Transient resonances in the inspirals of point particles into black holes
TTransient resonances in the inspirals of point particles into black holes ´Eanna ´E. Flanagan
Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA
Tanja Hinderer
Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA
We show that transient resonances occur in the two body problem in general relativity, for spinningblack holes in close proximity to one another when one black hole is much more massive than theother. These resonances occur when the ratio of polar and radial orbital frequencies, which is slowlyevolving under the influence of gravitational radiation reaction, passes through a low order rationalnumber. At such points, the adiabatic approximation to the orbital evolution breaks down, andthere is a brief but order unity correction to the inspiral rate. The resonances cause a perturbationto orbital phase of order a few tens of cycles for mass ratios ∼ − , make orbits more sensitiveto changes in initial data (though not quite chaotic), and are genuine non-perturbative effectsthat are not seen at any order in a standard post-Newtonian expansion. Our results apply to animportant potential source of gravitational waves, the gradual inspiral of white dwarfs, neutronstars, or black holes into much more massive black holes. Resonances effects will increase thecomputational challenge of accurately modeling these sources. Introduction:
The dynamics of a two-body system emit-ting gravitational radiation is an important problem ingeneral relativity. Binary systems of compact bodiesundergo a radiation-reaction-driven inspiral until theymerge. There are three different regimes in the parame-ter space of these systems: (i) The weak field, Newtonianregime r (cid:29)
1, where r is the orbital separation in unitswhere G = c = M = 1 and M is the total mass. Thisregime can be accurately modeled using post-Newtoniantheory, which consists of an expansion in the small pa-rameter 1 /r [1]. (ii) The relativistic, equal mass regime r ∼ ε ∼ ε = µ/M is the mass ratio with µ being the reduced mass), which must be treated usingnumerical relativity. Numerical relativists have recentlysucceeded for the first time in simulating binary blackhole mergers [2]. (iii) The relativistic, extreme mass ratioregime r ∼ ε (cid:28)
1, which is characterized by long, grad-ual inspirals on a timescale ∼ ε − , and for which compu-tational methods are currently under development.In this Letter, we show that in the relativistic, extrememass ratio regime, there are qualitatively new aspects tothe two-body problem in general relativity, namely theeffects of transient resonances. While resonances are acommon phenomenon in celestial mechanics when threeor more objects are involved [3], they can occur with justtwo objects in general relativity, due to its nonlinearity.They are a non-perturbative effect for highly relativisticsources and are not seen at any order in standard post-Newtonian expansions of inspiral solutions. Their exis-tence is closely related to the onset of chaotic dynamics,which has previously been shown to occur in general rel-ativity in other, cosmological contexts [4].The resonances have direct observational relevance:Compact objects (1 (cid:46) µ/M (cid:12) (cid:46)
10, where M (cid:12) is theSolar mass) inspiraling into much larger black holes areexpected to be a key source for gravitational wave detec- tors. Advanced LIGO will potentially observe 3 −
30 suchevents per year, with 50 (cid:46)
M/M (cid:12) (cid:46) (cid:46) M/M (cid:12) (cid:46) out to cosmological dis-tances at a rate of ∼
50 per year [6]. The observed grav-itational wave signal will be rich in information. For ex-ample, one will be able to extract a map of the spacetimegeometry of the central object and test if it is consistentwith general relativity’s predictions for a black hole [5, 7].Such mapping will require accurate theoretical models ofthe gravitational waveforms, which remain phase coher-ent with the true waveforms to an accuracy of ∼ ∼ ε − ∼ M/µ ∼ − of cyclesof inspiral. Over the past decade there has been a sig-nificant research effort aimed at providing such accuratemodels [8]. Resonances will complicate this enterprise,as we discuss below. Method of Analysis:
Over timescales short compared tothe dephasing time ∼ ε − / , inspirals can be accuratelymodeled using black hole perturbation theory, with ε asthe expansion parameter. The leading order motion isgeodesic motion on the background Kerr metric. At thenext order the motion is corrected by the particle’s self-force or radiation reaction force, for which a formal ex-pression is known [9], and which has been computed ex-plicitly in special cases; see, e.g., the review [8]. Overthe longer inspiral timescale ∼ ε − , it is necessary toaugment these methods with two-timescale expansionswhich are currently under development [10, 11]. In thisframework the leading order motion is an adiabatic in-spiral, and there are various post-adiabatic corrections.Geodesic motion in the Kerr spacetime is an integrabledynamical system, and it is useful to use the the corre-sponding generalized action-angle variables to parame- a r X i v : . [ g r- q c ] J u l terize the inspiral. The resulting equations are [17]: dq α dτ = ω α ( J ) + εg (1) α ( q θ , q r , J ) + O ( ε ) , (1a) dJ ν dτ = εG (1) ν ( q θ , q r , J ) + ε G (2) ν ( q θ , q r , J ) + O ( ε ) . (1b)Here τ is Mino time [13] and J ν are the con-served integrals of geodesic motion given by J ν =( E/µ, L z /µ, Q/µ ), where E is the energy, L z is the angu-lar momentum, and Q the Carter constant. The variables q α = ( q t , q r , q θ , q φ ) are generalized angle variables conju-gate to Mino time [10]. The right hand sides at O ( ε )describe geodesic motion, with fundamental frequencies ω r , ω θ and ω φ . The forcing functions g (1) α , G (1) ν and G (2) ν are due to the first order and second order self-forces, andare 2 π -periodic in q θ and q r . The piece of G (1 , ν that iseven under q θ → π − q θ , q r → π − q r , and the pieceof g (1) α that is odd, are the dissipative self-force, and theremaining piece is the conservative self-force [10].In the limit ε →
0, the leading order solutions to Eqs.(1) are given by the following adiabatic prescription [10]:Drop the forcing terms g (1) α and G (2) ν , and replace G (1) ν byits average over the 2-torus parameterized by q θ , q r . It isnow known how to evaluate this averaged force explicitly[12, 13], although generic adiabatic inspirals have not yetbeen computed numerically.Consider now post-adiabatic effects. The dynamicalsystem (1) consists of a perturbed, integrable Hamilto-nian system. Resonances in this general type of systemhave been studied in detail and are well understood [14],and we can apply the general theory to the present con-text. The existence of resonances in this context has pre-viously been suggested by Refs. [15, 16]. We will presentthree different treatments of the resonances: (i) An intu-itive, order of magnitude discussion, which is sufficient todeduce their key properties; (ii) A numerical treatment;and (iii) A sketch of a formal analytic derivation. A moredetailed treatment will be presented in Ref. [17]. Order of Magnitude Estimates:
Suppose that we have anadiabatic solution, which will be of the form q α ( τ, ε ) = ψ α ( ετ ) /ε , J ν ( τ, ε ) = J ν ( ετ ). Consider now the post-adiabatic correction terms in Eqs. (1), near some ar-bitrarily chosen point τ = 0. We expand q θ as q θ = q θ + ω θ τ + ˙ ω θ τ + O ( τ ), where subscripts 0 denoteevaluations at τ = 0, and we expand q r similarly. We alsoexpand G (1) ν as a double Fourier series: G (1) ν ( q θ , q r , J ) = (cid:80) k,n G (1) ν kn ( J ) e i ( kq θ + nq r ) , where the 00 term is the adia-batic approximation, and the remaining terms drive post-adiabatic effects. Inserting the expansions of q θ and q r ,we find for the phase of the ( k, n ) Fourier component(constant) + ( kω θ + nω r ) τ + ( k ˙ ω θ + n ˙ ω r ) τ + . . . (2)Normally, the second term is nonzero and thus the forceoscillates on a timescale ∼
1, much shorter than the in-spiral timescale ∼ /ε , and so the force averages to zero. However, when the resonance condition kω θ + nω r = 0is satisfied, the ( k, n ) force is slowly varying and can-not be neglected, and so gives an order-unity correc-tion to the right hand side of Eq. (1b). The durationof the resonance is given by the third term in (2) to be τ res ∼ / √ v ˙ ω ∼ / √ vε , where v = | k | + | n | is the order ofthe resonance; after times longer than this the quadraticterm causes the force to oscillate and again average tozero. The net change in the action variables J ν is there-fore ∆ J µ ∼ ˙ Jτ res ∼ ετ res ∼ (cid:112) ε/v . After the resonance,this change causes a phase error ∆ φ that accumulatesover an inspiral, of order the total inspiral phase ∼ /ε times ∆ J/J ∼ (cid:112) ε/v , which gives ∆ φ ∼ / √ vε. This discussion allows us to deduce several key prop-erties of the resonances. First, corrections to the gravita-tional wave signal’s phase due to resonance effects scaleas the square root of the inverse of mass of the small body.These corrections thus become large in the extreme-mass-ratio limit, dominating over all other post-adiabatic ef-fects, which scale as ε ∼ ω r /ω θ is a low order rationalnumber. There is a simple geometric picture correspond-ing to this condition [16, 18]: the geodesic orbits do notergodically fill out the ( q θ , q r ) torus in space as genericgeodesics orbits do, but instead form a 1 dimensionalcurve on the torus. This implies that the time-averagedforces for these orbits are not given by an average overthe torus, unlike the case for generic orbits.Third, they occur only for non-circular, non-equatorialorbits about spinning black holes. For other cases, theforcing terms G (1) ν depend only on q θ , or only on q r , butnot both together, and thus the Fourier coefficient G (1) ν kn will vanish for any resonance.Fourth, they are driven only by the spin-dependentpart of the self-force, for the same reason: spherical sym-metry forbids a dependence on q θ in the zero-spin limit.Fifth, they appear to be driven only by the dissipa-tive part of the self-force, and not by the conservativepart, again because the forcing terms do not depend onboth q θ and q r . We have verified that this is the case upto the post-Newtonian order that spin-dependent termshave been computed [19], and we conjecture that it is trueto all orders. The reason that this occurs is that the con-servative sector of post-Newtonian theory admits threeindependent conserved angular momentum components;the ambiguities in the definition of angular momentumare associated with radiation, in the dissipative sector.As a consequence, the perturbed conservative motion isintegrable to leading order in ε , and an integrable per-turbation to a Hamiltonian cannot drive resonances.Sixth, although the resonance is directly driven onlyby dissipative, spin dependent self-force, computing res-onance effects requires the conservative piece of the firstorder self-force and the averaged, dissipative piece of thesecond order self-force. Those pieces will cause O (1) cor-rections to the phases over a complete inspiral [10], andthe kicks ∆ J µ produced during the resonance depend onthe O (1) phases at the start of the resonance.Seventh, resonances give rise to increased sensitivityto initial conditions, analogous to chaos but not as ex-treme as chaos, because at a resonance information flowsfrom a higher to a lower order in the perturbation ex-pansion. For example, we have argued that changes tothe phases at O (1) prior to the resonance will affect thepost-resonance phasing at O (1 / √ ε ). Similarly changes tothe phases at O ( √ ε ) before resonance will produce O (1)changes afterwards. With several successive resonances,a sensitive dependence on initial conditions could arise. Numerical Integrations:
The scaling relation ∆ φ ∝ / √ ε suggests the possibility of phase errors large compared tounity that impede the detection of the gravitational wavesignal. To investigate this possibility, we numerically in-tegrated the exact Kerr geodesic equations supplementedwith approximate post-Newtonian forcing terms. Whileseveral such approximate inspirals have been computedpreviously [20], none have encountered resonances, be-cause resonances require non-circular, non-equatorial or-bits about a spinning black hole with non orbit-averagedforces, which have not been simulated before.For the numerical integrations we use instead of q α thevariables ¯ q α = (¯ q t , ¯ q r , ¯ q θ , ¯ q φ ) = ( t, ψ, χ, φ ) where ψ and χ are the angular variables for r and θ motion defined inRef. [13]. The equations of motion (1) in these variablesare t ,τ = ¯ ω t (¯ q θ , ¯ q r , J ) , φ ,τ = ¯ ω φ (¯ q θ , ¯ q r , J ) , (3a)¯ q θ,τ = ¯ ω θ (¯ q θ , J ) + εh (1) θ (¯ q θ , ¯ q r , J ) + O ( ε ) , (3b)¯ q r,τ = ¯ ω r (¯ q r , J ) + εh (1) r (¯ q θ , ¯ q r , J ) + O ( ε ) , (3c) J ν,τ = εH (1) ν (¯ q θ , ¯ q r , J ) + O ( ε ) . (3d)Here τ is Mino time [13], the frequencies ¯ ω are given in[13], and h (1) α and H (1) ν are given in terms of the compo-nents of the 4-acceleration in [21].We parameterize the three independent components ofthe acceleration in the following way: a α = a ˆ r e α ˆ r + a ˆ θ e α ˆ θ + a ⊥ (cid:15) αβγδ u β e γ ˆ r e δ ˆ θ +( a ˆ r u ˆ r + a ˆ θ u ˆ θ ) u α , where (cid:126)u is the 4-velocityand (cid:126)e ˆ r and (cid:126)e ˆ θ are unit vectors in the directions of ∂ r and ∂ θ . We compute the dissipative pieces of a ˆ r , a ˆ θ and a ⊥ from the results of [22], as functions of ˜ r = r + a / (4 r ), E n = E −
1, and ¯ K = Q + a L z + a E n , and then expandto O ( a ) and to the leading post-Newtonian order at eachorder in a [17]. We also add the conservative component,expressed similarly and computed to O ( a ) and to theleading post-Newtonian order, taken from Refs. [23]; seeRef. [17] for details.We numerically integrate Eqs. (3) twice, once usingthe adiabatic prescription, and once exactly, and thensubtract at fixed t to obtain the post-adiabatic effects.The adiabatic prescription involves numerically integrat-ing the right hand sides over the torus parameterized FIG. 1: [Top] The adiabatic inspiral computed from our ap-proximate post-Newtonian self-force, for a mass ratio ε = µ/M = 3 × − , with black hole spin parameter a = 0 . p = 9 . M , eccentric-ity e = 0 .
7, and orbital inclination θ inc = 1 .
20. The bottomcurve is e , the middle curve is θ inc , and the top curve is ratio offrequencies ω θ /ω r , shown as functions of p . [Middle] The fluc-tuating, dissipative part of the first order self-force causes astrong resonance when ω θ /ω r = 3 / p = 8 . E , angular momentum L z andCarter constant Q , as functions of p , scaled to their values atresonance, and divided by the square root (cid:112) µ/M of the massratio. The sudden jumps at the resonance are apparent, withthe largest occurring for the Carter constant. [Bottom] Thelower curve is the correction to the number of cycles φ/ (2 π )of azimuthal phase of the inspiral caused by the fluctuating,dissipative part of the first order self-force. The sharp down-ward kick due to the resonance at p = 8 .
495 can be clearlyseen. The resonant corrections to the number of cycles of r and θ motion are similar. These phase shifts scale as (cid:112) M/µ .The upper curve is the post-adiabatic phase correction dueto the conservative piece of the first order self-force, which isconsiderably smaller and is independent of the mass ratio. by q θ , q r at each time step, where q r = F r (¯ q r ) /F r (2 π ), F r (¯ q r ) = (cid:82) ¯ q r d ¯ q r / [¯ ω r (¯ q r , J )] , with a similar formula for q θ . This is numerically time consuming, but the adiabaticintegration can take timesteps on the inspiral timescale ∼ /ε rather than the dynamical timescale ∼ ε = 3 × − with a =0 .
95, in terms of the relativistic eccentricity e , semilatusrectum p and orbital inclination θ inc , which are functionsof E , L z and Q [24]. This example has a strong resonanceat ω θ /ω r = 3 /
2, that generates jumps in the conservedquantities of order a few percent times √ ε , and causesphase errors over the inspiral of order 20 cycles. Phaseerrors of this magnitude will be a significant impedimentto signal detection with matched filtering. We find thatthe resonance effects are dominated by the O ( a ) terms,and the effect of the O ( a ) terms are small. Analytic Derivation:
In terms of the slow time variable˜ τ = ετ , the solutions of the dynamical system (1) awayfrom resonances can be expressed as an asymptotic ex-pansion in ε at fixed ˜ τ [10, 14]: q α ( τ, ε ) = 1 ε (cid:104) ψ (0) α (˜ τ ) + √ εψ (1 / α (˜ τ ) + O ( ε ) (cid:105) , (4a) J ν ( τ, ε ) = J (0) ν (˜ τ ) + √ ε J (1 / ν (˜ τ ) + O ( ε ) . (4b)The leading order terms give the adiabatic approximationdescribed above, and satisfy [10] ψ (0) α, ˜ τ = ω α [ J (0) ], J (0) ν, ˜ τ = (cid:104) G (1) ν (cid:105) [ J (0) ], where the angular brackets denote an aver-age over the ( q r , q θ ) torus. The subleading, post-1 / J (1 / ν, ˜ τ − (cid:104) G (1) ν (cid:105) ,J µ J (1 / µ =∆ J (1 / ν δ ( τ ), ψ (1 / α, ˜ τ = ω α,J µ J (1 / µ , where the δ -functionsource term arises at a resonance, taken to occur at τ = 0.Near the resonance we use an ansatz for the solutionswhich is an asymptotic expansion in √ ε at fixed ˆ τ = √ ετ ,and then match these solutions onto pre-resonance andpost-resonance solutions of the form (4) [14]. To linearorder in the force Fourier coefficients (an approximationwhich is valid here to within a few percent [17]), thejumps in the action variables for a resonance ( k, n ) canbe computed by substituting the adiabatic solutions intothe right hand side of Eqs. (1) and solving for the per-turbation to the action variables. The result is∆ J (1 / ν = (cid:88) s (cid:54) =0 (cid:115) π | αs | exp (cid:20) sgn( αs ) iπ isχ res (cid:21) G (1) ν sk,sn , where χ res = kq θ + nq r , α = kω θ, ˜ τ + nω r, ˜ τ and all quan-tities are evaluated at the resonance τ = 0 using theadiabatic solution. In Ref. [17] we give the exact ex-pression for this quantity that does not linearize in theforce Fourier coefficients. We note that evaluating thephase χ res requires knowledge of the second subleading, O (1) phase in Eq. (4a), which in turn requires knowl-edge of the force components g (1) α , G (1) ν and (cid:104) G (2) ν (cid:105) in Eq.(1) [10]. In addition, to obtain the phase to O (1) accu-racy after the resonance, it is necessary to also computethe subleading, O ( ε ) jumps in J ν and O (1) jumps in q α ,which are given in [17]. Discussion:
The dynamics of binary systems in generalrelativity is richer than had been appreciated. Transientresonances occurring during the inspiral invalidate theadiabatic approximation and give rise to corrections tothe orbital phase that can be large compared to unity. Itwill be necessary to incorporate resonances into theoreti-cal models of the gravitational waveforms for inspirals of compact objects into massive black holes, an importantgravitational wave source. This will require knowledgeof the second order gravitational self force and will bechallenging.
Acknowledgments:
This work was supported by NSFGrants PHY-0757735 and PHY-0457200, by the Johnand David Boochever Prize Fellowship in TheoreticalPhysics to TH at Cornell, and by the Sherman FairchildFoundation. We thank Scott Hughes, Marc Favata, SteveDrasco and Amos Ori for helpful conversations. [1] L. Blanchet, Living Reviews in Relativity , 3 (2002).[2] F. Pretorius (2007), arXiv:0710.1338.[3] C. D. Murray and S. F. Dermott, Solar System Dynamics (Cambridge University Press, 2000).[4] N.J. Cornish and J. Levin, Phys. Rev. Lett. , 998(1997); J.D. Barrow and J. Levin, Phys. Rev. Lett. ,656 (1998).[5] D. A. Brown et al., Phys. Rev. Lett. , 201102 (2007);I. Mandel et. al., Astrophys. J. , 1431 (2008).[6] P. Amaro-Seoane et al., Class. Quant. Grav. , R113(2007).[7] L. Barack and C. Cutler, Phys. Rev. D ,042003 (2007).[8] L. Barack, Class. Quant. Grav. , 213001 (2009).[9] Y. Mino et al., Phys. Rev. D55 , 3457 (1997); T. C. Quinnand R. M. Wald, Phys. Rev.
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