Fourth post-Newtonian effective-one-body Hamiltonians with generic spins

In a compact binary coalescence, the spins of the compact objects can have a significant effect on the orbital motion and gravitational-wave (GW) emission. For generic spin orientations, the orbital plane precesses, leading to characteristic modulations of the GW signal. The observation of precession effects is crucial to discriminate among different binary formation scenarios, and to carry out precise tests of General Relativity. Here, we work toward an improved description of spin effects in binary inspirals, within the effective-one-body (EOB) formalism, which is commonly used to build waveform models for LIGO and Virgo data analysis. We derive EOB Hamiltonians including the complete fourth post-Newtonian (4PN) conservative dynamics, which is the current state of the art. We place no restrictions on the spin orientations or magnitudes, or on the type of compact object (e.g., black hole or neutron star), and we produce the first generic-spin EOB Hamiltonians complete at 4PN order. We consider multiple spinning EOB Hamiltonians, which are more or less direct extensions of the varieties found in previous literature, and we suggest another simplified variant. Finally, we compare the circular-orbit, aligned-spin binding-energy functions derived from the EOB Hamiltonians to numerical-relativity simulations of the late inspiral. While finding that all proposed Hamiltonians perform reasonably well, we point out some interesting differences, which could guide the selection of a simpler, and thus faster-to-evolve EOB Hamiltonian to be used in future LIGO and Virgo inference studies.


I. INTRODUCTION
The observation of gravitational waves (GWs) from coalescing binaries [1][2][3][4] using a continually improving network of GW detectors [5][6][7][8] is a milestone in fundamental physics and astrophysics. As the detectors increase their sensitivity, we will observe more events, with larger signal-to-noise ratios, spanning a larger region of the parameter space. Thus, to faithfully recover the sources' properties, it is important to improve the accuracy of models of waveforms from binaries of compact objects (black holes and/or neutron stars) on generic orbits and with generic spin orientations. In the generic case, the orbital plane and the objects' spins precess about the direction of the system's total angular momentum, leading to modulations of the GW signal which are a smokinggun signature of the dynamical influence of the spins. Including such precession effects in GW template models, as opposed to restricting to the simpler aligned-spin case with no precession, is important for more inclusive GW searches, more accurate inference studies and tests of General Relativity.
The effective-one-body (EOB) framework [9,10] aims at providing a synergy between multiple analytical approximations and numerical-relativity (NR) simulations of relativistic inspiraling binaries. The core ingredient of the EOB approach is the EOB Hamiltonian, a canonical Hamiltonian describing the binary's (conservative) * mohammed.khalil@aei.mpg.de † jan.steinhoff@aei.mpg.de ‡ justin.vines@aei.mpg.de § alessandra.buonanno@aei.mpg.de orbital dynamics, which both (i) agrees, in its post-Newtonian (PN) expansion, 1 with known results for arbitrary mass ratios from PN calculations (in the weak-field and low-speed regime), and (ii) becomes, in the extrememass-ratio limit, an exact Hamiltonian for a test (or probe) particle in an exact black-hole spacetime, valid for arbitrary separations and speeds. The EOB Hamiltonian is naturally expressed as a deformation of the zeromass-ratio test-particle Hamiltonian, with the deformation determined by finite-mass-ratio results from the PN approximation. For example, the original (nonspinning) EOB Hamiltonian [9] becomes, as the mass ratio goes to zero, the exact Hamiltonian for a test mass undergoing geodesic motion in a Schwarzschild (nonspinning black hole) spacetime. In generalizing to spinning black holes, the first natural replacement for the Schwarzschild-geodesic Hamiltonian is the Hamiltonian for geodesic (test-mass) motion in an exact Kerr (spinning black hole) spacetime. 2 A spinning EOB (SEOB) Hamiltonian incorporating the Kerrgeodesic limit was first constructed in Ref. [12] including leading-order (LO) spin-orbit and LO spin-squared effects in the PN expansion. This was later extended to the next-to-leading (NLO) [13] and next-to-NLO (NNLO) [14] spin-orbit levels, and to the NLO spin-squared level for aligned spins [15,16] and then for generic (precessing) spins [17]. The Kerr-geodesic-based approach for aligned spins has been further developed in Refs. [18][19][20][21][22][23][24], e.g., by including matter effects (for neutron stars) and calibration to NR simulations. A second category of SEOB Hamiltonians is based on the Hamiltonian for a spinning test-body (test spin) in a Kerr background [25,26], first developed with NLO [27] and then NNLO [28] spin-orbit terms and with LO spin-squared terms. Such Hamiltonians have always been applicable for generic (precessing) spins. They have been generalized to include tidal effects in Refs. [29,30], they have been used for studies of extreme-mass-ratios binaries in Ref. [31] and periastron advance in Ref. [32], and they have been refined and calibrated to NR simulations in Refs. [33][34][35][36][37][38]. EOB Hamiltonians have also been constructed to include information from gravitational self-force calculations [39][40][41][42] (for extreme mass ratios) and from the post-Minkowskian approximation [43][44][45] (assuming weak fields but allowing arbitrary speeds). A recent comparison of various SEOB waveform models is given in Ref. [46]. Waveform models constructed with the SEOB Hamiltonians based on a spinning test-body in a Kerr background [34][35][36][37][38] have been employed in template banks of LIGO and Virgo, and inference studies of binary black holes [1,2,4]. For parameter-estimation of binary neutron stars both classes of SEOB Hamiltonians have been employed in Ref. [4].
The goal of the present paper is to construct SEOB Hamiltonians for compact binaries (black holes or neutron stars) that include all known PN results to 4PN order for generic orbits and spin orientations. Beyond the up-to-NNLO spin-orbit and spin-squared contributions, the 4PN level includes also the LO cubic and quartic in spin terms. Previous work is not complete to 4PN order for generic spins [17,27,28] or complete to 4PN but valid for aligned spins only [19,22,23,47]. We construct three SEOB Hamiltonians in this paper: (i) a Hamiltonian based on Ref. [17], which uses the idea of "centrifugal radius" r c [19], while recovering the Kerr-geodesic limit; (ii) a simplified version of the Hamiltonian from Ref. [17] that does not use a centrifugal radius and has a different factorization for the PN spin corrections, similarly recovering the Kerr-geodesic limit; and (iii) one Hamiltonian following Refs. [25,27,28] which recovers the dynamics of a spinning test-body in the Kerr spacetime in the small-mass-ratio limit (see Table I for a summary of the differences between these Hamiltonians). As we wish to somewhat fairly compare different treatments of spin effects in the EOB formalism, we have modified some details of the original proposals of Refs. [17,28] such that all the Hamiltonians agree in the zero-spin limit. We compare the aligned-spin circular-orbit binding energy functions from the different Hamiltonians with NR simulations and with the aligned-spin Hamiltonian from Refs. [19,23,47]. This enables one to assess compromises between accuracy and simplicity of the SEOB Hamiltonians.
The paper is organized as follows. In Sec. II we provide an overview of SEOB Hamiltonians and their construc-tion. Sections III and IV present the ansätze of the SEOB Hamiltonians, with explicit results matched to 4PN in Appendix B. We then compare the aligned-spin circularorbit binding energy of the Hamiltonians against NR in Sec. V. Our conclusions are given in Sec. VI. Appendix A corrects an omission at NLO S 2 in the Hamiltonian of Ref. [15].

Notation
We use geometric units such that the speed of light c and the Newton constant G are equal to 1. We utilize various combinations of the masses m 1 , m 2 of the binary's components, For the spins S 1 , S 2 , we define the dimensionless versions along with the intermediate a 1 , a 2 . The relative position and momentum are denoted by r and p, respectively. Using an implicit Euclidean background, it holds where n = r/r with r = |r|, and L is the orbital angular momentum with magnitude L. For convenience, we also introduce rescaled dimensionless variables, and similarly for the magnitudesr = |r|, etc.; here, H is any of several Hamiltonians encountered below, and a = S Kerr /M is the rescaled spin of an effective Kerr black hole.
We consider a spinning binary in the center-of-mass frame. The orbital dynamics is described by the relative separation r and linear momentum p vectors, and the internal dynamics is assumed to be captured by the spins S 1 and S 2 of each body. The Poisson brackets between these dynamical variables are the standard ones, The utility of this energy map was demonstrated, e.g., in Refs. [9,43,55]. For instance, if for H eff one just takes the Hamiltonian of geodesics in Schwarzschild spacetime, then H EOB correctly describes both the 1PN and first post-Minkowskian dynamics [9,43].

A. The effective Hamiltonian
The central idea of the EOB Hamiltonian is to combine the dynamics in the test-body limit (with no restriction on the speed or field strength) with the PN dynamics (not restricted in the mass ratio). In this way, one might overcome some of the limitations of the individual approximations. This can be achieved by making an ansatz for H eff as a deformation of the test-body-limit Hamiltonian (deforming it such that PN results are recovered), which is the purpose of this section. Note that in the test-body limit H EOB ≈ H eff + const.
Let us review the Hamiltonian of a spinning test-body in Kerr spacetime [25,26]. One can easily specialize this to the nonspinning (geodesic) case, which is the basis of some SEOB models. These test-body Hamiltonians are the basis for all SEOB models. The (inverse) Kerr metric g µν Kerr in Boyer-Lindquist coordinates (x µ ) = (t, r, θ, φ) is given by the line element where M is the mass of the black hole, σ = M a is its spin, and The Hamiltonian of a spinning test-body H Kerr can be obtained as a solution of the mass-shell constraint (see, e.g., Ref. [26]) where p µ = (−H Kerr , p r , p θ , p φ ), µ is the mass of the testbody, S ab * = −S ab * is its spin tensor in a local Lorentz frame (e a µ e aν = g µν Kerr ), ω µab = e bν ∇ µ e a ν are the Ricci rotation coefficients, and ∇ µ is the covariant derivative. The canonical spin vector of the test-body S * is given by S i * = 1 2 ijk S jk * and the components S 0i are fixed by the supplementary condition S ab (e b µ p µ + µ 2 δ 0 b ) = 0 + O(S 2 * ), all in the local frame.
Let us split H Kerr into a part dependent on the testspin S * and the remaining S * -independent terms into parts even and odd in the Kerr spin a, Following the procedure outlined above, and choosing the local frame from Ref. [27], this leads to and with explicit expressions for the fictitious gravitomagnetic (frame-dragging) force interacting with the test-spin S * given in Ref. [32] in terms of the vectors F µ (reproduced here in Sec. IV). A simplified version of this Hamiltonian for aligned spins and motion in the equatorial plane can be found in Ref. [56]. Simplifications for the generic-spin case are possible by making a different choice for the local frame which may simplify the Ricci rotation coefficients, see, e.g., Appendix C of Ref. [26]. The Hamiltonian above is written in terms of components instead of vectors, which is a disadvantage for some purposes. Following Ref. [17], we transform to a 3-vector notation (with an implicit flat Euclidean background) by treating (r, θ, φ) as spherical coordinates, with r = (x, y, z) = r(sin θ cos φ, sin θ sin φ, cos θ) and a = (0, 0, a). This is accompanied by a transformation of the momenta p r , p θ , p φ to the new momenta p, which makes it an overall canonical transformation. Noting that a 2 p 2 φ /r 2 = (n × p · a) 2 , a cos θ = n · a, and a 2 sin 2 θ = a 2 − (n · a) 2 , this results in the even-in-a Hamiltonian with and Similarly, the odd-in-a part reads We now have all ingredients in order to discuss how an ansatz for the effective Hamiltonian can be built. In general, one takes the effective Hamiltonian to be a deformation of the Kerr Hamiltonian (the deformation parameter being the symmetric mass ratio ν), either for a test spin or a test mass. While all EOB models agree on the identification of the masses between the test-body and comparable mass case (M = m 1 + m 2 , µ = m 1 m 2 /M ), different choices are made for mapping the spins a and S * to S 1 and S 2 . Let us consider a simple explicit example. We could write the even-in-a part of the effective Hamiltonian as where the momentum-independent potentials A, B p , B np , B npa are the Kerr potentials given above modified by PN corrections (to be determined). The quantity Q is a momentum-dependent potential introduced in Ref. [55], which may accommodate PN terms that do not fit into the momentum-independent potentials. (In cases where Q vanishes, the deformed Hamiltonian can be interpreted as describing geodesic motion in a ν-deformed Kerr metric.) The mentioned potentials should all be of even order in spin, while terms of odd order in spin should be included via a deformation of H Kerr odd . More explicit ansätze for the PN-corrected SEOB Hamiltonians and their potentials are discussed below.

B. Matching to post-Newtonian results
To fix the potentials in the ansatz for an effective Hamiltonian, one demands that the EOB Hamiltonian H EOB agrees with the Hamiltonian in the PN approximation H PN up to a canonical transformation. This will eventually not uniquely fix the potentials, but leave some (gauge) freedom.
Here we use the spinning PN Hamiltonian derived in the framework and gauges introduced in Ref. [57], since it is available to 4PN order in the spinning sector [49]. Broken up into leading order (LO), next-to-LO (NLO), next-to-NLO (NNLO) PN parts and into powers of spin, it reads in Refs. [92][93][94][95][96], and H LO S 4 in Refs. [92][93][94]96]. These Hamiltonians are valid for both black holes and neutron stars. They depend on coefficients (C (ES 2 ) ,C (BS 3 ) , C (ES 4 ) ) which are the proportionality constants between the spin-induced multipoles (quadrupole, octupole, hexadecapole) and symmetric-tracefree tensors built out of (two, three, four) spin vectors (respectively). The proportionality constants depend on the type of compact object (and on the equation of state in case of a neutron star); here they are normalized to 0 for black holes (in the original paper [49] they are normalized to 1 and denoted without a tilde, see also Appendix B). This normalization makes sense here since we base the EOB Hamiltonian on a deformation of the Kerr one. Of course, the PN Hamiltonian H PN = H PN ns + H PN spin must be supplemented by its nonspinning (ns) part H PN ns , which we only need to 2PN order here in order to construct the canonical transformation of the spin sector; it can be derived, e.g., from the Lagrangian in Ref. [97].
The condition that the EOB Hamiltonian H EOB must coincide with results for the PN-approximate binary Hamiltonian H PN up to a canonical transformation reads where G is the generating function of the canonical transformation. If G is small in the PN approximation, then the series in Eq. (21) terminates after a finite number of terms at a given PN order. In practice, one makes a PNapproximate and manifestly rotation invariant ansatz for G in terms of the canonical variables; we provide an explicit expression for G as Mathematica code in the supplementary material. Equation (21) then leads to constraints on the coefficients in the ansatz for G and H eff . The remaining freedom in the coefficients is a gauge freedom within the EOB formalism. Let us note some general considerations about how part of this gauge freedom can be fixed in SEOB models. Since binaries are expected to be on almost circular orbits during their last orbits, it makes sense to fix the gauge freedom of the EOB Hamiltonian such that it simplifies for circular orbits, for which p r ≡ n · p = 0 [55]. Taking the ansatz in Eq. (19) as an example, this means thatusing the canonical transformation discussed above-one should transform as many PN terms as possible into a form such that they can be included in the potential B Kerr np , which drops out of the Hamiltonian for circular orbits. In the nonspinning case, it is additionally possible to require that the potential Q depends on the momentum only via p r , and this uniquely fixes all EOB gauge freedom [9,55]. For the example in Eq. (19), following the structure of the nonspinning Hamiltonian, it is natural to require that: (i) the momentum-dependence of H Kerr odd is expressed in terms of p r whenever possible [13], (ii) B npa = B Kerr npa [17], and (iii) terms in Q have a power in p r that is as high as possible. The last requirement ensures that Q vanishes for circular orbits, as in the nonspinning case.
These considerations still leave some remaining gauge freedom in the spinning case, which we fix such to simplify the EOB Hamiltonian also for aligned spins. For example, it is possible to choose PN corrections in the potential B p such that it only depends on terms of the form n · S but not S · S. Any remaining gauge freedom beyond that may be chosen arbitrarily.

III. SPINNING EFFECTIVE-ONE-BODY HAMILTONIANS WITH TEST MASS
In this section we present different ansätze for effective Hamiltonians based on the Kerr geodesic one. That is, we do not include the Kerr test-spin Hamiltonian H Kerr S * here and instead make an ansatz of the form The explicit lengthy results from the matching at 4PN order against PN results (and fixing of the remaining gauge freedom) are given in Appendix B. We start with an extension of the SEOB Hamiltonian from Ref. [17], which we call SEOB rc TM , to 4PN order, here including spin effects at LO S 3 , and NNLO S 2 . We also extend that Hamiltonian from black holes to generic compact objects, e.g., neutron stars. We proceed with a simplified version of the SEOB rc TM Hamiltonian to 4PN order that does not make use of the "centrifugal radius" introduced in Ref. [19]. For completeness, we also summarize the SEOB rc,align TM Hamiltonian from Refs. [19,22,23,47] which is valid for aligned spins only. We do not include additional PN terms in the SEOB rc,align TM Hamiltonian since it is already 4PN complete for generic bodies. For convenience we summarize the Hamiltonians in Table I. A. Effective-one-body Hamiltonian with test-mass limit and centrifugal radius: SEOB rc TM Reference [17] was the first to construct an SEOB Hamiltonian with NLO spin-squared terms for generic spin orientations (but omitting a subtle contribution, included here, see Appendix A). Here we extend the Hamiltonian to include NNLO spin-squared and LO spin-cubed terms, and add multipole constants to make it applicable to generic bodies like neutron stars.
For the even-in-spin part of the SEOB rc TM Hamiltonian, we use the ansatz in Eq. (19), where the Kerr spin is mapped according to This ensures that the Hamiltonian reproduces leadingorder PN results at all even orders in spin [98]. The effective Hamiltonian further uses the "centrifugal radius" r c , which was introduced in Ref. [19] and is defined such that the Kerr Hamiltonian for aligned spins and equatorial orbits can be written as H Kerr even = A Kerr µ 2 + p 2 φ /r 2 c + p 2 r /B(r) , which implies the definition The centrifugal radius was generalized to generic spin orientations in Ref. [17]. In terms of r c , the Kerr potential A Kerr from Eq. (16a) can be written equivalently as In the nonspinning limit, only the first term above remains, which reduces to the Schwarzschild A-potential. This is the reason why Ref. [17] adds the zero-spin PN corrections to 1 − 2M/r c . However, in this paper we intend to investigate spin effects across different Hamiltonian descriptions, so we need to make sure that the nonspinning Hamiltonians are identical. That is, we need to choose a method for adding zero-spin corrections that can be applied to all four EOB Hamiltonians considered here. We simply multiply the Kerr potential A Kerr by zero-spin PN corrections denoted A 0 below (without performing a Padé or log resummation 3 of A 0 ). For the spin-squared corrections, we follow Ref. [17] and add spin-squared corrections of the form n · S to the term 1 + (n · a) 2 /r 2 , and add corrections of the form S · S to 1 + 2M/r c , since it has an expansion of the form 1 + 2M/r − M a 2 /r 3 + . . . . One employs similar considerations for adding PN corrections to the B-potentials in Eq. (23), leading to the following ansatz: Note that we use the gauge choice from Ref. [17], i.e., there are no corrections of the form S · S in the potential B p , which simplifies the Hamiltonian for aligned spins and circular orbits.
The 4PN corrections to the nonspinning effective Hamiltonian were obtained in Ref. [99]. Since we factor the PN corrections in A 0 (r c ), we choose it such that the PN expansion of the A potential agrees, in the nonspinning limit, with the results of Ref. [99]. Writing the PN corrections using scaled variables (4) to simplify notation, we obtain where the corrections are expressed in terms of the centrifugal radius r c , withr c = r c /M , andp r = p r /µ. Note that here and in the SEOB models discussed below, we are using Taylor-expanded and not resummed versions of these potentials-we want to compare the different ansätze of the Hamiltonians irrespective of possible resummations for the potentials (see also footnote 3). Spin-squared contributions, up to NNLO, are added to the Hamiltonian using the following ansatz where we followed Ref. [17] in expressing the corrections in terms of r c . We employ notation such that, e.g., Each c n stands for an independent undetermined coefficient in our ansatz, i.e., we use the same symbol c n for all coefficients to simplify notation. The full expressions after matching to PN results are provided in Appendix B. Note that we added LO S 2 corrections to the A-potential above (which vanish for black holes) to account for the multipole constants of neutron stars.
The NLO S 2 contributions were included in the ef-fective Hamiltonian in Ref. [17], however the authors missed a contribution in matching the EOB Hamiltonian to PN results, namely from the LO S 2 generating function applied to the LO SO Hamiltonian, i.e., from the In Appendix A, we write the matching results for NLO S 2 , using the notation of Ref. [17], after taking into account the missing Poisson bracket.
The leading-order quartic-in-spin terms A S 4 are zero for black holes, since the Kerr Hamiltonian, with the mapping a = a 1 + a 2 , automatically reproduces them, but they are nonzero for other types of compact objects. We take the most generic expression for the S 4 corrections where a summation over the spins of the two bodies is implied, and terms symmetric under the exchange of the two bodies' labels are only included once. The spin-orbit and spin-cubed PN corrections are added to the odd-in-spin part of the Kerr Hamiltonian H Kerr odd . For the SO part we use the ansatz in Refs. [17,19], and we add to it S 3 corrections, where Note that an inverse-Taylor resummation is used for G S and G S * , which improves the description of the binary dynamics for aligned spins [19]. In the spin-cubed corrections G S 3 andG S 3 , a gauge freedom exists which we chose such that terms of the form p 2 r χ i · χ j or L 2 χ i · χ j are not included. Explicit results after matching at 4PN can be found in Appendix B 1.

B. A simplified effective-one-body Hamiltonian with test-mass limit: SEOB TM
Since it is important to have fast and simple EOB waveform models, in this section, we consider a simplified version of the SEOB rc TM Hamiltonian that uses r instead of r c for the PN corrections. In order to assess the effect of this simplification, we also avoid resummations that are not motivated by the structure of the interactions, i.e., we factorize spin corrections to the Kerr potentials and do not use an inverse-Taylor resummation for the spin-orbit part.
The potentials of the effective Hamiltonian are simply taken to be where the zero-spin corrections A 0 (r), D 0 (r) and Q 0 (r) are given by Eqs. (28) but in terms of r instead of r c .
The ansätze for the S 2 and S 4 corrections are given by the corresponding expressions from the previous section, i.e., Eqs. (29) and (30), but using r instead of r c (and with different coefficients c n ). For H Kerr odd , we modify the odd-in-a part of the Kerr Hamiltonian by the SO and S 3 PN corrections, that iŝ with the ansätze for the coefficients given in Eqs. (32a) and (32b), but again written with r instead of r c (and different c n ). Explicit results after matching at 4PN can be found in Appendix B 2.
C. Aligned effective-one-body Hamiltonian with test-mass limit and centrifugal radius: SEOB rc,align

TM
In this section, we consider the aligned-spin EOB Hamiltonian proposed by Damour and Nagar in Ref. [19] and extended in Refs. [22,23,47], which we denote SEOB rc,align TM . That Hamiltonian is similar to the alignedspin limit of the SEOB rc TM Hamiltonian from above, except that the even-in-spin PN corrections are added to the centrifugal radius.
The even-in-spin effective Hamiltonian is given bŷ The EOB potentials A and B are given in Ref. [19], but we do not use Padé resummation and we modify how the zero-spin PN corrections are added such that they agree with the other Hamiltonians in this paper, that is where A 0 , D 0 and Q 0 are given by Eq. (28). Note that Refs. [23,46,47] use Q ≡ 2ν(4 − 3ν)p 4 r /r 2 c instead of Q 0 , and use p r * ≡ p r A/B instead of p r .
The spin-squared and spin-quartic corrections are added to the centrifugal radius, which is here defined bŷ and whereâ Q depends on the compact object's multipolar constantŝ whereâ i = a i /M ,â = |a|/M , and recalling Eq. (24). The spin-orbit part was obtained in Ref. [19] with NNNLO ν-independent spinning-test-body contributions and ν-dependent contributions calibrated to NR. We do not include those higher-order corrections here, but we follow Ref. [19] in using an inverse-Taylor resummation/calibration of the coefficients G S and G S * in see Eqs. (B7a), and where G S 3 andG S 3 for aligned spins take the simple form Including spin-cubic contributions was discussed in Appendix A of Ref. [47], which we implement here so that the effective Hamiltonian includes all PN information at the same order as the other Hamiltonians considered in this paper. Explicit results after matching at 4PN can be found in Appendix B 3.

IV. EFFECTIVE-ONE-BODY HAMILTONIAN WITH TEST-SPIN LIMIT: SEOB TS
The SEOB Hamiltonian proposed in Refs. [27,28] is based on the Hamiltonian of a spinning test body in the background of a Kerr black hole, which we here denote by SEOB TS (see Table I). In this section, we extend that Hamiltonian to 4PN order; compared to previous results, we add NLO S 2 , NNLO S 2 LO S 3 , and LO S 4 PN corrections, for generic compact objects and spin orientations.
The SEOB TS Hamiltonian, as expressed in Ref. [32], is given bŷ where andŜ * ≡ S * /M µ is a rescaling of the spin of the test body. The spins are mapped according to where the functions f * and g * are given by Eqs. (50)-(52) of Ref. [28], and that σ = M a is the spin of the background Kerr metric; it does not hold a = a 1 + a 2 as for the models discussed above. The spin maps are analogous to the mapping of the masses M , µ according to Eq. (I), with the difference that the spin maps relate dynamical variables. The deformed metric is obtained by substituting ∆, Σ, and Λ in the Kerr metric by Σ =r 2 +σ 2 cos 2 θ, as in The potential D 0 (r) is given by Eq. (28b), and the potential A is given by where A 0 is given by Eq. (28a), andr H,± are the scaled inner and outer radii of a Kerr black hole, i.e., Finally, the vectorsF t ,F r ,F θ , andF φ describe the fictitious force acting on the test-body spinŜ * (frame dragging) in the deformed Kerr metric. They are given by Eq. (6) in Ref. [32], which we rewrite here for convenience, Here, the unit vectors (n,ξ,v) are defined byn = x r ,ξ =ê σ Z ×n, andv =n ×ξ, whereê σ Z = σ/σ denotes the direction of the (deformed) Kerr spin.
Out of 38 possible terms in the most general expression for Q S 4 , 16 terms were removed via a gauge choice. We started by removing the three terms that do not vanish for aligned spins, but the other 13 terms were chosen arbitrarily. Explicit results after matching at 4PN can be found in Appendix B 4.

V. COMPARISON WITH NUMERICAL RELATIVITY
In this section, we compare the four SEOB Hamiltonians considered in this paper to NR simulations through the binding energy for circular orbits and aligned spins. The NR binding energy data we use here were extracted from the Simulating eXtreme Spacetimes (SXS) catalog [100] in Ref. [101]. Hereafter, in this section, we use the term "aligned spins" to mean spins parallel to, and in Binding energy (left panel) and fractional binding energy (right panel) versus the "velocity" parameter v for nonspinning binary-black-hole configurations with different mass ratios. The four SEOB Hamiltonians considered here are identical for zero spin. The relative NR error shown in the right panel is a conservative 1.1% estimate. The initial value of v (the left end of the plots' domain) here is determined by the beginning of the NR simulation with q = 10; those with lower mass ratios have several cycles at lower frequencies not shown here. We stress that the SEOB Hamiltonians at 4PN order are not calibrated to NR simulations. the same direction as, the orbital angular momentum L, but we use the term "antialigned spins" to mean spins opposite to the direction of L.
The binding energy is calculated by evaluating the EOB Hamiltonian for circular orbits (p r = 0) and solving numericallyṗ r = −∂H EOB /∂r = 0 for the angular momentum p φ at some radius. The orbital frequency Ω is obtained from We then calculate the binding energy and orbital frequency as r goes from the beginning of the NR simulation to the innermost-stable circular orbit (ISCO) of the Hamiltonian, which marks the end of the inspiral phase of the binary coalescence and the beginning of the plunge. The ISCO is calculated by setting both the first and second derivatives of the Hamiltonian with respect to r to zero, i.e., ∂H EOB /∂r = 0 = ∂ 2 H EOB /∂r 2 . It should be noted that the binding energy is extracted from NR simulations from an evolving binary, tracking the radiated energy in GWs. From the EOB Hamiltonians, however, we obtain the binding energy here by assuming exact circular orbits at different orbital separations, neglecting the orbital decay (radiation-reaction) due to the emitted GWs. The NR and EOB binding energies are thus not expected to agree exactly here during the last few orbits (see discussions in Ref. [45]).
In Fig. 1, we plot the binding energy for nonspinning configurations with different mass ratios q as a function where the numbers in brackets refer to the values of the dimensionless spins of the two bodies (χ 1 , χ 2 ). The spinsquared contributions to the binding energy E S 2 refer to both S 2 i and S 1 S 2 interactions. Similarly, spin-cubic contributions E S 3 refer to both S 3 i and S 2 i S j . We see that the spin-orbit contribution is about an order of magnitude larger than the spin-squared contribution, which in turn is an order of magnitude larger than the spin-cubic contribution. All SEOB Hamiltonians give comparable results for the spin-orbit part, however for the spin-squared contribution, the SEOB TS and SEOB TM Hamiltonians give better agreement with NR than the other two Hamiltonians. For the cubic-in-spin-contributions, the NR error is larger than the EOB values for the binding energy, and hence we cannot conclude which Hamiltonian is better in terms of S 3 contributions.
In Fig. 3, we compare the fractional energy difference at four GW cycles (i.e., two orbits) before merger for various spin configurations with mass ratio q = 3. We see that, for all configurations at that frequency, the relative difference with NR is around 1%. For aligned spins, all Hamiltonians give comparable re- sults, but the SEOB TS Hamiltonian gives better agreement with NR for antialigned spins.
We also compare the binding energy as a function of velocity for some configurations with aligned spins (Fig. 4) and antialigned spins (Fig. 5). The curves in these figures start at the beginning of the available NR simula-tions and end at the ISCO of the EOB Hamiltonians. All effective Hamiltonians considered here have an ISCO for arbitrary spins, except that the SEOB TS Hamiltonian does not have an ISCO for large aligned spins 0.92. From the three panels at the top of Fig. 4, we see that for large aligned spins ( 0.8), the SEOB TS Hamiltonian shows slightly better agreement with NR than the other Hamiltonians. However, for smaller spins ( 0.6), all Hamiltonians give very similar results. This is also the case when the two spins are both large but in opposite directions. For antialigned spins, the difference between the four Hamiltonians is smaller than in the aligned-spin case; the SEOB TS Hamiltonian gives better agreement with NR than the other Hamiltonians, but the difference is small, even for spin magnitudes of 0.97, and becomes negligible for smaller spins.
Finally, in Fig. 6, we compare the effect of adding spin PN orders to the effective Hamiltonian for a configuration with mass ratio q = 3 and spins χ 1 = χ 2 = 0.85. For all Hamiltonians, adding higher spin orders improves agreement with NR, except for the SEOB TS Hamiltonian where adding LO S 3 and LO S 4 gives slightly worse agreement. We checked that using different spin configurations gives qualitatively similar behavior.
Overall, beside the small differences pointed out above, all Hamiltonians perform reasonably well compared to NR simulations. One should expect that the differences that accumulate during the last orbits can be compensated by a calibration of the Hamiltonians, applying also further resummations to the potentials, which we leave for future work. This would be of particular interest for the simplified Hamiltonian, in order to prepare and evaluate it as a possible starting point for an EOB waveform model.

VI. CONCLUSIONS
In this paper, we built spinning EOB Hamiltonians that include the complete fourth post-Newtonian conservative dynamics for generic (precessing) spins. These Hamiltonians are also valid for generic compact objects (e.g., black holes or neutron stars) since we included multipole constants that parametrize the deformation of the compact object due to its rotation.
In particular, we considered and extended four SEOB Hamiltonians: (i) an extension of the SEOB Hamiltonian from Ref. [17] by adding NNLO S 2 and LO S 3 contributions, in addition to adding the multipole constants; (ii) a simplified version of that Hamiltonian that differs in how the spin corrections are added to the Kerr metric, and that does not use the concept of centrifugal radius; (iii) the aligned-spin Hamiltonian from Refs. [19,23,47], which already includes complete 4PN information for generic compact objects, but considered here for comparison with the other Hamiltonians; (iv) an extension of the SEOB Hamiltonian from Refs. [27,28], which uses a test spin, by adding NLO S 2 , NNLO S 2 , LO S 3 , and LO S 4 contributions, in addition to adding the multipole constants. Since our goal in this paper was to improve the description of spin effects in the EOB formalism, we modified the zero-spin part of the above Hamiltonians such that they are identical in that limit. Furthermore, we did not include NR calibration parameters or resummations of the PN corrections (e.g., with Padé or log resummations of the zero-spin part) since they constitute an implicit calibration to NR that improves the performance of EOB Hamiltonians only in certain models.
We compared the four SEOB Hamiltonians considered here with NR simulations by calculating the binding energy for circular orbits and aligned spins. We found that all Hamiltonians show good agreement with NR, and that the difference between the Hamiltonians is quite small up to moderate values of the spins and a handful number of GW cycles before merger. For large spins, the SEOB TS Hamiltonian performs better at large frequencies, but since all Hamiltonians have an error of about 1% compared to NR at about four GW cycles before merger, the simplest SEOB Hamiltonian SEOB TM could be an excellent candidate for building an improved EOB waveform model with precessing spins. The simplicity will allow one to have a fast-to-evolve set of equations of motion, and could help in calibrating the EOB waveforms built with SEOB TM to NR simulations. However, more analyses, which include dissipative effects, and a careful study of how the GW frequency approaches merger, are needed to pin down the more suitable SEOB Hamiltonian. Indeed, as several studies have shown [33][34][35][36][37][38], to attach robustly the merger-ringdown waveform to the inspiralplunge one in the EOB formalism, dynamical quantities, such as the orbital frequency, radial separation and momentum vectors, have to behave regularly around and beyond the EOB photon orbit.
We leave for future work to complete the SEOB Hamiltonians to a gravitational waveform model, i.e., provide resummed expressions for GW modes and associated radiation-reaction forces. Once radiation-reaction forces are included in the model, it is important to perform comparisons with NR for precessing spins and to use those comparisons to study different resummation options and to add calibration parameters in order to improve the accuracy of EOB waveforms toward merger.

ACKNOWLEDGMENTS
We thank Sergei Ossokine for providing us with the numerical-relativity data for the binding energy used in this paper.

22
, and b np,nχ 12 , which vanish in Ref. [17]. The six symmetries between the coefficients in Ref. [17] are also absent in our solution (assuming the gauge conditions b p,χ ij = 0).

Appendix B: Hamiltonian coefficients after matching to PN results
In this Appendix, we present the results of matching the SEOB Hamiltonians using the procedure described in Sec. II B. Here, we express the multipole constants as (B1) such that the black hole results are easily obtained by settingC ... = 0.
The expressions for the Hamiltonians and the potentials given in this Appendix are provided as Supplemental Meterial [103] in the form of Mathematica files.

Hamiltonian
The coefficients of the SEOB rc,align

TM
Hamiltonian are given in Ref. [47], but we rewrite them here for convenience in the notation used in the rest of the paper.
The spin-orbit and spin-cubic correction in Eq. (39) The spin-squared and spin-quartic corrections in Eq. (37) are given by