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Analytical effective-one-body formalism for extreme-mass-ratio inspirals: eccentric orbits

Extreme-mass-ratio-inspiral (EMRI) is one of the most important sources for the future space-borne gravitational wave detectors. In such kind of systems, the compact objects usually orbit around the central supermassive black holes with complicated trajectories. Usually, the trajectory is approximated as geodesic of a test-particle in Kerr space-time, and the orbital evolution are simulated with the help of adiabatic approximation. However, this omits the influence of the compact object on the back ground. In the present paper, employing effective-one-body formalism, we analytically calculate out the trajectories of a compact object around a massive Kerr black hole in equatorial-eccentric orbit, and express the fundamental orbital frequencies in explicit forms. Our formalism include the first-order corrections of mass-ratio in the conservative orbital motion. Furthermore, we insert the mass-ratio related terms in the first post-Newtonian energy fluxes. By calculating the gravitational waves from the Teukolsky equations, we quantitatively reveal the influence of the mass of the compact object on the data analysis. We find that the shrinking of geodesic motion by taking the small objects as test particles may be not appropriate for the detection of EMRIs.


I. INTRODUCTION
The successful detection of gravitational waves (GWs) by Advanced LIGO and Virgo [1][2][3][4][5][6] announces that the era of GW Astronomy is coming. This kind of ground-based detectors observe the GWs in high frequency band. LISA, a space-borne gravitational wave (GW) detector which proposed by Europe and USA [7], at the same time, two Chinese space projects Taiji [8] and Tian-Qin [9], will be planed to launch after 2030. All these detectors focus on GWs at low frequency (about 0.1 mHz to 1 Hz). Extreme-mass-ratio inspirals (EMRIs) composed by compact objects (stellar black holes, neutron stars, white drawfs and etc.) and supermassive black holes (SMBHs), are expected as one of the most important sources for these space-borne detectors [10][11][12].
The signals from EMRIs usually are very weak, but with one years' observation, the signal-to-noise ratio can be enough to be detected by matched filtering technology [10]. For detecting this kind of long duration signals, the requirement of accuracy of waveform templates is very high. Typically, after 10 5 cycles, the dephaseing should be less than few radians [11,13]. Nowadays, there are a few of EMRI templates, like as AK [14], AAK [15], NK [16], XSPEG [17] and so on. All of them take the small object as a test particle and omit the mass in their conservation dynamics part. Some works considered the correction due the small mass by using effective-one-body (EOB) formalism, but only in circular orbits [18,19] or for eccentric orbits with data fitted parameters [20]. There are also intermediatemass-ratio inspirals (IMRIs) composed with stellar compact objects and intermediate massive black holes(IMBHs) or IMBHs orbiting SMBHs, and the mass-ratio of IMRI is around 10 −3 [21]. In this situation, the mass-ratio correction on the conservative orbital dynamics should be more important.
The EOB formalism, by including the mass-ratio corrections in post-Newtonian (PN) expansions, can well describe the dynamical evolution of binary black holes [22,23], and are widely used to construct the waveform templates for LIGO [24][25][26][27][28][29][30][31]. Most of these models only considered the circular orbit cases. Recently, Hinderer et. al. gave an analytical eccentric EOB dynamics for Schwarzschild BHs [32]. Cao and Han built an eccentric EOBNR waveform template (SEOBNRE) for spinning black holes [33], but the orbits did not be geometrized and the orbital parameters did not be well defined.
It is well known that the orbits of EMRIs could be highly eccentric [11], and the supermassive black hole in the center should be spinning in general. In the present paper, we extend the previous work by Hinderer and Babak to the Kerr black holes. As a start, for equatorial-eccentric EMRIs, we analytically transfer the original EOB dynamical equations to geometric kinetic motion with orbital parameters: semilatus rectum p and the eccentricity e together with two phase variables associated with the spatial geometry of the radial and azimuthal motion denoted by (ξ, φ).
The EOB formalism was originally introduced in [22,23] to describe the evolution of binary system. We start by considering an EMRI system with central Kerr black hole m 1 and inspiraling object m 2 (assume it is nonspinning for simplicity) which is restricted on the equatorial plane of m 1 (m 2 m 1 ). For the moment, we neglect any radiation reaction effects and focus on purely geodesic motion. The conservative orbital dynamics is derived via Hamilton's equations using the EOB Hamiltonian andĤ eff = H eff /µ . The deformed-Kerr metric is given by [41] The quantities Σ, ∆ t , ∆ r , Λ t and ω fd in Eqs. (2.1a)-(2.1e) are given by where a is the effective Kerr parameter and u = M/r. The values of ω fd 1 and ω fd 2 given by a preliminary comparison of EOB model with numerical relativity results are about −10 and 20, respectively. The metric potentials A and D for the EOB model are given in the Appendix A. The effective Hamiltonian associated with the metric (2.1a)-(2.1e) has the form [41] where H S is the Hamiltonian caused by the spin of the effective particle. Considering the effective spin s ∼ µa/M is very small for EMRIs, we omit this term in the present work for simplification and will include it in the next work. The most dominant part, H NS is the Hamiltonian for a non-spinning test-particle of mass µ, given by The energy of the system is given by which implies the relationĤ the canonical EOB dynamics without radiation reactioṅ 14) The effective Hamiltonian associated with the deformed-Kerr metric has the form Solving Eq.(2.15) for P r in terms of (E, P φ , r) leads tô where we have defined the reduced momentaP r = P r /µ andP φ = P φ /µ The conservative EOB equations of equatorial motion without radiation reaction can then be written aṡ Eqs. (2.17) are more convenient than the original canonical EOB equations (2.14), because the dependence onP r has been eliminated with the energy E which is a constant in conservative system and changes only due to radiation reaction if GW fluxes are considered.

B. Re-parameterization of the constants and equation of motion
The constants of motion and the dynamical equations in the last subsection can be written in the geometrized orbital elements semilatus rectum p and eccentricity e. This will make the description of system more intuitive. For an eccentric orbit, it exists periastron and apastron points which can be expressed as where r 1,2 are the turning points of the radial motion, i.e., apastron and periastron respectively. By setting the radial equation of motion (2.17a) equals to zero withP r = 0, we can solve out the two points. Furthermore, taking r 1 , r 2 into Eq. (2.16) to makeP r = 0 again, finally we get the constants of motion (E,P φ ) in terms of (p, e) where the coefficients are (2.20d) (2.20e) (2.20f) The above formalism for Kerr black hole are much more complicated than the Schwarzschild ones in [32]. Obviously, for the test-particle limit ν → 0, the above results will go back the geodesic motion of test particle in Kerr spacetime. The orbital radius at arbitrary moment is expressed by the semilatus rectum p and the eccentricity e together with the phase variable associated with the spatial geometry of the radial motion denoted by ξ. These variables are defined by expressing the radial motion as so that the periastron and apastron correspond to ξ = (0, π) mod 2π respectively. For conservative systemṗ =ė = 0. If we take the radiation reaction of GWs into account, by differentiating Eqs. (2.19), we express the evolution of (p, e) by the energy and angular momentum fluxes of gravitational radiatioṅ The final set of EOB equations of motion with radiation reaction are Eqs. (2.23a) together with the evolution of the phases described by Eq. 2.22 and Eq. 2.17b, and the radius of motion at arbitrary moment is given by Eq. 2.21. Now all the equations of motion are expressed in terms of only the geometric parameters (p, e, ξ) and effective Kerr parameter a.
Finally we can get the orbital coordinates of the effective test particle in terms of only ξ: where P(e, p, ξ) is the first term in the right hand of Eq. (2.22) (2.24e)

C. Quantitative influences of mass-ratio on the conservative dynamics
Now we calculate the fundamental orbital frequencies in terms of the geometric parameters. This was done in the test particle limit due to the analytical integrals of geodesic in Kerr spacetime. While in the EOB formalism with the mass-ratio correction, the situation becomes complicated. Firstly, we express the radial frequency ω r which reflect the period of the radial motion from the periastron to apastron and back to periastron again, and the orbital period T r can be calculated by taking ξ from 0 to 2π, then In Eq. (2.17b), we have already written the variation of φ with coordinate time t. Rigidly, the radial motion is the real periodic motion. When the particle passes through the periastron twice, ∆φ will be larger than 2π because of the periastron procession caused by relativistic effect. We can then define the frequency of the azimuthal motion by ∆φ/T r . So, we compute the azimuthal frequency from the orbit-average of the φ motion as Whereṙ is given in Eq. (2.17a), and P r , E, P φ are expressed in Eqs. (2.16,2.19), just replace r in these equation with ξ, the above two integrals only contain argument ξ and can be integrated easily. Here we do not write down the fully expanded expressions of these two integrals, it is direct and trivial.
With the expressions of two fundamental frequencies at hand, now we investigate how the effective Kerr parameter a and symmetric mass-ratio ν affect on the features of the radial and azimuthal frequencies from the test particle limit in conservative dynamics. The effect of spin parameter on these frequencies for various mean orbital separation and eccentricity is shown in Fig. 1. Obviously, as the semilatus rectum increases,in other words, mean orbital radius correspondingly increases, this leads to a decrease of the periastron procession ω φ /ω r −1, approaches to the Newtonian limit ω r = ω φ . Interestingly, as the spin of central SMBH increases, the periastron procession decreases. The effect of eccentricity on the procession becomes obvious only when the separation p and spin a are both small (see the top three curves in Fig. 1). In this extreme mass-ratio case (ν = 10 −3 ), the behaviour of periastron procession depending on the spin, orbital separation and eccentricity are similar.
Next, we consider the radial and azimuthal frequencies' shifts due to the mass-ratio of the binary in the absence of radiation reaction. As we all known, the test-particle orbit already has a precise analytical model. For the binary system which consider mass radio, now we have the analytical EOB orbital solution with eccentricity for spinning BHs. In order to observe the impact of mass-ratio on radial and azimuthal frequencies in different conditions with Effect of spin parameter on periastron procession for an binary with symmetric mass-ratio ν = 10 −3 . At a fixed spin parameter, an orbit with bigger semilatu rectums has a smaller periastron procession than those with smaller semilatus rectums. For the orbits with the same eccentricity and semilatus rectum, the higher spin parameter induces a smaller periastron precession.
TABLE I. The relative differences of orbital frequencies ωr, ω φ between test-particle and EOB models with small mass-ratios. various orbital parameters, we compare the test-particle results and the EOB results. This comparison is shown in Table I. The right-most column of this table is the relative frequency shift divided by mass-ratio ∆ω/(ων), where ∆ω/ω is the relative difference of radial/azimuthal frequency between the test-particle frequency (ω r0 , ω φ0 ) and the EMRI ones, i.e., (ω r − ω r0 )/ω r0 , (ω φ − ω φ0 )/ω φ0 .
The frequency shift ∆ω/(ων) due to the mass-ratio is almost independent with mass-ratio itself, and the relative shift ∆ω/ω is in a range about [0.1ν, 10ν] based on Table I. This indicates that we must consider the influence of mass-ratio in the conservative dynamics for EMRIs, because the cycles of an typical EMRI waves ∼ 1/ν in LISA band, and if the relative error of frequency reaches ∼ ν, the dephase will accumulate to a few of radians at the end   of evolution. This may induce a failure of detection of EMRIs with the test particle model. Figure 2 illustrates the effect of eccentricity on the radial frequency shift(left panel) and the azimuthal one (right panel) with various semilatus rectum p and spin parameter a. It shows that the shifts of ω r and ω φ due to mass-ratio have different performances versus eccentricity. The results for different mass-ratios are also plotted (triangles, solid line and points represent ν = 10 −6 , 10 −4 and 10 −2 respectively), clearly show that ∆ω/(ων) is not sensitive to massratio except for the radial frequency while the trajectory approaches the vicinity of the innermost stable orbit (ISO) and mass-ratio becomes 0.01 (blue lines in the left panel). Figure 3 demonstrates the effect of semilatus rectum p on the radial frequency shift(left panel) and the azimuthal one (right panel) with various spin a and eccentricity e. When p becomes smaller, the frequency shifts become larger. The sudden growth of frequency shift is due to the orbit approaching to the ISO. Figure 4 illustrates the effect of BH's spin on the radial frequency shift (left panel) and the azimuthal one (right panel) with various semilatus rectum p and eccentricity e. For the cases of p = 5, when a becomes small, the orbits will be very close to the ISO, then the frequency shifts grow very fast.
The semilatus rectum p of ISO is the separatrix of bound orbits and in test particle limit is given by the analytic expression from Eq. (24)   However, the above equation is approximate of ISO even for test particles. Considering the influence of the small mass on the background, the ISO of EMRIs should deviate from the test particle one. Figure 5 shows the effect of mass-ratio on the boundary curve p s (e). We can see that the deviation of EOB's ISO can be 10% of the test particle model. In addition, the error of approximate expression (2.27) becomes large for fast spinning BH.

III. THE ORBITAL EVOLUTION AND WAVEFORMS
In this section we introduce the gravitational wave fluxes of energy and angular momentum in previous literature, and calculate the gravitational wave strain by the Teukolsky equation which is a perturbation theory of Kerr black hole [34]. We include the mass-ratio into the 1PN terms of energy and angular momentum fluxes, and compare the performance of approximate fluxes from 2PN to 4PN.

A. Radiation fluxes
The analytic 4PN O(e 6 ) formulae of energy and angular momentum fluxes in Boyer-Lindquist coordinates are given by [43] in parameter υ ≡ 1/p. For convenience, we use p and transfer their expressions as follows Here for simplicity, we just write the 2PN formalism. In the adiabatic limit, the evolution of energy and angular momentum is driven by the orbit-averaged radiation reaction forces so thaṫ

7)
Then the evolution of e and p due to the gravitational radiation is given bẏ We fully expand these partial derivatives above for convenience . (3.12) The explicit forms of the functions F 1 −F 6 and f 1 −f 20 are given in the Appendix B. The evolution for auxiliary phase of radial motion now can be calculated bẏ ξ = (1 + e cos ξ) 2 epM sin ξṙ + cot ξ eė − 1 + e cos ξ ep sin ξṗ .
(3.13) Figure 6 demonstrates the evolution of eccentricity and semilatus rectum of with different orbital parameters and mass-ratios. We can find that the evolution of ν = 10 −3 deviate the other two evolution with smaller mass-ratio when the evolution is close to the end. Figure 7 illustrates the evolution of orbits with 2PN (left panels) and 3PN fluxes (right panels). It looks like that the results of 3PN are not very reasonable when the orbits are extremely relativistic. Figure 8 shows the performances of various PN fluxes on orbital evolution with very extreme spin. Similar with [44], the 2PN formalism may be still the best choice. Therefore, we will use 2PN fluxes to calculate the orbital evolution.

B. Waveform
In this subsection, we calculate the waveforms by solving the Teukolsky equations [34]. Our method is based on frequency-domain decomposition, and has been developed in previous works [20,[35][36][37], in which the gravitational   truncated at 2PN, 2.5PN, 3PN, 3.5PN and 4PN. waveform from an eccentric EMRI with total mass M at distance R , latitude angle Θ and azimuthal angle Φ of could be written as where l, m, k are the harmonic numbers, φ mk ≡ ω mk (t)dt, −2 S aω lmk (Θ) denotes spin-weighted spheroidal harmonics which depend on the polar angles Θ of the observer's direction of sight and the direction of orbital angular momentum of the source. Z H lmk describes the amplitude of each mode, which could be calculated by the radial component of the Teukolsky equation (see Appendix D for details). In this article, we set Θ = 0 ("face on") and Φ = 0 without losing the generality, and ω mk is where ω r and ω φ denote the orbital frequencies of radial and azimuthal direction respectively which are given in Eqs. (2.25,2.26). Due to our analytical solution of orbits and frequencies in the previous section, the calculation of Teukolsky-base waveform becomes very convenient and accurate.
As an example, figure 9 illustrates the numerical waveforms of four evolution stages of an EMRI with inital parameters p 0 = 10 M, e 0 = 0.6 and mass-ratio ν = 10 −3 . We can find that as long as the time elapses, due to the semilatus rectum p becomes smaller, the GW frequency becomes higher. At the same time, because of the decreasing of eccentricity e, the GW strains no longer vary strongly when the small object passing through the periastron and apastron.
Matched filtering [45] is widely used in GW detection in LIGO and Virgo data analysis and also will be used in the future space-borne detectors. We employ this technology to quantitatively analyze the influence of mass-ratio on the EMRI waveforms. Figure 10 illustrates the match results of the waveforms with mass-ratio correction both in conservative dynamics and fluxes and the waveforms by test particle model. We employ six EMRIs with mass-ratio from 10 −6 to 10 −3 . The results show that for mass-ratio as low as 10 −6 , the test particle model may be still valid in waveform template calculations. However, for mass-ratio ∼ 10 −5 , if the small body at the extremely relativistic orbit around the central BH, the waveform templates of test-particle approximation will be invalid after a few of month's evolution. For mass-ratio ∼ 10 −3 , even for large separation of orbit, the match of two waveforms drops just after two months. We may conclude that the mass-ratio correction in EMRI waveform model should be important in the detection of such kind of systems.

IV. CONCLUSIONS AND OUTLOOK
In the present paper, based on the EOB theory, we give analytical orbital solutions of elliptic EMRIs with spinning black holes. The solutions are derived with geometric parameters p and e instead of the EOB coordinates and momenta. The fundamental properties of the motion due to mass ratio and black hole's spin are discussed. We also give the expressions for two orbital frequencies. With these formalism at hand, it is convenient to combine with the frequency-domain Teukolsky equation, and generate accurate numerical waveforms. In addition, we express the forms of orbital evolution under gravitational radiation. We also insert the 1PN mass-ratio correction into the energy and angular momentum fluxes and the performance of fluxes with different PN orders on orbital evolution are shown.
Using matched filtering, we reveal the influence of mass ratio on the detection of EMRI GWs. We indicate that for mass-ratio ν 10 −5 , the conservative gravitational self-force of small objects should be considered into the construction of EMRI waveform templates. Considering the main EMRI waveform models like AK, AAK, NK and XSPEG etc. take the small objects as test particles, our model may make progress on the development of EMRI templates.
As we mentioned before, in the present model, we temporarily omit the effective spin of the small object. In the EOB theory, this spin of effective test particle is ∼ µa/M even if the small object does not really rotate. This is why we state that our present model only works for EMRIs and still is an improvement comparing to test particle approximation. Furthermore, the present model can not use to inclined orbits. We will solve these two problems in the next work.
One of the scientific targets of EMRIs is to detect the spacetime geometry of the SMBH. For this target, an accurate and efficient waveform template is needed. However, this is still a challenge now. The analytical orbital solution including mass-ratio and eccentricity given in this paper is more accurate and efficient description of the EMRI orbits. The combination of the analytical orbit and the Teukolsky equation can generate accurate waveforms. We hope our work is useful to the development of EMRI waveform template for space-borne detectors. The log-resummed, calibrated A-potential is given by the expression from APPENDIX A of [46] A =∆ u ∆ 0 ν + ν log ∆ 5 u 5 + ∆ 4 u 4 + ∆ 3 u 3 + ∆ 2 u 2 + ∆ 1 u + 1 + 1 , where K is a calibration parameter tuned to numerical-relativity simulations whose most recently updated value was determined in Eq. (4.8) of Ref. [47] K = 267.788ν 3 − 126.687ν 2 + 10.2573ν + 1.7336 (A2h) The D-potential is Appendix B: functions that appear in the solution forė andṗ The quantities repeated  [32] for an equal-mass binary without spin. It shows that our formalism coincide with the Schwarzschild case when a = 0 but with mass-ratio ν = 0.25. Combining the both results, our formalism and codes for orbital evolution are validated.

Appendix D: The Teukolsky equation
The gravitational perturbation of Kerr space-time is described by the Teukolsky equation by the Weyl curvature (complex) scalar ψ 4 , decomposed in frequency domain ψ 4 = ρ 4 +∞ −∞ dω lm R lmω (r) −2 S aω lm (θ)e imφ e −iωt with spinweighted spheroidal harmonics −2 S aω lm , obeys [34]: where T lmω (r) is the source term, which is connected by the stress-energy tensor of the perturbation source, and the potential is where K = (r 2 + a 2 )ω − ma, λ = E lm + a 2 ω 2 − 2amw − 2 and ∆ = r 2 − 2M r + a 2 . First, we consider the homogeneous Teukolsky equation where the source term is zero. We can solve it by analytical expansion, as discussed in [48,49] and here we don't go into technical details of it. The homogeneous Teukolsky equation allows two independent solutions R H lmω , which is purely ingoing at the horizon, and R ∞ lmω , which is purely outgoing at infinity: R H lmω = B hole lmω ∆ 2 e −ipr * , r → r + R H lmω = B out lmω r 3 e iωr * + r −1 B in lmω e −iωr * , r → ∞; (D3) R ∞ lmω = D out lmω e ipr * + ∆ 2 D in lmω e −ipr * , r → r + R ∞ lmω = r 3 D ∞ lmω e iωr * , r → ∞, where p = ω − ma 2M r+ , r + = M + √ M 2 − a 2 and r * is the tortoise coordinate related to r by dr * /dr = (r 2 + a 2 )/∆ Then, using the homogeneous solutions and proper boundary conditions, we can construct the solution to radial Teukolsky equation with source term. By imposing BH boundary condition, i.e. wave being purely outgoing at infinity and purely ingoing at horizon, the radial function is: The asymptotic behavior of this solution near horizon and infinity is: R BH lmω (r → r + ) = Z ∞ lmω ∆ 2 e −ipr * .
By taking the limit at r → ∞ and r → r + of the solution (Eq. D5), with the asymptotic behavior of homogeneous solutions (Eq. D3, D4), one can find the amplitudes Z H,∞ lmω :