The Effective One Body description of the Two-Body problem
aa r X i v : . [ g r- q c ] J un The Effective One Body description of theTwo-Body problem
Thibault Damour and Alessandro Nagar
Abstract
The Effective One Body (EOB) formalism is an analytical approach whichaims at providing an accurate description of the motion and radiation of coalescingbinary black holes with arbitrary mass ratio. We review the basic elements of thisformalism and discuss its aptitude at providing accurate template waveforms to beused for gravitational wave data analysis purposes.
A network of ground-based interferometric gravitational wave (GW) detectors(LIGO/VIRGO/GEO/ . . . ) is currently taking data near its planned sensitivity [1].Coalescing black hole binaries are among the most promising, and most exciting,GW sources for these detectors. In order to successfully detect GWs from coalesc-ing black hole binaries, and to be able to reliably measure the physical parametersof the source (masses, spins, . . . ), it is necessary to know in advance the shape ofthe GW signals emitted by inspiralling and merging black holes. Indeed, the detec-tion and subsequent data analysis of GW signals is made by using a large bank of templates that accurately represent the GW waveforms emitted by the source.Here, we shall introduce the reader to one promising strategy toward having anaccurate analytical description of the motion and radiation of binary black holes, Thibault DamourInstitut des Hautes Etudes Scientifiques, 35 route de Chartres,F-91440 Bures-sur-Yvette, France, e-mail: [email protected] NagarInstitut des Hautes Etudes Scientifiques, 35 route de Chartres,F-91440 Bures-sur-Yvette, France,INFN, Sezione di Torino, Italy, e-mail: [email protected]. Here we use the adjective “analytical” for methods that solve explicit (analytically given) ordi-nary differential equations (ODE), even if one uses standard (Runge-Kutta-type) numerical tools1 Thibault Damour and Alessandro Nagar which covers all its stages (inspiral, plunge, merger and ring-down): the
EffectiveOne Body approach [2, 3, 5, 4]. As early as 2000 [3] this method made several quan-titative and qualitative predictions concerning the dynamics of the coalescence, andthe corresponding GW radiation, notably: (i) a blurred transition from inspiral toa ‘plunge’ that is just a smooth continuation of the inspiral, (ii) a sharp transition,around the merger of the black holes, between a continued inspiral and a ring-downsignal, and (iii) estimates of the radiated energy and of the spin of the final blackhole. In addition, the effects of the individual spins of the black holes were inves-tigated within the EOB [4, 6] and were shown to lead to a larger energy releasefor spins parallel to the orbital angular momentum, and to a dimensionless rotationparameter J / E always smaller than unity at the end of the inspiral (so that a Kerrblack hole can form right after the inspiral phase). All those predictions have beenbroadly confirmed by the results of the recent numerical simulations performed byseveral independent groups [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,23, 24, 25, 26, 27, 28, 29] (for a review of numerical relativity results see also [30]).Note that, in spite of the high computer power used in these simulations, the cal-culation of one sufficiently long waveform (corresponding to specific values of themany continuous parameters describing the two arbitrary masses, the initial spinvectors, and other initial data) takes on the order of two weeks. This is a very strongargument for developing analytical models of waveforms.Those recent breakthroughs in numerical relativity (NR) open the possibility ofcomparing in detail the EOB description to NR results. This EOB/NR comparisonhas been initiated in several works [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. Thelevel of analytical/numerical agreement is unprecedented, compared to what hasbeen previously achieved when comparing other types of analytical waveforms tonumerical ones. In particular, Refs. [40, 41] have compared two different kind ofanalytical waveforms, computed within the EOB framework, to the most accurategravitational waveform currently available from the Caltech-Cornell group, findingthat the phase and amplitude differences are of the order of the numerical error.If the reader wishes to put the EOB results in contrast with other (Post-Newtonianor hybrid) approaches he can consult, e.g. , [27, 28, 42, 43, 44, 45, 46, 47].Before reviewing some of the technical aspects of the EOB method, let us indi-cate some of the historical roots of this method. First, we note that the EOB approachcomprises three, rather separate, ingredients:1. a description of the conservative (Hamiltonian) part of the dynamics of two blackholes;2. an expression for the radiation-reaction part of the dynamics;3. a description of the GW waveform emitted by a coalescing binary system.For each one of these ingredients, the essential inputs that are used in EOB worksare high-order post-Newtonian (PN) expanded results which have been obtained by to solve them. The important point is that, contrary to 3D numerical relativity simulations, nu-merically solving ODE’s is extremely fast, and can therefore be done (possibly even in realtime) for a dense sample of theoretical parameters, such as orbital ( n = m m / M , . . . ) or spin( ˆ a = S / Gm , q , j , . . . ) parameters.he Effective One Body description of the Two-Body problem 3 many years of work, by many researchers (see references below). However, one ofthe key ideas in the EOB philosophy is to avoid using PN results in their original“Taylor-expanded” form ( i.e. c + c v + c v + c v + · · · + c n v n ) , but to use theminstead in some resummed form ( i.e. some non-polynomial function of v , definedso as to incorporate some of the expected non-perturbative features of the exact re-sult). The basic ideas and techniques for resumming each ingredient of the EOB aredifferent and have different historical roots. Concerning the first ingredient, i.e. theEOB Hamiltonian, it was inspired by an approach to electromagnetically interactingquantum two-body systems introduced by Br´ezin, Itzykson and Zinn-Justin [48].The resummation of the second ingredient, i.e. the EOB radiation-reaction force F , was originally inspired by the Pad´e resummation of the flux function introducedby Damour, Iyer and Sathyaprakash [49]. Recently, a new and more sophisticatedresummation technique for the radiation reaction force F has been introduced byDamour, Iyer and Nagar [50] and further employed in EOB/NR comparisons [40].It will be discussed in detail below.As for the third ingredient, i.e. the EOB description of the waveform emitted bya coalescing black hole binary, it was mainly inspired by the work of Davis, Ruffiniand Tiomno [51] which discovered the transition between the plunge signal and aringing tail when a particle falls into a black hole. Additional motivation for theEOB treatment of the transition from plunge to ring-down came from work on the,so-called, “close limit approximation” [52].Let us finally note that the EOB approach has been recently improved [37, 50, 40]by following a methodology consisting of studying, element by element, the physicsbehind each feature of the waveform, and on systematically comparing variousEOB-based waveforms with ‘exact’ waveforms obtained by NR approaches. Amongthese ‘exact’ NR waveforms, it has been useful to consider the small-mass-ratiolimit n ≡ m m / ( m + m ) ≪
1, in which one can use the well controllable ‘lab-oratory’ of numerical simulations of test particles (with an added radiation-reactionforce) moving in black hole backgrounds [35, 36].
Before discussing the various resummation techniques used in the EOB approach,let us briefly recall the ‘Taylor-expanded’ results that have been obtained by pushingto high accuracies the post-Newtonian (PN) methods.Concerning the orbital dynamics of compact binaries, we recall that the 2.5PN-accurate equations of motion have been derived in the 1980’s [53, 54, 55, 56]. Beware that the fonts used in this chapter make the greek letter n (indicating the symmetric massratio) look very similar to the latin letter v = n indicating the velocity. As usual ‘ n -PN accuracy’ means that a result has been derived up to (and including) terms whichare ∼ ( v / c ) n ∼ ( GM / c r ) n fractionally smaller than the leading contribution. Thibault Damour and Alessandro Nagar Pushing the accuracy of the equations of motion to the 3PN ( ∼ ( v / c ) ) level provedto be a non-trivial task. At first, the representation of black holes by delta-functionsources and the use of the (non diffeomorphism invariant) Hadamard regularizationmethod led to ambiguities in the computation of the badly divergent integrals thatenter the 3PN equations of motion [57, 58]. This problem was solved by using the(diffeomorphism invariant) dimensional regularization method ( i.e. analytic contin-uation in the dimension of space d ) which allowed one to complete the determina-tion of the 3PN-level equations of motion [59, 60]. They have also been derived byan Einstein-Infeld-Hoffmann-type surface-integral approach [61]. The 3.5PN termsin the equations of motion are also known [62, 63, 64].Concerning the emission of gravitational radiation, two different gravitational-wave generation formalisms have been developed up to a high PN accuracy: (i)the Blanchet-Damour-Iyer formalism [65, 66, 67, 68, 69, 70, 71] combines amultipolar post-Minkowskian (MPM) expansion in the exterior zone with a post-Newtonian expansion in the near zone; while (ii) the Will-Wiseman-Pati formalism[72, 73, 74, 62] uses a direct integration of the relaxed Einstein equations. These for-malisms were used to compute increasingly accurate estimates of the gravitationalwaveforms emitted by inspiralling binaries. These estimates include both normal,near-zone generated post-Newtonian effects (at the 1PN [66], 2PN [75, 76, 72],and 3PN [77, 78] levels), and more subtle, wave-zone generated (linear and non-linear) ‘tail effects’ [69, 79, 80, 71]. However, technical problems arose at the 3PNlevel. Similarly to what happened with the equation of motion, the representationof black holes by ‘delta-function’ sources causes the appearance of dangerouslydivergent integrals in the 3PN multipole moments. The use of Hadamard (par-tie finie) regularization did not allow one to unambiguously compute the needed3PN-accurate quadrupole moment. Only the use of the (formally) diffeomorphism-invariant dimensional regularization method allowed one to complete the 3PN-levelgravitational-radiation formalism [82].The works mentioned in this Section (see [83] for a detailed account and morereferences) finally lead to PN-expanded results for the motion and radiation of bi-nary black holes. For instance, the 3.5PN equations of motion are given in the form( a = , i = , , d z ia dt = A i cons a + A iRRa , (1)where A cons = A + c − A + c − A + c − A , (2)denotes the ‘conservative’ 3PN-accurate terms, while A RR = c − A + c − A , (3)denotes the time-asymmetric contibutions, linked to ‘radiation reaction’.On the other hand, if we consider for simplicity the inspiralling motion of aquasi-circular binary system, the essential quantity describing the emitted gravita-tional waveform is the phase f of the quadrupolar gravitational wave amplitude he Effective One Body description of the Two-Body problem 5 h ( t ) ≃ a ( t ) cos ( f ( t ) + d ) . PN theory allows one to derive several different func-tional expressions for the gravitational wave phase f , as a function either of timeor of the instantaneous frequency. For instance, as a function of time, f admits thefollowing explicit expansion in powers of q ≡ n c ( t c − t ) / GM (where t c denotes aformal ‘time of coalescence’, M ≡ m + m and n ≡ m m / M ) f ( t ) = f c − n − q / + (cid:229) n = ( a n + a ′ n ln q ) q − n / ! , (4)with some numerical coefficients a n , a ′ n which depend only on the dimensionless(symmetric) mass ratio n ≡ m m / M . The derivation of the 3.5PN-accurate ex-pansion (4) uses both the 3PN-accurate conservative acceleration (2) and a 3.5PNextension of the (fractionally) 1PN-accurate radiation reaction acceleration (3) ob-tained by assuming a balance between the energy of the binary system and thegravitational-wave energy flux at infinity (see, e.g. , [83]).Among the many other possible ways [84] of using PN-expanded results to pre-dict the GW phase f ( t ) , let us mention the semi-analytic T4 approximant [42, 32].The GW phase defined by the T4 approximant happens to agree well during theinspiral with the NR phase in the equal mass case [27]. However, this agreementseems to be coincidental because the T4 phase exhibits significant disagreementwith NR results for other mass ratios [39] (as well as for spinning black-holes [47]). The PN-expanded results briefly reviewed in the previous Section are expected toyield accurate descriptions of the motion and radiation of binary black holes onlyduring their early inspiralling stage, i.e. as long as the PN expansion parameter g e = GM / c R (where R is the distance between the two black holes) stays signifi-cantly smaller than the value ∼ where the orbital motion is expected to becomedynamically unstable (‘last stable circular orbit’ and beginning of a ‘plunge’ leadingto the merger of the two black holes). One needs a better description of the motionand radiation to describe the late inspiral (say g e & ), as well as the subsequent plunge and merger . One possible strategy for having a complete description of themotion and radiation of binary black holes, covering all the stages (inspiral, plunge,merger, ring-down), would then be to try to ‘stitch together’ PN-expanded analyticalresults describing the early inspiral phase with 3D numerical results describing theend of the inspiral, the plunge, the merger and the ring-down of the final black hole,see, e.g. , Refs. [86, 32].However, we wish to argue that the EOB approach makes a better use of all theanalytical information contained in the PN-expanded results (1)-(3). The basic claim(first made in [2, 3]) is that the use of suitable resummation methods should allow Thibault Damour and Alessandro Nagar one to describe, by analytical tools, a sufficiently accurate approximation of the en-tire waveform , from inspiral to ring-down, including the non-perturbative plungeand merger phases. To reach such a goal, one needs to make use of several tools: (i)resummation methods, (ii) exploitation of the flexibility of analytical approaches,(iii) extraction of the non-perturbative information contained in various numericalsimulations, (iv) qualitative understanding of the basic physical features which de-termine the waveform.Let us start by discussing the first tool used in the EOB approach: the systematicuse of resummation methods. Essentially two resummation methods have been em-ployed (and combined) and some evidence has been given that they do significantlyimprove the convergence properties of PN expansions. The first method is the sys-tematic use of
Pad´e approximants . It has been shown in Ref. [49] that near-diagonalPad´e approximants of the radiation reaction force F seemed to provide a goodrepresentation of F down to the last stable orbit (which is expected to occur when R ∼ GM / c , i.e. when g e ≃ ). In addition, a new route to the resummation of F has been proposed very recently in Ref. [50]. This approach, that will be discussedin detail below, is based on a new multiplicative decomposition of the metric mul-tipolar waveform (which is originally given as a standard PN series). In this case,Pad´e approximants prove to be useful to further improve the convergence propertiesof one particular factor of this multiplicative decomposition.The second resummation method is a novel approach to the dynamics of compactbinaries, which constitutes the core of the Effective One Body (EOB) method.For simplicity of exposition, let us first explain the EOB method at the 2PNlevel. The starting point of the method is the 2PN-accurate Hamiltonian describing(in Arnowitt-Deser-Misner-type coordinates) the conservative, or time symmetric,part of the equations of motion (1) ( i.e. the truncation A cons = A + c − A + c − A of Eq. (2)) say H ( qqq − qqq , ppp , ppp ) . By going to the center of mass of the sys-tem ( ppp + ppp = ) , one obtains a PN-expanded Hamiltonian describing the relativemotion , qqq = qqq − qqq , ppp = ppp = − ppp : H relative2PN ( qqq , ppp ) = H ( qqq , ppp ) + c H ( qqq , ppp ) + c H ( qqq , ppp ) , (5)where H ( qqq , ppp ) = m ppp + GM m | qqq | (with M ≡ m + m and m = m m / M ) correspondsto the Newtonian approximation to the relative motion, while H describes 1PNcorrections and H H ( qqq , ppp ) can be thought of as describing a ‘test particle’ of mass m orbiting aroundan ‘external mass’ GM . The EOB approach is a general relativistic generalization of this fact. It consists in looking for an ‘external spacetime geometry’ g ext mn ( x l ; GM ) such that the geodesic dynamics of a ‘test particle’ of mass m within g ext mn ( x l , GM ) is We henceforth denote by F the Hamiltonian version of the radiation reaction term A RR , Eq. (3),in the (PN-expanded) equations of motion. It can be heuristically computed up to (absolute) 5.5PN[77, 81, 82] and even 6PN [85] order by assuming that the energy radiated in gravitational wavesat infinity is balanced by a loss of the dynamical energy of the binary system.he Effective One Body description of the Two-Body problem 7 equivalent (when expanded in powers of 1 / c ) to the original, relative PN-expandeddynamics (5).Let us explain the idea, proposed in [2], for establishing a ‘dictionary’ betweenthe real relative-motion dynamics, (5), and the dynamics of an ‘effective’ particle ofmass m moving in g ext mn ( x l , GM ) . The idea consists in ‘thinking quantum mechan-ically’ . Instead of thinking in terms of a classical Hamiltonian, H ( qqq , ppp ) (such as H relative2PN , Eq. (5)), and of its classical bound orbits, we can think in terms of thequantized energy levels E ( n , ℓ ) of the quantum bound states of the Hamiltonianoperator H ( ˆ qqq , ˆ ppp ) . These energy levels will depend on two (integer valued) quan-tum numbers n and ℓ . Here (for a spherically symmetric interaction, as appropriateto H relative ), ℓ parametrizes the total orbital angular momentum ( LLL = ℓ ( ℓ + ) ¯ h ),while n represents the ‘principal quantum number’ n = ℓ + n r +
1, where n r (the ‘ra-dial quantum number’) denotes the number of nodes in the radial wave function. Thethird ‘magnetic quantum number’ m (with − ℓ ≤ m ≤ ℓ ) does not enter the energylevels because of the spherical symmetry of the two-body interaction (in the centerof of mass frame). For instance, a non-relativistic Coulomb (or Newton!) interaction H = m ppp + GM m | qqq | (6)gives rise to the well-known result E ( n , ℓ ) = − m (cid:18) GM m n ¯ h (cid:19) , (7)which depends only on n (this is the famous Coulomb degeneracy). When consider-ing the PN corrections to H , as in Eq. (5), one gets a more complicated expressionof the form E relative2PN ( n , ℓ ) = − m a n (cid:20) + a c (cid:16) c n ℓ + c n (cid:17) + a c (cid:16) c n ℓ + c n ℓ + c n ℓ + c n (cid:17) (cid:21) , (8)where we have set a ≡ GM m / ¯ h = G m m / ¯ h , and where we consider, for simplic-ity, the (quasi-classical) limit where n and ℓ are large numbers. The 2PN-accurateresult (8) had been derived by Damour and Sch¨afer [87] as early as 1988. The di-mensionless coefficients c pq are functions of the symmetric mass ratio n ≡ m / M ,for instance c = ( − n + n ) . In classical mechanics ( i.e. for large n and ℓ ),it is called the ‘Delaunay Hamiltonian’, i.e. the Hamiltonian expressed in terms ofthe action variables J = ℓ ¯ h = p H p j d j , and N = n ¯ h = I r + J , with I r = p H p r dr .The energy levels (8) encode, in a gauge-invariant way, the 2PN-accurate relativedynamics of a ‘real’ binary. Let us now consider an auxiliary problem: the ‘effec-tive’ dynamics of one body, of mass m , following a geodesic in some ‘external’ This is related to an idea emphasized many times by John Archibald Wheeler: quantum mechan-ics can often help us in going to the essence of classical mechanics. We consider, for simplicity, ‘equatorial’ motions with m = ℓ , i.e. , classically, q = p . Thibault Damour and Alessandro Nagar (spherically symmetric) metric g ext mn dx m dx n = − A ( R ) c dT + B ( R ) dR + R ( d q + sin q d j ) . (9)Here, the a priori unknown metric functions A ( R ) and B ( R ) will be constructed inthe form of expansions in GM / c R : A ( R ) = + a GMc R + a (cid:18) GMc R (cid:19) + a (cid:18) GMc R (cid:19) + · · · ; B ( R ) = + b GMc R + b (cid:18) GMc R (cid:19) + · · · , (10)where the dimensionless coefficients a n , b n depend on n . From the Newtonian limit,it is clear that we should set a = −
2. By solving (by separation of variables) the‘effective’ Hamilton-Jacobi equation g mn eff ¶ S eff ¶ x m ¶ S eff ¶ x n + m c = , S eff = − E eff t + J eff j + S eff ( R ) , (11)one can straightforwardly compute (in the quasi-classical, large quantum numberslimit) the Delaunay Hamiltonian E eff ( N eff , J eff ) , with N eff = n eff ¯ h , J eff = ℓ eff ¯ h (where N eff = J eff + I eff R , with I eff R = p H p eff R dR , P eff R = ¶ S eff ( R ) / dR ). This yields a result ofthe form E eff ( n eff , ℓ eff ) = m c − m a n (cid:20) + a c (cid:18) c eff11 n eff ℓ eff + c eff20 n (cid:19) + a c (cid:18) c eff13 n eff ℓ + c eff22 n ℓ + c eff31 n ℓ eff + c eff40 n (cid:19) (cid:21) , (12)where the dimensionless coefficients c eff pq are now functions of the unknown coeffi-cients a n , b n entering the looked for ‘external’ metric coefficients (10).At this stage, one needs (as in the famous AdS/CFT correspondence) to define a‘dictionary’ between the real (relative) two-body dynamics, summarized in Eq. (8),and the effective one-body one, summarized in Eq. (12). As, on both sides, quantummechanics tells us that the action variables are quantized in integers ( N real = n ¯ h , N eff = n eff ¯ h , etc.) it is most natural to identify n = n eff and ℓ = ℓ eff . One then stillneeds a rule for relating the two different energies E relativereal and E eff . Ref. [2] proposedto look for a general map between the real energy levels and the effective ones(which, as seen when comparing (8) and (12), cannot be directly identified because It is convenient to write the ‘external metric’ in Schwarzschild-like coordinates. Note that theexternal radial coordinate R differs from the two-body ADM-coordinate relative distance R ADM = | qqq | . The transformation between the two coordinate systems has been determined in Refs. [2, 5].he Effective One Body description of the Two-Body problem 9 Fig. 1
Sketch of the correspondence between the quantized energy levels of the real and effectiveconservative dynamics. n denotes the ‘principal quantum number’ ( n = n r + ℓ +
1, with n r = , , . . . denoting the number of nodes in the radial function), while ℓ denotes the (relative) orbital angularmomentum ( LLL = ℓ ( ℓ + ) ¯ h ) . Though the EOB method is purely classical, it is conceptuallyuseful to think in terms of the underlying (Bohr-Sommerfeld) quantization conditions of the actionvariables I R and J to motivate the identification between n and ℓ in the two dynamics. they do not include the same rest-mass contribution ), namely E eff m c − = f (cid:18) E relativereal m c (cid:19) = E relativereal m c + a E relativereal m c + a (cid:18) E relativereal m c (cid:19) + · · · ! . (13)The ‘correspondence’ between the real and effective energy levels is illustrated inFig. 1Finally, identifying E eff ( n , ℓ ) / m c to f ( E relativereal / m c ) yields six equations, relat-ing the six coefficients c eff pq ( a , a ; b , b ) to the six c pq ( n ) and to the two energycoefficients a and a . It is natural to set b = + M = m + m ).One then finds that there exists a unique solution for the remaining five unknowncoefficients a , a , b , a and a . This solution is very simple: a = , a = n , b = − n , a = n , a = . (14) Indeed E totalreal = Mc + E relativereal = Mc + Newtonian terms + / c + · · · , while E effective = m c + N + / c + · · · .0 Thibault Damour and Alessandro Nagar Note, in particular, that the map between the two energies is simply E eff m c = + E relativereal m c (cid:18) + n E relativereal m c (cid:19) = s − m c − m c m m c (15)where s = ( E totreal ) ≡ ( Mc + E relativereal ) is Mandelstam’s invariant = − ( p + p ) .Note also that, at 2PN accuracy, the crucial ‘ g ext00 ’ metric coefficient A ( R ) (whichfully encodes the energetics of circular orbits) is given by the remarkably simplePN expansion A ( R ) = − u + n u , (16)where u ≡ GM / ( c R ) and n ≡ m / M ≡ m m / ( m + m ) .The dimensionless parameter n ≡ m / M varies between 0 (in the test masslimit m ≪ m ) and (in the equal-mass case m = m ). When n →
0, Eq. (16)yields back, as expected, the well-known Schwarzschild time-time metric coeffi-cient − g Schw00 = − u = − GM / c R . One therefore sees in Eq. (16) the rˆole of n as a deformation parameter connecting a well-known test-mass result to a non trivialand new 2PN result. It is also to be noted that the 1PN EOB result A ( R ) = − u happens to be n -independent, and therefore identical to A Schw = − u . This is re-markable in view of the many non-trivial n -dependent terms in the 1PN relativedynamics. The physically real 1PN n -dependence happens to be fully encoded inthe function f ( E ) mapping the two energy spectra given in Eq. (15) above.Let us emphasize the remarkable simplicity of the 2PN result (16). The 2PNHamiltonian (5) contains eleven rather complicated n -dependent terms. After trans-formation to the EOB format, the dynamical information contained in these elevencoefficients gets condensed into the very simple additional contribution + n u in A ( R ) , together with an equally simple contribution in the radial metric coefficient: ( A ( R ) B ( R )) = − n u . This condensation process is even more drastic whenone goes to the next (conservative) post-Newtonian order: the 3PN level, i.e. ad-ditional terms of order O ( / c ) in the Hamiltonian (5). As mentioned above, thecomplete obtention of the 3PN dynamics has represented quite a theoretical chal-lenge and the final, resulting Hamiltonian is quite complicated. Even after going tothe center of mass frame, the 3PN additional contribution c H ( qqq , ppp ) to Eq. (5) in-troduces eleven new complicated n -dependent coefficients. After transformation tothe EOB format [5], these eleven new coefficients get “condensed” into only three additional terms: (i) an additional contribution to A ( R ) , (ii) an additional contribu-tion to B ( R ) , and (iii) a O ( ppp ) modification of the ‘external’ geodesic Hamiltonian.For instance, the crucial 3PN g ext00 metric coefficient becomes A ( R ) = − u + n u + a n u , (17)where u = GM / ( c R ) , a = − p ≃ . , (18) he Effective One Body description of the Two-Body problem 11 u A P [ A ] P [ A ( a = 0)] P [ A ( a = 0 , a = 0)] Fig. 2
Various approximations and Pad´e resummation of the EOB radial potential A ( u ) , where u = GM / ( c R ) , for the equal-mass case n = /
4. The vertical dashed lines indicate the corresponding(adiabatic) LSO location [2] defined by the condition d E / dR = d E / dR =
0, where E is theeffective energy along the sequence of circular orbits ( i.e. , when P eff R = while the additional contribution to B ( R ) gives D ( R ) ≡ ( A ( R ) B ( R )) = − n u + ( n − ) n u . (19)Remarkably, it is found that the very simple 2PN energy map Eq. (15) does not needto be modified at the 3PN level.The fact that the 3PN coefficient a in the crucial ‘effective radial potential’ A ( R ) , Eq. (17), is rather large and positive indicates that the n -dependent non-linear gravitational effects lead, for comparable masses ( n ∼ ), to a last stable (cir-cular) orbit (LSO) which has a higher frequency and a larger binding energy thanwhat a naive scaling from the test-particle limit ( n → ) would suggest. Actually,the PN-expanded form (17) of A ( R ) does not seem to be a good representationof the (unknown) exact function A EOB ( R ) when the (Schwarzschild-like) relativecoordinate R becomes smaller than about 6 GM / c (which is the radius of the LSOin the test-mass limit). In fact, by continuity with the test-mass case, one a prioriexpects that A ( R ) always exhibits a simple zero defining an EOB “effective hori-zon” that is smoothly connected to the Schwarzschild event horizon at R = GM / c when n →
0. However, the large value of the a coefficient does actually prevent A to have this property when n is too large, and in particular when n = /
4, as itis visually explained in Fig. 2. The black curves in the figure represent the A func-tion at 1PN (solid line), 2PN (dashed line) and 3PN (dash-dot line) approximation:while the 2PN curve still has a simple zero, the 3PN does not, due to the large value of a . It was therefore suggested [5] to further resum A ( R ) by replacing it by asuitable Pad´e ( P ) approximant. For instance, the replacement of A ( R ) by A ( R ) ≡ P [ A ( R )] = + n u + d u + d u + d u (20)ensures that the n = case is smoothly connected with the n = .The use of Eq. (20) was suggested before one had any (reliable) non-perturbativeinformation on the binding of close black hole binaries. Later, a comparison withsome “waveless” numerical simulations of circular black hole binaries [89] hasgiven some evidence that Eq. (20) is physically adequate. In Refs. [4, 89] it was alsoemphasized that, in principle, the comparison between numerical data and EOB-based predictions should allow one to determine the effect of the unknown higherPN contributions to Eq. (17). For instance, one can add a 4PN-like term + a n u or a 5PN-like term + a n u in Eq. (17), and then Pad´e the resulting radial function.The new resummed A potential will exhibit an explicit dependence on a (at 4PN)or ( a , a ) (at 5PN), that is A ( R ; a , n ) = P h A ( R ) + n a u i , (21)or A ( R ; a , a , n ) = P h A ( R ) + n a u + n a u i . (22)Comparing the predictions of A ( R ; a , n ) or A ( R ; a , a , n ) to numerical data mightthen determine what is the physically preferred “effective” value of the unknown co-efficient a (if working at 4PN effective accuracy) or of the doublet ( a , a ) (whenincluding also 5PN corrections). For illustrative purposes, Fig. 2 shows the effectof the Pad´e resummation with a = a = n = /
4. Note that the Pad´e re-summation procedure is injecting some “information” beyond that contained in thenumerical values of the PN expansion coefficients a n ’s of A ( R ) . As a consequence,the operation of Pad´eing and of restricting a and a to the (3PN-compatible) val-ues a = = a do not commute: A ( R ; 0 , / ) = A ( R ; 0 , , / ) = A ( R , / ) .In this respect, let us also mention that the 4PN a -dependent Pad´e approximant A ( R ; a , n ) exactly reduces to the 3PN Pad´e approximant A ( R ; n ) when a is re-placed by the following function of n a ( n ) ≡ n ( − p ) ( n − ) . (23) The PN-expanded EOB building blocks A ( R ) , B ( R ) , . . . already represent a resummation of thePN dynamics in the sense that they have “condensed” the many terms of the original PN-expandedHamiltonian within a very concise format. But one should not refrain to further resum the EOBbuilding blocks themselves, if this is physically motivated. We recall that the coefficient n and ( d , d , d ) of the Pad´e approximant are determined bythe condition that the first four terms of the Taylor expansion of A in powers of u = GM / ( c R ) coincide with A .he Effective One Body description of the Two-Body problem 13 Note that the value of the A -reproducing effective 4PN coefficient a ( n ) in theequal mass case is a ( / ) ≃ − . a = − .
16 quoted in Ref. [28] (but note that the correct A -reproducing4PN coefficient depends on the symmetric mass ratio n ). Similarly, when work-ing at the 5PN level, A ( R ; a , a , n ) exactly reduces to the 4PN Pad´e approximant A ( R ; a , n ) when a is replaced by the following function of both n and a : a ( n , a ) ≡ n (cid:0) a + (cid:0) − p (cid:1) a + (cid:0) − p (cid:1) (cid:0) ( n + ) − p (cid:1)(cid:1) [( − p ) n − ] . (24)The use of numerical relativity data to constrain the values of the higher PN param-eters ( a , a ) is an example of the useful flexibility [88] of analytical approaches:the fact that one can tap numerically-based, non-perturbative information to im-prove the EOB approach. The flexibility of the EOB approach related to the use ofthe a -dependent radial potential A ( R ; a , n ) has been exploited in several recentworks [33, 37, 38, 39, 28, 41] focusing on the comparison of EOB-based wave-forms with waveforms computed via numerical relativity simulations. Collectively,all these studies have shown that it is possible to constrain a (together with otherflexibility parameters related to the resummation of radiation reaction, see below)so as to yield an excellent agreement (at the level of the published numerical errors)between EOB and numerical relativity waveforms. The result, however, cannot besummarized by stating that a is constrained to be in the vicinity of a special nu-merical value. Rather, one finds a strong correlation between a and other parame-ters, notably the radiation reaction parameter v pole introduced below. More recently,Ref. [40] could get rid of the flexibility parameters (such as v pole ) related to theresummation of radiation reaction, and has shown that one can get an excellentagreement with numerical relativity data by using only the flexibility in the doublet ( a , a ) (the other parameters being essentially fixed internally to the formalism).We shall discuss this result further in Sec. 5 below.The same kind of n -continuity argument discussed so far for the A function needsto be applied also to the D ( R ) function defined in Eq. (19). A straightforwardway to ensure that the D function stays positive when R decreases (since it is D = n →
0) is to replace D ( R ) by D ( R ) ≡ P [ D ( R )] , where P indicatesthe ( , ) Pad´e approximant and explicitly reads D ( R ) = + n u − ( n − ) n u . (25)The resummation of A (via Pad´e approximants) is necessary for ensuring the exis-tence and n -continuity of a last stable orbit (see vertical lines in Fig. 2), as well asthe existence and n -continuity of a last unstable orbit , i.e. of a n -deformed analogof the light ring R = GM / c when n →
0. We recall that, when n =
0, the lightring corresponds to the circular orbit of a massless particle, or of an extremely rel- ativistic massive particle, and is technically defined by looking for the maximumof A ( R ) / R , i.e. by solving ( d / dR )( A ( R ) / R ) =
0. When n = n -deformed light ring” (technically defined by solving ( d / dR )( A ( R : n ) / R ) =
0) is to entail, in its vicinity, the existence of a maximumof the orbital frequency W = d j / dt (the resummation of D ( R ) plays a useful rolein ensuring the n -continuity of this plunge behavior). In the previous Section we have described how the EOB method encodes the con-servative part of the relative orbital dynamics into the dynamics of an ’effective’particle. Let us now briefly discuss how to complete the EOB dynamics by definingsome resummed expressions describing radiation reaction effects. One is interestedin circularized binaries, which have lost their initial eccentricity under the influenceof radiation reaction. For such systems, it is enough (as shown in [3]) to include aradiation reaction force in the p j equation of motion only. More precisely, we areusing phase space variables r , p r , j , p j associated to polar coordinates (in the equa-torial plane q = p ). Actually it is convenient to replace the radial momentum p r bythe momentum conjugate to the ‘tortoise’ radial coordinate R ∗ = R dR ( B / A ) / , i.e.P R ∗ = ( A / B ) / P R . The real EOB Hamiltonian is obtained by first solving Eq. (15)to get E totalreal = √ s in terms of E eff , and then by solving the effective Hamiltonian-Jacobi equation to get E eff in terms of the effective phase space coordinates qqq eff and ppp eff . The result is given by two nested square roots (we henceforth set c = H EOB ( r , p r ∗ , j ) = H realEOB m = n q + n ( ˆ H eff − ) , (26)where ˆ H eff = vuut p r ∗ + A ( r ) + p j r + z p r ∗ r ! , (27)with z = n ( − n ) . Here, we are using suitably rescaled dimensionless (effective)variables: r = R / GM , p r ∗ = P R ∗ / m , p j = P j / m GM , as well as a rescaled time t = T / GM . This leads to equations of motion ( r , j , p r ∗ , p j ) of the form Completed by the O ( ppp ) terms that must be introduced at 3PN.he Effective One Body description of the Two-Body problem 15 d j dt = ¶ ˆ H EOB ¶ p j ≡ W , (28) drdt = (cid:18) AB (cid:19) / ¶ ˆ H EOB ¶ p r ∗ , (29) d p j dt = ˆ F j , (30) d p r ∗ dt = − (cid:18) AB (cid:19) / ¶ ˆ H EOB ¶ r , (31)which explicitly read d j dt = Ap j n r ˆ H ˆ H eff ≡ W , (32) drdt = (cid:18) AB (cid:19) / n ˆ H ˆ H eff (cid:18) p r ∗ + z Ar p r ∗ (cid:19) , (33) d p j dt = ˆ F j , (34) d p r ∗ dt = − (cid:18) AB (cid:19) / n ˆ H ˆ H eff ( A ′ + p j r (cid:18) A ′ − Ar (cid:19) + z (cid:18) A ′ r − Ar (cid:19) p r ∗ ) , (35)where A ′ = dA / dr . As explained above the EOB metric function A ( r ) is definedby Pad´e resumming the Taylor-expanded result (10) obtained from the matchingbetween the real and effective energy levels (as we were mentioning, one uses asimilar Pad´e resumming for D ( r ) ≡ A ( r ) B ( r ) ). One similarly needs to resum ˆ F j ,i.e., the j component of the radiation reaction which has been introduced on ther.h.s. of Eq. (30). During the quasi-circular inspiral ˆ F j is known (from the PNwork mentioned in Section 2 above) in the form of a Taylor expansion of the formˆ F Taylor j = − nW r w ˆ F Taylor ( v j ) , (36)where v j ≡ W r w , and r w ≡ r [ y ( r , p j )] / is a modified EOB radius, with y beingdefined as y ( r , p j ) = r (cid:18) dA ( r ) dr (cid:19) − + n vuut A ( r ) + p j r ! − , (37)which generalizes the 2PN-accurate Eq. (22) of Ref. [90]. In Eq. (36) we have de-fined ˆ F Taylor ( v ) = + A ( n ) v + A ( n ) v + A ( n ) v + A ( n ) v + A ( n , log v ) v + A ( n ) v + A ( n = , log v ) v , (38) Fig. 3
The extreme-mass-ratio limit ( n = ℓ =
6) obtained via perturbationtheory. where we have added to the known 3.5PN-accurate comparable-mass result thesmall-mass-ratio 4PN contribution [91]. We recall that the small-mass contributionto the Newton-normalized flux is actually known up to 5.5PN order, i.e. to v in-cluded. The standard Taylor expansion of the flux, (38), has rather poor convergenceproperties when considered up to the LSO. This is illustrated in Fig. 3 in the small-mass limit n =
0. The convergence of the PN-expanded flux can be studied in detailin the n = F ℓ max = ℓ max (cid:229) ℓ = ℓ (cid:229) m = F ℓ m , (39)where F ℓ m = F ℓ | m | already denotes the sum of two equal contributions correspondingto + m and − m ( m = F ℓ vanishes for circular orbits). To be precise, the “exact”result exhibited in Fig. 3 is given by the rather accurate approximation F ( ) obtainedby choosing ℓ max =
6; i.e., by truncating the sum over ℓ in Eq. (39) beyond ℓ =
6. Inaddition, one normalizes the result onto the “Newtonian” (i.e., quadrupolar) result F N = / ( m / M ) v . In other words, the solid line in Fig. 3 represents the quantityˆ F ≡ F ( ) / F N .For clarity, we selected only three Taylor approximants: 3PN ( v ), 3.5PN ( v ) and 5.5PN ( v ). These three values suffice to illustrate the rather large scatter he Effective One Body description of the Two-Body problem 17 among Taylor approximants, and the fact that, near the LSO, the convergence to-wards the exact value (solid line) is rather slow, and non monotonic. [See also Fig. 1in Ref. [94] and Fig. 3 of Ref. [49] for fuller illustrations of the scattered and nonmonotonic way in which successive Taylor expansions approach the numerical re-sult.] The results shown in Fig. 3 elucidate that the Taylor series (38) is inadequateto give a reliable representation of the energy loss during the plunge. That is the rea-son why the EOB formalism advocates the use of a “resummed” version of F j , i.e.a nonpolynomial function replacing Eq. (38) at the r.h.s. of the Hamilton’s equation(and coinciding with it in in the v / c ≪ n →
0) and thengeneralized to the comparable mass case. ˆ F Taylor using a one-parameter family of Pad´eapproximants: tuning v pole Following [49], one resums ˆ F Taylor by using the following Pad´e resummation ap-proach. First, one chooses a certain number v pole which is intended to represent thevalue of the orbital velocity v j at which the exact angular momentum flux wouldbecome infinite if one were to formally analytically continue ˆ F j along unstable circular orbits below the Last Stable Orbit (LSO): then, given v pole , one defines theresummed ˆ F ( v j ) asˆ F resummed ( v j ) = (cid:18) − v j v pole (cid:19) − P (cid:20)(cid:18) − v j v pole (cid:19) ˆ F Taylor ( v j ; n = ) (cid:21) , (40)where P denotes a ( , ) Pad´e approximant.If one first follows the reasoning line of [49], and fixes the location of the pole inthe resummed flux at the standard Schwarzschild value v ( n = ) pole = / √
3, one gets theresult in Fig. 4. By comparison to Fig. 3, one can appreciate the significantly better
Fig. 4
The extreme-mass-ratio limit ( n = v pole = / √
3. The sequence of Pad´e approximants isless scattered than the corresponding Taylor ones and closer to the exact result. (and monotonic) way in which successive
Pad´e approximants approach (in L ¥ normon the full interval 0 < x < x LSO ) the numerical result. Ref. [49] also showed that theobservationally relevant overlaps (of both the “faithfulness” and the “effectualness”types) between analytical and numerical adiabatic signals were systematically betterfor Pad´e approximants than for Taylor ones. Note that this figure is slightly differentfrom the corresponding results in panel (b) of Fig. 3 in [49] (in particular, the presentresult exhibits a better “convergence” of the v curve). This difference is due to thenew treatment of the logarithmic terms (cid:181) log x . Instead of factoring them out infront as proposed in [49], we consider them here (following [37]) as being part ofthe “Taylor coefficients” f n ( log x ) when Pad´eing the flux function.A remarkable improvement in the ( L ¥ ) closeness between ˆ F Pad´e-resummed ( v ) andˆ F Exact ( v ) can be obtained, as suggested by Damour and Nagar [37] (following ideasoriginally introduced in Ref. [97]), by suitably flexing the value of v pole . As pro-posed in Ref. [37], v pole is tuned until the difference between the resummed and theexact flux at the LSO is zero (or at least smaller than 10 − ). The resulting closenessbetween the exact and tuned-resummed fluxes is illustrated in Fig. 5. It is so good(compared to the previous figures, where the differences were clearly visible) thatwe need to complement the figure with Table 1. This table compares in a quantitativeway the result of the “untuned” Pad´e resummation ( v pole = / √
3) of Ref. [49] to theresult of the “ v pole -tuned” Pad´e resummation described here. Defining the function D ˆ F ( v ; v pole ) = ˆ F Resummed ( v ; v pole ) − ˆ F Exact ( v ) measuring the difference between aresummed and the exact energy flux, Table 1 lists both the values of D ˆ F at v = v LSO and its L ¥ norm on the interval 0 < v < v LSO for both the untuned and tuned cases. he Effective One Body description of the Two-Body problem 19
Fig. 5
The extreme mass ratio limit ( n = flexing the value of the parameter v pole so to improve the agreement with the exact result. Table 1
Errors in the flux of the two (untuned or tuned) Pad´e resummation procedures. Fromleft to right, the columns report: the PN-order; the difference between the resummed and the exactflux, D ˆ F = ˆ F Resummed − ˆ F Exact , at the LSO, and the L ¥ norm of D ˆ F , || D ˆ F || ¥ (computed over theinterval 0 < v < v LSO ), for v pole = / √
3; the flexed value of v pole used here; ˆ D F at the LSO andthe corresponding L ¥ norm (over the same interval) for the flexed value of v pole .PN-order D ˆ F / √ || D ˆ F || / √ ¥ v pole D ˆ F v pole LSO || D ˆ F || v pole ¥ v ) -0.048 0.048 0 . . × − v ) -0.051 0.051 0 . . × − v ) -0.022 0.022 0 . . × − Note, in particular, how the v pole -flexing approach permits to reduce the L ¥ normover this interval by more than an order of magnitude with respect to the untunedcase. Note that the closeness between the tuned flux and the exact one is remarkablygood (4 . × − ) already at the 3PN level.It has recently been shown in several works [37, 38, 39, 41] that the flexibility in the choice of v pole could be advantageously used to get a close agreement withNR data (at the level of the numerical error). We will not comment here any furtheron this parameter-dependent resummation procedure of the energy flux and addressthe reader to the aforementioned references for further details. In this section we shall introduce the reader to the new resummation technique forthe multipolar waveform (and thus for the energy flux) introduced in Ref. [36, 37]and perfected in [50]. The aim is to summarize here the main ideas discussed in [50]as well as to collect most of the relevant equations that are useful for implementa-tion in the EOB dynamics. To be precise, the new results discussed in Ref. [50] aretwofold: on the one hand, that work generalized the ℓ = m = new resumma-tion procedure which consists in considering a new theoretical quantity, denoted as r ℓ m ( x ) , which enters the ( ℓ, m ) waveform (together with other building blocks, seebelow) only through its ℓ -th power: h ℓ m (cid:181) ( r ℓ m ( x )) ℓ . Here, and below, x denotes theinvariant PN-ordering parameter x ≡ ( GM W / c ) / .The main novelty introduced by Ref. [50] is to write the ( ℓ, m ) multipolar wave-form emitted by a circular nonspinning compact binary as the product of severalfactors, namely h ( e ) ℓ m = GM n c R n ( e ) ℓ m c ℓ + e ( n ) x ( ℓ + e ) / Y ℓ − e , − m (cid:16) p , F (cid:17) ˆ S ( e ) eff T ℓ m e i d ℓ m r ℓℓ m . (41)Here e denotes the parity of ℓ + m ( e = p ( ℓ + m ) ), i.e. e = ℓ + m even), and e = ℓ + m odd); n ( e ) ℓ m and c ℓ + e ( n ) are numerical coefficients; ˆ S ( e ) eff is a m -normalizedeffective source (whose definition comes from the EOB formalism); T ℓ m is a re-summed version [36, 37] of an infinite number of “leading logarithms” entering the tail effects [69, 103]; d ℓ m is a supplementary phase (which corrects the phase effectsnot included in the complex tail factor T ℓ m ), and, finally, ( r ℓ m ) ℓ denotes the ℓ -thpower of the quantity r ℓ m which is the new building block introduced in [50]. Notethat in previous papers [36, 37] the quantity ( r ℓ m ) ℓ was denoted as f ℓ m and we willmainly use this notation below. Before introducing explicitly the various elementsentering the waveform (41) it is convenient to decompose h ℓ m as h ℓ m = h ( N , e ) ℓ m ˆ h ( e ) ℓ m , (42)where h ( N , e ) ℓ m is the Newtonian contribution and ˆ h ( e ) ℓ m ≡ ˆ S ( e ) eff T ℓ m e i d ℓ m f ℓ m represents aresummed version of all the PN corrections. The PN correcting factor ˆ h ( e ) ℓ m , as wellas all its building blocks, has the structure ˆ h ( e ) ℓ m = + O ( x ) .Entering now in the discussion of the explicit form of the elements enteringEq. (41), we have that the n -independent numerical coefficients are given by he Effective One Body description of the Two-Body problem 21 n ( ) ℓ m = ( i m ) ℓ p ( ℓ + ) !! s ( ℓ + )( ℓ + ) ℓ ( ℓ − ) , (43) n ( ) ℓ m = − ( i m ) ℓ p i ( ℓ + ) !! s ( ℓ + )( ℓ + )( ℓ − m )( ℓ − )( ℓ + ) ℓ ( ℓ − ) , (44)while the n -dependent coefficients c ℓ + e ( n ) (such that | c ℓ + e ( n = ) | = n (as in Ref. [99, 101]), although they are more convenientlywritten in terms of the two mass ratios X = m / M and X = m / M in the form c ℓ + e ( n ) = X ℓ + e − + ( − ) ℓ + e X ℓ + e − = X ℓ + e − + ( − ) m X ℓ + e − . (45)In the second form of the equation we have used the fact that, as e = p ( ℓ + m ) , p ( ℓ + e ) = p ( m ) .Let us turn now to discussing the structure of the ˆ S ( e ) eff and T ℓ m factors. To this aim,following Ref. [50], we recall that the along the sequence of EOB circular orbits,which are determined by the condition ¶ u (cid:8) A ( u )[ + j u ] (cid:9) =
0, the effective EOBHamiltonian (per unit m mass) readsˆ H eff = H eff m = q A ( u )( + j u ) (circular orbits) . (46)where the squared angular momentum is given by j ( u ) = − A ′ ( u )( u A ( u )) ′ (circular orbits) , (47)with the prime denoting d / du . Inserting this u -parametric representation of j in Eq. (46) defines the u -parametric representation of the effective Hamiltonianˆ H eff ( u ) . In the even-parity case (corresponding to mass moments), since the lead-ing order source of gravitational radiation is given by the energy density, Ref. [50]defined the even-parity “source factor” asˆ S ( ) eff ( x ) = ˆ H eff ( x ) ℓ + m even , (48)where x = ( GM W / c ) / . In the odd-parity case, they explored two, equally mo-tivated, possibilities. The first one consists simply in still factoring ˆ H eff ( x ) ; i.e., indefining ˆ S ( , H ) eff = ˆ H eff ( x ) also when ℓ + m is odd. The second one consists in fac-toring the angular momentum J . Indeed, the angular momentum density e i jk x j t k enters as a factor in the (odd-parity) current moments, and J occurs (in the small- n limit) as a factor in the source of the Regge-Wheeler-Zerilli odd-parity multipoles.This leads us to define as second possibilityˆ S ( , J ) eff = ˆ j ( x ) ≡ x / j ( x ) ℓ + m odd , (49) where ˆ j denotes what can be called the “Newton-normalized” angular momentum,namely the ratio ˆ j ( x ) = j ( x ) / j N ( x ) with j N ( x ) = / √ x . In Ref. [50] the relativemerits of the two possible choices were discussed. Although the analysis in theadiabatic n = J -factorization in the following, that we willtreat as our standard choice.The second building block in our factorized decomposition is the “tail factor” T ℓ m (introduced in Refs. [36, 37]). As mentioned above, T ℓ m is a resummed versionof an infinite number of “leading logarithms” entering the transfer function betweenthe near-zone multipolar wave and the far-zone one, due to tail effects linked toits propagation in a Schwarzschild background of mass M ADM = H realEOB . Its explicitexpression reads T ℓ m = G ( ℓ + − k ) G ( ℓ + ) e p ˆˆ k e k log ( kr ) , (50)where r = GM and ˆˆ k ≡ GH realEOB m W and k ≡ m W . Note that ˆˆ k differs from k by arescaling involving the real (rather than the effective ) EOB Hamiltonian, computedat this stage along the sequence of circular orbits.The tail factor T ℓ m is a complex number which already takes into account someof the dephasing of the partial waves as they propagate out from the near zone toinfinity. However, as the tail factor only takes into account the leading logarithms,one needs to correct it by a complementary dephasing term, e i d ℓ m , linked to sublead-ing logarithms and other effects. This subleading phase correction can be computedas being the phase d ℓ m of the complex ratio between the PN-expanded ˆ h ( e ) ℓ m and theabove defined source and tail factors. In the comparable-mass case ( n = d phase correction to the leading quadrupolar wave was originally computed inRef. [37] (see also Ref. [36] for the n = n -dependent waveform [101] have been obtained in [50].The last factor in the multiplicative decomposition of the multipolar waveformcan be computed as being the modulus f ℓ m of the complex ratio between the PN-expanded ˆ h ( e ) ℓ m and the above defined source and tail factors. In the comparable masscase ( n = f modulus correction to the leading quadrupolar wave was com-puted in Ref. [37] (see also Ref. [36] for the n = f ℓ m ’s to the highest possible PN-accuracy by starting from the currently known 3PN-accurate n -dependent wave-form [101]. In addition, as originally proposed in Ref. [37], to reach greater accu-racy the f ℓ m ( x ; n ) ’s extracted from the 3PN-accurate n = n = f case discussed in [37], this amounted to adding 4PN and 5PN n = not equivalent to the straightforward hybrid sum ansatz, he Effective One Body description of the Two-Body problem 23 ˜ h ℓ m = ˜ h known ℓ m ( n ) + ˜ h higher ℓ m ( n = ) (where ˜ h ℓ m ≡ h ℓ m / n ) that one may have thoughtto implement.In the even-parity case, the determination of the modulus f ℓ m is unique. In theodd-parity case, it depends on the choice of the source which, as explained above,can be connected either to the effective energy or to the angular momentum. We willconsider both cases and distinguish them by adding either the label H or J to thecorresponding f ℓ m . Note, in passing, that, since in both cases the factorized effectivesource term ( H eff or J ) is a real quantity, the phases d ℓ m ’s are the same.The above explained procedure defines the f ℓ m ’s as Taylor-expanded PN seriesof the type f ℓ m ( x ; n ) = + c f ℓ m ( n ) x + c f ℓ m ( n ) x + c f ℓ m ( n , log ( x )) x + . . . (51)Note that one of the virtues of our factorization is to have separated the half-integerpowers of x appearing in the usual PN-expansion of h ( e ) ℓ m from the integer powers,the tail factor, together with the complementary phase factor e i d ℓ m , having absorbedall the half-integer powers. In Ref. [39] all the f ℓ m ’s (both for the H and J choices)have been computed up to the highest available ( n -dependent or not) PN accuracy.In the formulas for the f ℓ m ’s given below we “hybridize” them by adding to theknown n -dependent coefficients c f ℓ m n ( n ) in Eq. (51) the n = c f ℓ m n ′ ( n = ) . The 1PN-accurate f ℓ m ’s for ℓ + m even and and alsofor ℓ + m odd can be written down for all ℓ . The complete result for the f ℓ m ’s thatare known with an accuracy higher than 1PN are listed in Appendix B of Ref. [39].Here, for illustrative purposes, we quote only the lowest f even ℓ m and f odd , J ℓ m up to ℓ = f ( x ; n ) = + ( n − ) x + (cid:0) n − n − (cid:1) x + (cid:18) n − n + p n − n − ( x ) + (cid:19) x + (cid:18) ( x ) − (cid:19) x + (cid:18) ( x ) − (cid:19) x + O ( x ) , (52) f J ( x ; n ) = + (cid:18) n − (cid:19) x + (cid:18) n − n − (cid:19) x + (cid:18) − ( x ) (cid:19) x + (cid:18) ( x ) − (cid:19) x + O ( x ) , (53) f ( x ; n ) = + (cid:18) n − (cid:19) x + (cid:18) n − n − (cid:19) x + (cid:18) −
787 eulerlog ( x ) (cid:19) x + (cid:18)
39 eulerlog ( x ) − (cid:19) x + O ( x ) , (54) f J ( x ; n ) = + n − n + ( n − ) x + n − n + n − ( n − ) x + (cid:18) − ( x ) (cid:19) x + O ( x ) , (55) f ( x ; n ) = + (cid:18) − n − (cid:19) x + (cid:18) − n − n + (cid:19) x + (cid:18) − ( x ) (cid:19) x + (cid:18) ( x ) − (cid:19) x + O ( x ) . (56)For convenience and readability, we have introduced the following “eulerlog” func-tions eulerlog m ( x ) eulerlog m ( x ) = g E + log2 +
12 log x + log m , where g E = . . . . is Euler’s constant.The decomposition of the total PN-correction factor ˆ h ( e ) ℓ m into several factors isin itself a resummation procedure which has already improved the convergence ofthe PN series one has to deal with: indeed, one can see that the coefficients en-tering increasing powers of x in the f ℓ m ’s tend to be systematically smaller thanthe coefficients appearing in the usual PN expansion of ˆ h ( e ) ℓ m . The reason for this isessentially twofold: (i) the factorization of T ℓ m has absorbed powers of m p whichcontributed to make large coefficients in ˆ h ( e ) ℓ m , and (ii) the factorization of either ˆ H eff or ˆ j has (in the n = x = / x n in any PN-expandedquantity to grow as 3 n as n → ¥ . To prevent some potential misunderstandings,let us emphasize that we are talking here about a singularity entering the analyticcontinuation (to larger values of x ) of a mathematical function h ( x ) defined (forsmall values of x ) by considering the formal adiabatic circular limit. The point isthat, in the n → n of the PN coefficients of h ( x ) (Taylor-expanded at x = h ( x ) which is nearest to x = x -plane. In the n → x -plane comesfrom the source factor ˆ H eff ( x ) or ˆ j ( x ) in the waveform and is located at the light-ring x LR ( n = ) = /
3. In the n = d ˆ H eff / dx and d ˆ j / dx have inverse square-root singularities located at what he Effective One Body description of the Two-Body problem 25 is called [3, 31, 33, 38, 37] the (Effective) “EOB-light-ring”, i.e., the (adiabatic)maximum of W , x adiabELR ( n ) ≡ (cid:0) M W adiabmax (cid:1) / & / f ℓ m ( x ) ’s quoted above does not seem to be good enough, especially nearor below the LSO, in view of the high-accuracy needed to define gravitational wavetemplates. For this reason, Refs. [36, 37] proposed to further resum the f ( x ) func-tion via a Pad´e (3,2) approximant, P { f ( x ; n ) } , so as to improve its behavior in thestrong-field-fast-motion regime. Such a resummation gave an excellent agreementwith numerically computed waveforms, near the end of the inspiral and during thebeginning of the plunge, for different mass ratios [36, 38, 39]. As we were mention-ing above, a new route for resumming f ℓ m was explored in Ref. [50]. It is based onreplacing f ℓ m by its ℓ -th root, say r ℓ m ( x ; n ) = [ f ℓ m ( x ; n )] /ℓ . (57)The basic motivation for replacing f ℓ m by r ℓ m is the following: the leading “Newtonian-level” contribution to the waveform h ( e ) ℓ m contains a factor w ℓ r ℓ harm v e where r harm isthe harmonic radial coordinate used in the MPM formalism [66, 68] . When com-puting the PN expansion of this factor one has to insert the PN expansion of the(dimensionless) harmonic radial coordinate r harm , r harm = x − ( + c x + O ( x )) , asa function of the gauge-independent frequency parameter x . The PN re-expansion of [ r harm ( x )] ℓ then generates terms of the type x − ℓ ( + ℓ c x + .... ) . This is one (thoughnot the only one) of the origins of 1PN corrections in h ℓ m and f ℓ m whose coefficientsgrow linearly with ℓ . The study of [50] has pointed out that these ℓ -growing termsare problematic for the accuracy of the PN-expansions. Our replacement of f ℓ m by r ℓ m is a cure for this problem. More explicitly, the the investigation of 1PN correc-tions to GW amplitudes [66, 68, 99] has shown that, in the even-parity case (but seealso Appendix A of Ref. [50] for the odd-parity case), c f ℓ m ( n ) = − ℓ (cid:16) − n (cid:17) + + c ℓ + ( n ) c ℓ ( n ) − b ℓ ( n ) c ℓ ( n ) − c ℓ + ( n ) c ℓ ( n ) m ( ℓ + ) ( ℓ + )( ℓ + ) , (58)where c ℓ ( n ) is defined in Eq. (45) and b ℓ ( n ) ≡ X ℓ + ( − ) ℓ X ℓ . (59)Focusing on the n = n dependence of c f ℓ m ( n ) isquite mild [50]), the above result shows that the PN expansion of f ℓ m starts as f even ℓ m ( x ; 0 ) = − ℓ x (cid:18) − ℓ + m ( ℓ + ) ℓ ( ℓ + )( ℓ + ) (cid:19) + O ( x ) . (60) Beware that this “Effective EOB-light-ring” occurs for a circular-orbit radius slightly largerthan the purely dynamical (circular) EOB-light-ring (where H eff and J would formally becomeinfinite).6 Thibault Damour and Alessandro Nagar Fig. 6
Performance of the new resummation procedure described in Ref. [50]. The total GW fluxˆ F (up to ℓ max =
6) computed from inserting in Eq. (62) the factorized waveform (41) with theTaylor-expanded r ℓ m ’s (with either 3PN or 5PN accuracy for r ) is compared with the “exact”numerical data. The crucial thing to note in this result is that as ℓ gets large (keeping in mind that | m | ≤ ℓ ), the coefficient of x will be negative and will approximately range between − ℓ/ − ℓ . This means that when ℓ ≥ f ℓ m would byitself make f ℓ m ( x ) vanish before the ( n =
0) LSO x LSO = /
6. For example, for the ℓ = m = f ( x ; 0 ) = − x ( + / ) ≈ − x ( + . ) whichmeans a correction equal to − x = / .
57 and larger than − f ( /
6; 0 ) ≈ − . = − .
26. This value is totally incompatiblewith the “exact” value f exact22 ( x LSO ) = . radiation reaction force F j is defined as F j = − W F ( ℓ max ) , (61)where the (instantaneous, circular) GW flux F ( ℓ max ) is defined as F ( ℓ max ) = p G ℓ max (cid:229) ℓ = ℓ (cid:229) m = | R ˙ h ℓ m | = p G ℓ max (cid:229) ℓ = ℓ (cid:229) m = ( m W ) | Rh ℓ m | . (62)As an example of the performance of the new resummation procedure based on thedecomposition of h ℓ m given by Eq. (41), let us focus, as before, on the computationof the GW energy flux emitted by a test particle on circular orbits on Schwarzschild he Effective One Body description of the Two-Body problem 27 spacetime. Figure 6 illustrates the remarkable improvement in the closeness betweenˆ F New-resummed and ˆ F Exact . The reader should compare this result with the previousFig. 3 (the straightforward Taylor approximants to the flux), Fig. 4 (the Pad´e re-summation with v pole = / √
3) and Fig. 5 (the v pole -tuned Pad´e resummation). Tobe fully precise, Fig. 6 plots two examples of fluxes obtained from our new r ℓ m -representation for the individual multipolar waveforms h ℓ m . These two examplesdiffer in the choice of approximants for the ℓ = m = r its 3PN Taylor expansion, T [ r ] , while the other one uses its 5PNTaylor expansion, T [ r ] . All the other partial waves are given by their maximumknown Taylor expansion . Note that the fact that we use here for the r ℓ m ’s somestraightforward Taylor expansions does not mean that this new procedure is not aresummation technique. Indeed, the defining resummation features of our procedurehave four sources: (i) the factorization of the PN corrections to the waveforms intofour different blocks, namely ˆ S ( e ) eff , T ℓ m , e i d ℓ m and r ℓℓ m in Eq. (41); (ii) the fact theˆ S ( e ) eff is by itself a resummed source whose PN expansion would contain an infinitenumber of terms; (iii) the fact that the tail factor is a closed form expression (seeEq. (50) above) whose PN expansion also contains an infinite number of terms and(iv) the fact that we have replaced the Taylor expansion of f ℓ m ≡ r ℓℓ m by that of its ℓ -th root, namely r ℓ m .In conclusion, Eqs. (41) and (62) introduce a new recipe to resum the ( n -dependent) GW energy flux that is alternative to the ( v pole -tuned) one given byEq. (40). The two main advantages of the new resummation are: (i) it gives a betterrepresentation of the exact result in the n → parameter-free : the only flexibility that one has in the definition of the wave-form and flux is the choice of the analytical representation of the function f , like,for instance, P { f } , ( T [ r ]) , ( T [ r ]) , etc., (although Ref. [50] has pointedout the good consistency among all these choices). Note, that when n =
0, the GWenergy flux will depend on the choice of resummation of the radial potential A ( R ) through the Hamiltonian (for the even-parity modes) or the angular momentum (forthe odd-parity modes). At the practical level, this means that the EOB model, im-plemented with the new resummation procedure of the energy flux (and waveform)described so far, will essentially only depend on the doublet of parameters ( a , a ) ,that can in principle be constrained by comparison with (accurate) numerical rela-tivity results. Contrary to the previous v pole -resummation of the radiation reaction,this route to resummation is free of radiation-reaction flexibility parameters. Wewill consider it as our “standard” route to the resummation of the energy flux in thefollowing Sections discussing in details the properties of the EOB dynamics andwaveforms. We recall that Ref. [50] has also shown that the agreement improves even more when the Taylorexpansion of the function r is further suitably Pad´e resummed.8 Thibault Damour and Alessandro Nagar In this section we marry together all the EOB building blocks described in the pre-vious Sections and discuss the characteristic of the dynamics of the two black holesas provided by the EOB approach. In the following three subsections we discussin some detail: (i) the set up of initial data for the EOB dynamics with negligibleeccentricity (Sec. 5.1); (ii) the structure of the full Effective One Body waveform,covering inspiral, plunge, merger and ringdown, with the introduction of suitableNext-to-Quasi-Circular (NQC) effective corrections to it (and thus to the energyflux) (Sec. 5.2); (iii) the explicit structure of the EOB dynamics, discussing the so-lution of the dynamical equations.
In this section we discuss in detail the so-called post-post-circular dynamical initialdata (positions and momenta) as introduced in Sec. III B of [37]. This kind of (im-proved) construction is needed to have initial data with negligible eccentricity. Sincethe construction of the initial data is analytical, including the correction is useful tostart the system relatively close and to avoid evolving the EOB equation of motionfor a long time in order to make the system circularize itself.To explain the improved construction of initial data let us introduce a formalbook-keeping parameter e (to be set to 1 at the end) in front of the radiation reactionˆ F j in the EOB equations of motion. One can then show that the quasi-circularinspiralling solution of the EOB equations of motion formally satisfies p j = j ( r ) + e j ( r ) + O ( e ) , (63) p r ∗ = ep ( r ) + e p ( r ) + O ( e ) . (64)Here, j ( r ) is the usual circular approximation to the inspiralling angular momen-tum as explicitly given by Eq. (47) above. The order e (“post-circular”) term p ( r ) isobtained by: (i) inserting the circular approximation p j = j ( r ) on the left-hand side(l.h.s) of Eq. (10) of [34], (ii) using the chain rule d j ( r ) / dt = ( d j ( r ) / dr )( dr / dt ) ,(iii) replacing dr / dt by the right-hand side (r.h.s) of Eq. (9) of [34] and (iv) solvingfor p r ∗ at the first order in e . This leads to an explicit result of the form (using thenotation defined in Ref. [34]) ep ( r ) = " n ˆ H ˆ H eff (cid:18) BA (cid:19) / (cid:18) d j dr (cid:19) − ˆ F j , (65)where the subscript 0 indicates that the r.h.s. is evaluated at the leading circularapproximation e →
0. The post-circular EOB approximation ( j , p ) was intro-duced in Ref. [3] and then used in most of the subsequent EOB papers [6, 31, he Effective One Body description of the Two-Body problem 29
32, 33, 34, 35]. The post-post-circular approximation (order e ), introduced inRef. [37] and then used systematically in Ref. [38, 39, 40], consists of: (i) for-mally solving Eq. (35) with respect to the explicit p j appearing on the r.h.s., (ii)replacing p r ∗ by its post-circular approximation, Eq. (65), (iii) using the chain rule d p ( r ) / dt = ( d p ( r ) / dr )( dr / dt ) , and (iv) replacing dr / dt in terms of p (to lead-ing order) by using Eq. (33). The result yields an explicit expression of the type p j ≃ j ( r )[ + e k ( r )] of which one finally takes the square root. In principle, thisprocedure can be iterated to get initial data at any order in e . As it will be shownbelow, the post-post-circular initial data ( j p + e k , p ) are sufficient to lead tonegligible eccentricity when starting the integration of the EOB equations of motionat radius r ≡ R / ( GM ) = At this stage we have essentially discussed all the elements that are needed to com-pute the EOB dynamics obtained by solving the EOB equation of motion, Eqs. (32)-(35). The dynamics of the system yields a trajectory ( qqq ( t ) , ppp ( t )) ≡ ( j ( t ) , r ( t ) , p j ( t ) , p r ∗ ( t )) in phase space. The (multipolar) metric waveform during the inspiral and plungephase, up to the EOB “merger time” t m (that is defined as the maximum of the orbitalfrequency W ,) is a function of this trajectory, i.e. h insplunge ℓ m ≡ h insplunge ℓ m ( qqq ( t ) , ppp ( t )) .Focusing only on the dominant ℓ = m = • The insplunge waveform : h insplunge ( t ) , computed along the EOB dynamics up tomerger, which includes (i) the resummation of the “tail” terms described aboveand (ii) some effective parametrization of Next-to-Quasi-Circular effects. The ℓ = m = (cid:18) Rc GM (cid:19) h insplunge22 ( t ) = n n ( ) c ( n ) x ˆ h ( n ; x ) f NQC22 Y , − (cid:16) p , F (cid:17) , (66)where the argument x is taken to be (following [90]) x = v j = ( r w W ) (where r w was introduced in Eq. (36) above). The resummed version of f enter-ing in ˆ h ( x ) used here is given by the following Pad´e-resummed function f Pf22 ≡ P [ f Taylor22 ( x ; n )] . In the waveform h above we have introduced (follow-ing [40]) a new ingredient, a “Next-to-Quasi-Circular” (NQC) correction factorof the form f NQC22 ( a , a ) = + a p r ∗ ( r W ) + a ¨ rr W , (67) Note that one could also similarly improve the subleading higher-multipolar-order contributionsto F j . In addition, other (similar) expressions of the NQC factors can be found in the literature [38,39, 41].0 Thibault Damour and Alessandro Nagar where a and a are free parameters that have to be fixed. A crucial facet of thenew EOB formalism presented here consists in trying to be as predictive as pos-sible by reducing to an absolute minimum the number of “flexibility parameters”entering our theoretical framework. One can achieve this aim by “analytically”determining the two parameters a , a entering (via the NQC factor Eq. (67)) the(asymptotic) quadrupolar EOB waveform ˆ Rh EOB22 (where ˆ R = R / M ) by imposing:(a) that the modulus | ˆ Rh EOB22 | reaches, at the EOB-determined “merger time” t m ,a local maximum , and (b) that the value of this maximum EOB modulus is equalto a certain (dimensionless) function of n , j ( n ) . In Ref. [40] we calibrated j ( n ) (independently of the EOB formalism) by extracting from the best current Nu-merical Relativity simulations the maximum value of the modulus of the Numer-ical Relativity quadrupolar metric waveform | ˆ Rh NR22 | . Using the data reported in[29] and [39], and considering the “Zerilli-normalized” asymptotic metric wave-form Y = ˆ Rh / √
24, we found j ( n ) ≃ . n ( − . ( − n )) . Our re-quirements (a) and (b) impose, for any given A ( u ) potential, two constraints onthe two parameters a , a . We can solve these two constraints (by an iterationprocedure) and thereby uniquely determine the values of a , a correspondingto any given A ( u ) potential. In particular, in the case considered here where A ( u ) ≡ A ( u ; a , a , n ) this uniquely determines a , a in function of a , a and n . Note that this is done while also consistently using the “improved” version of h given by Eq. (66) to compute the radiation reaction force via Eq. (62). • a simplified representation of the transition between plunge and ring-downby smoothly matching (following Refs. [36]), on a ( p + ) -toothed “comb” ( t m − p d , . . . , t m − d , t m , t m + d , . . . , t m + p d ) centered around a matching time t m ,the inspiral-plus-plunge waveform to a ring-down waveform, made of the super-position of several quasi-normal-mode complex frequencies, (cid:18) Rc GM (cid:19) h ringdown22 ( t ) = (cid:229) N C + N e − s + N ( t − t m ) , (68)with s + N = a N + i w N , and where the label N refers to indices ( ℓ, ℓ ′ , m , n ) , with ( ℓ, m ) = ( , ) being the Schwarzschild-background multipolarity of the consid-ered (metric) waveform h ℓ m , with n = , , . . . being the ‘overtone number’of the considered Kerr-background Quasi-Normal-Mode, and ℓ ′ the degree ofits associated spheroidal harmonics S ℓ ′ m ( a s , q ) . As discussed in [3] and [36],and already mentioned above, the physics of the transition between plunge andring-down (which was first understood in the classic work of Davis, Ruffini andTiomno [51]) suggests to choose as matching time t m , in the comparable-masscase, the EOB time when the EOB orbital frequency W ( t ) reaches its maximum value.Finally, one defines a complete, quasi-analytical EOB waveform (covering thefull process from inspiral to ring-down) as: Refs. [36, 38] use p = i.e. a 5-teethed comb, and, correspondingly, 5 positive-frequency KerrQuasi-Normal Modes.he Effective One Body description of the Two-Body problem 31 h EOB22 ( t ) = q ( t m − t ) h insplunge22 ( t ) + q ( t − t m ) h ringdown22 ( t ) , (69)where q ( t ) denotes Heaviside’s step function. The final result is a waveform thatonly depends on the two parameters ( a , a ) which parametrize some flexibility onthe Pad´e resummation of the basic radial potential A ( u ) , connected to the yet uncal-culated (4PN, 5PN and) higher PN contributions. We conclude this section by discussing the features of the typical EOB dynamics ob-tained by solving the EOB equation of motion Eqs. (32)-(35) with post-post-circularinitial data. The resummation of the radiation reaction force uses the multiplicativedecomposition of h ℓ m given by Eq. (41) with NQC correction to the ℓ = m = a = a = −
20 (see below why) while a and a are obtained consistently according tothe iteration procedure discussed above. The system is started at r =
15 and j = p j = . p r ∗ = − . t m = p r ∗ tends to a finite value after the merger(contrary to p r , that would diverge), yielding a more controllable numerical treat-ment of the late part of the EOB dynamics. So far we have seen that (at least) two different EOB models (of dynamics andwaveforms) are available. They differ, essentially, in the way the resummation of theGW energy flux yielding the radiation reaction force is performed. The first EOBmodel, that we will refer to as the “old” one, basically uses a Pad´e-resummationof the energy flux with an external parameter v pole that must be fixed in some way.The second EOB model, that we will refer to as the “improved” one, uses a moresophisticated resummation procedure of the energy flux, multipole by multipole,in such a way that the final result depends explicitly only on the same parameters ( a , a ) that are used to parametrize higher PN contribution to the conservative partof the dynamics.In the last three years, the power of the “old” EOB model has been exploited invarious comparisons with numerical relativity data, aiming at constraining in someway the space of the EOB flexibility parameters (notably represented by a and v pole ) by looking at regions in the parameter space where the agreement between the Fig. 7
EOB dynamics for a = a = −
20. Clockwise from the top left panel,the panelsreport: the trajectory, the radial separation r ( t ) , the radial momentum p r ∗ (conjugate to r ∗ ), theorbital frequency W ( t ) , the angular momentum p j ( t ) and the orbital phase j ( t ) . numerical and analytical waveforms is at the level of numerical error. For example,after a preliminary comparison done in Ref. [31], Buonanno et al. [33] compared restricted EOB waveforms to NR waveforms computed by the NASA-Goddardgroup, showing that it is possible to tune the value of a so as to have a goodagreement between the two set of data. In particular, for a =
60 and v pole givenaccording to the (nowadays outdated) suggestion of Ref. [49], in the equal-masscase ( n = / ± . The terminology “restricted” refers to a waveform which uses only the leading
Newtonian ap-proximation, h ( N , e ) ℓ m , to the waveformhe Effective One Body description of the Two-Body problem 33 GW cycles over 14 GW cycles. In the case of a mass ratio 4 : 1 ( n = . ± .
035 GW cycles over 9 GW cycles.Later, the resummed factorized EOB waveform of Eq. (66) above within the “old”EOB model has been compared to several set of equal-mass and unequal-mass NRwaveforms: (i) in the comparison with the very accurate inspiralling simulation ofthe Caltech-Cornell group [27] the dephasing stayed smaller than ± .
001 GW cy-cles over 30 GW cycles (and the amplitudes agreed at the ∼ − level) [37]; (ii)in the comparison [38] with a late-inspiral-merger-ringdown NR waveform com-puted by the Albert Einstein Institute group, the dephasing stayed smaller than ± .
005 GW cycles over 12 GW cycles; (iii) in the (joint) comparison [39] betweenEOB and very accurate equal-mass inspiralling simulation of the Caltech-Cornellgroup [27] and late-inspiral-merger-ringdown waveform for 1:1, 2:1 and 4:1 massratio data computed by the Jena group it was possible to tune the EOB flexibilityparameters (notably a and v pole ) so that the dephasing stayed at the level of thenumerical error. The same “old” model, with resummed factorized waveform, andthe parameter-dependent (using v pole ) resummation of radiation reaction force, wasrecently extended by adding 6 more flexibility parameters to the ones already intr-duced in Refs. [37, 39], and was “calibrated” on the high-accuracy Caltech-Cornellequal-mass data [41]. This calibration showed that only 5 flexibility parameters ( a , v pole and three parameters related to non-quasi-circular corrections to the waveformamplitude) actually suffice to make the “old” EOB and NR waveform agree, bothin amplitude and phase, at the level of the numerical error (this multi-flexed EOBmodel brings in an improvement with respect to the one of Refs. [37, 39] espe-cially for what concerns the agreement between the waveform amplitude around themerger).Recently, Ref. [40] has introduced and fully exploited the possibilities of the “im-proved” EOB formalism described above, taking advantage of: (i) the multiplica-tive decomposition of the (resummed) multipolar waveform advocated in Eq. (41)above, (ii) the effect of the NQC corrections to the waveform (and energy flux)given by Eq. (66), and, most importantly, (iii) the parameter-free resummation ofradiation reaction F j . In Ref. [40] the ( a , a ) -dependent predictions made bythe “improved” formalism were compared to the high-accuracy waveform froman equal-mass BBH ( n = /
4) computed by the Caltech-Cornell group [29], (andnow made available on the web). It was found that there is a strong degeneracybetween a and a in the sense that there is an excellent EOB-NR agreement foran extended region in the ( a , a ) -plane. More precisely, the phase difference be-tween the EOB (metric) waveform and the Caltech-Cornell one, considered betweenGW frequencies M w L = .
047 and M w R = .
31 (i.e., the last 16 GW cycles beforemerger), stays smaller than 0.02 radians within a long and thin banana-like regionin the ( a , a ) -plane. This “good region” approximately extends between the points ( a , a ) = ( , − ) and ( a , a ) = ( − , + ) . As an example (which actuallylies on the boundary of the “good region”), we have followed [40] in consideringhere the specific values a = , a = −
20 (to which correspond, when n = / a = − . , a = . M as time unit.
500 1000 1500 2000 2500 3000 3500 4000−0.3−0.2−0.100.10.20.3 t ℜ [ Ψ ] / ν Numerical Relativity (Caltech-Cornell)EOB ( a = 0; a =-20) Fig. 8
This figure illustrates the comparison between the “improved” EOB waveform (quadrupolar( ℓ = m =
2) metric waveform (66) with parameter-free radiation reaction (61) and with a = a = −
20) with the most accurate numerical relativity waveform (equal-mass case) nowadays available.The phase difference between the two is Df ≤ ± .
01 radians during the entire inspiral and plunge.Ref. [40] has shown that this agreement is at the level of the numerical error.
This result relies on the proper comparison between NR and EOB time series,which is a delicate subject. In fact, to compare the NR and EOB phase time-series f NR22 ( t NR ) and f EOB22 ( t EOB ) one needs to shift, by additive constants, both one of thetime variables, and one of the phases. In other words, we need to determine t and a such that the “shifted” EOB quantities t ′ EOB = t EOB + t , f ′ EOB22 = f EOB22 + a (70)“best fit” the NR ones. One convenient way to do so is first to “pinch” the EOB/NRphase difference at two different instants (corresponding to two different frequen-cies). More precisely, one can choose two NR times t NR1 , t NR2 , which determine twocorresponding GW frequencies w = w NR22 ( t NR1 ) , w = w NR22 ( t NR2 ) , and then findthe time shift t ( w , w ) such that the shifted EOB phase difference, between w and w , Df EOB ( t ) ≡ f ′ EOB22 ( t ′ EOB2 ) − f ′ EOB22 ( t ′ EOB1 ) = f EOB22 ( t EOB2 + t ) − f EOB22 ( t EOB1 + t ) is equal to the corresponding (unshifted) NR phase difference Df NR ≡ f NR22 ( t NR2 ) − f NR22 ( t NR1 ) . This yields one equation for one unknown ( t ), and (uniquely) determinesa value t ( w , w ) of t . [Note that the w → w = w m limit of this procedure yieldsthe one-frequency matching procedure used in [27].] After having so determined t ,one can uniquely define a corresponding best-fit phase shift a ( w , w ) by requiringthat, say, f ′ EOB22 ( t ′ EOB1 ) ≡ f EOB22 ( t ′ EOB1 ) + a = f NR22 ( t NR1 ) . Alternatively, one can start by giving oneself w , w and determine the NR instants t NR1 , t NR2 atwhich they are reached.he Effective One Body description of the Two-Body problem 35 t Merger time
Fig. 9
Close up around merger of the waveforms of Fig. 8. Note the excellent agreement between both modulus and phasing also during the ringdown phase.
Having so related the EOB time and phase variables to the NR ones we canstraigthforwardly compare all the EOB time series to their NR correspondants. Inparticular, we can compute the (shifted) EOB–NR phase difference D w , w f EOBNR22 ( t NR ) ≡ f ′ EOB22 ( t ′ EOB ) − f NR22 ( t NR ) . (71)Figure 8 compares (the real part of) our analytical metric quadrupolar waveform Y EOB22 / n to the corresponding (Caltech-Cornell) NR metric waveform Y NR22 / n . ThisNR metric waveform has been obtained by a double time-integration (following theprocedure of Ref. [39]) from the original, publicly available, curvature waveform y . Such a curvature waveform has been extrapolated both in resolution and inextraction radius. The agreement between the analytical prediction and the NR resultis striking, even around the merger. See Fig. 9 which closes up on the merger. Thevertical line indicates the location of the EOB-merger time, i.e., the location of themaximum of the orbital frequency.The phasing agreement between the waveforms is excellent over the full timespan of the simulation (which covers 32 cycles of inspiral and about 6 cycles ofringdown), while the modulus agreement is excellent over the full span, apart fromtwo cycles after merger where one can notice a difference. More precisely, the phase The two frequencies used for this comparison, by means of the “two-frequency pinching tech-nique” mentioned above, are M w = .
047 and M w = . Fig. 10
Comparison between Numerical Relativity and EOB metric waveform for the 2:1 massratio. difference, Df = f EOBmetric − f NRmetric , remains remarkably small ( ∼ ± .
02 radians) dur-ing the entire inspiral and plunge ( w = .
31 being quite near the merger). By com-parison, the root-sum of the various numerical errors on the phase (numerical trun-cation, outer boundary, extrapolation to infinity) is about 0 .
023 radians during theinspiral [29]. At the merger, and during the ringdown, Df takes somewhat largervalues ( ∼ ± . .
05 radians during ringdown, which is comparable to theEOB-NR phase disagreement. In addition, Ref. [40] compared the “improved” EOBwaveform to accurate numerical relativity data (obtained by the Jena group [39]) onthe coalescence of unequal mass-ratio black-hole binaries. Fig. 10 shows the re-sult of the EOB/NR waveform comparison for a 2:1 mass ratio, corresponding to n = /
9. When a = a = −
20 one finds a = − . a = . triple comparison between (i) the NR GW energy flux atinfinity (which was computed in [28]); (ii) the corresponding analytically predictedGW energy flux at infinity (computed by summing | ˙ h ℓ m | over ℓ, m ); and (iii) (mi-nus) the mechanical energy loss of the system, as predicted by the general EOB he Effective One Body description of the Two-Body problem 37 Fig. 11
The triple comparison between Numerical Relativity and EOB GW energy fluxes and theEOB mechanical energy loss. formalism, i.e. the “work” done by the radiation reaction ˙ E mechanical = W F j . Thiscomparison is shown in Fig. 11, which should be compared to Fig. 9 of [28]. Wekept here the same vertical scale as [28] which compared the NR flux to older ver-sions of (resummed and non-resummed) analytical fluxes and needed such a ± v of the differentiated metric waveform ˙ h .] By contrast, we seeagain the striking closeness (at the ∼ × − level) between the EOB and NR GWfluxes. As both fluxes include higher multipoles than the ( , ) one, this closenessis a further test of the agreement between the improved EOB formalism and NR re-sults. [We think that the ∼ s difference between the (coinciding) analytical curvesand the NR one on the left of the Figure is due to uncertainties in the flux computa-tion of [28], possibly related to the method used there of computing ˙ h .] Note that therather close agreement between the analytical energy flux and the mechanical en-ergy loss during late inspiral is not required by physics (because of the well-known“Schott term” [104]), but is rather an indication that ˙ h ℓ m can be well approximatedby − im W h ℓ m We have reviewed the basic elements of the Effective One Body (EOB) formalism.This formalism is still under development. The various existing versions of the EOBformalism have all shown their capability to reproduce within numerical errors thecurrently most accurate numerical relativity simulations of coalescing binary blackholes. These versions differ in the number of free theoretical parameters. Recentlya new “improved” version of the formalism has been defined which contains essen-tially only two free theoretical parameters.Among the successes of the EOB formalism let us mention:1. An analytical understanding of the non-adiabatic late-inspiral dynamics and ofits “blurred” transition to a quasi-circular plunge;2. The surprising possibility to analytically describe the merger of two black holesby a seemingly coarse approximation consisting of matching a continued inspiralto a ringdown signal;3. The capability, after using suitable resummation methods, to reproduce withexquisite accuracy both the phase and the amplitude of the gravitational wavesignal emitted during the entire coalescence process, from early-inspiral, to late-inspiral, plunge, merger and ringdown;4. The gravitational wave energy flux predicted by the EOB formalism agrees,within numerical errors, with the most accurate numerical-relativity energy flux;5. The ability to correctly estimate (within a 2% error) the final spin and mass ofnonspinning coalescing black hole binaries [this issue has not been discussed inthis review, but see Ref. [34]].We anticipate that the EOB formalism will also be able to provide an accuratedescription of more complicated systems than the nonspinning BBH discussed inthis review. On the one hand, we think that the recently improved EOB frame-work can be extended to the description of (nearly circularized) spinning blackhole systems by suitably incorporating both the PN-expanded knowledge of spineffects [105, 106, 108] and their possible EOB resummation [4, 107]. On the otherhand, the EOB formalism can also be extended to the description of binary neutronstars or mixed binary systems made of a black hole and a neutron star [109, 110].An important input for this extension is the use of the relativistic tidal properties ofneutron stars [111, 112, 113]Finally, we think that the EOB formalism has opened the realistic possibility ofconstructing (with minimal computational resources) a very accurate, large bank ofgravitational wave templates, thereby helping in both detecting and analyzing thesignals emitted by inspiralling and coalescing binary black holes. Though we havehad in mind in this review essentially ground-based detectors, we think that theEOB method can also be applied to space-based ones,i.e., to (possibly eccentric)large mass ratio systems.
Acknowledgments.
AN is grateful to Alessandro Spallicci, Bernard Whiting and all the organizersof the “Ecole th´ematique du CNRS sur la masse (origine, mouvement, mesure)”. he Effective One Body description of the Two-Body problem 39
Among the many colleagues whom we benefitted from, we would like to thank par-ticularly Emanuele Berti, Bernd Br¨ugmann, Alessandra Buonanno, Nils Dorband,Mark Hannam, Sascha Husa, Bala Iyer, Larry Kidder, Eric Poisson, Denis Pollney,Luciano Rezzolla, B.S. Sathyaprakash, Angelo Tartaglia and Loic Villain, for fruit-ful collaborations and discussions. We are also grateful to Marie-Claude Vergne forhelp with Fig. 1.
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