Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription
Chris Kavanagh, Donato Bini, Thibault Damour, Seth Hopper, Adrian C. Ottewill, Barry Wardell
SSpin-orbit precession along eccentric orbits for extreme mass ratio black hole binariesand its effective-one-body transcription
Chris Kavanagh, Donato Bini, Thibault Damour, Seth Hopper, Adrian C. Ottewill, and Barry Wardell
5, 6 Institut des Hautes ´Etudes Scientifiques, 91440 Bures-sur-Yvette , France. Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, 00185 Rome, Italy. CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico - IST,Universidade de Lisboa, 1049 Lisboa, Portugal School of Mathematics and Statistics and Institute for Discovery,University College Dublin, Belfield, Dublin 4, Ireland. School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland. Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
In this work we present an analytical gravitational self-force calculation of the spin-orbit pre-cession along an eccentric orbit around a Schwarzschild black hole, following closely the recentprescription of Akcay, Dempsey, and Dolan. We then transcribe this quantity within the Effective-One-Body (EOB) formalism, thereby determining several new, linear-in-mass-ratio, contributions inthe post-Newtonian expansion of the spin-orbit couplings entering the EOB Hamiltonian. Namely,we determine the second gyro-gravitomagnetic ratio g S ∗ ( r, p r , p φ ) up to order p r /r included. I. INTRODUCTION
The development of accurate waveform templates forcompact binaries is essential for the science of gravita-tional waves. For example, the extraction of physical in-formation from the first LIGO detections [1, 2] has madea key use of a bank of ∼ ,
000 semianalytical templates[3, 4], describing the inspiral, merger and ringdown oftwo comparable-mass black holes, that were developedwithin the Effective-One-Body (EOB) formalism [5–8].For future detectors such as LISA to reach their full po-tential one needs to describe systems with mass ratiosvarying from 1:1 to ∼ , evolving over long in-spirals into plunge, merger and ringdown phases. Whenattempting to model such orbital evolutions one typicallyrelies on several approximate ways of solving Einstein’sequations, valid in different asymptotic regimes: post-Newtonian (PN) theory in the slow-motion, weak-fieldregime; post-Minkowskian (PM) theory in the weak-fieldregime; gravitational self-force (SF) theory for small massratios; numerical relativity (NR) for strong-field compa-rable mass binaries; and EOB theory for analytically in-terpolating between various regimes.Recent years have witnessed a fruitful crossbreedingbetween these various methods. Notably, the EOB for-malism has provided, through its natural theoretical flex-ibility, a common ground for incorporating the results ofother approaches. Examples of recent works contributingto the crossbreeding between EOB theory and other ap-proximation methods are: EOB (cid:83) PN [9–12]; EOB (cid:83)
PM[13]; EOB (cid:83)
SF [14–17]; and EOB (cid:83)
NR [18–22].The primary focus of this paper is on the third of thesestrategies: the extraction of physical information fromSF results, and their EOB transcription. The SF ap-proach — in which Einstein’s equations are solved per-turbatively with the mass-ratio as small parameter — isideally suited to describing the motion of compact binarysystems with a large discrepancy in the masses. An im- portant theme (initiated in Refs. [23, 24]) within the SFcommunity over the past ten years has been the extrac-tion of physically meaningful quantities through the com-putation of gauge-invariant SF quantities. These quanti-ties are typically defined within the conservative sector,with dissipative effects of the self-force ignored or turnedoff. See Refs. [25, 26] for the first corresponding EOBtranscriptions of gauge-invariant SF quantities. In theliterature there now exists a wide array of gauge invari-ant quantities, each with varying dependencies on theperturbed metric and its derivatives. The utility of theseinclude: insights into the physical effects of the self-force(see, e.g., [27]); comparisons within SF theory betweencalculations in differing gauges (e.g. [28]); comparisonswith independent PN calculations (e.g.[29–31]) and withNR codes (e.g. [32]); and the extraction of high-PN-order contributions to the potentials of EOB theory (e.g.[33–35]).In a recent paper, Akcay, Dempsey and Dolan [36]presented a methodology for calculating the gauge in-variant self-force correction to the spin-orbit precessionof a spinning compact body along an eccentric orbit inSchwarzschild spacetime, as well as a numerical calcula-tion of the precession using a Lorenz gauge code. Theirpresentation is the first example of a gauge invariantquantity for an eccentric orbit binary which depends onderivatives of the metric perturbation, and gives accessfor the first time to spin-orbit effects along an eccentric orbit.The first aim of the present work is to complement the(mostly numerical) results of Ref. [36] by presenting an analytical calculation of their invariant as a PN expansionwithin SF theory. To do so we rely on a low eccentricityassumption in a manner following closely that of Ref.[15]. By contrast, both, with Ref. [15] (which used aRegge-Wheeler gauge), and with Ref. [36] (which useda Lorenz gauge), we will work in a so-called radiationgauge, which unlike the Regge-Wheeler gauge is readily a r X i v : . [ g r- q c ] J un extendible to a Kerr spacetime. This will provide anindependent check of the gauge invariance of the spin-precession quantity defined in [36].The second aim of the present work is to explicitlyderive the relationship between the spin precession in-variant along eccentric orbits, and the various potentialsparametrizing spin-orbit effects within the EOB formal-ism. [For the corresponding relationship in the simplercase of circular orbits see Ref. [34].] We shall then usethis relationship to show how the knowledge of the O ( e )[respectively, O ( e )] eccentric corrections to the spin pre-cession translates into new information about the termsquadratic (resp., quartic) in the radial momentum in thespin-orbit potentials of the EOB Hamiltonian.The organisation of this paper is as follows. InSec. II we review the formalism for calculating the ec-centric spin-precession, discuss eccentric geodesics inSchwarzschild spacetime and their perturbation and re-view the radiation gauge approach to reconstructing theperturbed metric. In Sec. III we describe the post-Newtonian approach we take to calculating the retardedmetric perturbation, the self-force, give our regulariza-tion and metric completion and finally the eccentric spinprecession. Then in Sec. IV, after briefly recalling theEOB parametrization of spin-orbit effects, we show howto transcribe the spin-precession invariant ∆ ψ ( p, e ) intoa knowledge of the O ( ν ) contribution to the second gyro-gravitomagnetic ratio g S ∗ ( u, p r , p φ ; ν ). II. GEODETIC SPIN-PRECESSIONA. Overview
We start by giving a brief summary of the prescriptionof Akcay, Dempsey and Dolan to calculate the gravita-tional self-force (SF) correction to the spin precession.For a more detailed description we refer the reader to[36].We wish to calculate the amount of precession angle atest spin vector accumulates over one radial period com-pared to the accumulated azimuthal angle. This preces-sion is conveniently measured by the quantity (in unitswhere G = c = 1) ψ ( m Ω r , m Ω ϕ ; q ) = Φ − ΨΦ . (2.1)Here, we consider a binary system with masses m and m (with q ≡ m m (cid:28) r , and Ω ϕ are, respectively the radial and (mean) azimuthal angularfrequencies. The question is then how to define Ψ.The spin vector s a is parallely transported alongan equatorial geodesic of the (regularized [37]) O ( q )-perturbed spacetime with four velocity u a and proper time τ : Ds a dτ = 0. Projecting this equation onto a partic-ular (polar-type) reference frame e aα , its spatial compo-nents satisfy d s dτ = ω × s (2.2)where( s ) i = e ai s a , ( ω ) i = − (cid:15) ijk ω jk , ω ij = − g ab e ai De bj dτ . (2.3)The spin precession is then entirely defined by the relativemotion of the tetrad throughout the orbit. Choosing thebasis so that only the ( θ -like) ( ω ) component is non-zero, one finds that the spin vector accumulates an angleΨ over one radial period (periapsis to periapsis)Ψ( m Ω r , m Ω ϕ ; q ) = (cid:73) ω ( τ ) dτ. (2.4)The aim is then to explicitly calculate the O ( q ), SF con-tributions to ψ , Eq. (2.1), and to Ψ, Eq.(2.4), i.e. thequantities ∆Ψ = Ψ(Ω r , Ω ϕ , q ) − Ψ(Ω r , Ω ϕ , , (2.5)and ∆ ψ = ψ (Ω r , Ω ϕ , q ) − ψ (Ω r , Ω ϕ ,
0) = − ∆ΨΦ , (2.6)where we recall that q = m /m denotes the small massratio, and where we used the fact thatΦ ≡ Ω ϕ T ≡ π Ω ϕ Ω r , (2.7)is the same on the perturbed ( q (cid:54) = 0) and background( q = 0) orbits. Here, differently from Ref. [36], we use theletter T = 2 π/ Ω r to denote the coordinate-time radialperiod.In practice, we shall work below with an intermedi-ate O ( q ) variation (denoted δ ) which does not keep fixedthe values of the two frequencies (Ω r , Ω ϕ ). We can thenrecover the correct value of ∆ ψ , Eq. (2.6), by first ‘sub-tracting’ the induced frequency shifts∆Ψ = δ Ψ − ∂ Ψ ∂ Ω r δ Ω r − ∂ Ψ ∂ Ω ϕ δ Ω ϕ , (2.8)and then computing ∆ ψ = − ∆ΨΦ . (2.9)With this broad outline in mind, the next few sectionswill focus on the explicit details of this calculation whenusing post-Newtonian (PN) expansions. B. Motion on the background ( q = 0 ):Schwarzschild spacetime In the usual Schwarzschild coordinates, the unper-turbed ( q = 0) metric takes the form ds = − f dt + f − dr + r ( dθ + sin θ dϕ ) (2.10)where f ≡ (1 − m /r ).
1. Background equatorial geodesics
We start be recalling some of the defining equationsfor a particle undergoing bound equatorial motion in aSchwarzschild spacetime. Here and henceforth we use asubscript p to denote evaluation at the position of theparticle. Such motion is parameterised by two constantsof motion, the specific energy and angular momentum( E , L ) respectively. The tangent four velocity is thengiven by u µp = (cid:18) E f p , u rp , , L r p (cid:19) . (2.11)Here we have made the standard restriction to equatorialmotion setting θ p = π , u θp = 0. The radial motion can beparametrised using Darwin’s relativistic anomaly χ [38] r p ( χ ) = pM e cos χ (2.12)where χ = 0 corresponds to periapsis, p is the semilatusrectum, and e the eccentricity. Using these parameters, E = ( p − − e p ( p − − e ) , L = p M p − − e (2.13)and dt p dχ = (cid:20) ( p − − e p − − e cos χ (cid:21) / × p M ( p − − e cos χ )(1 + e cos χ ) , (2.14) dϕ p dχ = (cid:20) pp − − e cos χ (cid:21) / , (2.15) dτ p dχ = M p / (1 + e cos χ ) (cid:20) p − − e p − − e cos χ (cid:21) / . (2.16)With these one can compute the unperturbed radialperiod as the coordinate time taken between successiveperiapses T ≡ (cid:90) π (cid:18) dt p dχ (cid:19) dχ . (2.17)The characteristic orbital frequencies are thenΩ r = 2 π T , Ω ϕ = Φ T , (2.18)with Φ = (cid:82) π (cid:16) dϕ p dχ (cid:17) dχ .
2. Background reference tetrad
As recalled above, in order to define the precession ofthe spin vector, one needs to choose a reference frame.Following [36], a suitable polar-type tetrad is that givenexplicitly (when q = 0) by Marck [39]: e a = u a = (cid:18) E f , u r , , L r (cid:19) , (2.19) e a = 1 f (cid:112) L /r ( u r , f E , , , (2.20) e a = (0 , , /r, , (2.21) e a = 1 r (cid:112) L /r (cid:18) EL f , L u r , , L r (cid:19) . (2.22)Using (2.3) the key frequency determining the precessionfunction is ω = EL r + L . (2.23)It is straightforward then to calculate the backgroundspin precession using (2.12),(2.13) with (2.1),(2.4). C. Motion and spin-precession in the perturbedspacetime ( q (cid:54) = 0 ) Ref. [36], generalizing previous results by Barack andSago [40], has derived an explicit integral expression forthe SF contribution δ Ψ to Ψ (using a specific variation δ which does not fix the frequencies). Their final resultreads δ Ψ = (cid:90) (cid:32) δ ˙Ψ˙Ψ − δu r u r (cid:33) ˙Ψ dτdχ dχ. (2.24)The term proportional to δu r appears since the propertime in (2.4) also needs to be varied. The evaluation ofthis expression requires further definitions. We have in-troduced ˙Ψ ≡ ω = ω [13] to be the integrand of (2.4)in favour of the ω of [36] to avoid overlap with the fre-quency notation for the frequency domain solutions ofthe Teukolsky equation. Its variation is given by Akcay et al as δ ˙Ψ = 12 ˙Ψ h + δ Γ [31]0 + ( c e b + c e b ) e a [3 ∇ b e a (2.25)where we have neglected a total derivative term whichvanishes upon integration over the orbit, δ Γ [31]0 is atetrad component of δ Γ µνρ = ( h µν,ρ + h µρ,ν − h νρ,µ )and c = 1 f (cid:112) L /r ( E δu rBS − u r δ E BS ) , (2.26) c = δ L BS r (cid:112) L /r (2.27)are terms arising from perturbing the tetrad. In this ex-pression the subscript BS refers to the perturbations inenergy, angular momentum and radial velocity definedby Barack and Sago in [40]. They differ from those usedby Akcay et al by the normalisation of the 4-velocity,because Barack and Sago normalise with respect to thebackground metric while Akcay et al normalise with re-spect to the perturbed spacetime. This leads to the fol-lowing relation valid to linear order in the mass-ratio δ E BS = δ E − E h , (2.28) δu rBS = δu r − u r h , (2.29) δ L BS = δ L − L h . (2.30)where h ≡ h µν e µ e ν . The O ( q ) perturbations δ E BS , δ L BS (which are denoted ∆ E and ∆ L by Barack andSago) of the energy and angular momentum u t ≡−E , u ϕ ≡ L entering the above equations have beenshown in Sec II. C of [40] to be determined by the fol-lowing quadratures δ E BS ( χ ) = δ E BS (0) − (cid:90) χ F cons t dτdχ dχ, (2.31) δ L BS ( χ ) = δ L BS (0) + (cid:90) χ F cons ϕ dτdχ dχ . (2.32)Here δ E (0) and δ L (0) are the energy and angular mo-mentum shift at periapsis, given in Eq. (37) and (38) of[40], and F cons µ is the conservative part of the self-forcewhich we discuss below (see Eq. (3.12)). [By contrast tothe Detweiler-Whiting formulation of SF we used in ourpresentation above, Barack and Sago use the formulationwhere the perturbed motion satisfies the forced-motionequation D u µ dτ = F µ ( τ ).] Using the normalisation con-dition of Akcay et al the perturbed radial velocity is thencalculated from the relation12 h − E δ E f p + u r δu r f p + L δ L r p = 0 . (2.33)Finally the gauge invariant precession function is calcu-lated using Eqs. (2.8) and (2.9). The frequency shifts cal-culated by perturbing (2.18) are given in Eq. (75)-(76) of [40]. We however do not require their α term, which theyadded to ensure the asymptotic flatness of their metricperturbation, since we will work in an asymptotically flatgauge. D. Radiation gauge metric perturbation
Our strategy for computing a post-Newtonian expan-sion of the retarded metric perturbation in many waysfollows closely that laid out in [15], in that we will usethe method of extended homogeneous solutions to con-struct a particular solution of a particular partial differ-ential equation whose solutions are related to the metricperturbation. The key difference however is that we shalluse the tetrad formalism and radiation gauge to constructthe metric, using the CCK procedure, so named after itsdevelopment by Chrzanowski [41] and Cohen and Kege-les [42, 43]. Specifically this involves building inhomoge-neous solutions to the Bardeen-Press-Teukolsky (BPT)equation ( a = 0 Teukolsky equation), from this a Hertzpotential and finally the metric perturbation and all itsfirst derivatives. Since much of the details are covered bya variety of authors e.g [44–48] we will give an abridgedoverview of the strategy and refer the reader to the givenreferences for details.
1. Bardeen-Press-Teukolsky (BPT) equation
The description of perturbations to a black hole space-time can be reduced to a single partial differential equa-tion for the tetrad components of the perturbed Weyltensor C µνρσ . In particular (essentially) all informationis simultaneously contained in the two quantities ψ and ψ defined by: ψ = − C αβγδ l α m β l γ m δ , (2.34) ψ = − C αβγδ n α ¯ m β n γ ¯ m δ , (2.35)where l µ , n ν m µ and ¯ m µ are the Kinnersley tetradlegs given in Appendix A. The dynamics of the per-turbed Weyl scalars on a Schwarzschild background aredescribed by the BPT equation: r ∆ ∂ ψ∂t − θ ∂ ψ∂ϕ − ∆ − s ∂∂r (cid:18) ∆ s +1 ∂ψ∂r (cid:19) − θ ∂∂θ (cid:18) sin θ ∂ψ∂θ (cid:19) − is cos θ sin θ ∂ψ∂ϕ − s (cid:20) m r ∆ − r (cid:21) ∂ψ∂t + ( s cot θ − s ) ψ = 4 πr T. (2.36)with ∆ ≡ r − m r. (2.37) Here, for s = +2 ψ = ψ , T = 2 T (2.38)and for s = − ψ = (cid:37) − ψ , T = 2 (cid:37) − T , (2.39)while the source terms are T = ( δ + ¯ (cid:36) − ¯ α − β − τ ) × [( D − (cid:15) − (cid:37) ) T − ( δ + ¯ (cid:36) − α − β ) T ]+ ( D − (cid:15) + ¯ (cid:15) − (cid:37) − ¯ (cid:37) ) × [( δ + 2 ¯ (cid:36) − β ) T − ( D − (cid:15) + 2¯ (cid:15) − (cid:37) ) T ] , (2.40) T = ( ∆ + 3 γ − ¯ γ + 4 µ + ¯ µ ) × (cid:2) ( ¯ δ − τ + 2 α ) T − ( ∆ + 2 γ − γ + ¯ µ ) T (cid:3) + ( ¯ δ − ¯ τ + ¯ β + 3 α + 4 (cid:36) ) × (cid:2) ( ∆ + 2 γ + 2¯ µ ) T − ( ¯ δ − ¯ τ + 2 ¯ β + 2 α ) T (cid:3) , (2.41)where D = l µ ∂ µ , ∆ = n µ ∂ µ and δ = m µ ∂ µ . Here T ij are the Kinersley-tetrad projections of the point particlesource. Now and henceforth we will focus on the s = 2solutions for ψ (a similar procedure could be followedwith ψ since it contains the same information as ψ ).This equation is fully separable by means of a Fouriertransform and projection over spin-weighted sphericalharmonics. Due to the double-periodicity of the eccentricorbits the Fourier transform reduces to a Fourier serieslabelled by the discrete frequencies ω = m Ω ϕ + n Ω r ψ ( t, r, θ, ϕ ) = (cid:88) (cid:96)m ψ (cid:96)m ( t, r ) Y (cid:96)m ( θ, ϕ ) , (2.42) ψ (cid:96)m ( t, r ) = ∞ (cid:88) n = −∞ e − iωt R (cid:96)mω ( r ) . (2.43)The radial functions s R (cid:96)mω ( r ) satisfy (cid:20) ∆ − s ddr (cid:18) ∆ s +1 ddr (cid:19) + r ω − is ( r − m ) r ω ∆+4 isωr − s λ (cid:96)m (cid:21) s R (cid:96)mω ( r ) = s T (cid:96)mω , (2.44)where s λ (cid:96)m = (cid:96) ( (cid:96) + 1) − s ( s + 1). Assuming a pair ofhomogenous solutions s ˆ R + (cid:96)mω , s ˆ R − (cid:96)mω to the above, thecorresponding inhomogeneous solution is written R (cid:96)mω ( r ) = c +2 ˆ R − (cid:96)mω ( r ) + c − ˆ R + (cid:96)mω ( r ) (2.45)where c + = 1 W (cid:96)mn (cid:90) r p r ∆ ˆ R + (cid:96)mω ( r ) T (cid:96)mω dr, (2.46) c − = 1 W (cid:96)mn (cid:90) rr p ∆ ˆ R − (cid:96)mω ( r ) T (cid:96)mω dr (2.47)and W (cid:96)mn = ∆ s +1 (cid:16) ˆ R + (cid:96)mω ˆ R −(cid:48) (cid:96)mω − ˆ R − (cid:96)mω ˆ R + (cid:48) (cid:96)mω (cid:17) .The source term here is T (cid:96)mω = 1 T (cid:90) π (cid:90) π (cid:90) π T e iωt Y ∗ (cid:96)m ( θ, ϕ ) sin θ dtdχ dθdϕdχ. (2.48)
2. Hertz potential and the retarded metric perturbation
As is standard we reconstruct the metric perturbationby means of an auxiliary function known as the Hertzpotential Ψ H , as in for example [45]. In the outgoingradiation gauge we work in, Ψ H satisfies the spin-2 BPTequation, as well as the angular equation ψ = 18 (cid:0) L ¯Ψ H + 12 m ∂ t Ψ H (cid:1) . (2.49)Here L = L L L − L − and L s = − (cid:0) ∂ θ − s cot θ + i csc θ∂ ϕ (cid:1) . Spectrally decomposing the Hertz potentialas Ψ H = (cid:88) (cid:96)mn e − iωt Ψ (cid:96)mω Y (cid:96)m ( θ, ϕ ) (2.50)Eq. (2.49) can be algebraically inverted to giveΨ (cid:96)mω = 8 ( − m D ¯ R (cid:96), − m, − ω + 12 im ω R (cid:96)mω D + 144 M ω (2.51)where D = (cid:96) ( (cid:96) + 1)( (cid:96) − (cid:96) + 2). The metric is thenobtained from Ψ H by applying a set of differential oper-ators: h αβ = − (cid:37) − { n α n β ( ¯ δ − α − ¯ β + 5 (cid:36) )( ¯ δ − α + (cid:36) )+ ¯ m α ¯ m β ( ∆ + 5 µ − γ − ¯ γ )( ∆ + µ − γ ) − n ( α n β ) (cid:2) ( ¯ δ − α + ¯ β + 5 (cid:36) + ¯ τ )( ∆ + µ − γ )+( ∆ + 5 µ − ¯ µ − γ − ¯ γ )( ¯ δ − α + (cid:36) ) (cid:3) } Ψ H + c.c. , (2.52)where c.c. denotes complex conjugation. The first deriva-tives of the metric which appear in the formula for thespin precession can then be written in terms of threederivatives of the inhomogeneous solution (2.45).Ultimately the decomposition in spin-weighted spheri-cal harmonics of ψ , Eq. (2.42), generates a correspond-ing decomposition of h αβ in tensorial spherical harmon-ics, with (cid:96) (together with the parity) labelling each ir-reducible representation of the rotation group. In turn,this generates a corresponding decomposition of the spin-precession ψ . We shall often refer to the irreducible piecesof these decompositions as “ (cid:96) -modes”. III. POST-NEWTONIAN APPROACH
We now wish to proceed with the calculation laid outin the previous sections analytically, using a PN assump-tion that the orbital separation between the two bodiesis large, i.e. p (cid:29)
1. For simplicity, we will additionallyassume small eccentricities ( e (cid:28) e and 1 /p .In practice, to achieve the required accuracy of the spinprecession invariant we will work to 7 orders in 1 /p and 6orders in e . This will yield the result to 5 PN orders andaccurate to order e (since the invariant is defined as theratio of two angles one loses two powers of e throughoutthe calculation, i.e. e accuracy is needed for e results).Our extra orders are kept to reduce potential systematicerrors. A. Background orbit
The various background orbital elements of II B canbe easily calculated in the PN regime, see e.g. [15]. Forexample the orbital period, T , can be calculated by ex-panding (2.14) dt p dχ = p / (cid:20)(cid:0) − e cos χ + 3 e cos χ + O ( e ) (cid:1) + 3 (cid:0) − e cos χ + e cos χ + O ( e ) (cid:1) p + O (cid:0) p − (cid:1) (cid:21) (3.1)and integrating order by order. This can be extendedwith ease to the desired orders in 1 /p and e . Repeatingthis for dϕ p dχ , we can obtain the two orbital frequencies m Ω r = (cid:18) − e p (cid:19) / (cid:20) − − e p + O ( p − ) (cid:21) (3.2) m Ω ϕ = (cid:18) − e p (cid:19) / (cid:20) e p + O ( p − ) (cid:21) (3.3)These are equivalent in the Newtonian limit. B. PN-expanded BPT equation
Before we compute the perturbed orbital elements weneed the self-force and thus the metric perturbation andits derivatives. For our set of homogeneous solutions weuse exactly those described in Ref. [47], with the rotationparameter a limiting to zero. That is, dropping the s = 2subscript and translating notation,ˆ R + / − (cid:96)mω = R up / in (cid:96)mω ( a → . (3.4)Note that while the computation of [47] is aimed at cir-cular orbits, the homogeneous solutions obtained thereinare derived with only the assumptions that the orbital ra-dius is large and that the frequency scales as ω ∼ r − / .Both of the assumptions are satisfied in our current study,as can be seen explicitly by Eq. (3.3). We would also like to emphasise the nature of these so-lutions as a function of (cid:96) . Since the regularised self forceis convergent in (cid:96) , in numerical studies of the self force afinite number of (cid:96) values are computed, which amountsto some corresponding accuracy when computing the fullsum over spherical harmonics, as in say Eq. (2.42). Inthe case of post-Newtonian expansions, the situation issomewhat different. It turns out one can compute homo-geneous solutions leaving (cid:96) as a parameter, the drawbackbeing that they typically breakdown for low (cid:96) . Thusthe strategy is to compute explicit PN expansions for (cid:96) = 2 . . .
6, the rest being captured by the general ex-pansions. Typically as one increases the order of our PNexpansion more low (cid:96) ’s are needed.As a particular example we will run through the pro-cedure with (cid:96) = 2. In Sec.II B of [47], the homogeneoussolutions to the radial Teukolsky equation are computed(similarly to Ref. [49]) as expansions in η = c with co-efficients in terms of the two variables X = GM/r , and √ X = ωr . These expansions are simplified by writingthem as a product of an exponential factor with radialdependence entirely contained in logarithmic terms, anda remaining series in η . After limiting a → R + (cid:96) =2 mω = − iX η √ X (cid:32) i (cid:112) X η + (cid:18) X − X (cid:19) η − iX / η + (cid:18) X X X X (cid:19) η (cid:33) + O ( η ) (3.5)ˆ R − (cid:96) =2 ,mω = 32 − i (cid:112) X η − X η + 328 iX / η + (cid:18) − X X
42 + 23 X (cid:19) η + O ( η ) (3.6)These solutions, while coming from usual Mano-Suzuki-Takasugi (MST) expansions, have been normalised to re-move certain radius independent factors that are unim-portant for constructing the inhomogeneous solution.The solutions can now be converted to series expansionsin 1 /p and e as a function of χ by using Eqs. (2.12) and(3.3) with the frequency ω = m Ω ϕ + n Ω r i.e. by eval-uating (3.5),(3.6) at the position of the particle for therelevant frequency values. In the above, the η factorsare simply an order counting tool and in practice can bedropped when converting to the expansion in 1 /p . Fornow we will hold off on fully switching to p, e and χ andinstead swap p for the dimensionless frequency variable y ≡ ( m Ω ϕ ) / . (3.7)The resulting double expansion in y and e for our exampleisˆ R + (cid:96) =2 ,mω = (cid:0) e cos χ + O ( e ) (cid:1) y + (cid:0) i ( m + n ) + 4 i ( m + n ) e cos χ + O ( e ) (cid:1) y / + (cid:18) −
12 ( m + n ) + (cid:18) −
32 ( m + n ) (cid:19) e cos χ + O ( e ) (cid:19) y − (cid:18) in + 16 i ( m + n ) + (cid:18) in + 13 i ( m + n ) (cid:19) e cos χ + O ( e ) (cid:19) y / + O ( y ) (3.8)ˆ R − (cid:96) =2 ,mω = 32 + (cid:0) − i ( m + n ) + i ( m + n ) e cos χ + O ( e ) (cid:1) y / + (cid:18) − m + n ) + 1114 ( m + n ) e cos χ + O ( e ) (cid:19) y + (cid:18) (cid:18) in + 114 i ( m + n ) (cid:19) + 32 (cid:18) − in − i ( m + n ) (cid:19) e cos χ + O ( e ) (cid:19) y / + O ( y ) (3.9)where we have removed a constant factor from R + (cid:96)mω . C. Hertz potential and retarded metricperturbation
The modes of the Hertz potential and thus the met-ric perturbation are then straightforwardly given usingEqs. (2.51) and (2.52). The main new feature in thisconstruction as compared to the circular case is the sumover the radial frequencies, i.e. the infinite sum over n .The sum is expected to be exponentially convergent forthe bound geodesics we are considering. This manifestsin the convergence of the small eccentricity expansion.What one finds is that for an expansion valid to e k , oneneeds only to sum n = − k, ..., k with h (cid:96)mnµν = O ( e k +1 ) for | n | > k . In other words, to capture higher eccentricityorbits one needs more and more n -modes.Computationally speaking the n -sum can be time con-suming and a potential bottleneck. We find it thereforemore convenient to sum the n -modes of the Hertz poten-tial to give Ψ (cid:96)m ( t, r ) = (cid:80) ∞ n = −∞ e − iωt Ψ (cid:96)mω ( r ). We mustalso compute the n -sum for each of the t derivatives thatthen appear in (2.52), for example ∂ t Ψ (cid:96)m ( t, r ) = − i ∞ (cid:88) n = −∞ ωe − iωt Ψ (cid:96)m ( r ) . (3.10)as well as the relevant r derivatives (which are obtainedfrom Eq. (2.51)). With these in hand the metric pertur-bation as a function of (cid:96) and m is more or less triviallycomputed. For instance, using Eq. (2.52) h (cid:96)mtt = 18 ( r − m ) (cid:18) ∂ θ Y (cid:96)m ( θ, ϕ ) + 2 m∂ θ Y (cid:96)m ( θ, ϕ )+ ( m − Y (cid:96)m ( θ, ϕ ) (cid:19) Ψ (cid:96)m ( t, r ) (3.11)(which can be recognized as being a pure scalar har-monic). Each metric component and its derivatives arein practice evaluated here at the position of the particle.At this stage we sum over m to get h (cid:96)µν = (cid:80) (cid:96)m = − (cid:96) h (cid:96)mµν . As will be discussed in more detail below, the singularnature of h µν ( t, r, θ, ϕ ) in the vicinity of the source world-line requires us to separately evaluate the two differentradial limits r → r ± p , from above or below, of the modes.These limits are indicated below by a ± subscript: e.g. h (cid:96)µν, ± , or F (cid:96)µ, ± .For ease of reading we omit explicit expressions foreach of the components of the metric and their first orderpartial derivatives. D. Perturbed geodesic and self-force
Before calculating the perturbed orbit quantities δ E , δ L and δu r we must explicitly compute the t and ϕ com-ponents of the conservative self-force. The self-force isgiven by [50] F µ = P µνλρ (2 h νλ ; ρ − h λρ ; ν ) , (3.12) P µνλρ = −
12 ( g µν + u µ u ν ) u λ u ρ . where the covariant derivatives here are taken with re-spect to the background metric. Formally this equationrequires the regularised metric perturbation; we will how-ever use it with the (cid:96) -modes of the retarded metric andleave regularization to later.At linear order in the mass ratio we can uniquely definethe dissipative and conservative parts of the self-force via F µ diss = 12 ( F µ [ h ret µν ] − F µ [ h adv µν ]) , (3.13) F µ cons = 12 ( F µ [ h ret µν ] + F µ [ h adv µν ]) . (3.14)Noting the symmetry relation of Eq. (2.80) of [51], theauthors of [52] rewrote these for the case of equatorialgeodesics purely in terms of the retarded solution: F µ diss = 12 ( F µ ret ( τ ) − (cid:15) ( µ ) F µ ret ( − τ )) (3.15) F µ cons = 12 ( F µ ret ( τ ) + (cid:15) ( µ ) F µ ret ( − τ )) , (3.16)where (cid:15) ( µ ) = ( − , , , − F µ ret ( τ → − τ ) ≡ F µ ret ( χ → − χ ). We compute this analyt-ically, e.g. for (cid:96) = 2 we have q − F cons t, + = (cid:0) e sin( χ ) + 3 e sin(2 χ ) + O ( e ) (cid:1) p − / + (cid:18) − e sin( χ ) − e sin(2 χ ) + O ( e ) (cid:19) p − / + O ( p − / ) , (3.17) q − F cons ϕ, + = (cid:18) − e sin( χ ) − e sin(2 χ ) + O ( e ) (cid:19) p − + (cid:18) e sin( χ ) + 1147 e sin(2 χ ) + O ( e ) (cid:19) p − + O ( p − ) . (3.18)In this double expansion the corresponding integrals needed for Eq. (2.32) are trivially computed order by order.Explicitly, δ E +BS = (cid:18)
32 + 3 e cos( χ ) + O ( e ) (cid:19) p − + (cid:18) − e cos( χ ) + O ( e ) (cid:19) p − + O ( p − ) , (3.19) δ L +BS = 32 √ p + (cid:18)
114 + 97 e cos( χ ) + O ( e ) (cid:19) p − / + O ( p − / ) , (3.20)and the corresponding δu r, +BS is found using the normalisation condition.Eq. (2.25) is now straightforwardly computed here togive the (cid:96) = 2 values δ ˙Ψ + = (cid:18)
32 + 3 cos χe + O ( e ) (cid:19) p − / − (cid:18) χe + O ( e (cid:19) p − / + O ( p − ) ,δ ˙Ψ − = − (cid:0) χe + O ( e ) (cid:1) p − / + 47 (cid:0) χe + O ( e (cid:1) p − / + O ( p − ) . (3.21)The integration of Eq. (2.24) is as usual applied order byorder.The only remaining perturbed quantities are the fre-quency shifts δ Ω r , δ Ω ϕ which appear in (2.9). UsingEq. (75)-(76) of [40] (without the α factor) we find m δ Ω + r = (cid:18) − e (cid:19) p − / + (cid:18) − e + O ( e ) (cid:19) p − / + O ( p − / ) m δ Ω + ϕ = (cid:18) − e (cid:19) p − / + (cid:18) − e + O ( e ) (cid:19) p − / + O ( p − / )(3.22)Using Eq. (2.9) we give illustrative sample expansions forthe retarded ∆ ψ for (cid:96) = 2:∆ ψ + (cid:96) =2 = 4348 p − − (cid:18) e (cid:19) p − + O ( p − ) , ∆ ψ − (cid:96) =2 = 4348 p − − (cid:18) e (cid:19) p − + O ( p − ) . (3.23)Let us also give sample expansions for the generic (cid:96) ex-pressions. This are valid for all (cid:96) greater than a valuedetermined by the given PN order (see Sec II B. ofRef. [35]). These are useful for clearly seeing the diver-gent behaviour discussed in the next sections∆ ψ + (cid:96) = 3 (cid:0) (cid:96) + 7 (cid:96) (cid:1) − (cid:96) )(3 + 2 (cid:96) ) p − + (cid:32) (cid:0) − (cid:96) − (cid:96) + 120 (cid:96) − (cid:96) − (cid:96) + 90 (cid:96) + 64 (cid:96) (cid:1) (cid:96) (1 + (cid:96) )( − (cid:96) )( − (cid:96) )(3 + 2 (cid:96) )(5 + 2 (cid:96) ) − (cid:0) − − (cid:96) − (cid:96) − (cid:96) + 71 (cid:96) + 435 (cid:96) + 145 (cid:96) (cid:1) e (cid:96) (1 + (cid:96) )( − (cid:96) )( − (cid:96) )(3 + 2 (cid:96) )(5 + 2 (cid:96) ) (cid:33) p − + O ( p − ) , ∆ ψ − (cid:96) = 3 (cid:0) (cid:96) + 7 (cid:96) (cid:1) − (cid:96) )(3 + 2 (cid:96) ) p − + (cid:32) − (cid:0) − (cid:96) + 1104 (cid:96) − (cid:96) − (cid:96) + 402 (cid:96) + 358 (cid:96) + 64 (cid:96) (cid:1) (cid:96) (1 + (cid:96) )( − (cid:96) )( − (cid:96) )(3 + 2 (cid:96) )(5 + 2 (cid:96) ) − (cid:0) − − (cid:96) − (cid:96) − (cid:96) + 71 (cid:96) + 435 (cid:96) + 145 (cid:96) (cid:1) e (cid:96) (1 + (cid:96) )( − (cid:96) )( − (cid:96) )(3 + 2 (cid:96) )(5 + 2 (cid:96) ) (cid:33) p − + O ( p − ) . (3.24) E. Regularization and completion of the metricperturbation
1. Metric completion (non-radiative multipoles (cid:96) ≤ ) The reconstructed retarded metric perturbationobtained by the CCK procedure laid out in the pre-vious section is well known to give the full metricperturbation modulo perturbations to the spacetimedue to the mass and angular momentum of the smallbody. In Schwarzschild spacetime this amounts to theabsence of the spherical harmonic (cid:96) = 0 , ψ + (cid:96) =0 = − + e p + − + e + O ( e ) p + O ( p − ) , ∆ ψ − (cid:96) =0 = − + e p + − + e + O (cid:0) e (cid:1) p + O ( p − ) , (3.25)∆ ψ + (cid:96) =1 = 1 p + − e p + − e + O ( e ) p + O ( p − ) , ∆ ψ − (cid:96) =1 = 1 p + + e p + + e + O ( e ) p + O ( p − ) . (3.26)
2. Regularization
So far in the calculation we have been constructingthe (cid:96) -modes of the spin precession invariant from theretarded metric perturbation. The full retarded metricperturbation is, however, a singular quantity due to thepoint particle delta function source term. This singularnature manifests itself as a direction-dependent divergentsum over (cid:96) -modes (each individual (cid:96) -mode being, how-ever, finite), which for our spin-precession invariant takesthe following form for large values of (cid:96) ∆ ψ (cid:96) →∞± = ± A ∞ (2 (cid:96) + 1) + B ∞ + O ( (cid:96) − ) . (3.27)Here, the sign of the A ∞ term is dependent on whetherone takes the limit r → r ± p from above or below. Bydefinition, the two coefficients A ∞ and B ∞ parametriz-ing the large- (cid:96) singular behavior Eq. (3.27) of ∆ ψ are independent of (cid:96) . [We have shown above (see, e.g., Eq.(3.24)) how our PN-expanded method allows us to ana-lytically extract the values of A ∞ and B ∞ .]Calculating the regularized finite value for ∆ ψ involvescalculating, and subtracting out, the (cid:96) -modes of the cor-responding singular piece ∆ ψ S of ∆ ψ , say∆ ψ S (cid:96), ± = ± A S (2 (cid:96) + 1) + B S . (3.28)Detweiler and Whiting [37] have shown how to definethe singular pieces of h S µν , h S µν,λ , and thereby ∆ ψ S , inthe Lorenz gauge. Recently, Ref. [53] emphasized that,when using (as we do here, following most of the analyt-ical work on SF corrections to gauge-invariant quantitiesstarting with Ref. [23]) a decomposition of h µν in tensor-harmonics (cid:96) -modes, there are subtleties concerning thevalue of the coefficient A S of (2 (cid:96) + 1) in Eq. (3.28). In-deed, while the value of B S was found to be independentof (cid:96) (and therefore equal to the large- (cid:96) value B ∞ enteringEq. (3.27)), Ref. [53] found that the value of A S stabi-lized to its asymptotic value A ∞ only when when (cid:96) ≥ (cid:96) = 0 , A S coefficientsthat differ from A ∞ . This difference comes from the pe-culiar low- (cid:96) dependence of certain radial derivatives of h µν , such as h tϕ,r or h ϕϕ,r , see notably Eqs. (6.11j) and(6.11l) in Ref. [53]. [Although the mentioned equationsdeal with the case of circular orbits, nothing fundamen-tally changes in the eccentric orbit case, and the sameconclusions hold.]In the Lorenz gauge, the correct regularized spin-precession invariant would then be given by∆ ψ = ∞ (cid:88) (cid:96) =0 (cid:16) ∆ ψ (cid:96) ± − ∆ ψ (cid:96), ± S (cid:17) = ∞ (cid:88) (cid:96) =0 (cid:0) ∆ ψ (cid:96) ± ∓ A S (2 (cid:96) + 1) − B S (cid:1) . (3.29)This result can be simplified by working with the averageover the limits r → r p from both sides, thereby avoidingthe need to keep track of the low (cid:96) dependence of A S ,and being able to use the value of B ∞ extracted fromthe large (cid:96) behavior of ∆ ψ , Eq. (3.27):∆ ψ = ∞ (cid:88) (cid:96) =0 (cid:18) (cid:0) ∆ ψ (cid:96), + + ∆ ψ (cid:96), − (cid:1) − B ∞ (cid:19) . (3.30)There is one remaining subtlety, which is that the regu-larization procedure explained above was derived in theLorenz gauge while our calculation of the tensorial (cid:96) modes was done in a radiation gauge. There are twoways to deal with this additional subtlety. On the onehand, as was shown by Barack and Ori in the case ofthe gravitational self-force [54], the procedure should also0work in a gauge which can be reached from Lorenz gaugeby a gauge transformation of a smooth enough nature.The radiation gauge we work in does not, however, fallwithin this class (for a discussion of the singular struc-ture of the various radiation gauges we refer the reader to[55]). Much work has been focussed recently on formallydefining the correct regularization procedure in radiationgauge. The strategy employed is to construct a gaugetransformation ξ µ which is defined locally in the vicinityof the worldline, and which can be used to transform fromthe radiation gauge to a “locally Lorenz” gauge. It is thenargued that this generates a correction to the mode-sumformula, Eq. (3.29), which has an overall sign that differson either side of the r → r p limit. A straightforwardsolution, then, is to take the average of both limits (aswe are doing), thus eliminating the troublesome singulargauge contribution.On the other hand, it is likely that a much simpler solution resolves the issue in our case. The quantity weare computing is a gauge invariant, and so we are freeto use the regularization procedure derived in the Lorenzgauge (with averaging to compensate for the fact thatwe are not decomposing into scalar spherical harmonics)and apply it to any other gauge in the same invarianceclass. We stress the point, however, for two reasons: (i)while we have partially checked the correctness of thisfact (notably by checking the continuity of ∆ ψ (cid:96) ± − ∆ ψ (cid:96), ± S across r = r p ), we have not analytically shown that ∆ ψ is invariant under the transformation from Lorenz to ra-diation gauge; and (ii) things would not be so straight-forward if one were interested in computing non-gauge-invariant quantities such as the self-force (in which casea more careful analysis along the lines of Ref. [55] maybe required).Finally, taking the large- (cid:96) limit of our expression (3.24)(to all orders we computed) we can read off the A ∞ and B ∞ coefficients: A ∞ = 38 p − + (cid:18) − e
32 + O ( e ) (cid:19) p − + (cid:18) − e
32 + O ( e ) (cid:19) p − + (cid:18) − e
512 + O ( e ) (cid:19) p − + (cid:18) − e O ( e ) (cid:19) p − + O ( p − ) (3.31) B ∞ = 2116 p − + (cid:18) − − e
512 + O ( e ) (cid:19) p − + (cid:18) − e O ( e ) (cid:19) p − + (cid:18) − e O ( e ) (cid:19) p − + (cid:18) − e O ( e ) (cid:19) p − + (cid:18) − e O ( e ) (cid:19) p − + O ( p − ) . (3.32)Although we do not use the A ∞ parameter due to ouraveraging, we have checked that both of these expres-sions agree with independently calculated A S and B S .These were computed from expansions in p and e of thesingular metric perturbation calculated using the meth-ods described in [53, 56, 57] with a projection onto scalarspherical harmonics. This provides a valuable check thatour radiative modes ( (cid:96) ≥
2) are demonstrating the cor- rect singular behaviour.
F. Spin-precession invariant
Upon regularization, our generic- (cid:96) terms can be seenanalytically to converge as (cid:96) − . This allows us to explic-itly compute the infinite series over (cid:96) , giving one of ourmain results∆ ψ = − p − + (cid:18)
94 + e (cid:19) p − + (cid:18)(cid:18) − π (cid:19) + (cid:18) − π (cid:19) e + O ( e ) (cid:19) p − + (cid:32)(cid:18) − γ
15+ 31697 π −
628 log( p )15 (cid:19) + (cid:18) − γ − π − −
268 log( p )5 (cid:19) e + O ( e ) (cid:33) p − + (cid:32)(cid:18) − − γ
35 + 2479221 π − p )35 (cid:19) + (cid:18) − γ − π − −
268 log( p )5 (cid:19) e + O ( e ) (cid:33) p − + (cid:32) π
315 + 319609 π e + O ( e ) (cid:33) p − / + (cid:32)(cid:18) − − γ π − π − p )3780 (cid:19) + (cid:18) − − γ
945 + 32088966503 π − π − − p )1890 (cid:19) e + O ( e ) (cid:33) p − + O (cid:16) p − / (cid:17) . (3.33)We could perform several checks on our result Eq.(3.33). First, the first three terms of our expansioncan be seen immediately to agree with the coefficientsof p − , p − , p − derived in Ref. [36].Second, we have compared the numerical values pre-dicted by our analytical result Eq. (3.33) to the numeri-cal estimates of ∆ ψ obtained in Ref. [36]. More precisely,Table II in Ref. [36] lists a sample of numerical estimates∆ ψ num ( p i , e k ) of ∆ ψ ( p, e ), together with estimates of thecorresponding numerical error σ num ( p i , e k ). This sampleincludes nineteen values, 10 ≤ p i ≤ p , and, for each value of p , nine values of e , namely: e k = 0 .
050 + 0 . k , k = 0 , , . . . ,
8. As ourmain aim was to compare the eccentricity dependence ofthese numerical data to the one predicted by our analyt-ical result Eq. (3.33), we extracted a numerical estimateof the O ( e ) contribution, say e ∆ ψ (2) ( p ), in ∆ ψ ( p, e ),∆ ψ ( p, e ) = ∆ ψ (0) ( p ) + e ∆ ψ (2) ( p ) + e ∆ ψ (4) ( p ) + O ( e ) , (3.34)in the following way. First, we used the analyti-cal knowledge of the leading-order O ( e ) contribution, e ∆ ψ e LO ( p ) = − e p − (derived in Ref. [36]) to workwith the O ( e )-corrected inclusion of numerical data∆ ψ num (cid:48) ( p i , e k ) ≡ ∆ ψ num ( p i , e k ) − e k ∆ ψ e LO ( p i ). Then,we fitted, for each value of p i the nine numerical data∆ ψ num (cid:48) ( p i , e k ); k = 0 , , . . . , e ,say ∆ ψ fit ( p i , e k ) = m i e k + q i . (3.35)We used a least-squares fit, weighted by the (in-verse squares of the) corresponding numerical errors σ num ( p i , e k ) listed in Table II of [36]. This fitting proce-dure gave us (for each p i ) estimates of m i ( ≈ ∆ ψ (2) ( p i ))and q i ( ≈ ∆ ψ (0) ( p i )), together with corresponding fittingerrors (obtained from the covariance matrix). In addi-tion, the goodness of each fit is measured by the corre-sponding reduced χ ( p i ) = χ / ( N data − N param ), with N data = 9 and N param = 2. We have checked that taking into account a next-to-leading-order O ( e ) contribution, e ∆ ψ e NLO ( p ) = c NLO e p − , with | c NLO | ≤
TABLE I. Comparison between the values m thy ( p ), predictedby our analytical result Eq. (3.33), of the coefficient of e in∆ ψ ( p, e ) = q ( p ) + m ( p ) e + O ( e ), to the numerical estimates m num ( p ) of m ( p ) obtained by least-squares fitting the numer-ical data for ∆ ψ ( p, e ) obtained in Ref.[36]. The last entry isthe ratio (3.36) between the difference m num − m thy and ourestimate of the total error σ tot m ≡ (cid:113) ( σ num m ) + ( σ thy m ) on m .See text for more details (notably about the estimates of theerrors σ thy m and σ num m ). p χ m num ( σ num m ) m thy ( σ thy m ) r m
10 0.221 2.83892(11) × − × − -0.5015 4.119 9.12787(61) × − × − -0.4520 2.591 4.40237(32) × − × − -0.4125 0.0357 2.561664(35) × − × − -0.3830 0.867 1.66508(23) × − × − -0.3735 0.646 1.16553(20) × − × − -0.3340 0.372 8.5943(13) × − × − -0.3045 0.102 6.58803(62) × − × − -0.3250 0.227 5.20356(82) × − × − -0.4855 0.481 4.2124(14) × − × − -0.5560 1.249 3.4798(15) × − × − -0.2965 3.630 2.9178(37) × − × − -0.7570 1.800 2.4793(43) × − × − -1.4075 3.113 2.1347(49) × − × − -1.1480 6.313 1.8583(90) × − × − -0.4185 4.832 1.6289(97) × − × − -0.5690 1.707 1.4552(39) × − × − × − × − × − × − -1.70 The results for our estimates of the e slope m i ≈ ∆ ψ (2) ( p i ) are given in Table I below.Namely, the first four entries of Table I are: p i , χ ( p i ), the numerical estimate m num i of the slope m i ≈ ∆ ψ (2) ( p i ) obtained from our fit, and the correspondingtheoretical prediction m thy i ≡ ∆ ψ (2)PN ( p i ), as obtainedfrom our PN-expanded analytical result Eq. (3.33). Inaddition, we have indicated in parentheses, both for thenumerical estimates of m i , and their theoretical ones, es-timates of the corresponding uncertainties in their val-ues. The estimates of the numerical uncertainty on m num i was obtained by renormalizing the fitting error2by a factor (cid:112) χ ( p i ). Indeed, though the goodness-of-fit parameters χ ( p i ) were always (as tabulated)of order unity, they were not always numerically closeto 1. In view of the difficulty, in numerical SF com-putations, to accurately estimate the numerical error,we considered that a value of χ ( p i ) different from 1indicated an inaccurate estimate of the numerical er-rors σ num ( p i , e k ), which we (coarsely) corrected by mul-tiplying σ num m ( p i , e k ) (and correlatively σ fit m i ) by a fac-tor (cid:112) χ ( p i ). Concerning the theoretical error σ thy m i ,it was estimated by the value of the last analyticallycomputed contribution to ∆ ψ (2) ( p i ) in Eq. (3.33), i.e. σ thy m ( p ) = | (18101 . . p ))1 /p | .Finally, the last entry in Table I displays the values ofthe ratios r m i ≡ m num i − m thy i σ tot m i , (3.36)where σ tot i ≡ (cid:113) ( σ num m i ) + ( σ thy m i ) is an estimate of thecombined numerical-analytical error on m i ≈ ∆ ψ (2) ( p i ).The most significant fact (for our purpose) in Table I isthat the values of the latter ratios are all of order unity.This is a valuable, independent check on our analyticalcomputations.Our fitting procedure has also given use numerical esti-mates of q i ≈ ∆ ψ (0) ( p i ) that we have satisfactorily com-pared (with corresponding ratios r q i ≡ ( q num i − q thy i ) /σ tot q i found to be of order unity) to the 9.5PN current analyti-cal knowledge of ∆ ψ (0) ( p ) (given in Appendix A of [36]).[We recall (see the end of section IV for more details) that∆ ψ (0) ( p ) = lim e → ∆ ψ ( p, e ) differs from ∆ ψ circ ( p ), andthat the difference ∆ ψ ( p, e → − ∆ ψ circ ( p ) was relatedin Section IIIB of [36] to the EOB function ρ ( x ) measur-ing periastron precession. While ∆ ψ circ ( p ) is known tovery high PN orders [35, 58], ρ ( x ) is currently known tothe 9.5-PN level [59].] IV. IMPROVING THE SPIN-ORBIT SECTOROF THE EOB HAMILTONIAN
In this section, we shall show how to transcribe thenew SF results contained in Eq. (3.33) above into animproved knowledge of the spin-orbit sector of the EOBHamiltonian. The inclusion of spin couplings in the EOBHamiltonian was initiated in Ref. [8] and developed inRefs. [3, 19, 20, 34, 60–70]. Here, we will focus on thecase of non-precessing spins (parallel or antiparallel tothe orbital angular momentum), and only consider effects linear in spins. Following the formulation of Refs. [60,66], the spin-orbit couplings are described by two phase-space-dependent gyrogravitomagnetic ratios g S and g S ∗ .We would like to emphasise that throughout this sec-tion all coordinate variables will be referring to EOB vari-ables which, despite overlapping labelling, are not to beconfused with those of the previous section. For example we will encounter an EOB eccentricity e , which is distinctto that used in Eq. (3.33) to parameterize the noncirculardependence of the SF spin precession invariant. To avoidissues when relating the EOB spin precession function toits previous SF version, we shall henceforth relabel allindependent variables from the previous section with anadditional overbar, i.e. the variables entering Eq. (3.33)will be now written as ¯ p and ¯ e . In addition, in order tobetter explicate the introduction of various dimensionlessquantities in the EOB formalism, we shall often return,in this section, to the use of physical units where G and c are not set to unity. A. EOB notation and reminders
Let us first recall the standard EOB results and nota-tion, which we shall follow here. The total Hamiltonianof the system is expressed as H ( R , P , S , S ) = M c (cid:115) ν (cid:18) H eff µc − (cid:19) ≡ M c h , (4.1)where (with the convention m < m , and m (cid:28) m inthe extreme-mass-ratio limit) M = m + m , µ = m m ( m + m ) , ν = µM = m m ( m + m ) , (4.2)and where the effective EOB Hamiltonian H eff is decom-posed as H eff = H Oeff + H SOeff . (4.3)Here H Oeff = c (cid:115) A (cid:18) µ c + P + (cid:18) B − (cid:19) P R + Q (cid:19) , (4.4)denotes the orbital part of the effective Hamiltonian, ex-pressed in terms of the squared linear momentum P = P R B + L R = P R B + P φ R , (4.5)with L = R × P denoting the orbital angular momentum(with magnitude L ≡ P φ ), and in terms of the EOB radialpotentials parametrizing the effective metric (specializedhere to equatorial motions) ds = g (eff) µν dX µ dX ν = − Ac dT + BdR + R dφ . (4.6)The last (quartic in momenta) contribution Q on theright-hand-side (rhs) of Eq. (4.4) will be defined below.In addition H SOeff = G phys S L · S + G phys S ∗ L · S ∗ (4.7)3denotes the spin-orbit part of the effective Hamiltonian,expressed in terms of the following two symmetric com-bination of the spin vectors S and S of the system S = S + S , S ∗ = m m S + m m S . (4.8)In the parallel-spin case that we consider here L · S = LS = P φ S and L · S ∗ = LS ∗ = P φ S ∗ . It is convenient towork with the following dimensionless, rescaled variables r = c RGM , u = GMc R ≡ r , j ≡ p φ = cP φ GM µ , p r = P R µc , (4.9)and quantitiesˆ H (eff) = H (eff) µc ≡ ˆ H O(eff) + ˆ H SO(eff) g S = R G phys S , g S ∗ = RR c G phys S ∗ (4.10)where R c = R + O (spin ) [66]. Here, as we work linearlyin spins, we can neglect the spin quadratic contributionto R c . In the following, we shall sometimes set, for sim-plicity, the velocity of light to 1. B. Present knowledge of the EOBgyrogravitomagnetic ratios g S and g S ∗ Let us describe the present knowledge of the two(phase-space-dependent) dimensionless gyrogravitomag-netic ratios g S and g S ∗ . First, from the PN-expandedpoint of view, g S and g S ∗ are known at the next-to-next-to-leading-order (NNLO) level [63, 64] g PN S ( u, p r , p φ ) = 2 + η (cid:20) − νu − νp r (cid:21) + η (cid:20) ν (cid:18) − u − up r + 58 p r (cid:19) + ν (cid:18) − u + 238 up r + 358 p r (cid:19)(cid:21) + O ( η ) ,g PN S ∗ ( u, p r , p φ ) = 32 + η (cid:20) − u − p r + ν (cid:18) − u − p r (cid:19)(cid:21) + η (cid:20) − u + 6916 up r + 3516 p r + ν (cid:18) − u − up r + 52 p r (cid:19) + ν (cid:18) − u + 5716 up r + 4516 p r (cid:19)(cid:21) + O ( η ) . (4.11)Here η ∼ /c is a place-holder for keeping track of thePN order, which we shall generally ignore in the follow-ing. The values of g S and g S ∗ cited above have been ex-pressed in the Damour-Jaranowski-Schaefer (DJS) spingauge [60, 71], which is defined so that these quantitiesdo not actually depend on p φ .Note that, at the PN order indicated above, g S and g S ∗ depend on the symmetric mass ratio ν in the followingway g S ( u, p r , p φ ) = 2 + νg ( ν ) S ( u, p r , p φ ) + ν g ( ν ) S ( u, p r , p φ )+ O ( ν ) ,g S ∗ ( u, p r , p φ ) = g ( ν ) S ∗ ( u, p r , p φ ) + νg ( ν ) S ∗ ( u, p r , p φ )+ ν g ( ν ) S ∗ ( u, p r , p φ ) + O ( ν ) . (4.12)Analytical gravitational self-force theory allowed oneto improve the knowledge on the first gyrogravitomag-netic ratio g S , to linear order in ν and for circular orbits.Namely, Ref. [70] derived (along circular orbits) the PNexpansion of g (circ) S ( u ) = 2 − νuδG resc S + O ( ν ) to the7.5PN level, see Eq. (4.3) there. For concreteness, let usquote here only the first few terms of this expansion g (circ) S ( u ) =2 − νu (cid:20) u + (cid:18) − π (cid:19) u + O ( u ) (cid:21) + O ( ν ) . (4.13) Concerning the second gyrogravitomagnetic ratio g S ∗ ,it was emphasized in Ref.[62] that the ν -independentpiece of g S ∗ , g ( ν ) S ∗ , could be exactly determined fromconsidering a spinning particle in an external background[61]. Taking as external background a Schwarzschild met-ric (consistently with our working linearly in spins) thisleads to the explicit expression g ( ν ) S ∗ ( u, p r , p φ ) = 11 + (cid:113) p φ u + (1 − u ) p r + 1 (cid:113) p φ u + (1 − u ) p r
21 + √ − u . (4.14)This exact expression for g ( ν ) S ∗ has introduced an ex-plicit dependence on p φ , corresponding to being in adifferent spin gauge than the DJS one used in the PN-expanded expressions (4.11).Gravitational self-force theory allowed, starting in2014, to acquire new knowledge on spin precession inextreme-mass-ratio binaries [34, 35, 68, 72]. The knowl-edge acquired in the latter references was limited to thecase of circular orbits and was transcribed within theEOB formalism in [34]. When considering circular orbits,it is natural to decompose g S ∗ (within self-force theory)in the following way g circ S ∗ ( u, ν ) = g circ ( ν ) S ∗ ( u ) + νg circ ( ν ) S ∗ ( u ) + O ( ν ) , (4.15)4where g circ ( ν ) S ∗ ( u ) ≡
31 + √ − u , (4.16)is defined by replacing p r by 0 and p φ by p (circ) φ ( u ) =[ u (1 − u )] − / in (4.14). The additional O ( ν ) correc-tion νg circ ( ν ) S ∗ ( u ) in Eq. (4.15) has been analytically de-termined as a PN expansion, up to the 9.5PN level in[34, 68]. [From the analytical results of [35] one couldfurther determine g circ ( ν ) S ∗ ( u ) to the 23PN level]. More-over, Ref. [34], combining analytical knowledge with afit to numerical SF data from Ref. [72] [together withnumerical SF data from [73]] derived a simple represen-tation of g circ ( ν ) S ∗ ( u ) as a rational function of u in theinterval 0 ≤ u < , see Eqs. (6.39)–(6.40) in Ref. [34].Let us quote here only the first terms of the PN expansionof g circ ( ν ) S ∗ ( u ) g circ ( ν ) S ∗ ( u ) = − u − u + (cid:18) π − (cid:19) u + (cid:18) −
24 ln( u ) − − − γ + 236632048 π (cid:19) u + O ( u ) . (4.17)See Eq. (A1) in the Appendix of Ref. [68] for the 9.5PNaccurate extension of this expression. C. Improving the analytical knowledge of g S and g S ∗ In the present paper we shall improve the SF knowl-edge of g S ∗ by computing the O ( p r ) corrections to Eq.(4.15). To this end, let us decompose g S ∗ in the followingway g S ∗ ( u, p r , p φ ; ν ) = g ( ν ) S ∗ ( u, p r , p φ ) + ν (cid:2) g S ∗ ( u )+ p r g S ∗ ( u ) + O ( p r ) (cid:3) + ν g ( ν ) S ∗ ( u )+ O ( ν p r ) + O ( ν ) . (4.18)Here g ( ν ) S ∗ ( u, p r , p φ ) is the phase-space function definedin Eq. (4.14) above, while we have written the addi-tional O ( ν ) and O ( ν ) contributions as p φ -independentfunctions of u and p r , expanded in powers of p r . [Weuse here the freedom allowed by DJS-type spin gaugeto eliminate any p φ dependence in the O ( ν ) terms.] Inthe following we shall use the SF result Eq. (3.33) todetermine the PN expansions of g S ∗ ( u ) and g S ∗ ( u ), namely g S ∗ ( u ) = g ∗ u + g ∗ u + g ∗ u + (cid:0) g c ∗ + g ln ∗ ln u (cid:1) u + . . .g S ∗ ( u ) = g ∗ + g ∗ u + g ∗ u + (cid:0) g c ∗ + g ln ∗ ln u (cid:1) u + (cid:0) g c ∗ + g ln ∗ ln u (cid:1) u + . . .g S ∗ ( u ) = g ∗ + g ∗ u + g ∗ u + (cid:0) g c ∗ + g ln ∗ ln u (cid:1) u + (cid:0) g c ∗ + g ln ∗ ln u (cid:1) u + . . . . (4.19)The first step towards the determination of the coeffi-cients in Eq. (4.19) is to relate the circular limit of Eq.(4.18) to the previous circular result, Eq. (4.15). Indeed,the circular limit of (4.18) reads g S ∗ ( u, p r , p φ ) (cid:12)(cid:12)(cid:12)(cid:12) circ = g ( ν ) S ∗ ( u, , p (circ) φ ( u, ν )) + ν (cid:2) g S ∗ ( u ) (cid:3) + O ( ν ) . (4.20)where the expression of the ν -dependent value of thesquare of p (circ) φ ( u, ν ), which is well known from EOBtheory [5, 25], reads[ p (circ) φ ( u, ν )] = − ∂ u A ( u ; ν ) ∂ u ( u A ( u ; ν ))= 1 u (1 − u ) (cid:20) − ν a ( u ) + (1 − u ) a (cid:48) ( u )2(1 − u ) (cid:21) + O ( ν ) . (4.21)Here A ( u ; ν ) = 1 − u + νa ( u ) + O ( ν ) is the ν -expansionof the main radial EOB potential A ( u ; ν ) = − g (eff)00 , seeEq. (4.6). This yields p (circ) φ ( u, ν ) = j (circ) ( u ) + νδj ( u ) + O ( ν ) , (4.22)with j (circ) ( u ) = 1 (cid:112) u (1 − u ) , δj ( u ) = − a ( u ) + (1 − u ) a (cid:48) ( u )4 √ u (1 − u ) / . (4.23)Inserting this result in Eq. (4.20), we see that the firstterm on the rhs contributes an additional O ( ν ) contribu-tion, namely g ( ν ) S ∗ ( u, , p (circ) φ ( u, ν )) =31 + √ − u + ν ∂g ( ν ) S ∗ ( u, , p φ ) ∂p φ (cid:12)(cid:12)(cid:12)(cid:12) p φ = j (circ) δj ( u ) + O ( ν ) . (4.24)This implies the following link g circ ( ν ) S ∗ ( u ) = ∂g ( ν ) S ∗ ( u, , p φ ) ∂p φ (cid:12)(cid:12)(cid:12)(cid:12) p φ = j (circ) δj ( u ) + g S ∗ ( u ) , (4.25)which determines the value of g S ∗ ( u ) from the previ-ously known results on g circ ( ν ) S ∗ ( u ), given to 4PN frac-tional accuracy in Eq. (4.17) and to 9.5PN accuracy in5Eq. (A1) of Ref. [68]. Let us cite explicitly here only thefirst coefficients in the PN expansion (4.19) of g S ∗ ( u ) g ∗ = − g ∗ = − g ∗ = 4132 π − g c ∗ = − − − γ + 269432048 π g ln ∗ = − , (4.26)leading to g S ∗ ( u ) = − u − u + (cid:18) π − (cid:19) u + (cid:18) −
24 ln( u ) − − − γ + 269432048 π (cid:19) u + O ( u ) . (4.27)In a second step, we can determine the coefficients g ∗ , . . . , g ln ∗ in the PN expansion, Eq.(4.19), of the O ( p r ) contribution to g S ∗ ( u, p r , p φ ), Eq. (4.18). Thetechnicalities of this determination will be explained inthe next subsection. For clarity, let us quote in advancethe result we shall obtain g ∗ = − g ∗ = − g c ∗ = − g ln ∗ = 0 g c ∗ = 1447441960 − π − − γg ln ∗ = − g c ∗ = − γ + 20974798192 π + 45416720 ln(3) − g ln ∗ = + 958135 . (4.28)Note that only the values of the first two coefficientswere known before, see Eq. (4.11). The results for g ∗ , g c ∗ , g ln ∗ , g c ∗ and g ln ∗ are new with this work. Atthe level O ( ν, p r ), instead, the only known coefficient[63, 64] is g ∗ = 52 , (4.29)as shown in Eqs. (4.11). D. EOB computation of Ω r and Ω φ as functions ofenergy and angular momentum We have seen above how SF theory led to a determi-nation of the functional link between the spin precessionquantity ψ and the two gauge-invariant frequencies of theorbital motion, Ω r and Ω φ : ψ ( Gm Ω r , Gm Ω φ ; q ) = ψ ( Gm Ω r , Gm Ω φ ) + q ∆ ψ ( Gm Ω r , Gm Ω φ ) + O ( q ) , (4.30)where q = m /m = ν + O ( ν ). In order to relatethe SF result, Eq. (3.33), on ∆ ψ ( Gm Ω r , Gm Ω φ ) tothe PN expansion of the EOB gyrogravitomagnetic ra-tio g S ∗ ( u, p r , p φ ; ν ), Eq. (4.18), we need, as a first task,to compute the functional link predicted by EOB theorybetween Ω r and Ω φ and the (gauge-invariant) total en-ergy E tot and orbital angular momentum L = P φ of thecorresponding motion of the binary system.Having in mind the link, Eq. (4.1), between E tot = H and the effective energy E eff = H eff (or, equivalently,ˆ E eff = E eff / ( µc ) and ˆ H eff = H eff / ( µc )), together withthe definition (4.9) of the rescaled angular momentum j ,our first task will be to compute Ω r and Ω φ as functionsof ˆ E eff and j .The two frequencies we are interested in can be writtenas Ω r = 2 π T , with T = (cid:73) dT , (4.31)and Ω φ = Φ T = (cid:72) dφ (cid:72) dT = (cid:72) ˙ φ dT T . (4.32)Here T denotes the physical time (to be distinguishedfrom the effective time T eff entering Eq. (4.6)), while (cid:72) denotes a periapsis-to-periapsis integral.Using the rescaled quantities (4.9) and (4.10), togetherwith the dimensionless, rescaled physical time t = c TGM , (4.33)the above frequencies become GM Ω r = 2 π (cid:72) dt , GM Ω φ = (cid:72) dφ (cid:72) dt . (4.34)The time integration in these integrals can be replacedby radial integration using Hamilton’s equations for therescaled radial variable drdt = 1 ν ∂h∂p r , (4.35) T is the standard Schwarzschild-like coordinate time, as observedat infinity. dφdt = 1 ν ∂h∂j . (4.36)In order to turn Eq. (4.35) into a relation of the type dt = f ( r ) dr we need the explicit expression of the(rescaled) radial momentum p r as a function of r . Thelatter relation is obtained by writing the law of conser-vation of energyˆ E eff = ˆ H eff ( u, p r , j ; S, S ∗ ) = (cid:114) A ( u ) (cid:16) j u + A ( u ) ¯ D ( u ) p r + ˆ Q (cid:17) + ju M ( g S S + g S ∗ S ∗ ) (4.37)where we replaced the metric potential B = g eff RR , Eq.(4.6), by ¯ D ≡ ( AB ) − .As our aim here is to compute the coupling coefficients g S , g S ∗ parametrizing effects linear in spins, it is easilyseen that it is enough to compute Ω r and Ω φ to zeroth order in spins. In other words, we can neglect the spin-dependent terms in the energy conservation law (4.37)and work with the simplified mass-shell conditionˆ E = A ( u ) (cid:16) j u + A ( u ) ¯ D ( u ) p r + ˆ Q (cid:17) + O (spin) . (4.38)The SF expansions of the EOB potential A , ¯ D andˆ Q = Q/µ read A ( u ) = 1 − u + νa ( u ) + O ( ν ) , ¯ D ( u ) = 1 + ν ¯ d ( u ) + O ( ν ) , ˆ Q ( u ) = νq ( u ) p r + νq ( u ) p r + νq ( u ) p r + O ( ν , p r ) . (4.39)The PN expansions of the first SF-order radial functions a ( u ), ¯ d ( u ), q ( u ), . . . entering the latter equations havebeen determined by SF theory to very high PN-orders:see Refs. [33, 35, 49, 74–77] for a ( u ), Refs. [14] for ¯ d ( u )and Refs. [14, 15, 59] for q ( u ), q ( u ) etc...For concreteness, let us quote here the beginning ofthese expansions A ( u ) = 1 − u + 2 νu + (cid:18) − π (cid:19) νu + (cid:20)(cid:18) π − γ + 2565 ln(2) (cid:19) ν + (cid:18) π − (cid:19) ν + 645 ν ln( u ) (cid:21) u + O ( u )¯ D ( u ) = 1 + 6 νu + (52 ν − ν ) u + (cid:20)(cid:18) − − π + 118415 γ − (cid:19) ν + (cid:18) π − (cid:19) ν + 59215 ν ln( u ) (cid:21) u + O ( u )ˆ Q ( u, p r ) = (cid:20) − ν ) νu + (cid:18)(cid:18) − − (cid:19) ν − ν + 10 ν (cid:19) u + O ( u ) (cid:21) p r + (cid:20)(cid:18) − − (cid:19) ν − ν + 6 ν (cid:21) u p r + O ( u , p r ) . (4.40)Inserting the SF expansion, Eqs. (4.39), into the mass-shell condition allows us to compute the functional de-pendence of p r on u , ˆ E eff and j . To the first SF-order,i.e., p r ( u, ˆ E eff , j ; S, S ∗ ) = p (0) r ( u, ˆ E eff , j )+ νp (1) r ( u, ˆ E eff , j )+ O ( ν , spin)(4.41) the explicit expressions of p (0) r and p (1) r read (when ne-glecting spins)7 p (0) r ( u, ˆ E eff , j ) = (cid:113) ˆ E − (1 − u )(1 + u j )1 − up (1) r ( u, ˆ E eff , j ) = − ˆ E p (0) r a ( u )(1 − u ) − p (0) r (cid:34) ¯ d ( u ) + a ( u )(1 − u ) + ( p (0) r ) q ( u ) + q ( u )( p (0) r ) + q ( u )( p (0) r ) + . . . − u (cid:35) . (4.42)We can then obtain an explicit expression for replacingtime integration by radial integration by inserting Eqs.(4.41) and (4.42) in the relation dt = (cid:113) ν ( ˆ E eff −
1) ˆ E eff Ap r ( A ¯ D + 2 p r νq + 3 p r νq + 4 p r νq + . . . ) dr , (4.43)which follows from the radial equation of motion (4.35).Inserting Eq. (4.43) in the t -integral expressions for Ω r and Ω φ , Eqs. (4.35) and (4.36), and formally expanding p r in powers of ν according to Eqs. (4.41) and (4.42),leads to r -integral expressions for the frequencies of thetype GM Ω r,φ ∼ (cid:88) n (cid:90) u (max) u (min) du f n ( u ) (cid:16) ˆ E − (1 − u )(1 + u j ) (cid:17) n + (4.44)where f n ( u ) involves a combination of (1 − u ) k , a ( u ),¯ d ( u ), etc. Note that these radial integrals are divergentfor n ≥
1. The appearance of singular integral is due toour formal replacement of the expansion (4.41) in orig-inally convergent integrals of the type (cid:72) drf ( r ) /p r . Asshown in Ref. [78], the correct result for these expansionsis obtained simply by taking Hadamard’s partie finie (Pf)of the singular integrals, Eq. (4.44).We are interested here in computing the PN-expansions of the frequencies. These PN-expansions canbe conveniently obtained: i) by replacing the various firstSF-order EOB potentials a ( u ), ¯ d ( u ), etc., by their PNexpansion illustrated in Eq. (4.40); and, using again thegeneral result of Ref. [78], ii) by formally expanding theradicals entering the denominators above, namelyˆ E − (1 − u )(1 + u j ) = ˆ E − u − j u + 2 j u , (4.45)around the Newtonian-like, quadratic radical R ( u ) = ˆ E − u − j u . (4.46)In other words, the combination of the ν -expansion andthe PN-expansion leads to integral expressions for the frequencies of the type GM Ω r,φ ∼ (cid:88) n Pf (cid:90) u (max)0 u (min)0 du ˜ f n ( u ) R n + ( u ) . (4.47)Here the symbol Pf denotes Hadamard’s partie finie.Note that the values of the end points of the u -integrations are different from the ones in Eq. (4.44)above and now denote the two roots of the Newtonian-like quadratic radical R ( u ), Eq. (4.46). Finally, we areleft with evaluating Newtonian-like radial integrals of thetype (4.47). When evaluating the partie finie of thesesingular integrals it is convenient to replace ˆ E eff and j bythe quantities u p and e defined so that the two roots of R ( u ) are u p (1 ± e ), namelyˆ E − − u p (1 − e ) , j = 1 u p . (4.48)As we are interested in slightly eccentric motions, wecan further expand the latter (singular) Newtonian-likeintegrals in powers of e .When doing so, we use the following Newtonian-likeparametrization of the inverse radius u = 1 /ru = u p (1 + e cos χ ) , (4.49)so that du = − u p e sin χ dχ , R ( u ) = u p e sin χ . (4.50)We have then shown that the e -expansion of the partiefinie of the integrals (4.47) is correctly obtained by takingthe (cid:15) term in the Laurent expansion in (cid:15) of the integralsof the type (cid:90) π − (cid:15)(cid:15) g n ( χ )sin n χ dχ , (4.51)that are generated by the expansions (4.47).Finally, the combined PN-, SF- and eccentricity-expansions of the frequencies yield GM Ω r = Ω (0) r ( u p , ν )+ e Ω (2) r ( u p , ν )+ e Ω (4) r ( u p , ν )+ O ( e ) , (4.52)with As said above, the so-defined EOB variables p ≡ /u p and e (0) r ( u , ν ) = u / p + (cid:18) ν − (cid:19) u / p + (cid:18) ν − (cid:19) u / p + (cid:20) − (cid:18) − π + 259148 (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) − π + 116815 γ + 58415 ln( u p ) + 14585 ln(3) − (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) γ − u p ) + 25523615 ln(2) + 48196 π (cid:19) ν (cid:21) u / p + 1177063 u p πν + O ( u / , ν )Ω (2) r ( u p , ν ) = − u / p + (cid:18) − ν + 154 (cid:19) u / p + (cid:18) − ν + 17116 (cid:19) u / p + (cid:20) (cid:18) π − (cid:19) ν (cid:21) u / p + (cid:20) (cid:18) − γ − u p ) − π − (cid:19) ν (cid:21) u / p + (cid:20) (cid:18) − π − γ − − u p )+ 21484375224 ln(5) − (cid:19) ν (cid:21) u / p − u p πν + O ( u / p , ν )Ω (4) r ( u p , ν ) = 38 u / p + (cid:18) − ν (cid:19) u / p + (cid:18) ν − (cid:19) u / p + (cid:20) − (cid:18) − π + 13927128 (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) u p ) − γ − π + 195312596 ln(5)+ 6647751160 ln(3) (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) − u p ) − − γ + 1832457724576 π − (cid:19) ν (cid:21) u / p − u p πν + O ( u / , ν ) . (4.53)Note that the dependence on ln u p starts at O ( u / p ). Similarly, the azimuthal frequency reads GM Ω φ = Ω (0) φ ( u p , ν ) + e Ω (2) φ ( u p , ν ) + e Ω (4) φ ( u p , ν ) + O ( e ) , (4.54)with, for example,Ω (0) φ ( u p , ν ) = u / p + (cid:18)
32 + 12 ν (cid:19) u / p + (cid:18) − ν + 1118 (cid:19) u / p + (cid:20) (cid:18) − π (cid:19) ν (cid:21) u / p + (cid:20) (cid:18) − π − γ − u p ) − (cid:19) ν (cid:21) u / p + (cid:20) (cid:18) − γ − − − u p ) − π (cid:19) ν (cid:21) u / p − u p πν + O ( u / p , ν )Ω (2) φ ( u p , ν ) = − u / p + (cid:18) − − ν (cid:19) u / p + (cid:18) − − ν (cid:19) u / p + (cid:20) − (cid:18) − π − (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) − γ − u p ) − π − (cid:19) ν (cid:21) u / p should be distinguished from the SF-defined variables ¯ p and ¯ e used as independent variables in Eq. (3.33) above. The parameter e defined in Eq. (4.48) does not measure the exacteccentricity of the corresponding EOB orbits that would vanish when p r vanishes. Nevertheless, when used in PN-expansion, itwill allow us to correctly evaluate the gauge-invariant quantitieswe are interested in. (cid:20) (cid:18) − π − γ − − u p ) − (cid:19) ν (cid:21) u / p − πνu p + O ( u / p , ν )Ω (4) φ ( u p , ν ) = 38 u / p + (cid:18) − ν (cid:19) u / p + (cid:18) ν − (cid:19) u / p + (cid:20) − (cid:18) − π + 30193128 (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) u p ) − γ − π + 634235 ln(3) (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) − u p ) − γ + 456443312288 π + 1301985547493225600 (cid:19) ν (cid:21) u / p + 2129328 u p πν + O ( u / p , ν ) . (4.55)Note again that the dependence on ln u p starts at O ( u / p ).In the expansions (4.53) and (4.55) above we haveonly displayed the beginning of the PN expansions ofΩ (0 , , r ( u p , ν ) and Ω (0 , , φ ( u p , ν ). Actually we have com-puted them to the highest PN order currently knownfrom SF calculations. Similarly, we have also computedthe eccentricity expansions (4.52) up to O ( e ).It will be convenient in the following to trade thetwo dimensionless frequencies GM Ω r and GM Ω φ by twoother related dimensionless (equally gauge-invariant) pa-rameters, namely the (fractional) periastron advance perorbit k ≡ Ω φ Ω r − π − , (4.56) and the dimensionless azimuthal frequency variable y ≡ ( Gm Ω φ ) / = (cid:18) − ν + O ( ν ) (cid:19) ( GM Ω φ ) / . (4.57)The combined eccentricity-, PN- and SF-expansions ofthese quantities read k ( u p , e ; ν ) = k (0) ( u p , ν )+ e k (2) ( u p , ν )+ e k (4) ( u p , ν )+ O ( e ) , (4.58)with k (0) ( u p , ν ) = 3 u p + (cid:18) − ν (cid:19) u p + (cid:20)
210 + (cid:18) −
205 + 12332 π (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − γ + 1916713072 π − − − u p ) − (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − π − u p ) + 17277335 ln(3) − γ − (cid:19) ν (cid:21) u p − πνu / p + O ( u p , ν ) k (2) ( u p , ν ) = (cid:18) − ν + 154 (cid:19) u p + (cid:20) (cid:18) −
106 + 123128 π (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − − − γ + 1117972048 π + 204125 ln(3) − u p ) (cid:19) ν (cid:21) u p + (cid:20) (cid:18) π − u p ) − − γ − − (cid:19) ν (cid:21) u p − πu / p ν + O ( u p , ν ) k (4) ( u p , ν ) = − u p ν (cid:20) (cid:18) − −
375 ln( u p ) − γ + 1352948 − π (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − u p ) − γ − π − (cid:19) ν (cid:21) u p − πνu / p + O ( u p , ν ) . (4.59)In the following it will be convenient to work with therescaled periastron advance per orbit ˆ k , defined byˆ k ≡ k . (4.60) Similarly y ( u p , e ; ν ) = y (0) ( u p , ν )+ e y (2) ( u p , ν )+ e y (4) ( u p , ν )+ O ( e ) , (4.61)with y (0) ( u p , ν ) = (cid:18) − ν (cid:19) u p + (cid:18) − ν + 1 (cid:19) u p + (cid:18) − ν (cid:19) u p + (cid:20)
83 + (cid:18) − π (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − π − γ − − u p ) (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − u p ) + 9717679216 π − − γ − − (cid:19) ν (cid:21) u p − πνu / p + O ( u p , ν ) y (2) ( u p , ν ) = (cid:18) ν − (cid:19) u p − u p ν + (cid:18) − − ν (cid:19) u p + (cid:20)
212 + (cid:18) − π − (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − π − u p ) + 2214445 ln(2) − γ − (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − − u p ) − − γ + 1212134608 π (cid:19) ν (cid:21) u p − πνu / p + O ( u p , ν ) y (4) ( u p , ν ) = ( − ν ) u p + (cid:18) −
638 + 11912 ν (cid:19) u p + (cid:20) − (cid:18) − π (cid:19) ν (cid:21) u p + (cid:20) − (cid:18) γ + 444445 ln( u p ) + 403385 ln(3) − π − (cid:19) ν (cid:21) u + (cid:20) − (cid:18) − π − u p ) + 140246105 γ (cid:19) ν (cid:21) u p + 3724671890 πνu / p + O ( u p , ν ) . (4.62)Finally, we will need in the following to invert the func-tional link between ( u p , e ) and (ˆ k, y ), i.e., to compute thefunctions u p = f u p (ˆ k, y ) , e = f e (ˆ k, y ) . (4.63)This inversion requires some care, because the Jacobian ∂ (ˆ k, y ) /∂ ( u p , e ) is of order u p near the origin of the u p , e plane. If we provisionally introduce the quantity (cid:15) ≡ e u p , the Jacobian ∂ (ˆ k, y ) /∂ ( u p , (cid:15) ) will be of order unitynear the origin of the u p , (cid:15) plane. This shows that we caninvert the link X i = ( u p , (cid:15) ) → Y i = (ˆ k, y ) by standardTaylor expansions of the symbolic type X i = A ij Y j + A ijk Y j Y k + . . . . (4.64)1When going back from the pair ( u p , (cid:15) ≡ e u p ) to the orig- inal pair ( u p , e ) we obtain the following transformation u p = ˆ k + (cid:20)(cid:18)
52 + 13 y ˆ k (cid:19) ν −
354 + 54 y ˆ k (cid:21) ˆ k + (cid:34)(cid:32) π y ˆ k − y ˆ k + 2512 (cid:18) y ˆ k (cid:19) + 503 − π (cid:33) ν − y ˆ k + 4516 (cid:18) y ˆ k (cid:19) + 4558 (cid:35) ˆ k + O (ˆ k , y ) e = 1 − (cid:18) ν (cid:19) y ˆ k + (cid:34) − y ˆ k + 14 (cid:18) y ˆ k (cid:19) + (cid:32) − y ˆ k + 16 (cid:18) y ˆ k (cid:19) (cid:33) ν (cid:35) ˆ k + (cid:34)(cid:32) − y ˆ k + 78 (cid:18) y ˆ k (cid:19) + 1516 (cid:18) y ˆ k (cid:19) − (cid:33) + (cid:32) − π y ˆ k + 41128 π (cid:18) y ˆ k (cid:19) + 122524 y ˆ k − (cid:18) y ˆ k (cid:19) + 94 (cid:18) y ˆ k (cid:19) + 18 (cid:33) ν (cid:35) ˆ k + O ( k , y ) . (4.65)Again we have only indicated, for concreteness, the be-ginning of these expansions. E. EOB computation of the spin precessionfrequency
When considering, as we do here, spin couplings tolinear order, i.e., an Hamiltonian of the form H = H orbital + Ω S · S + Ω S · S , Hamilton’s equations ofmotion for the spins ( a = 1 , d S a dt = { S a , H } (4.66)yields d S a dt = Ω S a × S a , (4.67)showing that Ω S a = ∂H∂ S a is the vectorial precession fre-quency of S a . Restricting to the case of interest of par-allel spins we conclude that the (algebraic) magnitude ofthe spin frequency of body 1 is given byΩ S = ∂H∂S = M νh ∂ ˆ H eff ∂S = (cid:18) GM (cid:19) νh ju (cid:18) g S + g S ∗ m m (cid:19) . (4.68) At first order in ν , this reads GM Ω S ( u, p r , j ; ν )= j (1 − ν )[1 − ν ( ˆ E eff − u [ νg S + (1 − ν ) g S ∗ ] + O ( ν )= j (1 − ν ˆ E eff ) u [ νg S + (1 − ν ) g S ∗ ] + O ( ν ) . (4.69)The averaged spin frequency is then given by (cid:104) Ω S (cid:105) ( ˆ E (eff) , j ; ν ) = 1 T (cid:73) Ω S dT = Ω r π (cid:73) Ω S dT . (4.70)As indicated, the averaged spin frequency (cid:104) Ω S (cid:105) is a func-tion of the conserved dynamical quantities, ˆ E (eff) and j .Replacing as above the T -integration by a radial inte-gration, say dT = f ( r ) dr = ˜ f ( u ) du , see Eq. (4.43), wesee that the computation of (cid:104) Ω S (cid:105) amounts to computingradial integrals of the type (cid:73) du ˜ f ( u ) u [ νg S + (1 − ν ) g S ∗ ( u, p r , j ; ν )] . (4.71)In the radial integral involving g S we can simply replace g S = 2 because of the ν prefactor. By contrast, in theradial integral involving g S ∗ we need to insert the ex-pression (4.18) which involves undetermined coefficientsat the p r and p r levels, such as g ∗ , g c ∗ , etc. This radialintegral can be computed by the same technique we usedabove for computing Ω r and Ω φ . Replacing as above ˆ E eff and j by the Newtonian-like quantities u p and e , definedby Eqs. (4.49), we found GM (cid:104) Ω S (cid:105) ( u p , e ; ν ) =Ω (0) S ( u p , ν ) + e Ω (2) S ( u p , ν ) + e Ω (4) S ( u p , ν ) + O ( e ) , (4.72)with, for example,2Ω (0) S ( u p , ν ) = (cid:18) − ν (cid:19) u / p + (cid:18) − ν (cid:19) u / p + (cid:18) − ν + 113116 (cid:19) u / p + (cid:18) − ν + 1271128 νπ + 90939128 (cid:19) u / p + (cid:18) νπ − ν ln(2) − νγ − ν + νg ∗ + 1946997256 − ν ln( u p ) (cid:19) u / p + O ( u p , ν )Ω (2) S ( u p , ν ) = (cid:18) −
94 + 32 ν (cid:19) u / p + (cid:18) − ν (cid:19) u / p + (cid:18) ν − (cid:19) u / p + (cid:18) νg ∗ − − ν − νπ (cid:19) u / p + (cid:18) νg ∗ + 6364074096 νπ − ν − νγ + 32 νg ∗ + 12 νg c ∗ − ν ln( u p ) + 12 νg ln ∗ ln( u p )+ 331875512 − ν ln(3) + 1895215 ν ln(2) (cid:19) u / p + O ( u p , ν )Ω (4) S ( u p , ν ) = (cid:18) − ν (cid:19) u / p + (cid:18) − ν (cid:19) u / p + (cid:18) ν − (cid:19) u / p + (cid:20) − (cid:18) − π − g ∗ (cid:19) ν (cid:21) u / p + (cid:20) − (cid:18) − g ∗ + 92 g ∗ + 38 g ln ∗ ln( u p ) + 66425 γ − − π + 33215 ln( u p ) + 50519720 ln(3) + 38 g c ∗ + 716 g ln ∗ (cid:19) ν (cid:21) u / p + O ( u p , ν ) . (4.73)As above, we have only indicated here the first terms inthe PN-expansions we computed.Let us now consider the ratio ψ ≡ (cid:104) Ω S (cid:105) Ω φ . (4.74)It is easily checked (see, e.g., Refs. [34, 72]) that the so-defined ratio is identical to the spin-precession mea-sure ψ introduced in Eq. (2.1) above. The combinedeccentricity-, PN- and SF-expansions of the function ψ ( u p , e ; ν ) is then found to be of the type ψ ( u p , e ) = ψ (0) ( u p , ν )+ e ψ (2) ( u p , ν )+ e ψ (4) ( u p , ν )+ O ( e ) , (4.75)where the beginnings of the expansions of ψ (0) ( u p , ν ) and ψ (2) ( u p , ν ) are given by ψ (0) ( u p , ν ) = (cid:18) − ν (cid:19) u p + ( 458 − ν ) u p + (cid:18) − ν + 66316 (cid:19) u p + (cid:18) − ν + 418 νπ + 47805128 (cid:19) u p + (cid:18) − ν ln(2) + νg ∗ − νγ + 1098971024 νπ − ν − ν ln( u p ) (cid:19) u p + O ( u p , ν ) ψ (2) ( u p , ν ) = (cid:18) − ν (cid:19) u p + (cid:18) − ν + 752 (cid:19) u p + (cid:20)(cid:18) − π + 12 g ∗ (cid:19) ν + 36398 (cid:21) u p + (cid:20) (cid:18) g ∗ + 12 g c ∗ − g ∗ − γ + 2826511024 π − g ln ∗ ln( u p ) − u p ) − (cid:19) ν (cid:21) u p + O ( u p , ν ) ψ (4) ( u p , ν ) = (cid:18) − ν + 316 (cid:19) u p + (cid:20) (cid:18) − π + 38 g ∗ + 38 g ∗ (cid:19) ν (cid:21) u p + (cid:20) (cid:18) − γ − π − u p ) + 196834 ln(3) + 14516 g ∗
3+ 9916 g ∗ + 34 g ln ∗ ln( u p ) + 38 g ln ∗ ln( u p ) + 34 g c ∗ + 38 g c ∗ + 716 g ln ∗ − (cid:19) ν (cid:21) u p + O ( u p , ν ) . (4.76)Though the so obtained function ψ ( u p , e ; ν ) is gauge-invariant ( u p and e being functions of the gauge-invariantdynamical quantities ˆ E eff and j ), it cannot be di-rectly compared with the (equally gauge-invariant) func-tion ψ ( m Ω r , m Ω φ ; q ) obtained in the SF computationabove.In order to compare our EOB-derived result with theSF result we first need to transform the dependence on u p and e into a dependence on Gm Ω r and Gm Ω φ , orequivalently, on y = ( Gm Ω φ ) / and ˆ k = ( Ω φ Ω r − ψ on (ˆ k, y ) is given, atfirst order in ν , by ψ (ˆ k, y ) = ψ (ˆ k, y ) + ν ∆ ψ (ˆ k, y ) + O ( ν ) , (4.77) where the beginnings of the expansions of ψ (ˆ k, y ) and∆ ψ (ˆ k, y ) read ψ (ˆ k, y ) = 32 ˆ k − (cid:18)
98 ˆ ky + 92 ˆ k (cid:19) + (cid:18) k y − y ˆ k + 272 ˆ k (cid:19) + (cid:18) − k y − k y −
458 ˆ ky − k (cid:19) + . . . ∆ ψ (ˆ k, y ) = − ˆ k + (cid:18) k −
54 ˆ ky (cid:19) + (cid:18) k π y − k π −
938 ˆ k y − y ˆ k + 694 ˆ k (cid:19) + (cid:16) c c + c ln6 ln(ˆ k ) (cid:17) ˆ k + . . . (4.78)with c c = − (cid:18) − g ∗ + 1770671440 − − (cid:19) (cid:18) y ˆ k (cid:19) + (cid:18) − g ∗ + 1090097915 ln(2) + 375 γ − g ∗ − π + 156078932 ln(3) (cid:19) (cid:18) y ˆ k (cid:19) + (cid:18) − g ∗ − g ∗ + 1111433768 + 111958192 π − − γ − (cid:19) y ˆ k − g ∗ + 716 g ∗ + 984260945 ln(2) + 5953 γ + 43812920 ln(3) + 158018549152 π c ln6 = 5956 − y ˆ k + 3710 (cid:18) y ˆ k (cid:19) . (4.79)Of most interest for our present work is the function∆ ψ (ˆ k, y ), Eq. (4.78). We can directly compare the latterfunction with the one computed with SF theory above inEq. (3.33), modulo the fact that the SF computed oneabove was expressed not in terms of ˆ k and y , but insteadin terms of eccentricity and semi-latus rectum parame-ters, ¯ e and ¯ p = 1 / ¯ u p , different from the ones, e and u p ,used here (recalling the notational change discussed atthe beginning of this section).The transformation between (¯ e, ¯ u p ) and ( e, u p ) is onlyneeded at order ν and can be obtained by identifying the µ -rescaled Schwarzschild energy and angular momentumused to define ¯ e and ¯ u p to the EOB quantities ˆ E eff and j .In other words, while (¯ e, ¯ u p ) were defined in Eqs. (2.12),(2.13), i.e., equivalently, by writingˆ E eff = (cid:115) (1 − u p ) − u p ¯ e − u p − ¯ u p ¯ e , (4.80) j = 1 (cid:112) ¯ u p (1 − u p − ¯ u p ¯ e ) , (4.81)the other pair ( e, u p ), used in our EOB computation, wasdefined by writingˆ E eff = (cid:113) − u p (1 − e ) , j = 1 √ u p . (4.82)The comparison between these two expressions impliesthe following transformation law u p = ¯ u p (1 − u p − ¯ u p ¯ e ) , − e = (1 − ¯ e ) 1 − u p (1 − u p − ¯ u p ¯ e ) . (4.83)Using either this transformation (together with inter-mediate equations given above) or directly the wellknown elliptic-integrals expressions giving Ω φ and k in a Schwarzschild background, consistently with Eqs.4(2.13)-(2.16) in Ref. [36], we obtainˆ k = ¯ u p + (cid:18)
92 + 14 ¯ e (cid:19) ¯ u p + (cid:18)
452 + 154 ¯ e (cid:19) ¯ u p + (cid:18) e + 9458 + 3158 ¯ e (cid:19) ¯ u p + (cid:18) e + 94564 ¯ e + 51038 (cid:19) ¯ u p + (cid:18) e + 9355532 ¯ e + 385256 ¯ e + 5613316 (cid:19) ¯ u p + (cid:18) e + 72972932 ¯ e + 31274116 + 15015256 ¯ e (cid:19) ¯ u p + (cid:18) e + 1094593564 ¯ e + 14073345128 + 675675512 ¯ e (cid:19) ¯ u p + O (¯ u p ) y = (1 − ¯ e )¯ u p + (2¯ e − e )¯ u p + (cid:18) −
238 ¯ e − e + 6¯ e (cid:19) ¯ u p + (cid:18) −
134 ¯ e + 24¯ e − e (cid:19) ¯ u p + (cid:18) − e −
14 ¯ e + 120¯ e (cid:19) ¯ u p + (cid:18) − e + 672¯ e + 51¯ e (cid:19) ¯ u p + (cid:18) e − e + 3936¯ e (cid:19) ¯ u p + (cid:18) − e + 8032¯ e + 23424¯ e (cid:19) ¯ u p + O (¯ u p ) . (4.84)Inserting these expressions in the EOB-derived func- tion ∆ ψ (ˆ k, y ) obtained above gives∆ ψ (¯ u p , ¯ e ) = ∆ ψ (0) (¯ u p )+ ¯ e ∆ ψ (2) (¯ u p )+ ¯ e ∆ ψ (4) (¯ u p )+ . . . (4.85)where ∆ ψ (0) (¯ u p ) will be discussed below, and where the O (¯ e ) contribution (which is new with this work) reads∆ ψ (2) (¯ u p ) = ¯ u p + (cid:18) − π (cid:19) ¯ u p + (cid:18) − u p ) − π + 12 g ∗ + 5365 γ + 117203 ln(2) − (cid:19) ¯ u p + (cid:18) π − − u p ) + 54 g ∗ + 97656251344 ln(5) + 12 g c ∗ + 12 g ln ∗ ln(¯ u p ) − γ − (cid:19) ¯ u p + 319609630 π ¯ u / p + (cid:18) g c ∗ − − g ∗ − g c ∗ + 12 ln(¯ u p ) g ln ∗ − g ln ∗ ln(¯ u p ) − − γ − π + 318425596072359296 π − u p ) (cid:19) ¯ u p + O (¯ u / p ) , (4.86)Here the terms up to O ( u p ) coincide with those givenin Eq. (5.4) of [36], and the undetermined coeffi-cients g ∗ , g c ∗ , etc. which parametrized the unknown O ( p r ) dependence of the EOB gyrogravitomagnetic ratio g S ∗ ( u, p r , p φ ) first enter ∆ ψ (2) (¯ u p ) at order O ( u ).Comparing with Eq. (3.33) above we then find uniquevalues for the so far undetermined EOB coefficients,namely5 g ∗ = − g c ∗ = 1447441960 − π − − γg ln ∗ = − g c ∗ = − γ + 20974798192 π + 45416720 ln(3) − g ln ∗ = + 958135 . (4.87)We also computed the O (¯ e ) contribution to ∆ ψ (¯ u p , ¯ e ) which we give below in its parametrized form∆ ψ (4) (¯ u p ) = −
12 ¯ u p + (cid:18) − g ∗ + 1953125192 ln(5) + 3710 ln(¯ u p ) − π + 38 g ∗ + 375 γ − (cid:19) ¯ u p + (cid:18) − π + 720900871537600 − u p ) + 38 g c ∗ + 2116 g ∗ + 5716 g ∗ + 722656255376 ln(5)+ 716 g ln ∗ + 34 g c ∗ + 34 g ln ∗ ln(¯ u p ) + 38 g ln ∗ ln(¯ u p ) − γ − (cid:19) ¯ u p + 10895815120 π ¯ u / p + (cid:18) g ln ∗ − g ∗ + 418 g c ∗ − g ∗ − g ln ∗ + 54 g c ∗ + 2116 g c ∗ + 38 g c ∗ + 54 ln(¯ u p ) g ln ∗ + 38 g ln ∗ ln(¯ u p ) + 2116 g ln ∗ ln(¯ u p )+ 418 g ln ∗ ln(¯ u p ) − − γ − π + 15116966065318874368 π − u p ) (cid:19) ¯ u p + O (¯ u / p ) . (4.88)As soon as some SF data on ψ at order O ( e ) becomeavailable, this result will allow one to determine the co-efficients parametrizing the O ( p r ) EOB gyrogravitomag-netic ratio g S ∗ .Concerning the quantity ∆ ψ (¯ u p , ¯ e → ψ (circ) (¯ u p ) [34, 35, 68]∆ ψ (circ) (¯ u p ) = ¯ u p − u p −
152 ¯ u p + (cid:18) − − − γ − u p ) + 204711024 π (cid:19) ¯ u p + . . . , (4.89)with the knowledge of the EOB function ρ ( x ) measuringperiastron precession at the 1SF-level [25, 59], namely∆ ψ (¯ u p , ¯ e →
0) = ∆ ψ (circ) (¯ u p ) −
12 (1 − u p ) / (1 − u p ) / − ¯ u p + ¯ u p k (¯ u p ) , (4.90)where k ( y ) = − ρ ( y ) − y − y ) / , (4.91)as obtained from the definition (1 + k (circ) ( x, ν )) − = 1 − x + νρ ( x )+ O ( ν ), i.e., using the link x = ( GM Ω φ ) / =(1 + ν ) y + O ( ν ) and k (circ) ( y, ν ) = − (cid:113) − ν ) y + νρ ( y )= k (0)(circ) ( y ) + νk ( y ) + O ( ν ) , (4.92)with k (0)(circ) ( y ) = − √ − y . (4.93)6 V. RESUMMATION OF g S ∗ SF technology allows one to reach high PN orderswhen working linearly in the symmetric mass ratio ν [14, 15, 33, 35, 49, 59, 74–77]. We have given here onemore example of this feature by deriving the O ( e ) (re-spectively, O ( p r )) piece in the spin precession function ψ (respectively, gyrogravitomagnetic ratio g S ∗ ) to manymore PN-orders than was previously known. Past workhas shown that an efficient way of using the high-PNinformation obtained by SF computations is to incorpo-rate it, together with known PN contributions which arehigher order in ν , within some suitably resummed cou-pling functions of the EOB formalism. Let us show howsuch an approach can be applied to the g S ∗ EOB couplingfunction.To blend and resum SF and PN information concerning g S ∗ let us start from the structure of the test-mass valueof g S ∗ , as written (when neglecting effects quadratic inspins) in Eq. (4.14). First, we note that this structurecan be rewritten as g ( ν ) S ∗ ( u, p r , p φ ) = (cid:112) A schw ( u )ˆ E schw + (cid:112) A schw ( u ) + 2 A schw ( u )ˆ E schw (1 + (cid:112) A schw ( u )) , (5.1)where A schw ( u ) = 1 − u and where ˆ E schw = (cid:113) (1 − u )(1 + p φ u + (1 − u ) p r ) is the conserved en-ergy of a test particle in a Schwarzschild background.Taking this reformulation as a model, let us define thefollowing EOB-compatible, ν -deformed g S ∗ coupling g (EOB − like) S ∗ ( u, p r , p φ ; ν ) = (cid:112) A P N ( u ; ν )ˆ H P N (orb , eff) + (cid:112) A P N ( u ; ν ) + 2 A P N ( u ; ν )ˆ H P N (orb , eff) (1 + (cid:112) A P N ( u ; ν )) , (5.2) where A P N ( u ; ν ) is defined as A P N ( u ; ν ) ≡ − u + 2 νu + (cid:18) − π (cid:19) νu , (5.3)and whereˆ H P N (orb , eff) ≡ (cid:112) A P N ( u ; ν ) × (cid:114)(cid:16) p φ u + A P N ( u ; ν ) ¯ D P N ( u ; ν ) p r + ˆ Q P N (cid:17) , (5.4)with ¯ D P N ( u ; ν ) = 1 + 6 νu + (52 ν − ν ) u , ˆ Q P N ( u ; ν ) = 2(4 − ν ) νu p r . (5.5)We propose to use the ν -deformed ratio g (EOB − like) S ∗ ,Eq. (5.2), as a reference value for the (unknown) ex-act g S ∗ , and to incorporate the current PN and SFinformation of g S ∗ in the form of a correcting factor r S ∗ ( u, p r , p φ ; ν ) of the type r S ∗ = 1+ O ( ν ). [We have alsoexplored the possibility of translating g S ∗ in terms of thegyrogravitomagnetic ratio of a test-mass in some spin-effective metric. However, this led to an effective metricwhose ν -deformation was drastically different (startingat the 1PN-order) from the one of the usual orbital EOBeffective metric, and which did not seem to offer a goodstarting point for resumming g S ∗ .] In other words, wepropose to define a resummed g S ∗ of the form g resum S ∗ ( u, p r , p φ ; ν ) = g (EOB − like) S ∗ ( u, p r , p φ ; ν ) r S ∗ ( u, p r , p φ ; ν ) . (5.6)When considering fast-spinning black holes, one can stilluse the factorized form (5.6), but with suitably spin-quadratic-extended values of A P N and ˆ H P N (orb , eff) in thedefinition (5.2) of g (EOB − like) S ∗ (e.g., as defined in Ref.[66]).We will suggest several possible estimates of the cor-recting factor r S ∗ in Eq. (5.6). All these estimates willbe constructed from the PN expansion of the correctingfactor r S ∗ = g SF + P NS ∗ /g (EOB − like) S ∗ , which is of the form r PN S ∗ ( u, p r , p φ ; ν ) = 1 + ν ˆ g ( u, p r , p φ ) + ν ˆ g ( u, p r , p φ ) , (5.7)whereˆ g ( u, p r , p φ ) = η (cid:18) − u − p r (cid:19) + η (cid:20) p r + (cid:18) − p φ u − u (cid:19) p r − p φ u − u (cid:21) + η (cid:20) ( − u + 1796 p φ u − p φ u ) p r + 17288 p φ u − u − p φ u + 4148 u π (cid:21) + η (cid:20)(cid:18) − p φ u − u ln(3) − u γ − u π + 9395245 u ln(2) − p φ u + 77576 p φ u + 1743637717280 u − u ln( u ) (cid:19) p r − u − p φ u + 113144 p φ u − p φ u − u ln(2) + 854219216 u π + 205576 p φ u π − u γ − u ln( u ) (cid:21) + η (cid:20)(cid:18) u ln(3) − u γp φ − p φ u − u ln(3) p φ + 2348827 u ln(2) p φ − u ln( u ) p φ − u ln(2) + 103604315 u γ + 86913824 p φ u − u + 225776912 p φ u + 836716120736 p φ u + 146005864 u π − p φ u π + 51802315 u ln( u ) (cid:19) p r + 524835 u γ − p φ u + 14537941472 p φ u − p φ u + 86941472 p φ u − u + 13059635110592 u π − u γp φ + 262435 u ln( u ) − u ln( u ) p φ + 353392945 u ln(2) − p φ u π − u ln(2) p φ − u ln(3) + 412673110592 p φ u π (cid:21) + O ( u ) , (5.8)and ˆ g ( u, p r , p φ ) = (cid:18) p r u − u + 158 p r (cid:19) η . (5.9)In Eq. (5.8) we have displayed only the beginning of thePN expansion of ˆ g , including in particular all the O ( p r )terms that have been derived in the present paper. Wekept in Eq. (5.8) only the terms that are fully known, i.e.,the terms that do not involve any yet undetermined co-efficient such as g ∗ up r η , etc. [More precisely, in ˆ g wesee that the p r contribution is fully known only at O ( η );by contrast, the p r one is now known from O ( η ) up to O ( η ); in ˆ g only terms at O ( η ) are known.] Beyond the contributions indicated, only the circular limit of ˆ g is known and it can be straightforwardly computed (inthe DJS gauge) by PN-expanding the prescription givenabove, using for instance either the explicit 9.5PN accu-rate g circ S ∗ result of Ref. [68] or the implicit 22PN-accurateresult of Ref. [35] on ψ circ .In the rest of this subsection we will motivate variousways (namely, Taylor-like or inverse-Taylor-like, at vari-ous PN-approximations) of defining the correcting factor r S ∗ in Eq. (5.7) by studying the circular limit of g resum S ∗ ,Eq. (5.6). For concreteness, let us exhibit the explicit4PN-accurate value of r S ∗ when setting p r → r P NS ∗ ( u, p φ ; ν ) = 1 − uνη + (cid:18) − νu − ν u − νp φ u (cid:19) η + (cid:18) − u + 4148 u π − p φ u + 17288 p φ u (cid:19) νη + (cid:18) − u − u p φ − p φ u + 113144 p φ u − u ln( uη ) − u ln(2) − u γ + 854219216 u π + 205576 u π p φ (cid:19) νη . (5.10)Note the explicit appearance of p φ in r P NS ∗ . We wish tostudy an estimate of g S ∗ that would be approximatelyvalid when considering (circularized) inspiralling and co-alescing binary black holes. Following the results of EOBtheory [5, 6] we can approximately replace p φ by a func-tion of u having the following properties.Above the last stable orbit we estimate p φ ≈ p (circ) φ ( u ) = (cid:112) − ∂ u A ( u ; ν ) /∂ u ( u A ( u ; ν )), which we sim-ply approximate in our present qualitative study by p φ ≈ [ u (1 − u )] − / . Beyond the last stable orbit we in-stead approximate p φ by a constant equals to its value at the last stable orbit (say p φ ≈ √ r S ∗ , and correlatively of g S ∗ , dur-ing coalescence, as a function of u . The results of this ex-ploratory study of the evolution of g S ∗ during coalescenceare illustrated in Fig. 1, which considers the equal-masscase, ν = 1 /
4. This figure compares the u -evolution ofvarious approximations to g S ∗ = g (EOB − like) S ∗ r S ∗ : 1) theone defined by using the Taylor-like 4PN approximantfor r S ∗ , Eq. (5.10) together with the just explained p φ replacements; 2) the one defined by using inverse resum-mation of r P NS ∗ , i.e., by taking the P [1 , Pad´e approxi-8
FIG. 1. Comparison of various estimates of the EOB gyro-gravitomagnetic ratio g S ∗ in the equal-mass case ( ν = 1 / g S ∗ defined by Eq. (5.6) is plotted asa function of u = GM/ ( c R ) for the four cases explained intext. mant of the PN-expansion of Eq. (5.10); 3) ditto with the9.5 PN-accurate generalization of (5.10); and finally 4) ditto with the inverse resummation of the latter r . P NS ∗ expansion.The main message of Fig. 1 is that, in confirma-tion of what had been already found at the next-to-leading order [60], and whatever be the approximationused, g S ∗ significantly decreases as the separation R = GM/u between the two bodies decreases. This decreasebecomes more pronounced when one uses higher PN-approximants. When using Taylor-approximants the de-crease of g S ∗ is so extreme that it formally changes a signbelow a certain separation R (crit) . For instance, at 4PNwe have R (crit) ≈ GM/ . ≈ . GM , while at 9.5PN wehave R (crit) ≈ GM/ . ≈ . GM . As this sign changeis physically unwarranted, we advise, when using Taylor-approximants, to replace g S ∗ by zero beyond the criticalseparation R (crit) (i.e., to replace g S ∗ → ( g S ∗ + | g S ∗ | )).If one considers that such a vanishing of g S ∗ is tooextreme a behaviour, one might consider using one ofthe inverse-resummed estimates of r S ∗ . At this stage thespread between the curves in Fig. 1 is a measure of ouruncertainty on the true value of g S ∗ in the strong-field domain. One will need comparisons between numericalrelativity simulations and EOB computations using var-ious g S ∗ functions (possibly including some free parame-ter parametrizing strong-field effects) to learn more aboutthe exact extent to which g S ∗ decreases in the strong-fielddomain. ACKNOWLEDGMENTS
CK thanks the authors of [36] for generously provid-ing an early draft of their work, and in particular SarpAkcay and Sam Dolan for many informative discussions.DB thanks ICRANet and the italian INFN for partialsupport and IHES for warm hospitality at various stagesduring the development of the present project. SH ac-knowledges financial support provided under the Euro-pean Union’s H2020 ERC Consolidator Grant “Matterand strong-field gravity: New frontiers in Einstein’s the-ory” grant agreement no. MaGRaTh-646597.
Appendix A: Kinnersley tetrad
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D81 , 084033(2010), arXiv:1002.0726 [gr-qc].[75] Leor Barack, Thibault Damour, and Norichika Sago,“Precession effect of the gravitational self-force ina Schwarzschild spacetime and the effective one-body formalism,” Phys. Rev.
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