A waveform model for eccentric binary black hole based on effective-one-body-numerical-relativity (EOBNR) formalism

A waveform model for eccentric binary black hole based on effective-one-body-numerical-relativity (EOBNR) formalism

Zhoujian Cao zjcao@amt.ac.cn Department of Astronomy, Beijing Normal University, Beijing 100875, China Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China    Wen-Biao Han Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai, 200030, People’s Republic of China School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
Abstract

Binary black hole systems are among the most important sources for gravitational wave detection. And also they are good objects for theoretical research for general relativity. Gravitational waveform template is important to data analysis. Effective-one-body-numerical-relativity (EOBNR) model has played an essential role in the LIGO data analysis. For future space-based gravitational wave detection, many binary systems will admit somewhat orbit eccentricity. At the same time the eccentric binary is also an interesting topic for theoretical study in general relativity. In this paper we construct the first eccentric binary waveform model based on effective-one-body-numerical-relativity framework. Our basic assumption in the model construction is that the involved eccentricity is small. We have compared our eccentric EOBNR model to the circular one used in LIGO data analysis. We have also tested our eccentric EOBNR model against to another recently proposed eccentric binary waveform model; against to numerical relativity simulation results; and against to perturbation approximation results for extreme mass ratio binary systems. Compared to numerical relativity simulations with eccentricity as large as about 0.2, the overlap factor for our eccentric EOBNR model is better than 0.98 for all tested cases including spinless binary and spinning binary; equal mass binary and unequal mass binary. Hopefully our eccentric model can be the start point to develop a faithful template for future space-based gravitational wave detectors.

pacs:
04.25.D-, 04.30.Db, 04.70.Bw, 95.30.Sf

I Introduction

The direct detection of gravitational waves (GW) has been announced recently by LIGO Abbott et al. (2016a, b, 2017) which opens the brand new window to our universe–gravitational wave astronomy. The success of LIGO is based on both the tremendous development of experiment technology and the improvement of theoretical research in the past decades. Matched filtering data analysis technique is very important to gravitational wave detection. As GW150914, GW151226 and GW170104 have witnessed, the matched filtering technique has improved the data quality and/or even make the noisy data detectable. Regarding to GW150914 and GW170104, we are some lucky. The signal is so strong that the matched filtering data analysis technology is not necessary to catch the signal, although the matched filtering data analysis can improve the signal to noise ratio (SNR) and confidence level strongly. Regarding to GW151226, the signal is much weaker than that of GW150914 and GW170104. Without the matched filtering data analysis technology, the signal is completely invisible. In contrast, the matched filtering data analysis technology digs out the signal from the strong noise with SNR 13 and confidence level 5. GW151226 is a good example showing that the detection of GW is the result of combination of experiment achievement and theoretical research progress Cao (2016).

In order to make the matched filtering technique work, the gravitational waveform template is essential Cao (2016). And the template strongly depends on the specific theoretical model of GW source. Currently there are two theoretical models which are ready for gravitational wave data analysis. They are effective-one-body-numerical-relativity (EOBNR) model Taracchini et al. (2014) and IMRPhenom model Abbott et al. (2016c). For example, all of GW150914, GW151226 and GW170104 depend on these two models strongly.

EOBNR model Buonanno et al. (2007) is a combination of effective-one-body theory of post-Newtonian approximation and numerical relativity. About the template bank of the binary black hole gravitational waveform, the related parameters are divided into intrinsic ones and extrinsic ones. The EOBNR model need only concern the intrinsic parameters including the total mass of the binary black hole , the mass ratio , the spins of the two black holes and the eccentricity of the orbit . While the extrinsic parameters, including luminosity distance , source location , the configuration of the orbit respect to the sight direction , the reaching time of the signal , the initial phase and the polarization angle respect to the detector , can be straightforwardly involved when we construct the template bank from EOBNR model.

To the quasi-circular and non-precession ( perpendicular to the orbit plane) binary black hole systems with mass ratio , the EOBNR model has done a quite good job Taracchini et al. (2014). (Note that the author in Pürrer (2016) mentioned the EOBNR model is valid for mass ratio range and spin range . But as pointed out by Husa et al. (2016); Khan et al. (2016); Chu et al. (2016); Kumar et al. (2016), current EOBNR model can at most be calibrated to numerical relativity only for the range and .) Regarding to precession binaries, a primary development of EOBNR model is available Pan et al. (2014a). Very recently, we have done an initial investigation to extend the EOBNR model to include gravitational wave memory in reference Cao and Han (2016). As to the mass ratio, there is no essential difficult to extend the EOBNR model to cover larger parameter range. But current simulation power of numerical relativity limits such development Lousto and Zlochower (2011). In principle, if only relevant numerical relativity results are available the EOBNR model can be calibrated to involved mass ratio. Regarding to eccentricity, the situation is different. Till now, the EOBNR model only works for case. Reference Bini and Damour (2012) touched this problem, but the authors only considered energy flux while left the relevant gravitational wave form alone. Although the EOBNR model admits kinds of limitations as described above, it provides a framework which is possible to extend the EOBNR model to treat these limitations. Recently the authors in Julié and Deruelle (2017) have extended EOB framework to scalar-tensor theory. Hopefully ones can treat gravitational waveform template for different gravitational theories Yunes et al. (2016) within one uniform framework, EOBNR model, in the future.

Due to the circularization effect of gravitational radiation Peters (1964), ones may expect that the binary black hole systems are always near circular when they enter the LIGO frequency band. But recent investigations show it is not absolutely true. The study of the galactic cluster M22 indicate that about 20% of binary black hole (BBH) mergers in globular clusters will have eccentricities larger than 0.1 when they first enter advanced LIGO band at 10Hz Strader et al. (2012), and that 10% may have eccentricities Antonini et al. (2016). Furthermore, a fraction of galactic field binaries may retain significant eccentricity prior to the merger event Samsing et al. (2014). BBHs formed in the vicinity of supermassive black holes (BH) may also merge with significant residual eccentricities VanLandingham et al. (2016). For space-based detectors such as eLISA Amaro-Seoane et al. (2012), LISA Bender et al. (1998); Audley et al. (2017), Taiji Gong et al. (2011) and Tianqin Luo et al. (2016), the orbit of the involved binary black hole systems may be highly eccentric due to recent perturbations by other orbiting objects Hils and Bender (1995); Shibata (1994). Recently there are many authors care about the binary black hole systems with eccentric orbit regarding to gravitational wave detection Yunes et al. (2009); Huerta et al. (2014); Sun et al. (2015); Huerta et al. (2017).

Assuming low eccentricity, the authors of Yunes et al. (2009) extended low order PN waveform model in frequency domain to including eccentricity. They called the corresponding model post-circular (PC) model. Later, the authors of Huerta et al. (2014) improved the PC model to EPC (enhanced post-circular) model which recovers the TaylorF2 model when the eccentricity vanishes. The EPC model is a phenomenologically extending of PC model. Its overall PN order is 3.5. Some numerical relativity simulations have been paid in the past to the eccentric binary black hole systems Sperhake et al. (2008); Hinder et al. (2010); Gold and Brügmann (2013). Along with the numerical relativity results, the x-model was proposed in Hinder et al. (2010). The x-model is a low order post-Newtonian (PN) model. Recently, this model was improved to include inspiral, merger and ringdown phases, and higher PN order terms for vanishing eccentricity part are included. This model was called ax-model by the authors of Huerta et al. (2017). All these models are valid for any mass ratio.

Regarding to large mass ratio binary black hole systems, ones may look the binary system as a perturbation of the big black hole. Then the gravitational wave problem is decomposed into the trajectory problem and the related waveform problem. In Han (2014), one of us investigated the eccentric binary using the Teukolsky equation to treat the waveform problem and combining the conserved EOB dynamics with numerical energy flux to treat the trajectory Han and Cao (2011). In Han and Cao (2011); Han (2014) the Teukolsky equation is solved numerically. Ones can also solve it through some analytical method Sasaki and Tagoshi (2003) or post-Newtonian approximation Forseth et al. (2016). In Drasco and Hughes (2006), the authors used geodesic equation to treat the eccentric orbit of a large mass ratio binary and used the Teukolsky equation to treat the waveform problem. Interestingly, people have used the method of geodesic equation and Teukolsky equation to find that the eccentricity may increase Apostolatos et al. (1993); Kennefick (1998) instead of always decay found through post-Newtonian approximation Peters (1964). And more, people used the method of geodesic equation and Teukolsky equation to find the interesting transient resonance phenomena Flanagan and Hinderer (2012); Berry et al. (2016). When a binary system passes through a transient resonance, the radial frequency and polar frequency become commensurate, and the orbital parameters will show a jump behavior. To our knowledge, the post-Newtonian approximation method can not yet give out the eccentricity increasing and the transient resonance results. Of course it is possible that the available post-Newtonian result is not accurate enough to get these two interesting phenomena. But it is also deserved to ask whether these two phenomena imply that the perturbation method breaks down. Ideally ones may use numerical relativity simulation to check this problem. But unfortunately, current numerical relativity techniques are far away to investigate this problem due to the huge computational cost for large mass ratio binary systems Lousto and Zlochower (2011) (but see Lewis et al. (2016)). Hopefully, effective-one-body-numerical-relativity (EOBNR) model may be used to check this problem. This is because on the side of almost equal mass cases, EOBNR framework can be and has been calibrated against numerical relativity; on the side of extreme mass ratio cases, EOB framework can also be and has also been used to describe the dynamics and the gravitational waveform Yunes et al. (2010). So we can expect EOBNR framework may play a bridge role to connect numerical relativity result with large mass ratio problem. In order to realize this kind of investigation, we need a EOBNR model being valid for eccentric binary systems, which is absent now. In current paper, we will go a little step to construct an EOBNR model for eccentric binary systems.

This paper is organized as follows. In the next section we will describe the extended EOBNR model including eccentricity. We call our model SEOBNRE (Spinning Effective-One-Body-Numerical-Relativity model for Eccentric binary). This model includes three essential parts which will be explained in detail respectively in the subsections of next section. The involved detail calculations and long equations are postponed to the Appendix. Then in Sec. III we check and test our SEOBNRE model against to quasi-circular EOBNR model; against to existing eccentric waveform model–ax model; against to numerical relativity simulation results and against to Teukolsky equation based waveform model for extreme mass ratio binary systems. Finally we give a summary and discussion in Sec. IV. Throughout this paper we will use units . Regarding to the mass of the binary we always assume .

Ii waveform model for eccentric binary based on EOBNR

Effective one body technique is a standard trick to treat the two body problem in the central force situation of classical mechanics, especially for Newtonian gravity theory Goldstein et al. (2001). In Buonanno and Damour (1999), Buonanno and Damour introduced the seminal idea of effective-one-body approach for general relativistic two body problem. The effective-one-body approach needs many inputs from post-Newtonian approximation, but it is more powerful than post-Newtonian approximation. Not like post-Newtonian approximation which will diverge before the late inspiral stage of the binary evolution, the effective-one-body approach works till the binary merger. And more it is convenient for the effective-one-body approach to adopt the result of perturbation method Yunes et al. (2010). At the same time, we can also combine the results of the effective-one-body approach and numerical relativity. As firstly done by Pan and his coworkers in Buonanno et al. (2007), such combination gives effective-one-body numerical relativity (EOBNR) model. Currently the most advanced EOBNR model is the SEOBNR which includes version 1 Taracchini et al. (2012), version 2 Taracchini et al. (2014) and version 3 Abbott et al. (2016d); Babak et al. (2017); Abbott et al. (2017). SEOBNR is valid only for quasi-circular orbit and black hole spin perpendicular to the orbital plane which means the precession is not presented. In this paper, we will extend SEOBNR model to treat eccentric orbit.

The EOB approach includes three building parts: (1) a description of the conservative part of the dynamics of two compact bodies which is represented by a Hamiltonian; (2) an expression for the radiation-reaction force which is added to the conservative Hamiltonian equations of motion; and (3) a description of the asymptotic gravitational waveform emitted by the binary system. The part 1 is independent of the character of the involved orbit. In another word, the part 1 is valid no matter the orbit is circular or eccentric. In current paper, we adopt the result from SEOBNRv1 model which will be summarized in the following. Regarding to parts 2 and 3, current SEOBNRv1 model is not valid to eccentric orbit. We will extend these two parts in current work. For convenience, we will refer our model SEOBNRE where the last letter E represents eccentricity.

ii.1 Conservative part for SEOBNRE model

The conservative part for SEOBNRE model is the same to that of SEOBNRv1 Taracchini et al. (2012). But the related equations are distributed in different papers. For reference convenience, we give a summary here.

The basic idea of EOB approach is reducing the conservative dynamics of the two body problem in general relativity to a geodesic motion (more precisely Mathisson-Papapetrou-Dixon equation Barausse et al. (2009)) on the top of a reduced spacetime which corresponds to the reduced one body. Roughly the reduced spacetime is a deformed Kerr black hole with metric form Barausse and Buonanno (2010)

(1)
(2)
(3)
(4)
(5)
(6)

where

(7)
(8)
(9)
(10)
(11)
(12)

We still call the coordinate used here Boyer-Lindquist coordinate. Following SEOBNRv1 we set and . Here and are respectively the mass and the Kerr spin parameter of the deformed Kerr black hole

(13)
(14)

and we have used notation and

(15)
(16)

where is the symmetric mass ratio of the binary with components mass , and Kerr parameter and .

Corresponding to the geodesic motion, or to say the Mathisson-Papapetrou-Dixon equations, the Hamiltonian can be written as Barausse and Buonanno (2011); Taracchini et al. (2012)

(17)
(18)

The detail expressions for the quantities , and involved in the above Hamiltonian are listed in the Appendix A.

Based on the above given Hamiltonian, we have then the equation of motion respect to the conservative part as

(19)
(20)

ii.2 Gravitational waveform part for SEOBNRE model

In the EOBNR framework, the gravitational wave form is described by spin-weighted spherical harmonic modes. This kind of modes is also extensively used in numerical relativity Bai et al. (2011). In SEOBNRv1, the modes are available. Note that only the positive modes are considered while the negative modes are produced through relation Pan et al. (2011a). Here means the complex conjugate.

In this work, we only consider mode although other modes can be extended straightforward. Our basic idea is decomposing the waveform into quasi-circular part and eccentric part. The strategy is following Huerta et al. (2017). We treat the eccentric part as perturbation by assuming that the eccentricity is small. Regarding to the quasi-circular part, we borrow the ones from SEOBNRv1 exactly. For convenience, we firstly review this part. Within EOBNR framework, the waveform is divided into two segments. One is after merger which is described with quasi-normal modes of some Kerr black hole. The other is inspiral-plunge stage which is described in the factorized form as Pan et al. (2011a)

(21)
(22)

where is the distance to the source; is the orbital phase; are the scalar spherical harmonics. Particularly for mode, we have Pan et al. (2011b, a); Taracchini et al. (2012)

(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)

where we have defined with being the Euler constant. In the equation of , the parameters , , , , are functions of , and parameters , , , and are functions of and . Following SEOBNRv1, we construct data tables for , , and based on the numerical relativity results of some specific cases for and . Then we interpolate to get the wanted values for the and in question. Then we solve the conditions (21)-(25) of Taracchini et al. (2012) for and .

For the eccentric part, the post-Newtonian (PN) result is valid till second PN order Will and Wiseman (1996)

(35)
(36)
(37)
(38)
(39)
(40)

In the above equations we have used the following notations. is the radial direction to the observer. lies along the line of nodes. and more notations include Will and Wiseman (1996); Kidder (1995)

(41)
(42)
(43)

We define the spin-weighted spherical harmonic modes as

(44)
(45)
(46)
(47)
(48)
(49)

Based on above results we express the (2,2) mode as

(50)

The involved notations such and are explained one by one in the Appendix B.

We assume the in the equation (50) includes quasi-circular part corresponding to and the left eccentric part. It is straightforward to check that is consistent to the Eq. (9.3) of Blanchet et al. (2008). So we define the eccentric correction as

(51)

where means the one given in the equation (50). In summary the inspiral-plunge waveform for SEOBNRE is

(52)

where is given in Eq. (21).

ii.3 Radiation-reaction force for SEOBNRE model

We have mentioned conservative part of the EOB dynamics in Eqs. (19) and (20). But that is only partial part of the whole EOB dynamics. The left part is related to the radiation-reaction force. Assume the radiation-reaction force is , then the whole EOB dynamics can be expressed as

(53)
(54)

In SEOBNRv1 model, the radiation-reaction force is related to the energy flux of gravitational radiation through Taracchini et al. (2012)

(55)
(56)

Here we need to note the sign of . Since here means the energy of the binary system, decreases due to the gravitational radiation, . So people call it dissipation sometimes. Corresponding to SEOBNRv1 code SEO (2012), since it treats quasi-circular cases without precession, which reduces

(57)

Regarding to the energy flux , SEOBNR model relates it to the gravitational wave form through Pan et al. (2011b)

(58)

And more the SEOBNR model assumes the dependence of on time is a harmonic oscillation. So with the orbital frequency of the binary. Then

(59)
(60)

Like EOBNR models, our SEOBNRE model is valid only for spin aligned binary black holes (spin is perpendicular to the orbital plane). In these systems, there is no precession will be involved. Most importantly, these systems admit a plane reflection symmetry respect to the orbital plane. Due to this symmetry of the corresponding spacetime and the symmetry of the spin-weighted spherical harmonic functions we have (Eqs. (44)-(46) of Brügmann et al. (2008))

(61)
(62)

Here we have used to represent the spherical coordinate respect to the gravitational wave source. Consequently we have Pan et al. (2011a). In the second equality of the above energy flux equation, SEOBNR model has taken this relation into consideration and has neglected the ‘memory’ modes Cao and Han (2016); Nichols (2017). We also note that some authors use the relation as an assumption in the cases where the plane reflection symmetry breaks down Taracchini et al. (2014); Pan et al. (2014a); Abbott et al. (2016d).

In our SEOBNRE model, we follow the steps of SEOBNRv1 model to construct the radiation reaction force. The only difference is replacing the waveform with our SEOBNRE waveforms (52).

Besides the above method to calculate the energy flux, one may also calculate based on post-Newtonian approximation together with results from conservative part of EOBNR model. This is the method taken by ax-model Huerta et al. (2017). Similar to the idea we taken to treat the waveform in the above sub-section A, we divide the energy flux into two parts which correspond to the circular part and the non-circular correction part. Then the over all energy flux can be written as

(63)

We give the detail calculations for the post-Newtonian energy fluxes including and for eccentric binary in the Appendix C.

Figure 1: Test for two identical spinless black holes with eccentricity . The four top panels are comparison of () mode of the gravitational wave form for SEOBNRE and SEOBNRv1. The bottom two panels are comparison of energy flux for the one generated with waveform (60), given in (63) and in (129). In the plot the energy flux generated with waveform (60) is marked with ‘flux based on waveform’. The given in (63) is marked with ‘with elliptic correction’. The in (129) is marked with ‘without elliptic correction’. SEOBNRE model is used in the energy flux comparison.
Figure 2: Quantitative comparison of waveform corresponding to Fig. 1. In the plot , .

ii.4 Initial data setting for the SEOBNRE dynamics

Within EOB framework, we solve the dynamical equations (53) and (54), then plug the evolved dynamical variables into the waveform expression (52). But firstly we need to setup the initial value for the dynamical variables . We take two steps to set the initial values. First, we look for the dynamical variable values for circular orbit (the authors in Buonanno et al. (2006) call it spherical orbit). Secondly, we adjust the momentum to achieve wanted eccentricity. In this work we consider binary black holes with spin perpendicular to the orbital plane. So the dynamics can be described with the test particle moves on the ecliptic plane of the central deformed Kerr geometry Pan et al. (2011a). Within the Boyer-Lindquist coordinate, we have . In order to get , and , we follow the Eqs. (4.8) and (4.9) of Buonanno et al. (2006) to solve

(64)
(65)
(66)

where is the speculated orbital frequency at initial time and the is the given frequency for gravitational wave at initial time. Assume the resulted solution for is , we adjust through

(67)

which means we put the test particle on the periastron of an elliptic orbit with eccentricity based on Newtonian picture Han (2014). If the approximation of test particle and Newtonian picture is not good enough, our initial data setting method can not work well. More sophisticated initial conditions for eccentric binary black hole system are possible Pan et al. (2014b). We leave such investigations to future study.

ii.5 Match the inspiral waveform to the merger-ringdown waveform

Like other EOBNR models, we assume the ringdown waveform can be described by the combination of quasi normal modes as

(68)

where are the complex eigen values of the corresponding quasi normal modes for a Kerr black hole, is the matching time point and are the combination coefficients for each mode. The same to SEOBNRv1 SEO (2012) we take .

In order to determine , we need to know the mass and spin of the final Kerr black hole. In principle the mass and the spin may be affected by the eccentricity. But here we neglect such dependence on the eccentricity as Huerta et al. (2017) based on the assumption that the eccentricity is small for the cases considered in current work. We specify the mass and the spin of the final black hole following Barausse and Rezzolla (2009); Tichy and Marronetti (2008)

(69)
(70)

Note that the above relations are valid only for the spins of the two black holes perpendicular to the orbital plane which are the cases considered in current work.

Regarding to the matching time point , we determine it based on inspiral dynamics as following. For the inspiral part, we solve the dynamics (53) and (54) till a time point which is called ‘merger time point’ for convenience. The criteria of the ‘merger time point’ is and the orbital frequency begins to decrease. Then we chose the matching time point corresponding to the time of the peak amplitude of the waveform .

At last we determine the coefficients base on the rule that the matching is smooth at through first order derivative of the waveform.

Iii Test results for SEOBNRE model

In this section we will compare our SEOBNRE model to several existing waveform results including SEOBNR model, ax model, numerical relativity simulation and Teukolsky equation results. In addition to the comparison between the waveforms directly, we use overlap factor to quantify the difference between our SEOBNRE model and these existing waveform results Cao et al. (2014). For two waveforms and , the overlap factor is defined as

(71)

with the inner product defined as

(72)

where means the Fourier transformation, represents frequency, is the one sided power spectral density of the detector noise, and (, ) is the frequency range of the detector. As a typical example, we consider advanced LIGO detector in the following investigations. More specifically, we take the sensitivity of LIGO-Hanford during O1 run as our which is got from the LSC webpage LSC (2016). Correspondingly we take Hz and Hz. Like other existing waveform models, the total mass of the binary black hole, the source location (, ), the angles between the eccentric orbit and the line direction (, ) and the polarization angle are free parameters Sun et al. (2015). But in order to let the waveform falls in LIGO’s frequency band, we choose M as an example to calculate the overlap factor . Regarding to the five angles, values are taken.

Figure 3: Test for two identical spinless black holes with eccentricity at . The comparison of orbital evolution for and . The result of is got through SEOBNRv1 code, and the result is got by SEOBNRE model. The inset of the plot is the blowup of the corresponding range.

iii.1 Comparison to SEOBNRv1

Firstly we compare the result of SEOBNRE model with against to SEOBNRv1. We consider two spinless black holes with equal mass. In Fig. 1 . The overlap factor between the two waveforms shown in Fig. 1 is . At the same time we have also checked the energy flux introduced in Huerta et al. (2017) which is shown in (129) and the elliptical orbit correction we calculated in (63). We find that the energy flux (129) can describe the real flux used by SEOBNRE dynamics (60) quite well in the early inspiral stage but fails at late inspiral and plunge stage. At later times, the energy flux (129) even becomes positive which is unphysical. This implies that the PN energy flux expression breaks down at late inspiral and plunge stage.

In Fig. 2 we compare the waveform more quantitatively, where the amplitude and phase are considered. Regarding to the elliptical orbit correction terms, as ones expect, they are ignorable before plunge in this quasi-circular case as shown in the bottom panel of Fig. 1. Near merger, our elliptic correction terms fail to distinguish real eccentric orbit with the plunge behavior in the quasi-circular orbit, so the difference for both amplitude and phase increase. At merger, the differences for amplitude and phase get maximal values, about 0.03 and 0.026rad respectively. As shown in Fig. 1 and Fig. 2, our SEOBNRE model can recover SEOBNRv1 result some well. In Fig. 1 and Fig. 2 we align the time at simulation start time () which corresponds to gravitational wave frequency . For two 10M black holes Hz. At this alignment time, we also set the phase of the gravitational wave 0 which makes the comparison easier.

Our second testing case is two identical spinless black holes with eccentricity at which corresponds to two 10M black holes with Hz. As shown in Fig. 3 we can see clearly the oscillation of radial coordinate respect to time which corresponds to the eccentric orbital motion. But this level of eccentricity is ignorable for waveform as shown in Fig. 4. Although some small, we can see the oscillation behavior of the energy flux with the same frequency to the motion in the bottom panel of Fig. 4. And again we find that the energy flux (129) can describe the real flux quite well in the early inspiral stage but fails at late inspiral and plunge stage. In Fig. 3 and Fig. 4, we have adjusted the time coordinate of result to align with the merger time of which is . The overlap factor between the two waveforms shown in Fig. 4 is .

Figure 4: Similar to Fig. 1 but for two identical spinless black holes with eccentricity at . The result of is got through SEOBNRv1 code, and the result is got by SEOBNRE model.

When we increase the eccentricity to for , the radial oscillation becomes stronger than that of Fig. 3. But the overall behavior is similar. Regarding to the waveform, phase difference to quasi-circular case appears as shown in Fig. 5. More quantitatively the amplitude difference and the phase difference are shown in Fig. 6. Along with the time, the two differences decrease. This is because the eccentricity is decreasing due to the circularizing effect of gravitational radiation. Near merger, such difference almost disappears. This result supports our assumption that we ignore the effect of eccentricity on the mass and spin of the final Kerr black hole in (69) and (70). Near merger the difference becomes larger again. But we note that the maximal difference is 0.03 and 0.02rad for amplitude and phase respectively. This level of difference is the same to the one we got for quasi-circular case shown in Fig. 2. So we believe this is resulted from the same reason as in quasi-circular case. Similar to Fig. 4, we here have adjusted the time coordinate of result to align with the merger time of