arX

iv:1

708.

0016

6v1

[gr

-qc]

1 A

ug 2

017

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

Zhoujian Cao1, 2, ∗ and Wen-Biao Han3, 4

1Department of Astronomy, Beijing Normal University, Beijing 100875, China2Institute of Applied Mathematics, Academy of Mathematics and Systems Science,

Chinese Academy of Sciences, Beijing 100190, China3Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai, 200030, People’s Republic of China

4School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China

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 waveformtemplate is important to data analysis. Effective-one-body-numerical-relativity (EOBNR) modelhas played an essential role in the LIGO data analysis. For future space-based gravitational wavedetection, many binary systems will admit somewhat orbit eccentricity. At the same time theeccentric binary is also an interesting topic for theoretical study in general relativity. In this paper weconstruct the first eccentric binary waveform model based on effective-one-body-numerical-relativityframework. 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. Wehave also tested our eccentric EOBNR model against to another recently proposed eccentric binarywaveform model; against to numerical relativity simulation results; and against to perturbationapproximation results for extreme mass ratio binary systems. Compared to numerical relativitysimulations with eccentricity as large as about 0.2, the overlap factor for our eccentric EOBNRmodel is better than 0.98 for all tested cases including spinless binary and spinning binary; equalmass binary and unequal mass binary. Hopefully our eccentric model can be the start point todevelop a faithful template for future space-based gravitational wave detectors.

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

I. INTRODUCTION

The direct detection of gravitational waves (GW) hasbeen announced recently by LIGO [1–3] which opensthe brand new window to our universe–gravitationalwave astronomy. The success of LIGO is based onboth the tremendous development of experiment tech-nology and the improvement of theoretical research inthe past decades. Matched filtering data analysis tech-nique is very important to gravitational wave detection.As GW150914, GW151226 and GW170104 have wit-nessed, the matched filtering technique has improved thedata quality and/or even make the noisy data detectable.Regarding to GW150914 and GW170104, we are somelucky. The signal is so strong that the matched filteringdata analysis technology is not necessary to catch the sig-nal, although the matched filtering data analysis can im-prove the signal to noise ratio (SNR) and confidence levelstrongly. Regarding to GW151226, the signal is muchweaker than that of GW150914 and GW170104. With-out the matched filtering data analysis technology, thesignal is completely invisible. In contrast, the matchedfiltering data analysis technology digs out the signal fromthe strong noise with SNR 13 and confidence level 5σ.GW151226 is a good example showing that the detec-tion of GW is the result of combination of experiment

∗Electronic address: zjcao@amt.ac.cn

achievement and theoretical research progress [4].

In order to make the matched filtering technique work,the gravitational waveform template is essential [4]. Andthe template strongly depends on the specific theoreti-cal model of GW source. Currently there are two the-oretical models which are ready for gravitational wavedata analysis. They are effective-one-body-numerical-relativity (EOBNR) model [5] and IMRPhenom model[6]. For example, all of GW150914, GW151226 andGW170104 depend on these two models strongly.

EOBNR model [7] is a combination of effective-one-body theory of post-Newtonian approximation and nu-merical relativity. About the template bank of the binaryblack hole gravitational waveform, the related parame-ters are divided into intrinsic ones and extrinsic ones.The EOBNR model need only concern the intrinsic pa-rameters including the total mass of the binary blackhole M , the mass ratio q ≡ m1

m2≥ 1, the spins of the

two black holes ~S1,2 and the eccentricity of the orbit e.While the extrinsic parameters, including luminosity dis-tance D, source location (θ, φ), the configuration of theorbit respect to the sight direction (ι, β), the reachingtime of the signal t0, the initial phase φ0 and the polar-ization angle respect to the detector ψ, can be straight-forwardly involved when we construct the template bankfrom EOBNR model.

To the quasi-circular e = 0 and non-precession (~S1,2

perpendicular to the orbit plane) binary black hole sys-tems with mass ratio q ∈ (1, 201 ), the EOBNR modelhas done a quite good job [5]. (Note that the author in

2

[8] mentioned the EOBNR model is valid for mass ratiorange 1 < q < 100 and spin range −1 < χ < 0.99. Butas pointed out by [9–12], current EOBNR model can atmost be calibrated to numerical relativity only for therange 1 < q < 20 and −0.85 < χ < 0.85.) Regarding toprecession binaries, a primary development of EOBNRmodel is available [13]. Very recently, we have done aninitial investigation to extend the EOBNR model to in-clude gravitational wave memory in reference [14]. As tothe mass ratio, there is no essential difficult to extend theEOBNR model to cover larger parameter range. But cur-rent simulation power of numerical relativity limits suchdevelopment [15]. In principle, if only relevant numer-ical relativity results are available the EOBNR modelcan be calibrated to involved mass ratio. Regardingto eccentricity, the situation is different. Till now, theEOBNR model only works for e = 0 case. Reference[16] touched this problem, but the authors only consid-ered energy flux while left the relevant gravitational waveform alone. Although the EOBNR model admits kindsof limitations as described above, it provides a frame-work which is possible to extend the EOBNR model totreat these limitations. Recently the authors in [17] haveextended EOB framework to scalar-tensor theory. Hope-fully ones can treat gravitational waveform template fordifferent gravitational theories [18] within one uniformframework, EOBNR model, in the future.

Due to the circularization effect of gravitational radia-tion [19], ones may expect that the binary black hole sys-tems are always near circular when they enter the LIGOfrequency band. But recent investigations show it is notabsolutely true. The study of the galactic cluster M22 in-dicate that about 20% of binary black hole (BBH) merg-ers in globular clusters will have eccentricities larger than0.1 when they first enter advanced LIGO band at 10Hz[20], and that ∼10% may have eccentricities e ∼ 1 [21].Furthermore, a fraction of galactic field binaries may re-tain significant eccentricity prior to the merger event [22].BBHs formed in the vicinity of supermassive black holes(BH) may also merge with significant residual eccentric-ities [23]. For space-based detectors such as eLISA [24],LISA [25, 26], Taiji [27] and Tianqin [28], the orbit of theinvolved binary black hole systems may be highly eccen-tric due to recent perturbations by other orbiting objects[29, 30]. Recently there are many authors care about thebinary black hole systems with eccentric orbit regardingto gravitational wave detection [31–34].

Assuming low eccentricity, the authors of [31] extendedlow order PN waveform model in frequency domain toincluding eccentricity. They called the correspondingmodel post-circular (PC) model. Later, the authors of[32] improved the PC model to EPC (enhanced post-circular) model which recovers the TaylorF2 model whenthe eccentricity vanishes. The EPC model is a phe-nomenologically extending of PC model. Its overall PNorder is 3.5. Some numerical relativity simulations havebeen paid in the past to the eccentric binary black holesystems [35–37]. Along with the numerical relativity re-

sults, the x-model was proposed in [36]. The x-modelis a low order post-Newtonian (PN) model. Recently,this model was improved to include inspiral, merger andringdown phases, and higher PN order terms for vanish-ing eccentricity part are included. This model was calledax-model by the authors of [34]. All these models arevalid for any mass ratio.

Regarding to large mass ratio binary black hole sys-tems, ones may look the binary system as a perturba-tion of the big black hole. Then the gravitational waveproblem is decomposed into the trajectory problem andthe related waveform problem. In [38], one of us investi-gated the eccentric binary using the Teukolsky equationto treat the waveform problem and combining the con-served EOB dynamics with numerical energy flux to treatthe trajectory [39]. In [38, 39] the Teukolsky equation issolved numerically. Ones can also solve it through someanalytical method [40] or post-Newtonian approximation[41]. In [42], the authors used geodesic equation to treatthe eccentric orbit of a large mass ratio binary and usedthe Teukolsky equation to treat the waveform problem.Interestingly, people have used the method of geodesicequation and Teukolsky equation to find that the ec-centricity may increase [43, 44] instead of always decayfound through post-Newtonian approximation [19]. Andmore, people used the method of geodesic equation andTeukolsky equation to find the interesting transient reso-nance phenomena [45, 46]. When a binary system passesthrough a transient resonance, the radial frequency andpolar frequency become commensurate, and the orbitalparameters will show a jump behavior. To our knowl-edge, the post-Newtonian approximation method can notyet give out the eccentricity increasing and the transientresonance results. Of course it is possible that the avail-able post-Newtonian result is not accurate enough to getthese two interesting phenomena. But it is also deservedto ask whether these two phenomena imply that the per-turbation method breaks down. Ideally ones may use nu-merical relativity simulation to check this problem. Butunfortunately, current numerical relativity techniques arefar away to investigate this problem due to the huge com-putational cost for large mass ratio binary systems [15](but see [47]). Hopefully, effective-one-body-numerical-relativity (EOBNR) model may be used to check thisproblem. This is because on the side of almost equalmass cases, EOBNR framework can be and has beencalibrated against numerical relativity; on the side ofextreme mass ratio cases, EOB framework can also beand has also been used to describe the dynamics and thegravitational waveform [48]. So we can expect EOBNRframework may play a bridge role to connect numericalrelativity result with large mass ratio problem. In orderto realize this kind of investigation, we need a EOBNRmodel being valid for eccentric binary systems, which isabsent now. In current paper, we will go a little step toconstruct an EOBNRmodel for eccentric binary systems.

This paper is organized as follows. In the next sec-tion we will describe the extended EOBNR model includ-

3

ing eccentricity. We call our model SEOBNRE (Spin-ning Effective-One-Body-Numerical-Relativity model forEccentric binary). This model includes three essentialparts which will be explained in detail respectively in thesubsections of next section. The involved detail calcula-tions and long equations are postponed to the Appendix.Then in Sec. III we check and test our SEOBNRE modelagainst to quasi-circular EOBNR model; against to ex-isting eccentric waveform model–ax model; against tonumerical relativity simulation results and against toTeukolsky equation based waveform model for extrememass ratio binary systems. Finally we give a summaryand discussion in Sec. IV. Throughout this paper we willuse units c = G = 1. Regarding to the mass of the binarywe always assume m1 ≥ m2.

II. WAVEFORM MODEL FOR ECCENTRICBINARY BASED ON EOBNR

Effective one body technique is a standard trick totreat the two body problem in the central force situ-ation of classical mechanics, especially for Newtoniangravity theory [49]. In [50], Buonanno and Damourintroduced the seminal idea of effective-one-body ap-proach for general relativistic two body problem. Theeffective-one-body approach needs many inputs frompost-Newtonian approximation, but it is more power-ful than post-Newtonian approximation. Not like post-Newtonian approximation which will diverge before thelate inspiral stage of the binary evolution, the effective-one-body approach works till the binary merger. Andmore it is convenient for the effective-one-body approachto adopt the result of perturbation method [48]. Atthe same time, we can also combine the results ofthe effective-one-body approach and numerical relativ-ity. As firstly done by Pan and his coworkers in [7], suchcombination gives effective-one-body numerical relativity(EOBNR) model. Currently the most advanced EOBNRmodel is the SEOBNR which includes version 1 [51], ver-sion 2 [5] and version 3 [3, 52, 53]. SEOBNR is valid onlyfor quasi-circular orbit and black hole spin perpendicularto the orbital plane which means the precession is notpresented. In this paper, we will extend SEOBNR modelto treat eccentric orbit.The EOB approach includes three building parts: (1)

a description of the conservative part of the dynamicsof two compact bodies which is represented by a Hamil-tonian; (2) an expression for the radiation-reaction forcewhich is added to the conservative Hamiltonian equationsof motion; and (3) a description of the asymptotic grav-itational waveform emitted by the binary system. Thepart 1 is independent of the character of the involved or-bit. In another word, the part 1 is valid no matter theorbit is circular or eccentric. In current paper, we adoptthe result from SEOBNRv1 model which will be sum-marized 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 con-venience, we will refer our model SEOBNRE where thelast letter E represents eccentricity.

A. Conservative part for SEOBNRE model

The conservative part for SEOBNRE model is thesame to that of SEOBNRv1 [51]. But the related equa-tions are distributed in different papers. For referenceconvenience, we give a summary here.The basic idea of EOB approach is reducing the conser-

vative dynamics of the two body problem in general rel-ativity to a geodesic motion (more precisely Mathisson-Papapetrou-Dixon equation [54]) on the top of a reducedspacetime which corresponds to the reduced one body.Roughly the reduced spacetime is a deformed Kerr blackhole with metric form [55]

ds2 = gttdt2 + grrdr

2 + gθθdθ2 + gφφdφ

2 + 2gtφdtdφ,

(1)

gtt = − Λt

∆tΣ, (2)

grr =∆r

Σ, (3)

gθθ =1

Σ, (4)

gφφ =1

Λt(−

ω2fd

∆tΣ+

Σ

sin2 θ), (5)

gtφ = − ωfd

∆tΣ(6)

where

Σ = r2 + a2 cos2 θ, (7)

∆t = r2(A(u) +a2u2

M2), (8)

∆r =∆t

D(u), (9)

Λt = ω4 − a2∆t sin2 θ, (10)

ω ≡√

r2 + a2, (11)

ωfd = 2Mar +Maη

r(ωfd

1 M2 + ωfd2 a2). (12)

We still call the coordinate (t, r, θ, φ) used here Boyer-Lindquist coordinate. Following SEOBNRv1 we set

ωfd1 = 0 and ωfd

2 = 0. Here M and a are respectively themass and the Kerr spin parameter of the deformed Kerrblack hole

M ≡ m1 +m2, (13)

M~a ≡ ~σ ≡ ~a1m1 + ~a2m2 (14)

and we have used notation u ≡ Mr and

A(u) ≡ 1− 2u+ 2ηu3 + η(94

3− 41

32π2)u4, (15)

4

D(u) ≡ 1/1 + log[1 + 6ηu2 + 2(26− 3η)ηu3], (16)

where η ≡ m1m2

(m1+m2)2is the symmetric mass ratio of the

binary with components mass m1, m2 and Kerr param-eter ~a1 and ~a2.Corresponding to the geodesic motion, or to say the

Mathisson-Papapetrou-Dixon equations, the Hamilto-nian can be written as [51, 56]

H =M

√

1 + 2η(Heff

Mη− 1), (17)

Heff = HNS +HS +HSC . (18)

The detail expressions for the quantities HNS , HS andHSC involved in the above Hamiltonian are listed in theAppendix A.Based on the above given Hamiltonian, we have then

the equation of motion respect to the conservative partas

~r =∂H

∂~p, (19)

~p = −∂H∂~r

. (20)

B. Gravitational waveform part for SEOBNREmodel

In the EOBNR framework, the gravitational waveform is described by spin-weighted −2 spherical har-monic modes. This kind of modes is also extensivelyused in numerical relativity [57]. In SEOBNRv1, themodes ℓ ∈ 2, 3, 4, 5, 6, 7, 8,m ∈ [−ℓ, ℓ] are available.Note that only the positivemmodes are considered whilethe negative m modes are produced through relationhℓm = (−1)ℓh∗ℓ,−m [58]. Here ∗ means the complex con-jugate.In this work, we only consider (ℓ,m) = (2, 2) 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 follow-ing [34]. We treat the eccentric part as perturbation byassuming that the eccentricity is small. Regarding tothe quasi-circular part, we borrow the ones from SEOB-NRv1 exactly. For convenience, we firstly review thispart. Within EOBNR framework, the waveform is di-vided into two segments. One is after merger which is de-scribed with quasi-normal modes of some Kerr black hole.The other is inspiral-plunge stage which is described inthe factorized form as [58]

h(C)ℓm = h

(N,ǫ)ℓm S

(ǫ)effTℓme

iδℓm(ρℓm)ℓNℓm, (21)

h(N,ǫ)ℓm =

Mη

Rn(ǫ)ℓmcℓ+ǫV

ℓΦY

ℓ−ǫ,−m(π

2,Φ), (22)

where R is the distance to the source; Φ is the orbitalphase; Y ℓm(Θ,Φ) are the scalar spherical harmonics.

Particularly for (2, 2) mode, we have [51, 58, 59]

ǫ = 0, (23)

V 2Φ = v2Φ, (24)

~vΦ = ~vp − ~n(~vp · ~n), ~vp ≡ ~r =∂H

∂~p, (25)

n(0)ℓm = (im)ℓ

8π

(2ℓ+ 1)!!

√

(ℓ+ 1)(ℓ+ 2)

ℓ(ℓ− 1), (26)

cℓ = (m2

m1 +m2)ℓ−1 + (−1)ℓ(

m1

m1 +m2)ℓ−1, (27)

S(0)eff =

Heff

Mη, (28)

Tℓm =Γ(ℓ+ 1− 2imΩH)

Γ(ℓ+ 1)eπmΩH+2imΩH ln(2mΩr0),

(29)

Ω ≡ v3Φ, r0 ≡ 2(m1 +m2)√e

, (30)

Nℓm = [1 +p2r

(rΩ)2(ahℓm

1 +ahℓm

2

r+ahℓm

3 + ahℓm

3S

r3/2

+ahℓm

4

r2+ahℓm

5

r5/2)] exp[i(

prrΩ

bhℓm

1

+p3rrΩ

(bhℓm

2 +bhℓm

3

r1/2+bhℓm

4

r))], (31)

δ22 =7

3v3 + (

428

105π − 4

3a)v6 + (

1712

315π2 − 2203

81)v9

− 24ηv5Φ +20

63av8Φ, (32)

ρ22 = 1 + (55

84η − 43

42)v2Φ

− 2

3[χS(1 − η) + χA

√

1− 4η]v3Φ

+ (a2

2− 20555

10584− 33025

21168η +

19583

42336η2)v4Φ − 34

21av5Φ

+ [1556919113

122245200+

89

252a2 − 48993925

9779616η − 6292061

3259872η2

+10620745

39118464η3 +

41

192ηπ2 − 428

105eulerlog2(v

2Φ)]v

6Φ

+ (18733

15876a+

1

3a3)v7Φ

+ [18353

21168a2 − 1

8a4 − (5.6 + 117.6η)η

− 387216563023

160190110080+

9202

2205eulerlog2(v

2Φ)]v

8Φ

− [16094530514677

533967033600− 439877

55566eulerlog2(v

2Φ)]v

10Φ ,

(33)

χS =a1/m1 + a2/m2

2, χA =

a1/m1 − a2/m2

2(34)

where we have defined eulerlogm(v2Φ) ≡ γE + ln(2mvΦ)with γE ≈ 0.5772156649015328606065120900824024 be-ing the Euler constant. In the equation of Nℓm, the pa-rameters ahℓm

1 , ahℓm

2 , ahℓm

3 , bhℓm

1 , bhℓm

2 are functions of

5

η, and parameters ahℓm

3S , ahℓm

4 , ahℓm

5 , bhℓm

3 and bhℓm

4 arefunctions of a and η. Following SEOBNRv1, we con-struct data tables for ahℓm

i , ahℓm

3S , bhℓm

1 and bhℓm

2 based onthe numerical relativity results of some specific cases fora and η. Then we interpolate to get the wanted valuesfor the a and η in question. Then we solve the conditions(21)-(25) of [51] for bhℓm

3 and bhℓm

4 .

For the eccentric part, the post-Newtonian (PN) resultis valid till second PN order [60]

hij = 2η(Qij + P1

2Qij + PQij + P3

2Qij

+ P3

2Qijtail), (35)

Qij = 2(vipvjp −

ninj

r), (36)

P1

2Qij = (m1 −m2)[3Nn

r(nivjp + vipn

j − rninj)

+Nv(ninj

r− 2vipv

jp)], (37)

PQij =1

3(1− 3η)[

N2n

r((3v2p − 15r2 +

7

r)ninj

+ 15r(nivjp + vipnj)− 14vipv

jp) +

NnNv

r(12rninj

− 16(nivjp + vipnj)) +N2

v (6vipv

jp −

2

rninj)]

+ [3(1− 3η)v2p − 2(2− 3η)

r]vipv

jp

+2

rr(5 + 3η)(nivjp + vipn

j)

+ [3(1− 3η)r2 − (10 + 3η)v2p +29

r]ninj

r, (38)

P3

2Qij = (m1 −m2)(1 − 2η)N3n

r[5

4(3v2p − 7r2 +

6

r)rninj

− 17

2rvipv

jp − (21v2p − 105r2 +

44

r)nivjp + vipn

j

12]

+1

4

N2nNv

r[58vipv

jp + (45r2 − 9v2p −

28

r)ninj

− 54r(nivjp + vipnj)] +

3

2

NnN2v

r(5(nivjp + vipn

j)− 3rninj)

+1

2N3

v (ninj

r− 4vipv

jp)+

δm

12

Nn

r(nivjp

+ vipnj)[r2(63 + 54η)− 128− 36η

r+ v2p(33− 18η)]

+ ninj r[r2(15− 90η)− v2p(63− 54η) +242− 24η

r]

− rvipvjp(186 + 24η)+ (m1 −m2)Nv

1

2vipv

jp[3− 8η

r

− 2v2p(1 − 5η)]−nivjp + vipn

j

2rr(7 + 4η)

− ninj

r[3

4(1− 2η)r2 +

1

3

26− 3η

r− 1

4(7 − 2η)v2p],

(39)

P3

2Qijtail = 4v5p[π(λ

iλj − ninj) + 6 ln vp(λinj + niλj)].

(40)

In the above equations we have used the follow-ing notations. N = (sin θ cosφ, sin θ sinφ, cosφ)is the radial direction to the observer. p =(cos θ cosφ, cos θ sinφ,− sin θ) lies along the line of nodes.

q = N × p and more notations include [60, 61]

Nn = N · ~n, (41)

Nv = N · ~vp, (42)

~λ =~vp − (~vp · ~n)~n|~vp − (~vp · ~n)~n|

. (43)

We define the spin-weighted spherical harmonic modesas

ǫ+ij =1

2(pipj − qiqj), (44)

ǫ×ij =1

2(piqj + qipj). (45)

h+ = ǫ+ijhij , (46)

h× = ǫ×ijhij . (47)

h = h+ − ih×, (48)

hℓ,m =

∫

h−2Y ∗

ℓ,mdΩ. (49)

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

h22 = 2η[Θij(Qij + P0Q

ij + P3

2Qijtail)

+ PnΘij(P1

2

n Qij + P

3

2

n Qij)

+ PvΘij(P1

2

v Qij + P

3

2

v Qij)

+ PnnΘijPnnQij

+ PnvΘijPnvQij

+ PvvΘijPvvQij

+ PnnnΘijP3

2

nnnQij

+ PnnvΘijP3

2

nnvQij

+ PnvvΘijP3

2

nvvQij

+ PvvvΘijP3

2

vvvQij ]. (50)

The involved notations such Θij and PnΘij are explainedone by one in the Appendix B.We assume the h22 in the equation (50) includes quasi-

circular part corresponding to h22|r=0 and the left eccen-tric part. It is straightforward to check that h22|r=0 isconsistent to the Eq. (9.3) of [62]. So we define the ec-centric correction as

h(PNE)22 = h22 − h22|r=0, (51)

where h22 means the one given in the equation (50). Insummary the inspiral-plunge waveform for SEOBNRE is

hinsp−plun22 = h

(C)22 + h

(PNE)22 , (52)

where h(C)22 is given in Eq. (21).

6

C. Radiation-reaction force for SEOBNRE model

We have mentioned conservative part of the EOB dy-namics in Eqs. (19) and (20). But that is only partialpart of the whole EOB dynamics. The left part is relatedto the radiation-reaction force. Assume the radiation-reaction force is ~F, then the whole EOB dynamics canbe expressed as

~r =∂H

∂~p, (53)

~p = −∂H∂~r

+ ~F. (54)

In SEOBNRv1 model, the radiation-reaction force ~F isrelated to the energy flux of gravitational radiation dE

dtthrough [51]

~F =1

MηωΦ|~r × ~p|dE

dt~p, (55)

ωΦ =|~r × ~r|r2

. (56)

Here we need to note the sign of dEdt . Since E here means

the energy of the binary system, E decreases due to thegravitational radiation, dE

dt < 0. So people call it dis-sipation sometimes. Corresponding to SEOBNRv1 code[63], since it treats quasi-circular cases without preces-

sion, |~r × ~p| ≈ pφ which reduces

~F =1

MηωΦ

dE

dt

~p

pφ. (57)

Regarding to the energy flux dEdt , SEOBNR model re-

lates it to the gravitational wave form through [59]

−dEdt

=1

16π

∑

ℓ

ℓ∑

m=−ℓ

|hℓm|2. (58)

And more the SEOBNR model assumes the dependenceof hℓm on time is a harmonic oscillation. So hℓm ≈mΩhℓm with Ω the orbital frequency of the binary. Then

−dEdt

=1

16π

∑

ℓ

ℓ∑

m=−ℓ

(mΩ)2|hℓm|2 (59)

=1

8π

∑

ℓ

ℓ∑

m=1

(mΩ)2|hℓm|2. (60)

Like EOBNR models, our SEOBNRE model is valid onlyfor spin aligned binary black holes (spin is perpendicularto the orbital plane). In these systems, there is no pre-cession will be involved. Most importantly, these systemsadmit a plane reflection symmetry respect to the orbitalplane. Due to this symmetry of the corresponding space-time and the symmetry of the spin-weighted sphericalharmonic functions we have (Eqs. (44)-(46) of [64])

h(t, π − θ, φ) = h∗(t, θ, φ), (61)

−2Yℓ,−m(π − θ, φ) = −2Y ∗

ℓm(θ, φ). (62)

Here we have used (θ, φ) to represent the spherical co-ordinate respect to the gravitational wave source. Con-sequently we have hℓm = (−1)ℓh∗ℓ,−m [58]. In the sec-ond equality of the above energy flux equation, SEOBNRmodel has taken this relation into consideration and hasneglected the ‘memory’ modes hℓ0 [14, 65]. We also notethat some authors use the relation hℓm = (−1)ℓh∗ℓ,−mas an assumption in the cases where the plane reflectionsymmetry breaks down [5, 13, 52].In our SEOBNRE model, we follow the steps of SEOB-

NRv1 model to construct the radiation reaction force.The only difference is replacing the waveform with ourSEOBNRE waveforms (52).Besides the above method to calculate the energy flux,

one may also calculate dEdt based on post-Newtonian ap-

proximation together with results from conservative partof EOBNR model. This is the method taken by ax-model[34]. Similar to the idea we taken to treat the waveformin the above sub-section A, we divide the energy flux intotwo parts which correspond to the circular part and thenon-circular correction part. Then the over all energyflux can be written as

dE

dt=dE

dt|(C) +

dE

dt|Elip −

dE

dt|Elip,r=0. (63)

We give the detail calculations for the post-Newtonianenergy fluxes including dE

dt |(C) and dEdt |Elip for eccentric

binary in the Appendix C.

D. Initial data setting for the SEOBNRE dynamics

Within EOB framework, we solve the dynamical equa-tions (53) and (54), then plug the evolved dynamical vari-ables into the waveform expression (52). But firstly weneed to setup the initial value for the dynamical vari-ables (~r, ~p). We take two steps to set the initial values.First, we look for the dynamical variable values for cir-cular orbit (the authors in [66] call it spherical orbit).Secondly, we adjust the momentum to achieve wantedeccentricity. In this work we consider binary black holeswith spin perpendicular to the orbital plane. So the dy-namics can be described with the test particle moves onthe ecliptic plane of the central deformed Kerr geometry[58]. Within the Boyer-Lindquist coordinate, we haveφ = 0, θ = π

2 , pθ = 0. In order to get r, pr and pφ, wefollow the Eqs. (4.8) and (4.9) of [66] to solve

dH

dr= 0, (64)

dH

dpθ= 0, (65)

dH

dpφ= ω0, (66)

where ω0 = πf0

is the speculated orbital frequency at

initial time and the f0 is the given frequency for gravita-

7

8e-5

4e-5

0 1000 2000 3000 4000 5000 6000 7000 8000

-dE

/dt

t [M]

without elliptic correctionwith elliptic correction

flux based on waveform

9000 9150

6e-4

0

t [M]

-0.4

-0.2

0

0.2

0.4

Re(

h 22

R/M

) SEOBNRv1SEOBNRE

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

Im(h

22 R

/M)

-0.2

0

0.2

0.4

FIG. 1: Test for two identical spinless black holes with eccentricity e0 = 0. The four top panels are comparison of (ℓ = 2, m = 2)mode of the gravitational wave form for SEOBNRE and SEOBNRv1. The bottom two panels are comparison of energy fluxfor the one generated with waveform (60), − dE

dtgiven in (63) and − dE

dt|(C) in (C1). In the plot the energy flux generated with

waveform (60) is marked with ‘flux based on waveform’. The − dEdt

given in (63) is marked with ‘with elliptic correction’. The

− dEdt

|(C) in (C1) is marked with ‘without elliptic correction’. SEOBNRE model is used in the energy flux comparison.

tional wave at initial time. Assume the resulted solutionfor r is r, we adjust r through

r0 = r/(1 + e0), (67)

which means we put the test particle on the periastron ofan elliptic orbit with eccentricity e0 based on Newtonianpicture [38]. If the approximation of test particle andNewtonian picture is not good enough, our initial datasetting method can not work well. More sophisticatedinitial conditions for eccentric binary black hole systemare possible [67]. We leave such investigations to futurestudy.

E. Match the inspiral waveform to themerger-ringdown waveform

Like other EOBNR models, we assume the ringdownwaveform can be described by the combination of quasi

normal modes as

hmerger-RDℓm =

N−1∑

n=0

Aℓmne−iσℓmn(t−tℓm

match), (68)

where σℓmn are the complex eigen values of the cor-responding quasi normal modes for a Kerr black hole,tℓmmatch is the matching time point and Aℓmn are the com-bination coefficients for each mode. The same to SEOB-NRv1 [63] we take N = 8.In order to determine σℓmn, we need to know the mass

and spin of the final Kerr black hole. In principle themass and the spin may be affected by the eccentricity.But here we neglect such dependence on the eccentricityas [34] based on the assumption that the eccentricity issmall for the cases considered in current work. We specifythe mass and the spin of the final black hole following[68, 69]

Mfinal =M [1 + 4(m0 − 1)η + 16m1η2(χ1 + χ2)], (69)

8

-400

-200

0

0 4000 8000

phas

e of

h22

t [M]

SEOBNRv1SEOBNRE

0

0.01

0.02

0 4000 8000

∆(p

hase

of h

22)

t [M]

0.1

0.2

0.3

0.4

0.5|h

22 R

/M|

SEOBNRv1SEOBNRE

1e-1

1e-5

1e-9

1e-13

∆|h

22 R

/M|

FIG. 2: Quantitative comparison of waveform h22 corresponding to Fig. 1. In the plot ∆|h22R/M | ≡ abs(|h22,SEOBNRER/M |−|h22,SEOBNRv1R/M |), ∆(phase of h22) ≡ abs(phase of h22,SEOBNRE − phase of h22,SEOBNRv1).

χfinal ≡afinalMfinal

= χ0 + ηχ0(t4χ0 + t5η + t0)

+ η(2√3 + t2η + t3η

2), (70)

m0 = 0.9515,m1 = −0.013,

q =m1

m2, χ0 =

χ1 + χ2q2

1 + q2, χ1,2 ≡ a1,2

m1,2,

t0 = −2.8904, t2 = −3.5171, t3 = 2.5763,

t4 = −0.1229, t5 = 0.4537.

Note that the above relations are valid only for the spinsof the two black holes perpendicular to the orbital planewhich are the cases considered in current work.Regarding to the matching time point tℓmmatch, we de-

termine it based on inspiral dynamics as following. Forthe inspiral part, we solve the dynamics (53) and (54)till a time point which is called ‘merger time point’ forconvenience. The criteria of the ‘merger time point’ isr < 6M and the orbital frequency begins to decrease.Then we chose the matching time point tℓmmatch corre-sponding to the time of the peak amplitude of the wave-

form hinsp−plunℓm .

At last we determine the coefficients Aℓmn base on therule that the matching is smooth at tℓmmatch through firstorder derivative of the waveform.

III. TEST RESULTS FOR SEOBNRE MODEL

In this section we will compare our SEOBNRE modelto several existing waveform results including SEOBNRmodel, ax model, numerical relativity simulation andTeukolsky equation results. In addition to the compari-son between the waveforms directly, we use overlap factorto quantify the difference between our SEOBNRE modeland these existing waveform results [70]. For two wave-forms h(t) and s(t), the overlap factor is defined as

O(h, s) ≡ 〈h|s〉√

〈h|h〉〈s|s〉, (71)

with the inner product defined as

〈h|s〉 ≡ 4Re

∫ fmax

fmin

h(f)s∗(f)

Sn(f)df (72)

9

0

5

10

15

20

25

30

35

40

-20000 0 20000 40000 60000 80000 100000 120000 140000

r [M

]

t [M]

e0=0e0=0.003

33

34

35

36

0 10000 20000 30000 40000

FIG. 3: Test for two identical spinless black holes with ec-centricity e0 = 0.003 at Mf0 ≈ 0.0015. The comparison oforbital evolution for e0 = 0.003 and e0 = 0. The result ofe0 = 0 is got through SEOBNRv1 code, and the e0 = 0.003result is got by SEOBNRE model. The inset of the plot isthe blowup of the corresponding range.

where ˜means the Fourier transformation, f representsfrequency, Sn(f) is the one sided power spectral den-sity of the detector noise, and (fmin, fmax) is the fre-quency range of the detector. As a typical example, weconsider advanced LIGO detector in the following inves-tigations. More specifically, we take the sensitivity ofLIGO-Hanford during O1 run as our Sn(f) which is gotfrom the LSC webpage [71]. Correspondingly we takefmin = 20Hz and fmax = 2000Hz. Like other existingwaveform models, the total mass M of the binary blackhole, the source location (θ, φ), the angles between theeccentric orbit and the line direction (ι, β) and the po-larization angle ψ are free parameters [33]. But in or-der to let the waveform falls in LIGO’s frequency band,we choose M = 20M⊙ as an example to calculate theoverlap factor O. Regarding to the five angles, valuesθ = φ = ι = β = ψ = 0 are taken.

A. Comparison to SEOBNRv1

Firstly we compare the result of SEOBNRE modelwith e0 = 0 against to SEOBNRv1. We consider twospinless black holes with equal mass. In Fig. 1 t22match ≈9048.1579M . The overlap factor between the two wave-forms shown in Fig. 1 is O = 0.99998. At the same timewe have also checked the energy flux − dE

dt |(C) introducedin [34] which is shown in (C1) and the elliptical orbitcorrection we calculated in (63). We find that the energyflux (C1) can describe the real flux used by SEOBNREdynamics (60) quite well in the early inspiral stage butfails at late inspiral and plunge stage. At later times,the energy flux (C1) even becomes positive which is un-physical. This implies that the PN energy flux expressionbreaks down at late inspiral and plunge stage.In Fig. 2 we compare the waveform h22 more quanti-

tatively, where the amplitude and phase are considered.Regarding to the elliptical orbit correction terms, as onesexpect, 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 distin-guish real eccentric orbit with the plunge behavior in thequasi-circular orbit, so the difference for both amplitudeand phase increase. At merger, the differences for am-plitude and phase get maximal values, about 0.03 and0.026rad respectively. As shown in Fig. 1 and Fig. 2, ourSEOBNRE model can recover SEOBNRv1 result somewell. In Fig. 1 and Fig. 2 we align the time at simula-tion start time (t = 0) which corresponds to gravitationalwave frequency Mf0 ≈ 0.004. For two 10M⊙ black holesf0 = 40Hz. At this alignment time, we also set the phaseof the gravitational wave 0 which makes the comparisoneasier.

Our second testing case is two identical spinless blackholes with eccentricity e0 = 0.003 atMf0 ≈ 0.001477647which corresponds to two 10M⊙ black holes with f0 =15Hz. As shown in Fig. 3 we can see clearly the oscilla-tion of radial coordinate r respect to time which corre-sponds to the eccentric orbital motion. But this level ofeccentricity is ignorable for waveform as shown in Fig. 4.Although some small, we can see the oscillation behaviorof the energy flux with the same frequency to the r mo-tion in the bottom panel of Fig. 4. And again we find thatthe energy flux (C1) can describe the real flux quite wellin the early inspiral stage but fails at late inspiral andplunge stage. In Fig. 3 and Fig. 4, we have adjusted thetime coordinate of e0 = 0 result to align with the mergertime of e0 = 0.003 which is t22match ≈ 128427.86M . Theoverlap factor between the two waveforms shown in Fig. 4is O = 0.99995.

When we increase the eccentricity to e0 = 0.03 forMf0 ≈ 0.001477647, the radial oscillation becomesstronger than that of Fig. 3. But the overall behavioris similar. Regarding to the waveform, phase differenceto quasi-circular case appears as shown in Fig. 5. Morequantitatively the amplitude difference and the phase dif-ference are shown in Fig. 6. Along with the time, thetwo differences decrease. This is because the eccentric-ity is decreasing due to the circularizing effect of gravi-tational radiation. Near merger, such difference almostdisappears. This result supports our assumption that weignore the effect of eccentricity on the mass and spin ofthe final Kerr black hole in (69) and (70). Near mergerthe difference becomes larger again. But we note thatthe maximal difference is 0.03 and 0.02rad for ampli-tude and phase respectively. This level of difference isthe same to the one we got for quasi-circular case shownin Fig. 2. So we believe this is resulted from the samereason as in quasi-circular case. Similar to Fig. 4, wehere have adjusted the time coordinate of e0 = 0 re-sult to align with the merger time of e0 = 0.03 which ist22match ≈ 101345.99M . The overlap factor between thetwo waveforms shown in Fig. 5 is O = 0.99300.

When we increase more the eccentricity to e0 = 0.3

10

8e-9

6e-9 0 1000 2000 3000 4000 5000 6000 7000 8000

-dE

/dt

t [M]

without elliptic correctionwith elliptic correction

flux based on waveform

128200 128500

6e-4

0

t [M]

-0.4

-0.2

0

0.2

0.4

Re(

h 22

R/M

) e0=0e0=0.003

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

Im(h

22 R

/M)

-0.2

0

0.2

0.4

FIG. 4: Similar to Fig. 1 but for two identical spinless black holes with eccentricity e0 = 0.003 at Mf0 ≈ 0.0015. The result ofe0 = 0 is got through SEOBNRv1 code, and the e0 = 0.003 result is got by SEOBNRE model.

for Mf0 ≈ 0.001477647, the radial oscillation becomesmuch stronger as expected while the overall behavior issimilar to that of Fig. 3. Regarding to the waveform, theoscillation behavior of the amplitude can be clearly seenas shown in Fig. 7. The amplitude difference and thephase difference are quantitatively shown in Fig. 8. Theoverall behavior is similar to that of Fig. 6. In this case,the maximal difference in the early inspiral stage is 0.03and 16rad for amplitude and phase respectively. Similarto Fig. 4, we here have adjusted the time coordinate ofe0 = 0 result to align with the merger time of e0 = 0.3which is t22match ≈ 11357.098M . The overlap factor be-tween the two waveforms shown in Fig. 7 is O = 0.46942.

When the initial eccentricity e0 becomes bigger than0.6 for Mf0 ≈ 0.001477647, we find that the correc-tion term of the waveform (51) becomes significant whichmeans the perturbation assumption of small eccentricitybreaks down. This situation may be improved by higherorder post-Newtonian result for eccentric orbit binary.But unfortunately the higher order PN results for ec-centric orbit binary is not available yet. Of course, thisdoes not mean our model can be applied to e0 < 0.6

cases. Cases with such large eccentricity need moretests against to, for example, numerical relativity sim-ulations. The tests done in this subsection indicate thatour SEOBNRE model can give consistent results com-pared to quasi-circular case.

B. Comparison to advanced x-model (ax model)

In [36] the authors proposed x-model to describe thegravitational waveform for eccentric orbital binary blackhole. For the early inspiral stage, the authors of [36]showed consistence between the x-model and the numer-ical relativity simulation result. Noting that the x-modelis based on low order post-Newtonian approximation,Huerta and his coworkers improved x-model with highorder PN results to get advanced x-model (ax model)in [34]. The full ax model includes inspiral, merger andringdown stages. Similar to our treatment about mergerand ringdown, ax model also takes the assumption thatthe eccentricity is small and has been dissipated awaybefore merger. So here we only compare our SEOBNRE

11

2e-8

6e-9 0 1000 2000 3000 4000 5000 6000 7000 8000

-dE

/dt

t [M]

without elliptic correctionwith elliptic correction

flux based on waveform

101200 101450

6e-4

0

t [M]

-0.4

-0.2

0

0.2

0.4

Re(

h 22

R/M

) e0=0e0=0.03

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

Im(h

22 R

/M)

-0.2

0

0.2

0.4

FIG. 5: Similar to Fig. 1 but for two identical spinless black holes with eccentricity e0 = 0.03 at Mf0 ≈ 0.0015. The result ofe0 = 0 is got through SEOBNRv1 code, and the e0 = 0.03 result is got by SEOBNRE model.

model to ax model for inspiral part.

In ax model, the adiabatic picture is taken. So theeccentricity itself is treated as a dynamical variable. Inall, ax model includes dynamical variables eccentricitye, reduced orbital frequency x ≈ (ωM)2/3 with ω theorbital frequency, the mean anomaly l and the relativeangular coordinate φ. In the comparison here, we takeinitial data x = 0.05, l = φ = 0 and vary e.

Firstly we compare the e0 = 0 case in Fig. 9. Over-all the two waveforms show very good consistence whichcorresponds to the result of fitting factor 0.95 respect toadvanced LIGO got in [34]. But we can still see some am-plitude difference near merger, and some phase differencewhen the evolution time becomes longer. Since we haveconfirmed our SEOBNRE model in Fig. 1 and Fig. 2 fore0 = 0 case, we attribute this difference to the relativelylow order PN approximation of ax model. Effectively theEOBNR model admits more than 3.5 PN order [7, 72],this is why we call ax model relatively low PN order. Theoverlap factor between the two waveforms (inspiral part,t < 0) shown in Fig. 9 is O = 0.99802.

In contrast to ax model, the eccentricity is not an

explicit dynamical variable in SEOBNRE model. Andlimited by the simplified method we taken to treat theinitial data described in the above section, we startour SEOBNRE simulation from somewhat low frequencyMf0 ≈ 0.001477647 and vary different e0 to fit ax modelresult. The fitting process has not been optimized, so thereal consistent result may be better than the ones pre-sented here. We have considered two concrete examples.The first example is e0 = 0.1 at x0 = 0.05 for ax modeland e0 = 0.15 at Mf0 ≈ 0.001477647 for SEOBNREmodel. The initial eccentricity for SEOBNRE model islarger. We suspect this is because the effective initialseparation implemented in SEOBNRE model is largerthan the one in ax model. After some evolution time,the circularization effect drives the eccentricity withinSEOBNRE model to the one presented in ax model. Thecomparison result is shown in Fig. 10. We can see theconsistence of amplitude and phase between ax modeland SEOBNRE model lasts more than 8000M. The over-lap factor between the two waveforms shown in Fig. 10is O = 0.93031. Our second example is the comparisonbetween e0 = 0.3 at x0 = 0.05 of ax model and e0 = 0.4

12

0

600

1200

0 50000 100000

phas

e of

h22

t [M]

e0=0e0=0.03

-1

-0.6

-0.2

0 50000 100000

∆(p

hase

of h

22)

t [M]

0.05

0.15

0.25

0.35

0.45|h

22 R

/M|

e0=0e0=0.03

0

0.01

0.02

0.03

∆|h

22 R

/M|

FIG. 6: Amplitude and phase comparison for h22 corresponding to the case e0 = 0.03 shown in Fig. 5. The result of e0 = 0 isgot through SEOBNRv1 code, and the e0 = 0.03 result is got by SEOBNRE model.

at Mf0 ≈ 0.001477647 of SEOBNRE model. The re-sult is shown in Fig. 11. The consistence of amplitudeand phase between ax model and SEOBNRE model lastsabout 5000M. The overlap factor between the two wave-forms shown in Fig. 11 is O = 0.66617.

C. Comparison to numerical relativity results

No matter how reasonable, we have taken several ap-proximations when we construct our SEOBNRE model.In contrast to this situation, numerical relativity solvesEinstein equation directly [73]. Up to the numerical er-ror, the results given by numerical relativity is the exactsolution to the Einstein equation. So we can use thesimulation results by numerical relativity to check andcalibrate the validity of SEOBNRE model. Since theEOBNR models have been well calibrated to numericalrelativity results for circular cases, and our SEOBNREmodel can recover the usual EOBNR model as shown inFig. 1 and Fig. 2 for e0 = 0 case, there is no surprisingthat our SEOBNRE model is consistent to quasi-circular

simulation results of numerical relativity.

For eccentric cases, there are some subtleties in defin-ing the eccentricity due to the ‘in’-spiral effect reduced bythe gravitational radiation. In this work we are not intentto touch this subtle problem involved in numerical rela-tivity [74, 75]. Instead, we take the similar recipe adoptedin the above subsection to do the comparison. For a givennumerical relativity simulation result, we vary the initialeccentricity e0 corresponding to Mf0 ≈ 0.001477647 forSEOBNRE to fit the numerical relativity result. Againthe fitting process is not optimized, so the consistence be-tween our SEOBNRE model and the numerical relativityresult may be better than the ones presented here.

We have done three comparisons in the current work.The numerical relativity simulation results come fromthe public data [76] which are calculated by SpEC code[77]. The first one is the waveform SXS:BBH:0091 [76]which corresponds to an equal mass, spinless binary blackhole with initial eccentricity e0 = 0.02181 starting evolveat orbital frequency Mf0 = 0.0105565727235. On theSEOBNRE side, the initial eccentricity is e0 = 0.1 start-ing evolve at orbital frequencyMf0 = 0.001477647 which

13

2e-6

1e-6

0 0 1000 2000 3000 4000 5000 6000 7000 8000

-dE

/dt

t [M]

without elliptic correctionwith elliptic correction

flux based on waveform

11200 11450

6e-4

0

t [M]

-0.4

-0.2

0

0.2

0.4

Re(

h 22

R/M

) e0=0e0=0.3

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

Im(h

22 R

/M)

-0.2

0

0.2

0.4

FIG. 7: Similar to Fig. 1 but for two identical spinless black holes with eccentricity e0 = 0.3 at Mf0 ≈ 0.0015. The result ofe0 = 0 is got through SEOBNRv1 code, and the e0 = 0.3 result is got by SEOBNRE model.

is the same as the ones shown in previous figures. Thecomparison is presented in Fig. 12. Although the eccen-tricity involved in this comparison is some small, the os-cillation of the gravitational waveform amplitude is clear.The consistence for both the amplitude and the phase isquite good. The overlap factor between the two wave-forms shown in Fig. 12 is O = 0.99201. At the sametime we also note that the overlap factor between thenumerical relativity waveform and the e0 = 0 SEOB-NRE waveform is 0.990. Our second comparison for nu-merical relativity simulation result is SXS:BBH:0106 [76].Both the numerical relativity simulations result and theSEOBNRE simulation admit exactly the same eccentric-ity parameters as the first comparison. The only dif-ference to the first comparison is the mass ratio for thebinary black hole which is 5 : 1 here. The comparisonis shown in Fig. 13. The overlap factor between the twowaveforms shown in Fig. 13 is O = 0.99739. In contrast,the overlapping factor between the numerical relativitywaveform and the e0 = 0 SEOBNRE waveform is 0.989.Interestingly we find that the consistence between numer-ical relativity result and SEOBNRE result is even better

than the first one. The SEOBNRE result can recover thenumerical relativity result for the whole inspiral-merger-ringdown process. We can understand this result as fol-lowing. As shown by Peters through post-Newtonian ap-proximation in [19], the decay of eccentricity and thelifetime for the binary can be estimated

de

dt= −304

15

M3η

a4(1− e2)5/2e(1 +

121

304e2), (73)

T (a0, e0) =768

425

5a40256M3η

(1− e20)7/2, (74)

where a means the separation (semimajor axis) of thebinary, and the subindex 0 means the initial quantities.Definitely this estimation can not be correct for the lateinspiral and merger stages considered in current paper.But this estimation can give us a qualitative picture.Compare the Fig. 12 and the Fig. 13, we can see thelifetime for mass ratio 5 : 1 binary is much shorter thanthe one for equal mass case. According to the above life-time estimation, we can deduce that the initial separationfor mass ratio 5 : 1 binary is shorter than that of equalmass one. The initial separations used in the numerical

14

4e02

2e02

0 0 5000 10000

phas

e of

h22

t [M]

e0=0e0=0.3

-20

-16

-12

-8

-4

0

0 5000 10000

∆(p

hase

of h

22)

t [M]

5e-1

3e-1

1e-1

|h22

R/M

|e0=0

e0=0.3

-0.03

-0.01

0.01

0.03

∆|h

22 R

/M|

FIG. 8: Amplitude and phase comparison for h22 corresponding to the case shown in Fig. 7. The result of e0 = 0 is got throughSEOBNRv1 code, and the e0 = 0.3 result is got by SEOBNRE model.

relativity simulations are 19 for equal mass binary and14 for mass ratio 5 : 1 binary respectively. Althoughthese separation values are gauge dependent, they showconsistence to the PN prediction. Then note that theeccentricity decay is proportional to the fourth power ofa, we can expect that the eccentricity decay involved inFig. 13 case is much faster than that in Fig. 12. Fastereccentricity decay results in smaller eccentricity duringthe later process. So our small eccentricity assumptionworks better.

Our third comparison investigates a larger eccentric-ity case for equal mass binary. On the numerical rel-ativity simulation side, the initial eccentricity is e0 =0.1935665 starting evolve at orbital frequency Mf0 =0.0146842176288. The eccentricity is about one orderlarger than the above two cases. The simulation datacorresponds to SXS:BBH:0323 [76]. The mass ratio ofthe two black holes in this simulation is 11:9. And thedimensionless spins for the big and small black hole are0.33 and −0.44 respectively. On the SEOBNRE side, theinitial eccentricity is e0 = 0.3 starting evolve at orbitalfrequencyMf0 = 0.001477647. The comparison is shown

in Fig. 14. During the inspiral stage, we can see the wave-form amplitude and phase are roughly consistent betweenthe numerical relativity result and the SEOBNRE result.The overlap factor between the two waveforms shown inFig. 14 is O = 0.98171. In contrast, the overlap factor be-tween the numerical relativity waveform and the e0 = 0SEOBNRE waveform is 0.849. Due to the larger eccen-tricity, the consistence is not as good as e0 ≈ 0.02 casesshown in Fig. 12 and Fig. 13. But we can note that theconsistence is quite good for merger and ringdown stage.We suspect this is because the eccentricity has decayedquite amount, so the consistence improves during thesestages.

D. Comparison to Teukolsky equation results forextreme mass ratio binary

The largest mass ratio of binary black holes investi-gated by numerical relativity is 100 : 1 [15]. But the sim-ulation only lasts two orbits which is too short to be usedfor gravitational waveform analysis. Till now the mass

15

-0.3

-0.1

0.1

0.3

-8000 -4000 0

Im(h

22 R

/M)

t[M]

-0.5

-0.3

-0.1

0.1

0.3R

e(h 2

2 R

/M)

ax modelSEOBNRE

FIG. 9: Comparison between ax model and SEOBNRE model for e0 = 0 case.

ratio of binary black hole investigated by numerical rela-tivity for gravitational waveform usage is less than 20 : 1[9–12]. This limitation of numerical relativity is due tothe computational cost for finite difference code, and/ordue to the complicated computational grid adjustmentfor spectral code. In the future the finite element codemay do some help on this problem [78–80]. But it is stillunder development. In contrast, our SEOBNRE modelis free of this kind of limitation. This is similar to allother EOBNR models.

When the mass ratio becomes quite large, the smallblack hole can be looked as a perturbation source respectto the spacetime of the large black hole. Consequentlythe Teukolsky formalism is reasonable to treat this kindof binary problem. In [39] we constructed such a modelto investigate extreme mass ratio binary system. In [38],one of us applied such model to investigate the eccentricbinary black holes. It is interesting to compare the resultsgot in [38] and the ones simulated with the SEOBNREmodel proposed here.

In all we have tested four cases. All of them are binaryblack holes with mass ratio about 1000 : 1. More accu-rately the symmetric mass ratio is η = 10−3. The dynam-

ical variables involved in the Teukolsky model [38, 39] arethe same to the ones in SEOBNRE model. So for thecomparison in this subsection, we set the initial data forSEOBNRE model exactly the same to the ones for theTeukolsky model. More concretely, within spherical co-ordinate we set r0 = p0

1+e0with p0 = 12, e0 = 0.3, φ0 = 0,

pr0 = 0 and pφ0similar to the Fig. 4 of [38].

Not like the above test cases which involve slowly spin-ning black holes considered in previous subsections, thefour test cases here admit high-spin black holes. We setthe big black hole spinning while leave the small blackhole spinless. The spin parameters χ for the four testcases are 0.9, 0.5, −0.5 and −0.9 respectively. Here thenegative value means the spin direction is anti-paralleledto the direction of the orbital angular momentum of thebinary. And the parameters pφ0

for the initial data are3.75295952324398, 3.86330280736881, 4.19862434393390and 4.35790829850906 respectively. Different to theFig. 4 of [38], here we have fixed p0 = 12 while variedpφ0

for corresponding χ. This setting makes us easier tocheck the effect of χ on the gravitational waveform.

The gravitational waveform description adopted in [38]used h+,× (note the y-axis label typos involved in the

16

-0.3

-0.1

0.1

0.3

-8000 -4000 0

Im(h

22 R

/M)

t[M]

-0.3

-0.1

0.1

0.3

Re(

h 22

R/M

)ax model

SEOBNRE

FIG. 10: Comparison between ax model (e0 = 0.1, x0 = 0.05) and SEOBNRE model (e0 = 0.15, Mf0 ≈ 0.001477647).

Fig. 4 of [38]). In order to make the comparison easierbetween the results in current paper and the ones in [38],we also adopt h+,× to describe the gravitational wave-form in this subsection. This is different to the sphericalharmonic modes description used in previous subsections.Here we ignore the higher than 22 spherical harmonicmodes and relate the h+,× to h22 through

h+ − ih× = h22−2Y22 + h∗22

−2Y2−2, (75)

where we have used the relation h2−2 = h∗22 with upperstar denotes the complex conjugate [58]. Then h+,× arefunctions of direction angles. Following the Fig. 4 of [38],we plot h+,×(

π2 , 0) in Fig. 15. Overall we can see that the

consistence between the results of Teukolsky model andthe ones of SEOBNRE model is good. When the com-parison time becomes longer, the phase difference showsup. In Fig. 16 we compare the phase of h+ − ih× corre-sponding to the cases shown in Fig. 15.As mentioned above, the four cases admit the same ini-

tial separation parameter p0 = 12. Based on Newtoniangravity, the same mass ratio, the same initial separationand the same eccentric setting may reduce roughly thesame orbital angular frequency. But we can see the larger

χ ones admit a little bit faster frequency of gravitationalwaveform, while correspondingly the smaller orbital an-gular momentum pφ0

(3.75 < 3.86 < 4.20 < 4.36). Weattribute this to the faster frame dragging effect of thebig black hole. In order to stay at the same radius posi-tion, corotating objects need smaller orbital angular mo-mentum, but the anti-rotating objects need larger or-bital angular momentum. And the frame dragging effectmakes the gravitational wave frequency bigger for coro-tating case while smaller for anti-rotating case.

The overlap factors between the two respective wave-forms for the cases shown in Fig. 15 are O = 0.985367,0.985209, 0.986240, and 0.985558 for χ = 0.9, 0.5,−0.5 and −0.9 respectively. When the comparison timeincreases to 20000M, the overlap factors decrease to0.690179, 0.672447, 0.549308, and 0.498645 respectively.One caution is in order here. Neither the SEOBNREmodel nor the Teukolsky model is guaranteed to be ac-curate at 1 post-adiabatic order. So the good overlap-ping does not mean either model is accurate enough forgravitational wave detection usage. Since the two mod-els make different approximations and therefore intro-duce different errors, the good overlapping does imply

17

-0.3

-0.1

0.1

0.3

-5500 -4500 -3500 -2500 -1500 -500

Im(h

22 R

/M)

t[M]

-0.5

-0.3

-0.1

0.1

0.3R

e(h 2

2 R

/M)

ax modelSEOBNRE

FIG. 11: Comparison between ax model (e0 = 0.3, x0 = 0.05) and SEOBNRE model (e0 = 0.4, Mf0 ≈ 0.001477647).

those differences are ignorable for the compared cases.Only when the 1 post-adiabatic results are available, ourSEOBNRE model can be checked more quantitatively forextreme mass ratio systems.

IV. DISCUSSION AND CONCLUSION

EOBNR model has contributed much to the gravita-tional wave detection. But the existing EOBNR modelsare limited to quasi-circular (e = 0) systems. With nodoubt, EOBNR model will continue to play an importantrole in the following LIGO observations. After about 20years, space-based detectors will begin to work. It isinteresting and important to ask whether the EOBNRmodel can still play an important role in the space-basedgravitational wave detection. Among the gravitationalwave sources for space-based detectors, binary systemsare important. And many of such binary systems ad-mit eccentric orbital motion [81]. Partial reason for thisfact is that the decay rate of the eccentricity is propor-tional to the symmetric mass ratio. So although it maybe not all of the issues limiting EOBNR model to work

for space-based detectors, the eccentric orbit problem isan important point blocking EOBNR model to work forspace-based detectors. So it is quite important to extendEOBNR model to describe eccentric binary systems. Weproposed the first such extending model in the currentpaper–SEOBNRE model.

Our idea for constructing SEOBNRE model is com-bining the existing excellent property of EOBNR mod-els for quasi-circular binary and the corrections comingfrom the eccentric orbit motion. The strategy is ex-panding involved quantities respect to eccentricity byassuming the smallness of the eccentricity. Then wecalculate the correction terms coming from eccentricitythrough post-Newtonian approximation. Although thepost-Newtonian order of the correction terms we got isonly to second order, we expect that such kind of correc-tion terms may work well. The reason is because whenthe eccentricity is large the separation of the binary isalso large, then relative low order post-Newtonian ap-proximation is necessary. Along with the decreasing ofthe separation of the binary, the eccentricity also de-creases due to the circularizing effect of the gravitationalradiation. Consequently the correction terms contribute

18

-0.2

0

0.2

-12000 -8000 -4000 0

Im(h

22 R

/M)

t[M]

-0.4

-0.2

0

0.2

Re(

h 22

R/M

)SXS:BBH:0091

SEOBNRE

FIG. 12: SXS:BBH:0091 one is the SpEC simulation result for equal mass spinless binary black hole with e0 = 0.02181 atorbital frequency 0.0105565727235. SEOBNRE one corresponds to e0 = 0.1 at Mf0 ≈ 0.001477647 for equal mass spinlessbinary black hole.

weaker and weaker. In contrast, although the high PN or-der approximation is needed due to the decreasing of theseparation, the existing excellent property of the EOBNRmodel can do the job.

Our SEOBNRE model includes Hamiltonian(Eq. (17)), waveform expression (Eqs. (52) and (68)) andthe related energy flux (Eq. (60)). We have comparedour SEOBNRE model against to quasi-circular EOBNRmodel for consistency check. For e = 0 the SEOBNREmodel can recover the existing EOBNR models well.When the eccentricity e increases, the difference betweenthe SEOBNRE model and quasi-circular EOBNR modelgrows. We have introduced an overlap factor as definedin (71) to quantify this difference. When e < 0.03 thedifference is small. The corresponding overlap factoris larger than 0.99. When e > 0.1 the overlap factorbecomes very low. As an example, the overlap factor fore = 0.3 becomes 0.47. This result added evidences tothe literature [33] that quasi-circular template will breakdown when the eccentricity becomes larger than 0.1.

We have also compared SEOBNRE model against to

another eccentric binary waveform model–ax model [34].When e < 0.15 the overlap factor between the ax modeland the SEOBNRE model is bigger than 0.9. Thisimplies the consistence between the ax model and theSEOBNE model. As an example of e > 0.2 cases, theoverlap factor for e = 0.3 between the ax model and theSEOBNRE model is as low as 0.67. This cautions usthat more investigations are needed for waveform modelabout highly eccentric binary.

Numerical relativity (NR) simulation results can belooked as the standard answer for the eccentric binary.We have tested three numerical relativity simulation re-sults. These three cases include spinless binary and spin-ning binary; equal mass binary and unequal mass binary.Compared to NR simulations with eccentricity 0.02, 0.02and 0.19, the overlap factor for SEOBNRE model is0.992, 0.997 and 0.982 respectively. In contrast, the over-lap factor between NR waveform and e = 0 SEOBNREwaveform is 0.990, 0.989 and 0.849 respectively.

Motivated by the gravitational wave sources for space-based detector, we have applied the SEOBNRE model

19

-0.2

0

0.2

-5000 -3000 -1000

Im(h

22 R

/M)

t[M]

-200 0 200

t[M]

-0.2

0

0.2R

e(h 2

2 R

/M)

SXS:BBH:0106SEOBNRE

FIG. 13: SXS:BBH:0106 one is the SpEC simulation result for two spinless black holes with mass ratio 1 : 5 start orbit withe0 = 0.02181 at orbital frequency 0.0105565727235. SEOBNRE one corresponds to e0 = 0.1 at Mf0 ≈ 0.001477647 for twospinless black holes with mass ratio 1 : 5.

to extreme mass ratio binaries. Specifically we consid-ered binaries with mass ratio 1 to 1000. Since numericalrelativity is not available yet for this kind of binaries,we compared the SEOBNRE model against to Teukolskyequation based model. Both spin aligned cases and anti-aligned cases are considered. All cases admit high eccen-tricity e = 0.3. If we only care about time lasting severalthousands M the overlap factor between the SEOBNREmodel and the Teukolsky equation based model is than0.9. For total mass M = 106 solar mass binaries severalthousandsM corresponds to the time of hours. If we con-sider lasting time with tens of thousands M , the overlapfactor drops below 0.7. Although the Teukolsky equa-tion based model does not represent the standard answeras numerical relativity, this result also reminds us morework is needed when long duration time is involved.

In the current paper, simple overlap factor is consid-ered to quantify the accuracy of the SEOBNRE model.For realistic gravitational wave detection, much more de-tail accuracy requirement [82–86] is needed. It is inter-esting to ask if our SEOBNRE model is ready or not for

particular detection projects such like eLISA [24], LISA[26], Taiji [27] and Tianqin [28]. We leave such investiga-tion as future works. On the other hand, there at leasttwo possible clues to improve SEOBNRE model. Thefirst one is calculating higher PN order terms for the ec-centric correction. The second one is the trick adoptedby Pan and his coworkers when they developed EOBNRmodel [7]. The trick is adding some tuning parametersinto the SEOBNRE model and then require these pa-rameters to fit the calibration waveform such as numeri-cal relativity simulation results. Noting that we did notput in any tunable parameters in the eccentric correctionterms, there are two strategies to apply the mentionedtrick. The first one is adjusting the existing parametersintroduced in the original EOBNR models. The secondone is adding more parameters into the eccentric correc-tion terms and adjusting them.

Regarding to the merger and ringdown parts waveform(68), the possible improvement is taking the effect of theeccentricity on the mass and the spin of the final Kerrblack hole into consideration. This point needs many

20

-0.4

-0.2

0

0.2

0.4

-4000 -2000

Im(h

22 R

/M)

t[M]

-200 0

t[M]

-0.4

-0.2

0

0.2

0.4R

e(h 2

2 R

/M)

SXS:BBH:0323SEOBNRE

FIG. 14: SXS:BBH:0323 one is the SpEC simulation result for two spinless black holes with equal mass start orbit withe0 = 0.1935665 at orbital frequency 0.0146842176288. SEOBNRE one corresponds to e0 = 0.3 at Mf0 ≈ 0.001477647 fortwo black holes with mass ratio 11:9 and spin χ1 = 0.33 and χ2 = −0.44 respectively which corresponds to the setting ofSXS:BBH:0323.

more numerical relativity simulations to extend the re-lations (69) and (70) to include eccentricity. We leavethese investigations to future study.

Acknowledgments

This work was supported by the NSFC (No. 11690023,No. 11622546, No. 11375260 and No. U1431120). Z Cao

was also supported by “the Fundamental Research Fundsfor the Central Universities”. W Han was also supportedby Youth Innovation Promotion Association CAS andQYZDB-SSW-SYS016 of CAS.

Appendix A: Detail expressions for the SEOBNRE Hamiltoanian

The SEOBNR Hamiltonian develops gradually, so the overall expressions are spreaded in the literature. In thisappendix section, we collect all the results for SEOBNR Hamiltonian together. Our major references are [51, 56].The involved terms corresponding to Eq. (18) can be expressed as following.

HNS

Mη= βipi + α

√

1 + γijpipj +Q4(p), (A1)

21

-0.1

0

0.1h +

/ηTeukolsky

SEOBNRE

-0.1

0

0.1

h x/η

-0.1

0

0.1

h +/η

-0.1

0

0.1

h x/η

-0.1

0

0.1

h +/η

-0.1

0

0.1

h x/η

-0.1

0

0.1

0 500 1000 1500 2000

h +/η

t[M]

-0.1

0

0.1

0 500 1000 1500 2000

h x/η

t[M]

FIG. 15: Comparison between Teukolsky equation based model (marked with ‘Teukolsky’) and SEOBNRE model. All caseshere admit mass ratio about 1 : 1000, initial eccentricity e0 = 0.3, and the initial separation parameter p0 = 12. From top rowto bottom row, the corresponding initial orbital angular momentum are χ = 0.9, pφ0

= 3.75; χ = 0.5, pφ0= 3.86; χ = −0.5,

pφ0= 4.20 and χ = −0.9, pφ0

= 4.36 respectively.

HS

Mη= ωSa + ωr

e−3µ−ν√∆r

2B(1 +√Q)

√Qξ2

[e2(µ+ν)p2ξSvr2 −Beµ+νpvpξSξr

2

+B2ξ2(e2µ(√

Q+Q)Sv + pnpvrSn

√

∆r − p2nSv∆r)]

+ ωcos θe−3µ−ν

√∆r

2B(1 +√Q)

√Q[Sn(−e2(µ+ν)p2ξr

2 +B2(p2vr2 − e2µ(

√

Q+Q)ξ2))−Bpn(BpvSv − eµ+νpξSξ)r√

∆r]

+e−µ+2ν

B2√Qξ2

(−B + eµ+ν)pξSar

+e−2µ+ν

B2(√Q+Q)ξ2

−Beµ+ννcos θpξr(1 + 2√

Q)Snξ2

+√

∆r[B(eµ+ννrpξr(1 + 2√

Q)Sv +BµrpvrSξ +BSξ(−µcos θpnξ2 +

√

Q(µrpvr − νrpvr + (νcos θ − µcos θ)pnξ2)))

−Breµ+νpξr(1 +

√

Q)Sv], (A2)

HSC = − η

2r3(S2

∗ − 3S2n) + dheffSS

(Mη)2

r4aSa, (A3)

α =1

√

−gtt, βi =

gti

gtt, γij = gij − gtigtj

gtt, i, j = (r, θ, φ) (A4)

22

0

500Teukolsky

SEOBNRE

0

500

0

500

0

500

0 5000 10000 15000 20000

t[M]

Pha

se o

f (h +

-ihx)

FIG. 16: Phase comparison of h+ − ih× between Teukolsky equation based model and SEOBNRE model. Cases correspond tothose of Fig. 15 respectively.

Q4(p) = 2η(4− 3η)p4nr2, Q = 1 +

p2vr2

Σξ2+

∆tΣ

Λt

p2ξr2

B2ξ2+ p2n

∆r

Σ, (A5)

S∗ = σ∗ +∆(1)σ∗

+∆(2)σ∗

+∆(3)σ∗

, σ∗ = a1m2 + a2m1, (A6)

∆(1)σ∗

=η

12rσ[3(Q− 1)r − 8− 36p2nr] + σ∗[4(Q− 1)r + 14− 30p2nr], (A7)

∆(2)σ∗

=σ

144r24(51η2 − 109η)(Q− 1)r − 16[7η(8 + 3η)] + 810η2p4nr

2 − 45η(Q− 1)2r2

− 6p2nr(16η + 147η2 + (39η2 − 6η)(Q− 1)r)

− σ∗

72r22η(27η − 353) + 2(103η − 60η2)(Q− 1)r − 360η2p4nr

2 + (23 + 3η)η(Q− 1)2r2

+ 6p2nr[54η2 − 47η + (21η2 − 16η)(Q− 1)r], (A8)

∆(3)σ∗

=dSOη

r3σ∗ (A9)

where S∗ is the spin of the test particle deduced in the effective-one-body reduction. Following SEOBNRv1 we setdSO = −69.5, dheffSS = 2.75. Our setting exactly follows the SEOBNRv1 code [63]. In the above equations we haveused notations

~n ≡ ~r

r, ~ξ ≡ ~σ

σ× ~n,~v ≡ ~n× ~ξ, (A10)

pn ≡ ~p · ~n, pξ ≡ ~p · ~ξ, pv ≡ ~p · ~v, (A11)

23

Sn ≡ ~S∗ · ~n, Sξ ≡ ~S∗ · ~ξ, Sv ≡ ~S∗ · ~v, (A12)

Sa ≡ ~S∗ ·~a

a, pn ≡ √

grrpn, pn ≡ ~n · ~p (A13)

ν =1

2log(

∆tΣ

Λt), µ =

1

2log(Σ), (A14)

ω ≡ ωfd

Λt, B ≡

√

∆t, (A15)

νr ≡ r

Σ+ω2(ω2∆′

t − 4r∆t)

2Λt∆t, µr ≡ r

Σ− 1√

∆r

, (A16)

ωr ≡ω′fdΛt − ωfdΛ

′t

Λ2t

, Br ≡√∆r∆

′t − 2∆t

2√∆t∆r

, (A17)

νcos θ ≡ a2ω2 cos θ(ω2 −∆t)

ΛtΣ, µcos θ ≡ a2 cos θ

Σ, (A18)

ωcos θ ≡ −2a2 cos θ∆tωfd

Λ2t

(A19)

where the prime denotes the derivatives with respect to r. (~r, ~p) are the canonical variable within Boyer-Lindquist

coordinate of the geodesic motion, while ~p is the momentum vector within tortoise coordinate. ~p and ~p are relatedthrough [87]

~p = ~p− ~n(~n · ~p)ξa − 1

ξa, (A20)

ξa ≡√∆t∆r

r2 + a2. (A21)

Within spherical coordinate ~r = (r, θ, φ) and we set ~a along θ = 0 direction, so ξ = sin θ in (A10).

Appendix B: Detail expressions for the eccentric part of SEOBNRE waveform

In this appendix section we show the detail calculation for the eccentric part of SEOBNRE waveform. We beginwith the notations introduced in Eq. (50) as following.

Θij =

∫

(ǫ+ij − iǫ×ij)−2Y ∗

22dΩ, (B1)

PnΘij =

∫

Nn(ǫ+ij − iǫ×ij)

−2Y ∗

22dΩ, (B2)

PvΘij =

∫

Nv(ǫ+ij − iǫ×ij)

−2Y ∗

22dΩ, (B3)

PnnΘij =

∫

N2n(ǫ

+ij − iǫ×ij)

−2Y ∗

22dΩ, (B4)

PnvΘij =

∫

NnNv(ǫ+ij − iǫ×ij)

−2Y ∗

22dΩ, (B5)

PvvΘij =

∫

N2v (ǫ

+ij − iǫ×ij)

−2Y ∗

22dΩ, (B6)

PnnnΘij =

∫

N3n(ǫ

+ij − iǫ×ij)

−2Y ∗

22dΩ, (B7)

PnnvΘij =

∫

N2nNv(ǫ

+ij − iǫ×ij)

−2Y ∗

22dΩ, (B8)

PnvvΘij =

∫

NnN2v (ǫ

+ij − iǫ×ij)

−2Y ∗

22dΩ, (B9)

PvvvΘij =

∫

N3v (ǫ

+ij − iǫ×ij)

−2Y ∗

22dΩ. (B10)

24

Based on Eqs. (35)-(49), direct calculation gives

Θij = (1

3

√

π

5,− i

3

√

π

5, 0,−1

3

√

π

5,

8

3√5π, 0), (B11)

PnΘij = (0, 0,−√

π5 (x1 − ix2)

3√

x21 + x22 + x23, 0,

√

π5 (ix1 + x2)

3√

x21 + x22 + x23, 0), (B12)

PvΘij = (0, 0,−1

3

√

π

5(v1 − iv2), 0,

1

3

√

π

5(iv1 + v2), 0), (B13)

PnnΘij = (

√

π5

(

−x21 + 8ix1x2 + 7x22 + x23)

21 (x21 + x22 + x23),−

i√

π5

(

3x21 + 3x22 + x23)

21 (x21 + x22 + x23),2√

π5 (x1 − ix2)x3

21 (x21 + x22 + x23),

√

π5

(

−7x21 + 8ix1x2 + x22 − x23)

21 (x21 + x22 + x23),−

2i√

π5 (x1 − ix2)x3

21 (x21 + x22 + x23),8√

π5 (x1 − ix2)

2

21 (x21 + x22 + x23)), (B14)

PnvΘij = (

√

π5 (−v1x1 + 4iv2x1 + 4iv1x2 + 7v2x2 + v3x3)

21√

x21 + x22 + x23,−

i√

π5 (3v1x1 + 3v2x2 + v3x3)

21√

x21 + x22 + x23,

√

π5 (v3(x1 − ix2) + (v1 − iv2)x3)

21√

x21 + x22 + x23,

√

π5 (−7v1x1 + 4iv2x1 + 4iv1x2 + v2x2 − v3x3)

21√

x21 + x22 + x23,

−√

π5 (v3(ix1 + x2) + (iv1 + v2)x3)

21√

x21 + x22 + x23,8√

π5 (v1 − iv2)(x1 − ix2)

21√

x21 + x22 + x23), (B15)

PvvΘij = (1

21

√

π

5

(

−v21 + 8iv1v2 + 7v22 + v23)

,− 1

21i

√

π

5

(

3v21 + 3v22 + v23)

,

2

21

√

π

5(v1 − iv2)v3,

1

21

√

π

5

(

−7v21 + 8iv1v2 + v22 − v23)

,

− 2

21i

√

π

5(v1 − iv2)v3,

8

21

√

π

5(v1 − iv2)

2), (B16)

PnnnΘij = (

√

π5 (x1 − ix2)

2x3

7 (x21 + x22 + x23)3/2

, 0,

−√

π5 (x1 − ix2)

(

2x21 + 5ix1x2 + 7x22 + 3x23)

21 (x21 + x22 + x23)3/2

,

√

π5 (x1 − ix2)

2x3

7 (x21 + x22 + x23)3/2

,

√

π5 (ix1 + x2)

(

7x21 − 5ix1x2 + 2x22 + 3x23)

21 (x21 + x22 + x23)3/2

,−2√

π5 (x1 − ix2)

2x3

7 (x21 + x22 + x23)3/2

), (B17)

PnnvΘij = (

√

π5 (x1 − ix2)(v3(x1 − ix2) + 2(v1 − iv2)x3)

21 (x21 + x22 + x23), 0,

−√

π5

(

(x1 − ix2)(2v1x1 + iv2x1 + 4iv1x2 + 7v2x2) + 2v3(x1 − ix2)x3 + (v1 − iv2)x23

)

21 (x21 + x22 + x23),

√

π5 (x1 − ix2)(v3(x1 − ix2) + 2(v1 − iv2)x3)

21 (x21 + x22 + x23),

√

π5

(

(x1 − ix2)(7iv1x1 + 4v2x1 + v1x2 + 2iv2x2) + 2v3(ix1 + x2)x3 + (iv1 + v2)x23

)

21 (x21 + x22 + x23),

−2√

π5 (x1 − ix2)(v3(x1 − ix2) + 2(v1 − iv2)x3)

21 (x21 + x22 + x23)), (B18)

PnvvΘij = (

√

π5 (v1 − iv2)(2v3(x1 − ix2) + (v1 − iv2)x3)

21√

x21 + x22 + x23, 0,

−√

π5

(

v23(x1 − ix2) + v21(2x1 + ix2) + v22(4x1 − 7ix2)− 2iv2v3x3 + 2v1(iv2x1 + 4v2x2 + v3x3))

21√

x21 + x22 + x23,

25

√

π5 (v1 − iv2)(2v3(x1 − ix2) + (v1 − iv2)x3)

21√

x21 + x22 + x23,

√

π5

(

v23(ix1 + x2) + v22(−ix1 + 2x2) + v21(7ix1 + 4x2) + 2v2v3x3 + v1(8v2x1 − 2iv2x2 + 2iv3x3))

21√

x21 + x22 + x23,

−2√

π5 (v1 − iv2)(2v3(x1 − ix2) + (v1 − iv2)x3)

21√

x21 + x22 + x23), (B19)

PvvvΘij = (1

7

√

π

5(v1 − iv2)

2v3, 0,

− 1

21

√

π

5(v1 − iv2)

(

2v21 + 5iv1v2 + 7v22 + 3v23)

,1

7

√

π

5(v1 − iv2)

2v3,

1

21

√

π

5(iv1 + v2)

(

7v21 − 5iv1v2 + 2v22 + 3v23)

,−2

7

√

π

5(v1 − iv2)

2v3) (B20)

In the above equations, we have listed by the components order 11, 12, 13, 22, 23, and 33 within Cartesian coordinate.The position and velocity components mean ~r = (x1, x2, x3), ~vp = (v1, v2, v3). And more we have

P1

2

n Qij =

3(m1 −m2)

r(nivjp + vipn

j − rninj), (B21)

P1

2

v Qij = (m1 −m2)(

ninj

r− 2vipv

jp), (B22)

P0Qij =

1

3[3(1− 3η)v2p − 2

(2− 3η)

r]vipv

jp +

2

rr(5 + 3η)(nivjp + vipn

j) + [3(1− 3η)r2 − (10 + 3η)v2p +29

r]ninj

r,

(B23)

PnnQij =

1− 3η

3r[(3v2p − 15r2 +

7

r)ninj + 15r(nivjp + vipn

j)− 14vipvjp], (B24)

PnvQij =

1− 3η

3r[12rninj − 16(nivjp + vipn

j)], (B25)

PvvQij =

1− 3η

3(6vipv

jp −

2

rninj), (B26)

P3

2

n Qij =

(m1 −m2)

12r(nivjp + vipn

j)[r2(63 + 54η)− 128− 36η

r+ v2p(33− 18η)]

+ ninj r[r2(15− 90η)− v2p(63− 54η) +242− 24η

r]− rvipv

jp(186 + 24η), (B27)

P3

2

v Qij = (m1 −m2)

1

2vipv

jp[3− 8η

r− 2v2p(1− 5η)]−

nivj + vipnj

2rr(7 + 4η)

− ninj

r[3

4(1− 2η)r2 +

1

3

26− 3η

r− 1

4(7− 2η)v2p], (B28)

P3

2

nnnQij =

(m1 −m2)(1 − 2η)

r[5

4(3v2p − 7r2 +

6

r)rninj − 17

2rvipv

jp − (21v2p − 105r2 +

44

r)nivjp + vipn

j

12], (B29)

P3

2

nnvQij =

(m1 −m2)(1 − 2η)

4r[58vipv

jp + (45r2 − 9v2p −

28

r)ninj − 54r(nivjp + vipn

j)], (B30)

P3

2

nvvQij =

3(m1 −m2)(1 − 2η)

2r(5(nivjp + vipn

j)− 3rninj), (B31)

P3

2

vvvQij =

(m1 −m2)(1 − 2η)

2(ninj

r− 4vipv

jp). (B32)

Appendix C: Post-Newtonian energy flux for eccentric binary

In this appendix section, we show the Post-Newtonian energy flux for eccentric binary. Following [34] for the circularpart we can combine the results involved in SEOBNRv1 [14, 34, 88] to get

−dEdt

|(C) =32

5η2x5[1− 1247 + 980η

336x+ 4πx3/2 + (−44711

9072+

9271η

504+

65η2

18)x2 − (

8191

672+

583η

24)πx5/2

26

+ (6643739519

69854400− 1712γE

105+

16π2

3− 134543η

7776+

41π2η

48− 94403η2

3024− 775η3

324− 856 log(16)

105− 856 log(x)

105)x3

+ (−16285

504+

214745η

1728+

27755η2

432)πx7/2

+ (−23971119313

93139200+

856γE35

− 8π2 − 59292668653η

838252800+ 5a0η +

856γη

315+

31495π2η

8064− 54732199η2

93312+

3157π2η2

144

+18929389η3

435456+

97η4

3888+

428 log(16)

35+

428

315η log(16) +

428 log(x)

35+

47468

315η log(x))x4

+ (−80213

768+

51438847η

48384− 205π2η

6− 42745411η2

145152− 4199η3

576)πx9/2

+ (−121423329103

82790400+

5778γE35

− 54π2 +4820443583363η

1257379200− 3715a0η

336+ 6a1η −

4066γEη

35− 31869941π2η

435456

− 2006716046219η2

3353011200− 55a0η

2

4+

214γEη2

105+

406321π2η2

48384+

2683003625η3

3359232− 100819π2η3

3456− 192478799η4

5225472

+33925η5

186624+

2889 log(16)

35− 2033

35η log(16) +

107

105η2 log(16) +

2889 log(x)

35− 391669

315η log(x)

− 122981

105η2 log(x))x5

+ (−623565

1792− 235274549η

241920+ 20a0η +

852595π2η

16128− 187219705η2

32256+

12915π2η2

64+

503913815η3

870912

− 24065η4

3456+

1792

3η log(x))πx11/2

+ (−1216355221

206976+

4815γE7

− 225π2 +45811843687349η

1149603840+

170515a0η

18144− 743a1η

56+ 7a2η + a3η

− 737123γEη

189− 84643435883π2η

670602240+

8774

63γEπ

2η − 410π4η

9− 37516325949517η2

603542016+

68305a0η2

2016− 33a1η

2

2

+6634γEη

2

63+

23084972185π2η2

5225472− 92455π4η2

1152+

6069288163291η3

2586608640+

295a0η3

18+

107γEη3

243− 114930545π2η3

1741824

− 145089945295η4

282175488+

141655π2η4

7776+

6942085η5

497664+

196175η6

3359232+

4815 log(16)

14− 737123

378η log(16)

+4387

63π2η log(16) +

3317

63η2 log(16) +

107

486η3 log(16) +

4815 log(x)

14+

13185899η log(x)

59535+ 7a3η log(x)

+4387

63π2η log(x) +

963937

189η2 log(x) +

6279367η3 log(x)

2430)x6

+ (−2χS − 3

4

√

1− 4ηχA)x3/2

+ [(−9

4+

136

9η)χS + (−23

16+

157

36η)√

1− 4ηχA]x5/2

+ (−8πχS − 17

6π√

1− 4ηχA)x3

+ ((476645

13608+

3086

189η − 1405

27η2)χS + (

180955

27216+

625

378η − 1117

108η2)

√

1− 4ηχA)x7/2], (C1)

with a0 = 153.8803, a2 = −55.13, a2 = 588, a3 = −1144. These a values are taken from [34, 88]. Here x ≡ √v. The

PN results for elliptic orbit read [60, 61]

−dEdt

|Elip =8

15

M4η2

r4[12v2 − 11r2

+1

28(−2r2v2(1487− 1392η) + v4(785− 852η) + 3r4(687− 620η) +

8Mr2(367− 15η)

r

+16M2(1− 4η)

r2− 160Mv2(17− η)

r)

+1

756(−24M3(253− 1026η+ 56η2)

r3− 36Mv4(4446− 5237η + 1393η2)

r

27

+108Mr2v2(4987− 8513η + 2165η2)

r+M2v2(281473+ 81828η + 4368η2)

r2+ 18v6(1692− 5497η + 4430η2)

− 3M2r2(106319 + 9798η + 5376η2)

r2− 54r2v4(1719− 10278η + 6292η2) + 54r4v2(2018− 15207η+ 7572η2)

− 18r6(2501− 20234η+ 8404η2)− 12Mr4(33510− 60971η+ 14290η2)

r)]

+8

15

M5η2

r5(n× ~v) · [(~χS +

√

1− 4η~χA)(27r2 − 37v2 − 12

M

r) + 4η~χS(12r

2 − 3v2 + 8M

r)]

+8

15

M6η3

r6[3(χ2

S − χ2A)(47v

2 − 55r2)− 3((n · ~χS)2 + (n · ~χA)

2)(168v2 − 269r2) + 71((~v · ~χS)2 + (~v · ~χA)

2)

− 342r[(~v · ~χS)(n · ~χS)− (~v · ~χA)(n · ~χA)]], (C2)

where the PN order for the non-spin part and spin-spin interaction part is second, but the PN order for the spin-orbitinteraction part is only 1.5.

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