, because there is a w ¼ ei½ðtÞþðtÞ; (82) dipolelike piece in two of the components. This can be ðoÞ 4R2 5 eliminated by making a gauge transformation, and one can then use Eq. (20) and the fact that R ¼ r M SAðtÞ to find that the Regge-Wheeler function is (at leading ¼ cos sin½ðtÞðtÞ: (77) 0 R2 order in r), sffiffiffiffiffiffiffi A small gauge transformation produces a change in the 2SAðtÞ 6 metric via ¼ ei½ðtÞþðtÞ: (83) ðoÞ R2 5 h^ ¼ h ; ;; (78) This means that on the matching surface, h^ sffiffiffiffiffiffiffi which, in this case, sets 0R ¼ 0. The remaining terms in 8S 6 the metric can then be expressed in terms of the odd-parity, ¼ ei½ðtÞþðtÞ: (84) vector spherical harmonics, ðoÞ AðtÞ 5

2;2 2;2 2;2 X ¼ðX ;X’ Þ We can then evolve the Regge-Wheeler equation, sffiffiffiffiffiffiffi Eq. (21), (with the odd-parity l ¼ 2 potential) using 1 15 Eq. (84) as the boundary condition along the matching ¼ sinei2’ði; sin cosÞ; (79) 2 2 surface. We will not take any radiation-reaction effects from the current-quadrupole perturbations into account sffiffiffiffi (since they are 1.5 PN orders below the leading-order 2;0 2;0 2;0 3 5 2 Newtonian radiation reaction of Sec. III C); as a result, X ¼ðX ;X’ Þ¼ sin cosð0; 1Þ: (80) 2 we will evolve the Regge-Wheeler function using the matching surface generated by the even-parity, mass- A short calculation shows that quadrupole perturbations alone. sffiffiffiffiffiffiffi SA t ð Þ 6 i½ðtÞþðtÞ 2;2 ðh^ ; h^ ’Þ¼2< e X 0 0 r2 5 B. Spin precession rffiffiffiffi 8SAðtÞ Before we discuss the evolution of the Regge-Wheeler cos½ðtÞðtÞX2;0: (81) and Zerilli functions, we will mention an effect that r2 5 is important for our recovering the correct qualitative

044035-17 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) behavior of the kick in superkick simulations. This effect The vector n^ is a unit vector from the center of mass. There was observed by Schnittman et al. in [54] and clarified to is an equivalent equation for the precession of SBðtÞ, us by Thorne [55]. In Schnittman et al.’s discussion of the identical to the equation above, under the interchange of superkick configuration, the authors observe that the spins A and B. Given the form of the equation above, the mag- precess in the orbital plane very rapidly during the merger, nitude of the spin does not change, and the spin precesses approaching the orbital frequency just before the ring- about the Newtonian angular momentum LNðtÞ. Moreover, down. We will give a heuristic argument of why this effect Bru¨gmann et al. found that for the superkick configuration, should occur before we explore two models that produce where the spins lie in the plane, precession of the spins spin precession (one based on the PN equations of motion does not produce a large component out of the plane (the z and the other based on geodetic precession in the component in this case). Schwarzschild spacetime). We will ultimately favor the For simplicity, therefore, we will just consider the com- latter. ponents of the spin in the orbital plane, which, at leading- order, will precess as a result of coupling to the Newtonian orbital angular momentum. The Newtonian angular mo- 1. Motivation for spin precession mentum is One can see the need for spin precession from the L t A t 2 t z; following simple argument. Just as the even-parity pertur- Nð Þ¼ ð Þ _ ð Þ^ (87) bations gave rise to a waveform that increased from twice where _ ðtÞ is the orbital frequency. With the assumption the orbital frequency to the quasinormal-mode frequency z z that SA ¼ SB ¼ 0, the spins precess via the equation, during the merger phase (see Fig. 8), so too must the odd- parity perturbations of the previous section give rise to a 7M_ ðtÞ S_ AðtÞ¼ ½z^ SAðtÞ; (88) part of the waveform that transitions from the orbital 8AðtÞ frequency to the same quasinormal-mode frequency as the even-parity perturbations. The quasinormal-mode fre- where we have also used the fact that this is an equal-mass M M M= M= quencies are the same, because both the Regge-Wheeler binary, ( A ¼ B ¼ 2 and ¼ 4). Taking the and Zerilli functions are generated by l ¼ 2, m ¼2 time derivative of Eq. (70), we obtain the expression for perturbations. Because the Zerilli function is generated the left-hand side of the equation above, ei2ðtÞ by a boundary condition proportional to and the S_ AðtÞ¼_ ðtÞ½z^ SAðtÞ: (89) Regge-Wheeler function produced by a boundary condi- tion that changes as ei½ðtÞþðtÞ, for the two perturbations Relating the two expressions, we arrive at the equation of to evolve in the same way, both ðtÞ, the orbital evolution, spin precession, and ðtÞ, the spin precession, should evolve in identical 7M _ ðtÞ¼ _ ðtÞ: (90) ways at the end of merger. Stated more physically, at the 8AðtÞ end of merger, the spins should precess at the orbital frequency. For the hybrid method, this expression will not lead to This rapid precession of the spins was observed by the spin-precession frequency approaching the orbital fre- Bru¨gmann et al. [56] in their study of black-hole super- quency, since AðtÞ2M for the entire evolution (and kicks. Using a combination of PN spin-precession and hence, the spin-precession frequency will not even be numerical-relativity data, they were able to match the half the orbital frequency at its maximum). In the next precession of the spin in their numerical simulations. section, we will put forward an equation of spin precession We will explain in the next section why this worked so based on geodetic precession in the external Schwarzschild well for their simulation, but why it will not work as well in spacetime, which will have the desired spin-precession the hybrid method. behavior. Before turning to the next section, we address the ques- tion of why PN spin precession worked so successfully for 2. Post-Newtonian spin precession Bru¨gmann et al. Their initial data begins in a gauge that is Bru¨gmann et al. begin from the well-known spin pre- identical to the 2 PN Arnowitt-Deser-Misner, Transverse- cession for a binary (see, e.g., [66]), Traceless (ADMTT) gauge, and they assume that it con- tinues to stay in that gauge throughout their evolution. As a M S_ t 1 3 B L t S t ; result, they use the puncture trajectories as the positions of Að Þ¼ 3 2 þ ½ Nð Þ Að Þ (85) AðtÞ 2MA the black holes, and the 2 PN ADMTT gauge expressions to relate the momenta of the black holes to their velocities. where we just write the leading-order effect from the Although the PN equations of spin precession are written in Newtonian angular momentum, harmonic gauge, they use the puncture results to calculate these expressions. This is reasonable, because the har- L t X t X t X_ t X_ t : Nð Þ¼ f½ Að Þ Bð Þ ½ Að Þ Bð Þg (86) monic and ADMTT gauge positions do not differ much

044035-18 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) A t M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi until separations of roughly ð Þ2 . Their puncture 2M separations do reach small values of AðtÞ proper time ). If we assume that _ does not superkick simulations. change much over an orbit (which is true during most of the evolution of the binary, as it changes only due to the radiation-reaction force), and we continue to denote the C. Numerical results and kick angle of the spin in the orbital plane by ðtÞ, then one can In the first part of this section, we describe how we write the solution to these equations [Eqs. (14.16a) and numerically solve the Regge-Wheeler equation (we con- (14.16b) of Hartle] as tinue to solve the Zerilli equation in the same way as

044035-19 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) described in Sec. IV), and we show a representative because the matching surface is driven by radiation- waveform obtained from the Regge-Wheeler function. reaction from the even-parity Zerilli function alone. Were We next describe how we calculate the linear-momentum we to include the radiation reaction arising from the spins, flux and the kick from the waveforms. Finally, we close however, we would need to evolve the equations for ðtÞ this section by studying the dependence of the kick on the and ðoÞ simultaneously, and in a manner identical to that initial angle between the spins and the linear momentum of described in Sec. III C. the PN point particles. We recover results that are qualita- Our initial conditions are identical to those described in tively similar to those seen in full numerical-relativity Sec. IVA, but we will set the dimensionless spin ¼ 1, simulations. and let ð0Þ vary over several values from 0 to 2, to study the influence of the initial angle on the kick. We first show 1. Numerical methods and waveforms the real part of the Regge-Wheeler function extracted at large constant v, in Fig. 11. The top panel is the full Regge- To calculate the Regge-Wheeler function, and thus the Wheeler function, whereas the bottom-left panel features radiated energy momentum in the gravitational waves, we the early part from the inspiral (so that one can see the first make the following observation. Because the odd- gradual increase in the amplitude and frequency that comes parity perturbation of the spins of the black holes is a from the combined effects of the binary inspiral, and the 1.5 PN effect, the corresponding radiation-reaction force increased rate of spin precession). In the bottom-right will also enter at 1.5 PN beyond the leading-order panel, we show the merger and ringdown phase, which is radiation-reaction force discussed in Sec. III C. obscured in the top panel. As the spins start precessing near Consequently, we do not take it into account in the the orbital frequency during merger, one can see the rapid leading-order treatment of the radiation-reaction force. growth of the Regge-Wheeler function. Moreover, we note that the spin-precession angle, ðtÞ, To see how this spin precession leads to a large kick, we does not enter into the evolution equations for the reduced plot both the even- and the odd-parity metric perturbations mass or for the Zerilli function. As a result, the evolution of extracted at large constant v in Fig. 12. We show the real ðtÞ and ðoÞ can be performed after the evolution of the binary without spin. In fact, the evolution of ðoÞ is carried ℜ[Ψ /M] out in the same manner as that described in Paper I, (e) 0.4 ℑ[Ψ /M] (o) 0.2 0.2 0.15 0 0.1

0.05 −0.2 ]/M

(o) 0 −0.4 Ψ [

ℜ −0.05 −0.6 500 1000 1500 2000 2500 3000 −0.1 t/M −0.15 0.2 0.5 −0.2 0 500 1000 1500 2000 2500 3000 0.1 t/M 0 0 0.02 0.2 −0.1 0.01 0.1 −0.2 −0.5 0 0 0 1000 2000 2800 2850 2900 2950 t/M −0.01 −0.1 t/M −0.02 −0.2 FIG. 12 (color online). The top panel shows the real part of the 0 500 1000 1500 2000 2500 2800 2900 3000 t/M t/M Zerilli function, the red (gray) curve, and the imaginary part of the Regge-Wheeler function, the black curve, throughout the FIG. 11. The top panel shows the real part of the Regge- inspiral, merger, and ringdown. The bottom-left panel shows Wheeler function throughout the entire evolution. The bottom- only the inspiral, where the Regge-Wheeler function is much left panel just focuses on the early times of inspiral, where the smaller than the Zerilli function, and oscillates at approximately Regge-Wheeler function slowly increases in frequency and half the frequency. In the bottom-right panel, during the merger amplitude because of the binary’s inspiral and the slow spin and ringdown, as the spins precess near the orbital frequency, the precession. In the bottom-right panel, one sees that as the spins Regge-Wheeler function increases in amplitude and frequency, begin to precess near the orbital frequency, the Regge-Wheeler and becomes in phase with the Zerilli function. This leads to a function dramatically increases in amplitude and frequency. large kick.

044035-20 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) part of the Zerilli function, ðeÞ, in red (gray) and the P t 1 _ _ : _ zð Þ¼ =½ðeÞðoÞ (103) imaginary part of the Regge-Wheeler function, ðoÞ,in black, for the angle ð0Þ that gives the maximum kick. When we discuss the kick velocity as a function of time, As we show below, in Eq. (103), it is the relative phase of we mean that we take minus the time integral of the the product of these components that is important in pro- momentum flux, normalized by the total mass, i.e., ducing the kick. During the early part of the evolution, the Z 1 t Regge-Wheeler function is quite small and oscillates with vkickðtÞ¼ P_ ðt0Þdt0: z M z (104) roughly half the period of the Zerilli function. This is t0 difficult to see in the upper panel of the full waveforms M in Fig. 12, but is more evident in the lower-left panel, We continue to normalize by the total mass , because showing just the early parts of the evolution. In the last numerical-relativity simulations have shown that it orbit before the merger and ringdown (shown in the lower- changes only by roughly 4% during a black-hole-binary merger (see, e.g., Campanelli et al. [52]); as a result, right panel), the spins start precessing rapidly, and, in the M case that produces the maximum kick, the real part of the normalizing by the total mass will not be a large source even-parity perturbation function, and the imaginary part of error. of the odd-parity function oscillate in phase during the merger and ringdown. (For the case with zero kick, the 3. Numerical results of the momentum flux and kick two functions are now out-of-phase by 90 degrees.) We now show the results of our numerical solutions for the superkick configuration. We first show in Fig. 13 the 2. Calculation of the kick momentum flux for several different initial angles of the spins, . In the plots, we subtract the value that gives zero We now discuss, more concretely, how we calculate kick, which we denote by ¼ 215=192. While the the kick emitted in gravitational waves. At radii much 0 shape of the pulse of momentum flux has a similar shape larger than the reduced gravitational wavelength, r to that seen in numerical-relativity simulations by =ð2Þ, one can relate the gravitational-wave polariza- GW Bru¨gmann et al. [56], the absolute magnitude is somewhat tions h and h to the Regge-Wheeler and Zerilli func- þ larger. tions via the expression, The increased overall magnitude of the kick becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v t X l more apparent when we plot kickð Þ in Fig. 14, where 1 ð þ 2Þ! l;m lm v t h ih ¼ ½ þ i Y ; (100) kickð Þ is defined by Eq. (104). As one can see, the largest þ r l ðeÞ ðoÞ 2 lm 2 l;m ð 2Þ!

Y −3 where 2 lm is a spin-weighted spherical harmonic. The x 10 energy radiated in gravitational waves is typically ex- pressed as 3 r2 I _ _ 2 P_ iðtÞ¼lim nijhþ ihj d; (101) r!1 16 2 n d where i is a radial unit vector and is the area element 1 on a 2-sphere. A somewhat lengthy calculation can then show that the momentum flux in the z direction is given by

(t)/dt 0 z β−β =π/2

dP 0 1 X ðl þ 2Þ! l;m P_ t ic _ l;m _ β−β =3π/4 zð Þ¼ ½ l;mðeÞ ðoÞ −1 0 16 l;m 2ðl 2Þ! β−β =7π/8 0 l ;m l ;m β−β =π l;m _ þ1 l;m _ þ1 0 d _ _ ; −2 þ lþ1;mððeÞ ðeÞ ðoÞ ðoÞ Þ (102) β−β =9π/8 0 β−β =5π/4 where cl;m ¼ 2m=½lðl þ 1Þ, and dl;m is a constant that also 0 −3 β−β =3π/2 depends upon l and m. The equations above appear in 0 several sources; these agree with those of Ruiz et al. [58] zero reference −4 [see their Eqs. (84), (11), (94), and (43), respectively]. 2700 2750 2800 2850 2900 2950 3000 In our case, however, we just treat the l ¼ 2 and t/M m ¼2 modes of the Regge-Wheeler and Zerilli func- FIG. 13 (color online). The momentum flux, P_ zðtÞ, as function tions, and the momentum flux coming from these modes = of time, for several values of 0, where 0 ¼ 215 192 is greatly simplifies. Because the m ¼2 modes are the value that gives zero kick. We also include a straight green complex conjugates of one another, we find that the mo- (light gray) dashed line at zero flux to indicate how the momen- mentum flux is tum flux varies around this point.

044035-21 DAVID A. NICHOLS AND YANBEI CHEN PHYSICAL REVIEW D 85, 044035 (2012) 0.1 0.08 v kick 0.084 sin(β−β ) 0.06 0 0.05 0.04

0.02 0

(t) β−β =π/2 0

kick 0 v β β π − =3 /4 kick

0 v β−β =7π/8 −0.05 0 −0.02 β−β =π 0 β−β =9π/8 −0.04 0 β−β =5π/4 −0.1 0 −0.06 β−β =3π/2 0 zero reference −0.08

2750 2800 2850 2900 2950 3000 0123456 β − β t/M 0

FIG. 14 (color online). The kick as a function of time, for FIG. 15 (color online). The kick calculated for several initial several different initial angles of the spin 0. There is also a values of , minus the angle that produces nearly zero kick, straight green (light gray) dashed line at zero velocity to indicate 0 ¼ 215=192. The data points are calculated from the nu- how the kick varies around this point. merical evolutions of this section, while the solid curve is a sinusoidal fit to the data. The hybrid model produces a sinusoi- dal dependence of the kick on the initial angle 0 very value of the kick is near 0.08 in dimensionless units, which precisely. is roughly a factor of 6 times larger than the estimated maximum from numerical-relativity simulations at lower We close this section by making the following observa- dimensionless spin parameters. This is largely because the tion, which may be known, though we have not seen in even-parity Zerilli function (proportional to the waveform) numerical-relativity results. Since the dependence on of is also significantly larger in amplitude than that of the kick is sinusoidal, then for each ; gives the numerical-relativity simulations. same kick and momentum flux pattern, P_ zðtÞ, just opposite Nevertheless, we then show, in this model, that the kick in sign. At the same time, though, because of the sinusoidal depends sinusoidally upon the initial orientation of the dependence there are two values that give rise to the same spins, as seen in numerical simulations by Campanelli kick in the same direction; however, the shape of the et al. v [52]. We plot the final value of the kick, kick momentum flux P_ zðtÞ is not the same for these two. One kick vz ðtfÞ, where tf is the last time in the simulation, as a can see this in Fig. 13, where the black dotted and dashed function of 0 in Fig. 15. The sinusoidal dependence curve and minus the red (gray) dotted and dashed curve in our model is exact up to numerical error. One can see give rise to the same kick; nevertheless, the pattern of the this must be the case from examining the form of our momentum flux is very different. A careful study of this expression for the momentum flux, Eq. (103). Because would reveal more about how the spins precess and would the evolution equations are not influenced by the orienta- be of some interest. tion of the spins, then the Zerilli function will be identical for different initial spin directions. The Regge-Wheeler function, however, will evolve in the same way, but be- VI. CONCLUSIONS cause the value on the matching surface is proportional to In this paper, we extended a hybrid method for head-on eiðtÞ [see Eq. (84)], the different evolutions will also mergers to treat inspiralling black-hole binaries. We differ by an overall phase, ei, where is the initial value introduced a way to include a radiation-reaction force in of the spin. Thus, when one takes the imaginary part of the hybrid method, and this led to a self-consistent set of product of the Regge-Wheeler and Zerilli functions to get equations that evolve the reduced-mass motion of the the momentum flux in Eq. (103), one will have sinusoidal binary and its gravitational waves. Using just PN and linear dependence. (In fact, we could have simply done one BHP theories, we were able to produce a full inspiral- evolution and changed the phasing as described above to merger-ringdown waveform that agrees well in phase find the above results; as a test of our method, however, we (though less well in amplitude) with those seen in full in fact performed the multiple evolutions to confirm this numerical-relativity simulations. Even though the dynam- idea.) ics during inspiral follow the modified dynamics of a point

044035-22 HYBRID METHOD FOR UNDERSTANDING BLACK-HOLE ... PHYSICAL REVIEW D 85, 044035 (2012) particle in Schwarzschild rather than the exact dynamics other hand, we would need to evolve the perturbations in of a black-hole binary, the phasing in the waveform a Kerr background. It would be simpler to choose the agrees well. Because we assume the background is a Kerr background to have the spin of the final, merged Schwarzschild black hole (rather than a Kerr, the true black hole, but one could also envision evolving pertur- remnant of black-hole binary inspirals), the merger and bations in an adiabatically changing Kerr-like background ringdown parts of the hybrid and numerical-relativity with a slowly varying mass and angular momentum pa- waveforms do not match as well. Nevertheless, the hybrid rameter that change in response to the emitted gravita- method does produce a waveform that is quite similar to tional waves. It would be of interest to see if such an that of numerical relativity. approach leads to an estimate of the spin of the final We also studied spinning black holes, particularly the black-hole similar to that proposed by Buonanno, Kidder, superkick configuration (antialigned spins in the orbital and Lehner [69]. Incorporating the PN corrections and a plane). We discussed a method to incorporate spin preces- new Hamiltonian would be the most straightforward im- sion, based on the geodetic precession of a spinning point provement, while those involving the Kerr background particle in the Schwarzschild spacetime. This caused the are technically more challenging, and computationally spins to lock to the orbital motion during the merger and more expensive. ringdown, which, in turn, helped to replicate the pattern of the momentum flux and the sinusoidal dependence of the ACKNOWLEDGMENTS merged black hole’s kick velocity seen in numerical simu- lations. Again, because the amplitude of the emitted gravi- We thank Jeandrew Brink, Tanja Hinderer, Lee tational waves does not match that of numerical-relativity Lindblom, Yasushi Mino, Mark Scheel, Bela´ Szila´gyi, simulations, the magnitude of the kick does not completely Kip Thorne, Huan Yang, and Aaron Zimmerman for dis- agree. Nevertheless, because the approximate method was cussing various aspects of this work with us. In particular, able to capture the pattern of the momentum flux, it gives we acknowledge Scheel for letting us use the waveform credence to the idea the locking of the spin-precession from the numerical-relativity simulation, Szilagyi for his frequency to the orbital frequency contributes to large input on numerical algorithms, Mino and Yang for their black-hole kicks. remarks on the validity of PN and BHP theories in the early It would be of interest to extend this approach to see if stage of this work, and Thorne for his constant encourage- it can recover the results of numerical relativity more ment and for reminding us that spin precession must lock precisely. To do this would involve a two-pronged ap- the spins to the orbital frequency. This work is supported proach: on the one hand, we would need to include higher by NSF Grants No. PHY-0653653 and No. PHY-1068881 PN terms in the metric in the interior while using a more and CAREER Grant No. PHY-0956189, and by the David accurate Hamiltonian to describe the conservative dynam- and Barbara Groce start-up fund. D. N. was supported by ics of the binary (such as the EOB Hamiltonian); on the the David and Barbara Groce Fund at Caltech.

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