Nutational resonances, transitional precession, and precession-averaged evolution in binary black-hole systems
NNutational resonances, transitional precession, and precession-averaged evolution inbinary black-hole systems
Xinyu Zhao, ∗ Michael Kesden, † and Davide Gerosa ‡ Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA TAPIR 350-17, California Institute of Technology,1200 E. California Blvd., Pasadena, California 91125, USA (Dated: August 21, 2017)In the post-Newtonian (PN) regime, the timescale on which the spins of binary black holes precessis much shorter than the radiation-reaction timescale on which the black holes inspiral to smallerseparations. On the precession timescale, the angle between the total and orbital angular momentaoscillates with nutation period τ , during which the orbital angular momentum precesses aboutthe total angular momentum by an angle α . This defines two distinct frequencies that vary onthe radiation-reaction timescale: the nutation frequency ω ≡ π/τ and the precession frequencyΩ ≡ α/τ . We use analytic solutions for generic spin precession at 2PN order to derive Fourier seriesfor the total and orbital angular momenta in which each term is a sinusoid with frequency Ω − nω forinteger n . As black holes inspiral, they can pass through nutational resonances (Ω = nω ) at whichthe total angular momentum tilts. We derive an approximate expression for this tilt angle and showthat it is usually less than 10 − radians for nutational resonances at binary separations r > M .The large tilts occurring during transitional precession (near zero total angular momentum) area consequence of such states being approximate n = 0 nutational resonances. Our new Fourierseries for the total and orbital angular momenta converge rapidly with n providing an intuitive andcomputationally efficient approach to understanding generic precession that may facilitate futurecalculations of gravitational waveforms in the PN regime. I. INTRODUCTION
The discovery of gravitational waves (GWs) emitted bybinary black holes (BBHs) [1–3] provides powerful moti-vation to better understand the generic behavior of suchsystems. BBH mergers can be divided into three stages:the inspiral during which the BBHs approach each otheras their orbit decays due to radiation reaction, the mergerproper in which the two BBH event horizons coalesce intothe single horizon of the final black hole, and the ring-down in which the final black hole settles down to anunperturbed Kerr solution describing an isolated spin-ning black hole [4]. The evolution during each of thesethree stages is best described by a different numericaltechnique. The post-Newtonian (PN) approximation pi-oneered by Einstein himself [5] works well during the in-spiral stage when the binary separation r is much greaterthan than the gravitational radius r g ≡ GM/c , where M = m + m is the sum of the BBH masses m i , G isNewton’s gravitational constant, and c is the speed oflight. Numerical relativity [6–8] is required to describethe final orbits and merger proper, while black-hole per-turbation theory [9–11] provides a good description of thelate ringdown when the spacetime is close to the Kerr so-lution describing the final black hole.This paper will focus on the inspiral stage of the mergerat binary separations r (cid:29) r g for which the PN approxi-mation is valid. This stage is important for several rea- ∗ [email protected] † [email protected] ‡ Einstein Fellow; [email protected] sons. BBHs with M (cid:46) M (cid:12) [such as the system respon-sible for GW151226 [2], the second detection by the LaserInterferometer Gravitational-wave Observatory (LIGO)]are well described by a PN inspiral when emitting GWsat the lower end of the LIGO sensitivity band. Althoughthe PN regime does not fall within the LIGO band formore massive systems like the one responsible for the firstLIGO detection GW150914 [1], in the future such sys-tems may be detectable in the PN regime at lower GWfrequencies by space-based observatories such as LISA[12]. Finally, the PN approximation is essential for evolv-ing BBHs from the wide separations at which they formto the smaller separations at which they emit detectableGWs [13–15]. This evolution is required for efforts to useBBH spins to distinguish between different astrophysicalmodels of BBH formation [16–21].In the PN regime, BBHs evolve on three distincttimescales: t orb ≡ (cid:18) r GM (cid:19) / , (1a) t pre ≡ c r / ( GM ) / = (cid:18) rr g (cid:19) t orb , (1b) t RR ≡ c r ( GM ) = (cid:18) rr g (cid:19) / t orb , (1c)where the direction of the binary separation vector r changes on the orbital timescale t orb , the directionsof the BBH spins S i and orbital angular momentum L change on the precession timescale t pre , and thebinary separation r shrinks on the radiation-reactiontimescale t RR . The validity of the PN approximation a r X i v : . [ g r- q c ] A ug ( r (cid:29) r g ) implies that these timescales obey the hier-archy t orb (cid:28) t pre (cid:28) t RR . This hierarchy suggests thatBBH dynamics can be understood through a multi-timescale analysis: the evolution on a given timescalecan be solved by holding constant quantities evolving onlonger timescales and time-averaging quantities evolvingon shorter timescales.In the case of BBH evolution, a multi-timescale analy-sis requires two different kinds of averaging: using Ke-plerian or higher PN-order solutions to the two-bodyproblem to orbit average when considering evolution onthe precession or radiation-reaction timescales, and usingPN solutions to the spin-precession equations to preces-sion average when considering evolution on the radiation-reaction timescale. Orbit averaging using either circu-lar or eccentric Keplerian orbits has been employed inmany previous studies of solutions to the spin-precessionequations [22–24]. In previous work [13, 14], we derivedanalytic solutions to the 2PN spin-precession equations,allowing us to precession average BBH dynamics at thisPN order for the first time. This precession averaging hasled to a vast increase in computational efficiency whenevolving BBH spins on the radiation-reaction timescaleas binaries inspiral from wide separations into the LIGOband. Readers can take advantage of these computa-tional savings by using the publicly available python module precession [15].In this paper, we make further use of precession av-eraging to derive a new series expansion for the genericevolution of the orbital angular momentum L on the pre-cession timescale. This expansion is highly analogous toa Fourier series, with amplitudes and frequencies varyingon the longer radiation-reaction timescale. This analysisis complicated by the fact that precession exhibits twodistinct frequencies. The nutation frequency ω ≡ π/τ is the frequency with which the angle θ L between L andthe total angular momentum J oscillates, where τ is theperiod of these oscillations. The precession frequencyΩ ≡ α/τ is the average rate at which L precesses in acone about J , where α is the precession angle over thenutation period τ . Each term in our series expansion cor-responds to simple precession of a vector in the plane per-pendicular to the precession-averaged (cid:104) J (cid:105) , with the mag-nitude of each vector fixed on the precession timescaleand the precession frequency given by Ω − nω for inte-ger n . The magnitude of the component of L parallel to (cid:104) J (cid:105) is chosen to maintain the proper normalization of L .This expansion converges rapidly with n , implying thatit may be useful in the construction of frequency-domainwaveforms for the inspiral portion of BBH mergers. Ouranalytic solutions to the spin-precession equations havealready been used for waveform construction in recentwork [25, 26]. We hope that the new precession-averagedexpansions for L and J developed later in this paper willbe similarly useful, as variation in the direction of J is amajor source of error for these efforts.Our new series expansion has also revealed the ex-istence of nutational resonances where Ω = nω . At such resonances, the precession-averaged rate (cid:104) d J /dt (cid:105) atwhich the total angular momentum is radiated is mis-aligned with (cid:104) J (cid:105) , implying that (cid:104) J (cid:105) is tilting on theprecession timescale. Although the resonance conditionΩ = nω is finely tuned at any given binary separation r ,generic BBHs often cross resonances as they inspiral fromwide separations towards merger. We derive approximateexpressions for the angle θ tilt through which (cid:104) J (cid:105) tilts ata resonance and show that such tilts are usually below10 − radians making them negligible for the purpose ofGW data analysis. An exception is the large tilts thatoccur during transition precession [22], which can be in-terpreted as an approximate n = 0 nutational resonancein much of the parameter space with near-vanishing totalangular momentum ( J (cid:39) L on the precession timescale. We showthat only a few terms in this expansion with the low-est values of | n | are required to produce excellent agree-ment with full numerical solutions of the orbit-averagedspin-precession equations, and explore the implicationsof this expansion for the evolution of the total angularmomentum J . In Section IV, we show that (cid:104) J (cid:105) tilts atnutational resonances where Ω = nω and derive an ap-proximate expression for the tilt angle θ tilt that we verifyagrees well with the tilts observed in full numerical so-lutions of the orbit-averaged spin-precession equations.In Section V, we examine how often generic binaries en-counter nutational resonances during their inspirals andthe distribution of tilt angles at these resonances. InSec. VI, we explore the connection between our newlydiscovered nutational resonances and transitional preces-sion near J (cid:39) G = c = 1. II. REVIEW OF SPIN PRECESSION
Consider binary black holes on a quasicircular orbitwith masses m and m , mass ratio q ≡ m /m ≤ M ≡ m + m , and symmetric mass ratio η ≡ m m /M = q/ (1 + q ) . Such a system will havean orbital angular momentum L with magnitude L = η ( M r ) / to lowest PN order and spins S i with mag-nitudes S i = χ i m i , where the dimensionless spins havemagnitudes 0 ≤ χ i ≤
1. The total spin S = S + S hasmagnitude S , and the total angular momentum J = L + S has magnitude J . Each of these quantities is either con-stant or evolves on one of the timescales given by Eq. (1).At the PN order we consider in this paper, the masses m i and dimensionless spin magnitudes χ i are constantthroughout the inspiral. The projected effective spin ξ ≡ M (cid:20) (1 + q ) S + (cid:18) q (cid:19) S (cid:21) · ˆ L , (2)referred to as χ eff in LIGO parameter estimation, is con-stant on the precession timescale t pre [27, 28] and is alsoconstant throughout the inspiral to the PN order weconsider. The magnitudes L and J of the orbital andtotal angular momenta evolve on the radiation-reactiontimescale t RR . It is sometimes convenient to define anadditional quantity κ ≡ J − L L = S · ˆL + S L , (3)because the limitlim r →∞ κ ≡ κ ∞ = S cos θ ∞ + S cos θ ∞ (4)is a finite constant that can be used to label BBHsthroughout their inspiral. In this expression, θ i ∞ is theangle between S i and L in the limit r → ∞ ; this an-gle is a constant since in this limit spin-orbit couplingdominates over spin-spin coupling and the two spins S i simply precess about the orbital angular momentum L .The total spin S , as well as the directions of S i , L , and J all evolve on the precession timescale t pre .This last point is somewhat subtle, since in the ab-sence of gravitational radiation, the magnitude and di-rection of the total angular momentum J are both con-served. In the case of simple precession, L and J precesson cones with opening angles θ Lz and θ J respectivelyabout a fixed direction ˆz in an inertial frame [22]. Thetimescale hierarchy t pre /t RR = ( r/r g ) − / (cid:28) θ J /θ Lz ∝ ( r/r g ) − / (cid:28)
1, but the frequency Ω withwhich L and J precess about their cones is the same andof order the inverse of the precession timescale t pre . Al-though generic spin precession is more complicated, thedirection of the total angular momentum J still evolves(by a small angle) on the precession timescale.In previous work [13, 14], we analyzed generic spin pre-cession under the approximation, valid in the absence ofradiation reaction, that the direction of J stays fixed. Inthis section, we summarize key results from that workwhich we will use in the following section where we relaxthe assumption that the direction of J stays fixed. Themany constants of motion on the precession timescalelisted above imply that there is only a single degree offreedom in the relative orientations of L , S , and S ,which we can conveniently specify by choosing the mag-nitude S of the total spin as a general coordinate. Forprecisely equal masses ( q = 1), S is constant and an al-ternative coordinate is required to specify this degree offreedom [29]. The angle θ L between L and J is given interms of S by the expressioncos θ L = J + L − S JL . (5) The hierarchy θ J (cid:28) θ Lz in the PN regime implies that θ Lz (cid:39) θ L to high accuracy. The relative orientationof S , L , and J in terms of this angle are shown inFig. 1. The total spin magnitude S oscillates in the range S − ≤ S ≤ S + , where the extrema S ± are the roots of theequation ξ = ξ ± ( S ), ξ is the projected effective spin givenby Eq. (2), and the two curves ξ ± ( S ) = { ( J − L − S )[ S (1 + q ) − ( S − S )(1 − q )] ± (1 − q ) A A A A } / (4 qM S L ) (6)form a closed loop we called the effective potential forspin precession. In this expression, we have used fourauxiliary functions A i which are defined as A ≡ [ J − ( L − S ) ] / , (7a) A ≡ [( L + S ) − J ] / , (7b) A ≡ [ S − ( S − S ) ] / , (7c) A ≡ [( S + S ) − S ] / . (7d)Fig. 1 shows that the oscillations of S correspond tonutation of the orbital angular momentum L , allowingus to define the nutation period τ = 2 (cid:90) S + S − dS | dS/dt | . (8)and nutation frequency ω ≡ π/τ . Note that in our ear-lier work [13–15], we referred to τ as the precession periodbecause we were focused on the relative orientations ofthe BBH spins and it has the precession timescale. Thenutation frequency only depends on quantities varyingon the radiation-reaction timescale. The time derivativeof the total spin magnitude S is dSdt = − − q )2 q S S S ( η M ) L (cid:18) − ηM ξL (cid:19) × sin θ sin θ sin ∆Φ , (9)where the angles θ i between L and S i are given bycos θ = 12(1 − q ) S (cid:20) J − L − S L − qM ξ q (cid:21) , (10a)cos θ = q − q ) S (cid:20) − J − L − S L + 2 M ξ q (cid:21) . (10b)The angle ∆Φ between the projections of S and S or-thogonal to L is given bycos ∆Φ = cos θ − cos θ cos θ sin θ sin θ , (11)where cos θ = S − S − S S S (12)is the cosine of the angle between S and S . L J = L + S ✓ L S = S + S ✓ J ✓ Lz S S ˆx ˆyˆx ˆy ˆz = ˆz FIG. 1. References frames useful for describing BBH spinprecession. The total spin S is the sum of the spins S and S of the more massive and less massive black holes. Thetotal angular momentum J is the sum of the orbital angularmomentum L and total spin S , and θ L is the angle between L and J . We define the xyz inertial frame such that ˆz pointsin the direction of the precession-averaged orbital and totalangular momenta (cid:104) L (cid:105) and (cid:104) J (cid:105) after many precession cycles.The basis vectors ˆx and ˆy complete the orthonormal triad.The angles between ˆz and L and J are given by θ Lz and θ J respectively; the hierarchy θ J (cid:28) θ Lz implies that θ Lz (cid:39) θ L .After a nutation period τ , L and J precess about ˆz by anangle α . We use these quantities to define the x (cid:48) y (cid:48) z (cid:48) rotatingframe in which ˆz (cid:48) = ˆz and ˆx (cid:48) and ˆy (cid:48) precess about ˆz withprecession frequency Ω = α/τ . Although L , S , and S return to their initial relativeorientation after a nutation period τ , in an inertial framethese vectors precess about ˆz by an angle α = 2 (cid:90) S + S − Ω z dS | dS/dt | , (13)whereΩ z = J (cid:18) η M L (cid:19) (cid:26) η (cid:18) − ηM ξL (cid:19) − q )2 qA A (cid:18) − ηM ξL (cid:19) [4(1 − q ) L ( S − S ) − (1 + q )( J − L − S )( J − L − S − ηM Lξ )] (cid:27) (14)is the instantaneous precession frequency. Note that inour earlier work, we identified ˆz with the instantaneous direction of the total angular momentum J rather thanits precession average (cid:104) J (cid:105) , because we were neglecting thesmall changes to the direction of J compared to that of L ( θ J (cid:28) θ L ). These results allow us to define the averageprecession frequency Ω ≡ α/τ . Although the nutationfrequency ω and precession frequency Ω are both of orderthe inverse precession timescale t pre , they generally differbecause α (cid:54) = 2 π . As shown in Fig. 1, we can definean orthonormal basis for our inertial frame by choosingvectors ˆx and ˆy perpendicular to ˆz . We can also definea frame rotating about ˆz = ˆz (cid:48) with precession frequencyΩ with rotating basis vectors ˆx (cid:48) = ˆx cos Ω t + ˆy sin Ω t , (15a) ˆy (cid:48) = − ˆx sin Ω t + ˆy cos Ω t . (15b)In the quadrupole approximation, GW emission re-moves angular momentum from the binary at a rate [30] d J dt = − (cid:16) rM (cid:17) − η L M . (16)The 1PN correction to this expression is also parallel tothe orbital angular momentum L [23]. This expressionimplies that the magnitudes of L and J evolve accordingto the equations dLdt = d J dt · ˆL = − (cid:16) rM (cid:17) − ηLM , (17a) dJdt = d J dt · ˆJ = dLdt cos θ L . (17b)This expression for dL/dt evolves on the radiation-reaction timescale, but the expression for dJ/dt evolveson the precession timescale because of the angular termcos θ L given by Eq. (5). We can precession average theright-hand side of Eq. (17b) using (cid:104) cos θ L (cid:105) = 2 τ (cid:90) S + S − cos θ L dS | dS/dt | (18)to obtain the precession-averaged loss of total angularmomentum (cid:104) dJ/dt (cid:105) = ( dL/dt ) (cid:104) cos θ L (cid:105) [13, 14]. Thisequation and Eq. (17a) can be numerically integratedwith a time step on the radiation-reaction timescale, pro-viding a vast savings in computational time compared toa time step on the precession timescale if one is only inter-ested in the relative orientations of L , S , and S speci-fied by Eqs. (10) and (11) [14, 15]. However, to determinethe directions of the vectors L and J in an inertial frame(perhaps for the purpose of calculating the emission ofGWs), one must integrate the instantaneous precessionfrequency Ω z given by Eq. (14) with a time step on theprecession timescale. In the next section, we derive newseries expansions for L and J in terms of quantities thatonly evolve on the radiation-reaction timescale which canin principle achieve similar computational savings to ourearlier expression for (cid:104) dJ/dt (cid:105) . III. A NEW EXPANSION
In the inertial (unprimed) frame defined in the previ-ous section, we can decompose the orbital angular mo-mentum L = L (cid:107) ˆz + L ⊥ (19)into components parallel and perpendicular to the di-rection ˆz of its precession average (cid:104) L (cid:105) . Without loss ofgenerality, we can choose L to lie in the xz plane at t = 0with total spin magnitude S = S − . With this choice, theperpendicular component of L is given by L ⊥ = L sin θ L (cos Φ L ˆx + sin Φ L ˆy ) . (20)The total spin magnitude S ( t ) is an even function of timewith period τ , implying that it is fully specified by itsvalues in the interval 0 ≤ t ≤ τ /
2. On this interval, S ( t )is the inverse of the function t ( S ) = (cid:90) SS − dS (cid:48) | dS/dt | , (21)where dS/dt is given by Eq. (9) and S − ≤ S ≤ S + .Eq. (5) indicates that θ L is similarly a periodic, evenfunction of time, while Eq. (14) requires Φ L ( t ) to be aperiodic, odd function of time defined by its valuesΦ L ( t ) = (cid:90) t Ω z dt (cid:48) = (cid:90) SS − Ω z dS (cid:48) | dS/dt | (22)in the interval 0 ≤ t ≤ τ /
2. The symmetry and peri-odicity of θ L and Φ L imply that we can Fourier expandthe perpendicular component of L in the rotating framegiven by Eq. (15) to obtain the series L ⊥ ( t ) = L + ∞ (cid:88) n =0 [ θ (cid:48) Lxn cos( nωt ) ˆx (cid:48) + θ (cid:48) Lyn sin( nωt ) ˆy (cid:48) ] . (23)Comparing Eqs. (20) and (23) and using Eq. (15) to re-late the rotating and inertial frames, we see that θ (cid:48) Lxn = 2 − δ n Lτ (cid:90) τ L ⊥ · ˆx (cid:48) cos( nωt ) dt = 2 τ (cid:90) S + S − cos(Φ L − Ω t ) sin θ L cos( nωt ) dS | dS/dt | (24)and θ (cid:48) Lyn = 2 − δ n Lτ (cid:90) τ L ⊥ · ˆy (cid:48) sin( nωt ) dt = 4 τ (cid:90) S + S − sin(Φ L − Ω t ) sin θ L sin( nωt ) dS | dS/dt | , (25) where the Kronecker delta δ ij equals unity for i = j andzero otherwise. We can use Eqs. (15) and (23) to obtainan equivalent series for L ⊥ in the inertial frame, L ⊥ ( t ) = L + ∞ (cid:88) n = −∞ θ Ln { cos[(Ω − nω ) t ] ˆx + sin[(Ω − nω ) t ] ˆy } , (26)where θ Ln = 1 + δ n θ (cid:48) Lx | n | − sgn( n ) θ (cid:48) Ly | n | ]= 2 τ (cid:90) S + S − cos(Φ L − Ω t + nωt ) sin θ L dS | dS/dt | . (27)One can obtain L from Eqs. (19) and (26) by recognizingthat the magnitude of L is conserved on the precessiontimescale implying that L (cid:107) = (cid:112) L − L ⊥ · L ⊥ . (28)Eq. (26) is an elegant expression for L ⊥ ; each termcorresponds to a vector with magnitude L | θ Ln | tracingout a circle with frequency Ω − nω in the plane orthog-onal to ˆz , the direction of the precession-averaged or-bital angular momentum (cid:104) L (cid:105) . These magnitudes andfrequencies are both evolving on the radiation-reactiontimescale t RR , implying that they can be numericallyevaluated throughout the inspiral with a time step oforder t RR leading to potentially large computational sav-ings. Eq. (26) also seems well suited for Fourier transfor-mation if one is interested in functions in the frequencydomain for GW analysis. We test its validity by compar-ing it to numerical integration of the full spin-precessionequations. We show this comparison in Fig. 2, includ-ing only the n = 0, ±
1, and ± − nω ) t ofthe sinusoids in Eq. (26) by the phases ψ n ( t ) = (cid:90) t (Ω − nω ) dt (cid:48) . (29)We see excellent agreement between our new precession-averaged series expansion in Eq. (26) and the tradi-tional numerical solutions of the orbit-averaged preces-sion equations, shown respectively by the green dashedand solid blue curves. The z -component of L (in the di-rection of its precession-averaged value) calculated in thetwo approaches agrees to a part in 10 , while residualsfor the perpendicular component L ⊥ grow to about the1% level by the time the binary inspirals from 600 M to500 M . These residuals result from numerical error inthe phasing given by Eq. (29); the neglected terms with | n | ≥ r/M . . . Residuals | ∆ L z | /M . . . . . . . L z / M q = 0 . χ = 0 . χ = 0 . ξ = 0 . κ ∞ = 0 . M Orbit − averagedPrecession − averaged 510512514516518520 r/M . . . . . . .
52 5805825845865885905 . . . . . . . r/M . . . Residuals | ∆ L y | /M − . − . − . − . . . . . . L y / M r/M − . − . − . . . . . − . − . − . . . . . r/M . . . Residuals | ∆ L x | /M − . − . − . − . . . . . . L x / M r/M − . − . − . . . . . − . − . − . . . . . FIG. 2. Comparison of the orbital angular momentum L ( t ) determined from a numerical integration of the orbit-averagedspin-precession equations and our new precession-averaged series expansion given by Eq. (26). The binary has mass ratio q = 0 .
7, dimensionless spin magnitudes χ = 0 . χ = 0 .
8, projected effective spin ξ = 0 .
3, and asymptotic projected totalspin κ ∞ = 0 . M . It inspirals from binary separation r = 600 M to r = 500 M . The top, middle, and bottom panels showthe components of L in an inertial frame in which ˆz points in the direction of (cid:104) L (cid:105) , the precession-averaged orbital angularmomentum. The solid blue curves show the orbit-averaged solution, the dashed green curves show our new precession-averagedsolution, and the red curves below each panel show the magnitude of the differences between the solutions. We have only usedthe five terms corresponding to n = 0 , ± , ± r = 590 M to r = 580 M and r = 520 M to r = 510 M . terms are the n = − n = 0 terms in the expansionof Eq. (26), but for a discrete interval between r = 600 M and r = 500 M , the two dominant terms are instead n = 0and n = +1. This results not from the continuous evo-lution of the coefficients θ Ln on the radiation-reactiontimescale, but from two discontinuities. At two pointsduring the inspiral from r = 600 M to r = 500 M , themagnitudes of the orbital and total angular momentum L and J attain values such that L and J are instanta-neously aligned once per nutation period at S = S − .This alignment implies that α , the angle by which L pre-cesses about J over a nutation period, cannot be defined[31]. This is purely a coordinate issue, analogous to theinability to define the total change in longitude on a tripthat passes directly over the North Pole. When an in-spiraling binary passes through values of L , J , and ξ forwhich alignment between L and J is possible, α changesdiscontinuously by ± π implying that the precession fre-quency Ω = α/τ changes discontinuously by the nutationfrequency ω = 2 π/τ . A shift Ω → Ω (cid:48) = Ω ± ω leads toa shift θ Ln → θ (cid:48) Ln = θ L,n ∓ according to Eq. (27). Thisshift will leave the infinite summation in Eq. (26) un-changed, merely relabeling the individual terms. Suchshifts occur twice during the inspiral from r = 600 M to r = 500 M of the binary shown in Fig. 2; α first increasesby 2 π , shifting the dominant terms from n = {− , } to n = { , +1 } , then decreases by 2 π , restoring n = {− , } as the dominant terms. The summation of the five terms n = { , ± , ± } shown in Fig. 2 always includes the twodominant terms and thus leaves no observable disconti-nuities in L .We show the five largest coefficients θ Ln for n =0 , ± , ± r = 10 M to 10 M in Fig. 3. The parameters for thisbinary, listed in the caption to the figure, were chosensuch that the binary passes through a nutational res-onance at r (cid:39) M . Such nutational resonances arethe focus of Sections IV and V; the same binary is alsoshown in Figs. 4 and 5. This binary differs from the oneshown in Fig. 2 in that it does not pass through any dis-continuities in α , but shares the common feature thatthe n = − , L can bemodeled to ∼
1% accuracy using just the two dominantterms in Eq. (26) whose coefficients vary smoothly onthe radiation-reaction timescale. This suggests that pre-cession averaging can provide computational savings forthe evolution of L during an inspiral similar to those ob-tained for the evolution of the total angular momentum J demonstrated in our previous work [13, 14].Our new expansion in Eq. (26) can also be used tocalculate the evolution of the total angular momentum J ( t ) in our inertial xyz frame. If the rate at which angularmomentum is radiated is related to the orbital angular r/M − − − − − − − | θ L n | n = − n = − n = 0 n = 1 n = 2 q = 0 . χ = 1 χ = 1 ξ = − . κ ∞ = − . M FIG. 3. The magnitudes of the five largest coefficients θ Ln in the series expansion of Eq. (26) as a function of binaryseparation r for a binary with mass ratio q = 0 .
5, maximaldimensionless spins χ = χ = 1, projected effective spin ξ = − .
33, and asymptotic projected total spin κ ∞ = − . M .The n = 0 and n = − n = +1 , +2 , − r = 10 M . momentum by Eq. (16), our expansion implies that d J ⊥ dt = − (cid:16) rM (cid:17) − η L ⊥ M = dLdt + ∞ (cid:88) n = −∞ θ Ln { cos[(Ω − nω ) t ] ˆx + sin[(Ω − nω ) t ] ˆy } . (30)If we integrate this expression on the precessiontimescale, holding fixed the amplitudes and frequenciesvarying on the longer radiation-reaction timescale, wefind a similar expansion for the perpendicular componentof the total angular momentum, J ⊥ ( t ) = J + ∞ (cid:88) n = −∞ θ Jn { sin[(Ω − nω ) t ] ˆx − cos[(Ω − nω ) t ] ˆy } , (31)where the coefficients in the two expansions of Eqs. (26)and (31) are proportional to each other: θ Jn θ Ln = 1 J dLdt (cid:18) − nω (cid:19) ∝ t pre t RR ∝ (cid:16) rM (cid:17) − / . (32)This agrees with the earlier finding that for simple pre-cession, the total angular momentum J precesses abouta cone with opening angle θ J ∝ ( r/M ) − much less thanthe opening angle θ L ∝ ( r/M ) − / of the cone aboutwhich the orbital angular momentum L precesses [22].Eq. (32) reveals that θ Jn diverges for Ω = nω , mathemat-ically equivalent to α = 2 πn from our definitions of theprecession and nutation frequencies in Sec. II. This con-dition, which we call a nutational resonance, has poten-tially profound implications for the evolution of J whichwe explore in the next section. IV. NUTATIONAL RESONANCES
At a nutational resonance, the arguments of the sinu-soids in the n = Ω /ω term in Eq. (30) vanish, implyingthat this term corresponds to constant emission of an-gular momentum in the x direction. This emission willcause the precession-averaged total angular momentum (cid:104) J (cid:105) to tilt towards the x axis and away from its initialdirection which defined the z axis. This tilting behaviorwill not continue indefinitely, because the precession fre-quency Ω and nutation frequency ω are both evolving onthe radiation-reaction timescale t RR . A generic binarywill not be in a nutational resonance (Ω /ω will not be aninteger), but as it inspirals towards merger it may passthrough one or more of such resonances. At each passagethrough a nutational resonance, the precession-averagedtotal angular momentum (cid:104) J (cid:105) will tilt by some angle θ tilt ,providing a randomly oriented “kick” of magnitude Jθ tilt to J ⊥ in an inertial frame. These kicks will accumulatethroughout the inspiral causing (cid:104) J (cid:105) to random walk awayfrom its initial direction at large separations set by bi-nary formation. Whether these tilts are astrophysicallyrelevant or lead to detectable GW signatures dependson both the magnitudes of the tilt angles θ tilt and thefrequency with which binaries encounter nutational res-onances. We will derive an analytic estimate of the tiltangle θ tilt in this section, then use this estimate to ex-plore the distribution of tilt angles as a function of binaryparameters in Sec. V.We show an example of BBHs passing through a n = 1nutational resonance in Fig. 4. We integrate Eq. (30)numerically backwards and forwards in time from theresonance at r (cid:39) M , defining the z axis to pointin the direction of the precession-averaged total angu-lar momentum (cid:104) J (cid:105) at this binary separation. We showthe dominant non-resonant n = 0 and n = − n = 1 term is shown by the green curve. On the preces-sion timescale, the non-resonant n = 0 and n = − xy plane with radii J | θ J | and J | θ J, − | and frequencies Ω and Ω + ω , consistent with the expan-sion for J ⊥ given in Eq. (31). On the radiation-reactiontimescale, these curves spiral outwards as θ Jn increasein magnitude as the binary separation r decreases from700 M to 270 M .The resonant n = 1 term exhibits qualitatively dif-ferent behavior, in addition to being much smaller in magnitude consistent with the hierarchy of coefficientsshown in Fig. 3. At large separations, where the preces-sion frequency Ω and nutation frequency ω have not quiteachieved resonance, the n = 1 term precesses in smallcircles with radii J | θ J | and very small frequency Ω − ω .This is shown in the top left corner of the bottom leftpanel of Fig. 4. As the binary approaches resonance, theangular momentum loss due to this term comes to pointin a fixed direction on the precession timescale (along the x axis). Next, the binary passes through resonance whenthe green curve reaches the origin at J x = J y = 0. Fi-nally, the n = 1 term resumes precession with frequencyΩ − ω (now negative) along circles with radii J | θ J | asshown in the bottom right corner of the bottom left panelof Fig. 4. The axes about which the n = 1 term precessesbefore and after resonance are displaced with respectto each other, corresponding to a tilt in the precession-averaged total angular momentum (cid:104) J (cid:105) .We can estimate the magnitude of this tilt by Taylorexpanding the resonant term in Eq. (30) about the res-onance and integrating analytically. We begin with thefrequency of the resonant term ,Ω − nω (cid:39) (cid:20) d (Ω − nω ) dL (cid:21) ( L − L ) = sD t , (33)where the total derivative of the frequency with respectto the magnitude of the orbital angular momentum isevaluated at resonance where L = L . In this expression,we have also defined the binary to pass through resonanceat t = 0 and two constants s ≡ sgn (cid:18) dαdL dLdt τ (cid:19) , (34) D ≡ (cid:12)(cid:12)(cid:12)(cid:12) dαdL dLdt τ (cid:12)(cid:12)(cid:12)(cid:12) / ∝ √ t RR t pre ∝ (cid:16) rM (cid:17) − / . (35)Eqs. (33) and (29) imply that the phase near resonanceis given by ψ n = (cid:90) t (Ω − nω ) dt (cid:48) (cid:39) sD t . (36)Inserting this phase into the arguments of the sinusoidsof the resonant term in Eq (30), we find that d J ⊥ n dt = dLdt θ Ln (cid:20) cos (cid:18) D t (cid:19) ˆx + s sin (cid:18) D t (cid:19) ˆy (cid:21) (37)Integrating Eq. (37) leads to J ⊥ n = √ D dLdt θ Ln (cid:20) C (cid:18) Dt √ (cid:19) ˆx + s S (cid:18) Dt √ (cid:19) ˆy (cid:21) (38)where C ( x ) and S ( x ) are the Fresnel integrals C ( x ) ≡ (cid:90) x cos t dt , (39a) S ( x ) ≡ (cid:90) x sin t dt . (39b) − . − . . . . J x /M [10 − ] − . − . . . . J y / M [ − ] − . − . − .
25 0 .
00 0 .
25 0 .
50 0 . J x /M [10 − ] − . − . − . . . . . J y / M [ − ] q = 0 . χ = 1 χ = 1 ξ = − . κ ∞ = − . M r/M = 700 → − . − . . . . J x /M [10 − ] − . − . . . . J y / M [ − ] . . . . . J x /M [10 − ] . . . . . J y / M [ − ] . . . . J x /M [10 − ] . . . . J y / M [ − ] n = 0 n = − n = 1 FIG. 4. The evolution of J ⊥ , the component of the total angular momentum in the xy plane, as binary black holes with massratio q = 0 .
5, maximal dimensionless spins χ = χ = 1, and projected effective spin ξ = − .
33 inspiral from r = 700 M to r = 270 M . The binary encounters a n = 1 nutational resonance (Ω = ω ) at r (cid:39) M when the magnitude of the total angularmomentum J (cid:39) . M . The direction of J at this point in the inspiral defines the z axis. The red, blue, and green curvesshow numerical integration of the n = { , − , +1 } terms respectively in the expansion of Eq. (30). The top left panel showsthe largest view, while the top right and middle right panels show two insets to this figure. The bottom left and bottom rightpanels show the two insets to the top right panel. The bottom left panel most clearly shows that the resonant n = 1 termshown by the green curve is tilted as it passes through resonance. − − − − J x /M [10 − ] − − − J y / M [ − ] q = 0 . χ = 1 χ = 1 ξ = − . κ ∞ = − . M r/M = 700 → Numerical integrationAnalytical approximation
FIG. 5. A comparison between numerical integration of theresonant term in Eq. (30) for the nutational resonance de-picted in Fig. 4 and our analytical approximation given byEq. (38). The agreement is excellent; the symmetric Eulerspiral shown by the dashed orange curve nearly perfectly de-scribes the numerical integration shown by the solid greencurve despite the significant changes in L and J as the binaryinspirals from r = 700 M to r = 270 M . Eq. (38) indicates that the resonant term J ⊥ n can beapproximated as an Euler spiral. We compare this Eulerspiral to a numerical integration of the resonant term inEq. (30) in Fig. 5.The Fresnel integrals have limiting valueslim x →±∞ C ( x ) , S ( x ) = ± (cid:112) π/ J ⊥ n ≡ J ⊥ n ( ∞ ) − J ⊥ n ( −∞ ) (41)in the precession-averaged total angular momentum rel-ative to its direction at resonance as a binary passesthrough a nutational resonance. This in turn impliesthat J tilts by an angle θ tilt = | ∆ J ⊥ n | J = (2 π ) / JD dLdt θ Ln ∝ (cid:18) t pre t RR (cid:19) / θ Ln ∝ (cid:16) rM (cid:17) − / . (42)For the nutational resonance shown in Fig. 5, the totalshift ∆ J ⊥ n predicted by Eq. (41) agrees with the nu-merical result obtained by integrating Eq. (30) to betterthan 1%. This justifies our use of Eq. (42) in the nextsection to estimate how the precession-averaged total an-gular momentum (cid:104) J (cid:105) tilts as BBHs encounter nutationalresonances during their inspirals. V. DISTRIBUTION OF NUTATIONALRESONANCES
In this section, we investigate how often BBHs en-counter nutational resonances as they inspiral towards merger from the large separations at which they form.As the condition α = 2 πn for integer n defines a nu-tational resonance, we begin by calculating α accordingto Eq. (13). Although the parameter space of all BBHswith given masses, spin magnitudes, and binary separa-tion is four dimensional (corresponding to the two BBHspin directions), two of these dimensions can be speci-fied by a global rotation of the system about J and theprecessional phase, neither of which affect α which varieson the radiation-reaction timescale. For these BBHs (forwhich L is fixed), α is purely function of J and ξ forallowed values of these parameters. We show a contourplot of α for these allowed values in Fig. 6, where the con-tour lines α = 2 πn identify nutational resonances. Thelargest allowed value of the magnitude of the total angu-lar momentum J is J max = L + S + S and occurs forthe “up-up” configuration in which both spins S and S are aligned with the orbital angular momentum L .Since L > S + S for these BBH masses and spins,the smallest allowed value of J is J min = L − S − S and occurs for the “down-down” configuration in which S and S are anti-aligned with L . The boundaries ofthe allowed region in the J − ξ plane are defined by twopaths connecting the “up-up” and “down-down” config-urations. The first of these paths, ξ max ( J ), connects themaxima of the effective potential ξ + ( S ) given by Eq. (6).This path includes the “down-up” configuration in whichthe spin S of the more massive black hole is anti-alignedwith L while the spin S of the less massive black hole isaligned. The second path ξ min ( J ) connects the minimaof the effective potential ξ − ( S ). The allowed region inFig. 6 consists of those BBHs for which J min ≤ J ≤ J max and ξ min ( J ) ≤ ξ ≤ ξ max ( J ).The n = 1 and n = 2 contours in Fig. 6 connectpoints on the ξ min ( J ) and ξ max ( J ) curves that consti-tute the boundaries of the allowed region. Because theseboundaries correspond to extrema of the effective poten-tial ξ ± ( S ) (what Schnittman [24] described as spin-orbitresonances), S does not oscillate, Ω z ( S ) given by Eq. (14)is a constant on the precession timescale, and the coef-ficients θ Ln given by Eq. (27) vanish for n (cid:54) = 0. Thetilt angle θ tilt given by Eq. (42) is proportional to θ Ln and thus must similarly vanish for n (cid:54) = 0. The n = 1and n = 2 contours in Fig. 6 are monotonic functionsof both J and ξ , so either of these quantities can beused to parametrize the curves. We show θ tilt ( J ) and θ tilt ( ξ ) in the top and right panels of Fig. 6. As ex-pected, θ tilt vanishes at the endpoints of these curves (theSchnittman spin-orbit resonances) for both nutationalresonances. The curves θ tilt ( J ) and θ tilt ( ξ ) are smoothfunctions for the n = 1 resonance, reaching a maximum θ tilt (cid:39) × − somewhere in the interior of the allowedregion. The corresponding curves for the n = 2 reso-nance show two sharp spikes where the tilt angle appearsto diverge. These spikes are artifacts of the approxima-tions used in Section IV and occur where dα/dL andthus D given by Eq. (35) vanish. Since D appears inthe denominator of Eq. (42) for θ tilt , the tilt angle cor-1 . . . . . . J/M − . − . . . . ξ q = 0 . χ = 1 χ = 1 r = 20 M up − updown − down up − downdown − up n = 1 n = 2 n = 3 α = 2 nπ J k L .
29 1 .
30 1 .
31 1 .
32 1 . . . . . . n =3 n =4 n =4 n =5 . . . . . . θ tilt [10 − ] n = 1 n = 2 n = 3 . . . . . . θ t il t [ − ] n = 1 n = 2 n = 3 FIG. 6.
Central panel:
A contour plot of the precession angle α as a function of the magnitude of the total angular momentum J and the projected effective spin ξ for BBHs with mass ratio q = 0 .
6, dimensionless spin magnitudes χ = χ = 1, and binaryseparation r = 20 M . The colored region shows the allowed values of J and ξ for these parameters, with the three cusps on theboundaries corresponding to the up-up, down-up, and down-down configurations. The thin dashed lines indicate nutationalresonances ( α = 2 πn for integer n ). The inset shows parameter space near the unstable up-down configuration where α → ∞ .The solid black lines show binaries for which J and the orbital angular momentum L are parallel at S = S − , implying that α is undefined. The value of α along contours crossing these lines changes by ± π . Top and right panels:
The tilt angle θ tilt along the α = 2 πn nutational resonance contours as functions of J and ξ respectively. The spikes on these curves correspondto parameters for which dα/dL → θ tilt → ∞ by Eqs. (35) and (42). respondingly diverges. Physically, points for which both α = 2 πn and dα/dL = 0 correspond to BBHs that arein nutational resonances and remain in these resonancesas they inspiral on the radiation-reaction timescale. Inpractice, the quadratic term in the Taylor expansion ofEq. (33) will be non-vanishing for these BBHs, implyingthat the phase ψ n given by Eq. (36) will by cubic ratherthan quadratic in t . An order-of-magnitude analysis forthese BBHs suggests that θ tilt will be proportional to( r/M ) − rather than ( r/M ) − / as in Eq. (42), implyingsomewhat larger but still finite tilts. The contours for the n > n = 3 contour beginson the ξ min ( J ) boundary, then curves up and to the rightuntil it encounters the solid black curve connecting the“up-up” and “up-down” configurations identifying thoseBBHs for which the total spin S and orbital angular mo-mentum L are aligned at S = S − . For these BBHs, thetotal angular momentum J = L + S is also aligned with L implying that α is undefined as was previously dis-cussed in Section III [31]. Crossing this solid black curvecauses α to change discontinuously by 2 π , transforming2our n = 3 contour into an n = 4 contour, another nuta-tional resonance. For these BBH masses and spins, the“up-down” configuration defining one endpoint of the α discontinuity curve lies in the interior rather than on the ξ min ( J ) boundary of the allowed region in the J − ξ plane.This occurs for binary separations r ud − < r < r ud+ ,where the limits r ud ± = ( √ χ ± √ qχ ) (1 − q ) M (43)define the range for which the “up-down” configurationis unstable to precession to large spin misalignments [32].For these unstable “up-down” configurations, the nu-tation period τ is infinite, just as it will take an in-finite amount of time for a particle moving in a one-dimensional potential to reach a local maximum (unsta-ble equilibrium point) which it has just enough energy toaccess. Since the precession frequency Ω remains finiteas one approaches the unstable “up-down” configurationwhile the nutation period τ diverges, the precession angle α = Ω τ also becomes infinite. This implies that as oneapproaches the point in the J − ξ plane corresponding tothe unstable “up-down” configuration, one will encounternutational resonances α = 2 πn for arbitrarily large val-ues of n . This is what we see in the inset to the centralpanel of Fig. 6: the contour lines corresponding to nuta-tional resonances spiral inwards towards the “up-down”configuration, with n increasing by an integer each timethe solid line marking the α discontinuity is crossed. Al-though n diverges along this spiral, the tilt angle θ tilt approaches zero because the BBH spends an increasingfraction of the nutation period with L closely alignedwith J , implying little tilt in (cid:104) J (cid:105) for radiation reactiondescribed by the quadrupole formula of Eq. (16).Now that we understand which BBHs are in nutationalresonances at a given binary separation (for example, r = 20 M in Fig. 6), we can examine when BBHs en-counter these resonances as their separation decreases asthey inspiral towards merger. In Fig. 7, we show α ( r ) for30 BBHs (10 each for mass ratios q = 0 .
8, 0.6, and 0.2)with randomly oriented maximal spins as they inspiralfrom r = 10 M to a final separation r = 10 M . At largeseparations, we see that the precession angles α asymp-tote to one of two different values for each of the threemass ratios; these asymptotic values are shown by thedashed black lines in Fig. 7. This surprising result canbe understood by recognizing that the lower PN orderspin-orbit coupling dominates over the high-order spin-spin coupling in the limit r → ∞ . In this limit, the anglesbetween the orbital angular momentum L and the BBHspins S and S are fixed to their asymptotic values θ ∞ and θ ∞ , L and the total angular momentum J are bothnearly aligned with the z axis, and the two spins precessabout this axis with respective frequencies [22, 23]Ω = (4 + 3 q ) η M (cid:16) rM (cid:17) − / , (44a)Ω = (4 + 3 /q ) η M (cid:16) rM (cid:17) − / . (44b) r/M π π π π π π π π π π π π π π α q = 0 . q = 0 . q = 0 . χ = 1 χ = 1 FIG. 7. The angle α by which the orbital angular momentum L and total angular momentum J precess about the z axisduring a nutation period τ as a function of binary separation r for 30 different binaries with randomly oriented maximalspins ( χ i = 1). The 10 green, red, and blue curves corre-spond to BBHs with mass ratios q = 0 .
8, 0.6, and 0.2 respec-tively. The dashed black lines show the asymptotic values α ∞± as r → ∞ for the three mass ratios. The dotted blacklines show the resonance condition α = 2 πn , while the col-ored circles indicate separations where the BBHs encounternutational resonances. Unless the BBHs masses are precisely equal, the massratio q < > Ω ). If the components of S and S perpendicular to the z axis are aligned at t = 0, theywill first realign (return to their initial relative orienta-tions) after a nutation period τ . Over this interval, thefaster spin S will precess about z by an additional 2 π radians compared to the slower spin S :(Ω − Ω ) τ = 2 π . (45)In order for J to remain nearly aligned with the z axis( θ J (cid:28) θ L ), L must have a component L ⊥ in the xy planeanti-aligned with the component S ⊥ of the total spin inthis plane. If S ⊥ = S sin θ ∞ > S ⊥ = S sin θ ∞ , S and thus L will precess about the z axis over the nutationperiod τ by an angle α ∞− = Ω τ = 2 π Ω Ω − Ω = 2 πq (4 + 3 q )3(1 − q ) , (46)which we have derived using Eqs. (44) and (45). If S sin θ ∞ < S sin θ ∞ , the asymptotic precession anglewill instead be given by α ∞ + = Ω τ = α ∞− + 2 π = 2 π (4 q + 3)3(1 − q ) . (47)3If the BBHs have isotropically oriented spins with mag-nitudes S and S , the fraction of binaries for which α asymptotes to α ∞ + for S > S is f + = | S − S | S S [sinh(2 cosh − C ) − − C ] , (48)while for S < S it is f + = | S − S | S S [sinh(2 sinh − C ) + 2 sinh − C ] , (49)where in both expressions C ≡ S / | S − S | / . For thethree mass ratios q = { . , . , . } in Fig. 7, Eqs. (46)through (48) imply α ∞− = { . π, . π, . π } , α ∞ + = { . π, . π, . π } , and f + = { . , . , . × − } . These values are consistent with the horizontaldashed lines in Fig. 7 and that { / , / , / } of thebinaries asymptote to α ∞ + for q = { . , . , . } .As the BBHs in Fig. 7 inspiral from large separationstowards merger, they encounter nutational resonancesmarked by small colored circles whenever α = 2 πn .BBHs with mass ratios for which α ∞± is close to aninteger multiple of 2 π are most likely to encounter nu-tational resonances at large binary separations. We alsosee several discontinuous jumps in α by ± π correspond-ing to configurations in which the orbital angular mo-mentum L and total angular momentum J are eitheraligned or anti-aligned at S = S + or S − . According toEq. (43), the BBHs with mass ratios q = { . , . , . } in Fig. 7 enter the regime where the “up-down” config-uration is unstable for binary separations r < r ud+ (cid:39){ M, M, . M } . The large peak values α > π occurring at r (cid:46) r ud+ for two of the q = 0 . J − ξ plane tothe unstable “up-down” configuration for which α → ∞ .The key point to take away from Fig. 7 is that most bina-ries with large spins and q (cid:38) . r < M where tilts are com-paratively large and GWs for solar-mass BBHs are emit-ted at frequency detectable by ground-based GW obser-vatories like LIGO.Having examined how α evolves with binary separa-tion for the 30 binaries show in Fig. 7, we now broadenour sample to 5 × binaries with a flat distribution ofmass ratios in the range 0 . < q < . < χ i <
1. In Fig. 8, we show all of the nu-tational resonances with | n | ≤ θ tilt > − encoun-tered by these binaries as they inspiral from r = 200 M to r = 10 M . No resonances with n ≤ × inspirals (an incidence of8.3%), with most occurring at r (cid:46) M as shown by thehistogram in the top panel of Fig. 8. The previous sam-ple shown in Fig. 7 suggests that BBHs with comparable masses should account for the majority of these nuta-tional resonances because the steeper slopes of their α ( r )curves should increase the probability that they cross an α = 2 πn line signaling a nutational resonance. There are2717 n = 1 resonances (65.4% of the total) with a broadrange of tilts, including a tail extending to θ tilt > − for r (cid:46) M as shown by the histogram in the rightpanel of Fig. 8. The largest tilt angles appears to scalewith binary separation as θ tilt ∝ ( r/M ) − / consistentwith the analytic estimate of Eq. (42). The 923 n = 2and 517 n = 3 resonances constitute smaller fractions ofthe total (22.2% and 12.4% respectively) and generallylead to smaller tilts θ tilt < − . Although there maybe finely tuned resonances missing from our sample witheven larger tilts (such as those with dα/dL = 0 indicatedby the spikes in the top and right panels of Fig. 6), theresults shown in Fig. 8 suggest that tilts from exact res-onances at binary separations r > M are too small tohave significant astrophysical consequences or detectableGW signatures. However, we will show in the next sec-tion that the large tilts associated with transitional pre-cession [22] can be interpreted as a consequence of anapproximate n = 0 nutational resonance. VI. TRANSITIONAL PRECESSION AS ANAPPROXIMATE NUTATIONAL RESONANCE
The tilt angles θ tilt shown in Fig. 8 are disappointinglysmall if we ever hope to measure their observational con-sequences. The coefficients θ Ln shown in Fig. 3 are sev-eral orders of magnitude larger for n = 0 and n = − n = 0 or n = − α = 2 πn contours in Fig. 6 suggests that if n = 0or n = − J − ξ plane. To testthis possibility, we plot these boundaries and the valueof α along them for BBHs with maximal spins, binaryseparations of r = 10 M , and three different mass ratios q = { . , . , . } in Fig. 9. These three mass ratiosprovide examples of the three alternative values of J min ,the minimum allowed magnitude of the total angular mo-mentum [14]. If L > S + S , the minimum allowed mag-nitude of J = L + S + S is L − S − S as is the casefor q = 0 . χ = χ = 1, and r = 10 M as seen in thetop panel of Fig. 9. This value of J min , indicated by thegreen square, corresponds to the “down-down” configura-tion indicated by one of the four circles showing the fourconfigurations in which the BBH spins S i are both eitheraligned or anti-aligned with L . The right side of thispanel shows α ( ξ ) as we circulate around the boundary ofthe allowed region in the J − ξ plane. The continuouscurve connecting ξ = ±
200 100 50 20 10 r/M − − − − θ t il t θ tilt ∝ ( r/M ) − / θ tilt ∝ ( r/M ) − / n = 1 n = 2 n = 3 n =1 n =2 n =3 n =1 n =2 n =3 FIG. 8. Distribution of nutational resonances as a function of binary separation r and tilt angle θ tilt for 5 × binarieswith isotropically oriented spins and flat distributions of mass ratios and dimensionless spins in the ranges 0 . < q < . < χ i <
1. The central panel is a scatter plot in which each blue circle, red triangle, and green square corresponds to a n = { , , } nutational resonance encountered by one of the BBHs. The dashed black line θ tilt ∝ ( r/M ) − / shows that thescaling of the largest tilt angles with binary separation agrees with the analytic prediction of Eq. (42). The top and rightpanels show histograms generated by binning this scatter plot of the nutational resonances as functions of binary separation r and tilt angle θ tilt for each value of n . The binaries encounter a total of N = { , , } resonances for n = { , , } with θ tilt > − in the range 200 M > r > M , implying that ∼ .
3% of the BBHs encounter such resonances. the three circles on the boundary of the allowed region(the “up-up”, “down-up”, and “down-down” configura-tions), the value of α changes discontinuously by ± π because of the coordinate discontinuity discussed previ-ously. As r ud − < r < r ud+ according to Eq. (43) for thischoice of parameters, the unstable “up-down” configura-tion lies in the interior of the allowed region and thusdoes not lead to a discontinuity in α along the bound-ary. It is important to note that the red curve connectingthe “down-up” and “down-down” configurations denot-ing BBHs for which α is undefined lies in the interior ofthe allowed region, although it is so close to boundaryas to appear indistinguishable from it in this figure. Thered and blue curves in the middle and bottom panels arealso in the interior of the allowed region despite theirclose proximity to the boundary.We now examine the middle panel of Fig. 9 which dif- fers from the top panel because the mass ratio has beenreduced to q = 0 .
32. For this mass ratio, | S − S |
Left panels:
Boundaries of the allowed region in the J − ξ plane for BBHs with binary separations of r = 10 M ,maximal spin magnitudes, and mass ratios of q = 0 . L aligned with the total angular momentum J once per nutationperiod, when S = S − = S min = | J − L | , while the BBHs along the blue curves have L aligned with J once per nutation periodwhen S = S + = S max = J + L . The four circles in each panel indicate the BBHs for which the spins S i are both either alignedor anti-aligned with L ; the circles labeled “UU”, “UD”, “DU”, and “DD” correspond to the up-up, up-down, down-up, anddown-down configurations respectively. The green squares indicate the minimum allowed value of J for each mass ratio. Rightpanels:
The precession angle α over a nutation period τ as a function of the projected effective spin ξ along the boundaries ofthe allowed regions. The horizontal dotted lines indicate values of ξ at which the red and blue curves intersect the boundaries,while the vertical dashed line indicates α = 0. ξ = ± r ud+ (cid:39) M for this mass ratio is barely above the binary separation r = 10 M , the unstable “up-down” configuration is veryclose to the right edge of the allowed region and α getsvery large near this configuration as was previously seenin Figs. 6 and 7. The three other discontinuous curves α ( ξ ) correspond to the three pieces of the left edge of theallowed region: the first piece connects the “down-down”and J = 0 configurations, the second piece connects the J = 0 and “down-up” configurations, and the long thirdpiece of the left boundary connects the “down-up” and“up-up” configurations. As in the top panel, we see that α experiences discontinuous jumps by ± π at the “up-up”, “down-up”, and “down-down” configurations. How-ever, as we trace along the left edge of the allowed regionand pass through the J = 0 configuration, we cross boththe red and blue curves leading to a discontinuity by ± π in α as seen in the right side of the panel (twice the sizeof the other discontinuities).Although α is undefined for the J = 0 configuration,we can consider the value of α in the neighborhood ofthis point of the J − ξ plane. Because the red and bluecurves are so close to the left edge of the allowed re-gion, the vast majority of this neighborhood will lie inbetween the red and blue curves where α has experi-enced only half of the ± π discontinuity that would re-sult from crossing both curves. Examining the midpointof the α discontinuity at J = 0 on the right side of themiddle panel of Fig. 9, we see that most of the neighbor-hood of this point has α (cid:39) n = 0 nutational resonance. The large size of the θ L coefficient in Fig. 3 compared to those with n ≥ n = 0 nutational resonanceshould lead to a much larger tilt than those found for the n = { , , } resonances shown in Fig. 8. In fact, thislarge tilt is already well known to the relativity commu-nity as the transitional precession described in Aposto-latos et al. [22]! Fig. 9 in that paper shows the evolutionof ˆL for a binary with q = 0 . S = 0, and maximalspin S nearly anti-aligned with L as the binary inspi-rals from r = 330 M to r = 6 M . This figure looks un-mistakably like the Euler spirals of Figs. 4 and 5 of ourpaper, although Apostolatos et al. were not able to ob-tain our analytic solution of Eq. (38). While the Eulerspirals at nutational resonances with n ≥ n = {− , } ,the large tilt resulting from the approximate n = 0 reso-nance during transitional precession can be seen withoutour expansion because it involves the dominant term. Wehave thus demonstrated that the well-known large tiltsduring transitional precession, illustrated for the specialcase S = 0 in Apostolatos et al. [22], can occur for J (cid:39) S (cid:54) = 0. They are in fact special cases of the moregeneral nutational resonances for arbitrary n that are farmore frequently encountered during generic misalignedinspirals. The middle panel of Fig. 9 suggests that most of the neighborhoods of the “down-up” and “up-up” con-figurations should similarly be approximate n = 0 reso-nances. However, the near alignment or anti-alignment ofboth BBH spins with L in these neighborhoods may leadto small values of the θ L coefficients and correspond-ing tilts. Future investigations searching for large tilts in (cid:104) J (cid:105) should consider configurations that are approximate n = {− , } nutational resonances.For completeness, we show the third possibility for J min in the bottom panel of Fig. 9. For these binarieswith the even smaller mass ratio q = 0 . S > L + S andtherefore J min = S − S − L >
0. The J = J min configu-ration coincides with the “down-up” configuration in thiscase as seen by the overlapping circle and green square.For this mass ratio and binary separation, r > r ud ± im-plying that the “up-down” configuration is stable andlies on the right edge of the allowed region in the J − ξ plane. Examining α along the boundary of the allowedregion as shown on the right side of the bottom panel,we see that there is no longer a continuous curve α ( ξ )connecting ξ = ± α discontinuity onthe right edge of the allowed region associated with thenow stable “up-down” configuration. The left and rightedges of the allowed region are each associated with twodiscontinuous α ( ξ ) curves, with four jumps in α by ± π corresponding to the four configurations with cos θ i = ± n = 0 or n = − α discontinuities jumpacross α = 0, while α never gets quite negative enoughfor an exact n = − VII. DISCUSSION
This paper seeks to provide qualitative and quanti-tative insight into the evolution of the orbital angularmomentum L and total angular momentum J in the PNregime for generic BBHs (unequal masses, two misalignedspins). We rely extensively on our earlier work [13, 14]in which we derived analytic solutions to the 2PN spin-precession equations for generic binaries in the absenceof radiation reaction. These solutions showed that therelative orientations of L and the BBH spins S i could befully specified by a single degree of freedom and that themagnitude S of the total spin S = S + S was a usefulcoordinate for describing this degree of freedom. In theabsence of radiation reaction, J is fixed (and thus equalto its precession-averaged value (cid:104) J (cid:105) ) and can be used todefine the z axis in an inertial reference frame. Withoutloss of generality, we can choose x and y axes in the planeperpendicular to ˆz . The direction of L in this frame canbe specified by the spherical coordinates θ L and Φ L . Forgeneric BBHs at 2PN order, L will both precess (evolu-tion of Φ L ) and nutate (evolution of θ L ). Over a nutation7period τ , S will oscillate back and forth between its ex-trema S ± set by the effective potential ξ ± ( S ), and θ L willsimilarly oscillate according to Eq. (5). While it nutates, L will precess at the time-dependent precession frequency d Φ L /dt = Ω z ( S ) given by Eq. (14), precessing by a totalangle α over a full nutation period τ .In this paper, we used τ and α to define the nutationfrequency ω ≡ π/τ and average precession frequencyΩ ≡ α/τ that characterize the evolution of L on theprecession timescale. We derived Eq. (26), a new seriesexpansion for the component of L in the xy plane, inwhich each term is a vector of length | θ Ln | that precessesabout the z axis with frequency Ω − nω . Fig. 2 demon-strates that just the two dominant terms in this series canvery accurately describe the evolution of L even when itexhibits seemingly complicated precession and nutation.Radiation reaction modifies the above analysis becausethe total angular momentum J no longer remains con-stant. However, with the z axis defined to point in thedirection of the precession-averaged total angular mo-mentum (cid:104) J (cid:105) , our new series expansion for L remainsapproximately valid because the angle θ J between theinstantaneous J and its precession average (cid:104) J (cid:105) is sup-pressed compared to θ L by the ratio of the precession andradiation-reaction timescales t pre /t RR ∝ ( r/M ) − / (cid:28) L and the two vectors J and (cid:104) J (cid:105) ( θ Lz (cid:39) θ L for theangles depicted in Fig. 1). To use our series expansionwith non-vanishing radiation reaction, we need only al-low the coefficients θ Ln and frequencies ω and Ω to varyon the radiation-reaction timescale, replacing the phasesof each term in the expansion by the time integrals ofthe now varying frequencies. The excellent agreementbetween our expansion and direct numerical integrationof the spin-precession equations shown in Fig. 2 was ob-tained with just this prescription for radiation reaction.Because the coefficients and frequencies of our expansiononly vary on the radiation-reaction timescale t RR (unlike L itself which evolves on the precession timescale t pre ),our expansion may provide vast computation savings if L ( t ) needs to be calculated over an entire inspiral to gen-erate gravitational waveforms.The proportionality between d J /dt and L for radia-tion reaction described by the quadrupole formula (whichalso holds for the 1PN corrections to this formula [23])implies that our new series expansion for L can also beused to describe d J /dt as seen in Eq. (30). To under-stand the evolution of J on the precession timescale, weneed only integrate this expansion while holding the co-efficients and frequencies (which vary on the radiation-reaction timescale) constant. This analytic integrationbreaks down whenever Ω − nω = 0 (mathematicallyequivalent to α = 2 πn ), since this combination of fre-quencies appears in the denominator of the integral. Weidentify this condition as a nutational resonance, sincethe average precession frequency Ω is an integer multipleof the nutation frequency ω . Physically, this breakdownoccurs because if α = 2 πn , the component of L in the xy plane will return to its initial value after a nutationperiod τ . The total angular momentum radiated in suc-cessive nutation periods will therefore point in the samedirection and add constructively, causing (cid:104) J (cid:105) to tilt intothe xy plane. Although an exact nutational resonancerequires a finely tuned value of α , the sample of 5 × BBHs shown in Fig. 8 shows that ∼
10% of BBHs en-counter a nutational resonance with n = { , , } as theyinspiral from r = 200 M to r = 10 M . However, the tiltangles θ tilt associated with these resonances are typicallyless than 10 − radians even at small r because the co-efficients θ Ln to which these tilts are proportional arehighly subdominant to the non-resonant n = − n = 0 terms in the series expansion for J .Although we have not found any exact n = 0 nuta-tional resonances, a careful examination of BBHs in theneighborhood of the J = 0 configuration as shown in themiddle panel of Fig. 9 reveals that most of these BBHsare in an approximate n = 0 resonance leading to largetilts. Our identification of this approximate nutationalresonance is in fact just a new description of the famil-iar phenomenon of transitional precession identified byApostolatos et al. [22]. In that paper, transitional pre-cession was derived in the limit that S = 0 and S (cid:39) − L ,but we have shown that it also applies for most configu-rations where the total spin S (cid:39) − L , even if both BBHsspins are near maximal. A more systematic investigationof other mass ratios and spin magnitudes could poten-tially discover other approximate n = − n = 0resonances where (cid:104) J (cid:105) experiences large tilts, a significantsource of error in the construction of gravitational wave-forms [26].We hope that the insights provided in this paper,particularly our elegant new series expansion for L inEq. (26), will prove useful for future calculations of GWemission and more general astrophysical studies of BBHs.Although the precession of L in an inertial frame has longbeen recognized for systems with misaligned spins, thenutation of L has received less attention. This nutation, aconsequence of multiple non-vanishing terms in Eq. (26),will likely generate distinctive observational signatures inboth gravitational waveforms and astrophysical phenom-ena like the jets and circumbinary disks associated withaccreting supermassive BBHs [33–35]. Whether thesesignatures are large and unambiguous enough to be de-tected remains an open question, but one we hope maybe addressed by the wealth of observations that will beprovided by upcoming GW and electromagnetic surveys. ACKNOWLEDGMENTS
M. K. is supported by the Alfred P. Sloan FoundationGrant No. FG-2015-65299 and NSF Grant No. PHY-1607031. D. G. is supported by NASA through EinsteinPostdoctoral Fellowship Grant No. PF6-170152 awardedby the Chandra X-ray Center, which is operated by theSmithsonian Astrophysical Observatory for NASA under8contract NAS8-03060. Computations were performed onthe Caltech computer cluster “Wheeler,” supported by the Sherman Fairchild Foundation and Caltech. Partialsupport is acknowledged by NSF Award CAREER PHY-1151197. [1] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber-nathy, F. Acernese, K. Ackley, C. Adams, T. Adams,P. Addesso, R. X. Adhikari, and et al., Phys. Rev. Lett. , 061102 (2016).[2] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber-nathy, F. Acernese, K. Ackley, C. Adams, T. Adams,P. Addesso, R. X. Adhikari, and et al., Phys. Rev. Lett. , 241103 (2016).[3] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber-nathy, F. Acernese, K. Ackley, C. Adams, T. Adams,P. Addesso, R. X. Adhikari, and et al., Phys. Rev. X ,041015 (2016).[4] R. P. Kerr, Phys. Rev. Lett. , 237 (1963).[5] A. Einstein, Sitzungsberichte der K¨oniglich PreußischenAkademie der Wissenschaften (Berlin) , 831 (1915).[6] F. Pretorius, Phys. Rev. Lett. , 121101 (2005).[7] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlo-chower, Phys. Rev. Lett. , 111101 (2006).[8] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, andJ. van Meter, Phys. Rev. Lett. , 111102 (2006).[9] T. Regge and J. A. Wheeler, Physical Review , 1063(1957).[10] F. J. Zerilli, Phys. Rev. D , 2141 (1970).[11] S. A. Teukolsky, Astrophys. J. , 635 (1973).[12] A. Sesana, Phys. Rev. Lett. , 231102 (2016).[13] M. Kesden, D. Gerosa, R. O’Shaughnessy, E. Berti, andU. Sperhake, Phys. Rev. Lett. , 081103 (2015).[14] D. Gerosa, M. Kesden, U. Sperhake, E. Berti, andR. O’Shaughnessy, Phys. Rev. D , 064016 (2015).[15] D. Gerosa and M. Kesden, Phys. Rev. D , 124066(2016).[16] M. Kesden, U. Sperhake, and E. Berti, Phys. Rev. D ,084054 (2010).[17] M. Kesden, U. Sperhake, and E. Berti, Astrophys. J. , 1006 (2010). [18] E. Berti, M. Kesden, and U. Sperhake, Phys. Rev. D ,124049 (2012).[19] D. Gerosa, M. Kesden, E. Berti, R. O’Shaughnessy, andU. Sperhake, Phys. Rev. D , 104028 (2013).[20] C. L. Rodriguez, M. Zevin, C. Pankow, V. Kalogera, andF. A. Rasio, Astrophys. J. , L2 (2016).[21] S. Stevenson, C. P. L. Berry, and I. Mandel, ArXiv e-prints (2017), arXiv:1703.06873 [astro-ph.HE].[22] T. A. Apostolatos, C. Cutler, G. J. Sussman, and K. S.Thorne, Phys. Rev. D , 6274 (1994).[23] L. E. Kidder, Phys. Rev. D , 821 (1995).[24] J. D. Schnittman, Phys. Rev. D , 124020 (2004).[25] K. Chatziioannou, A. Klein, N. Cornish, and N. Yunes,Phys. Rev. Lett. , 051101 (2017).[26] K. Chatziioannou, A. Klein, N. Yunes, and N. Cornish,Phys. Rev. D , 104004 (2017), arXiv:1703.03967 [gr-qc].[27] T. Damour, Phys. Rev. D , 124013 (2001).[28] ´E. Racine, Phys. Rev. D , 044021 (2008).[29] D. Gerosa, U. Sperhake, and J. Voˇsmera, Class. Quant.Grav. , 064004 (2017).[30] P. C. Peters and J. Mathews, Phys. Rev. , 435 (1963).[31] D. Trifir`o, R. O’Shaughnessy, D. Gerosa, E. Berti,M. Kesden, T. Littenberg, and U. Sperhake, Phys. Rev.D , 044071 (2016).[32] D. Gerosa, M. Kesden, R. O’Shaughnessy, A. Klein,E. Berti, U. Sperhake, and D. Trifir`o, Phys. Rev. Lett. , 141102 (2015).[33] D. Merritt and R. D. Ekers, Science , 1310 (2002).[34] M. C. Miller and J. H. Krolik, Astrophys. J. , 43(2013).[35] D. Gerosa, B. Veronesi, G. Lodato, and G. Rosotti,Mon. Not. R. Astron. Soc.451