# Phenomenological relationship between eccentric and quasi-circular orbital
binary black hole waveform

Hao Wang<EMAIL_ADDRESS>Department of Astronomy, School of Physics,
Huazhong University of Science and Technology, Wuhan 430074, China  Yuan-Chuan
Zou<EMAIL_ADDRESS>Department of Astronomy, School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China  Yu Liu
<EMAIL_ADDRESS>Department of Astronomy, School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China

###### Abstract

Eccentricity, an important parameter of gravitational waves, has been paid
more and more attention because it can reflect the dynamics of compact object
mergers. Obtaining an accurate and fast gravitational waveform template is of
great significance for the estimation of gravitational wave parameters. This
paper aims to do an extended study of the phenomenological fitting model
proposed by Setyawati and Ohme for adding eccentricity to quasi-circular
orbital waveforms. We extend the range of the waveform for mass ratio to
$[1,7]$, eccentricity to $[0,0.4]$, and time range from a fixed time range
$[-1500M,-29M]$ to any case. We also study higher-order modes, spin-aligned
and spin-precessing waveforms. We discover that, after expanding some fitting
parameters, the model can be applied to mass ratios $q\in[1,7]$ or higher, and
can be applied to almost the entire time range of numerical relativity even to
$t=12000M$ prior to merger. It can be applied to higher eccentricity up to
$e=0.4$. But the higher the eccentricity, the less the accuracy. For
$e\in[0,0.1]$, it gives an overlap of more than 99.99%. For $e\in[0.1,0.2]$,
it gives an overlap of more than 99.9%. For $e\in[0.2,0.3]$, it gives an
overlap of more than 99%. For $e\in[0.3,0.4]$, it gives an overlap of more
than 90%. The reason for these phenomena is that the larger the eccentricity,
the larger the deviation of the eccentricity estimator from the cosine
function due to the large change in the morphology of the eccentric waveform,
and the worse the fitting effect of the model. It can be applied to higher-
order modes and gives the same overlap behavior. After adding a shift
parameter, it can be applied to spin-aligned or spin-antialigned waveforms.
After obtaining spin-precessing effect, it can be applied to the spin-
precessing case. In summary, non-spining, spin-aligned, spin-antialigned or
spin-precessing waveforms with eccentricity can be constructed from quasi-
circular non-spining waveforms by the phenomenological model, which is not
only helpful for us to quickly construct phenomenological gravitational wave
templates, but also reveals a phenomenological and universal relationship
between eccentric waveform and quasi-circular orbital waveform.

††preprint: APS/123-QED

##  I Introduction

Since the first detection of the binary black hole merger event GW150914 in
2015, gravitational wave astronomy has ushered in a new era [2]. So far, 93
gravitational wave events have been detected by ground-based gravitational
wave detectors LIGO [3], Virgo [4] and KAGRA [5] including binary black holes
(BHHs), 2 black hole-neutron stars (BHNS), and 2 binary neutron stars (NSNS)
[6].

Current extraction of gravitational wave signals all use circular orbital
waveforms templates, because the evolution of isolated binary stars is
generally considered to circularize due to gravitational wave radiation, whose
eccentricity can be negligible when entering the gravitational wave detection
frequency band at about 10 Hz [7, 8, 9, 10]. However, there are many ways that
BBHs will gain eccentricity before the merger. In dense regions of stars like
globular clusters [11, 12, 13, 14, 15, 16, 17] and galactic nuclei [18, 19],
BBHs can acquire eccentricity due to double-single [20], double-double
interactions [21], and gravitational capture [18]. Or in a three-body system,
such as binary objects in the neighborhood of a supermassive black hole,
eccentricity of inner binary objects will oscillate due to the Kozai-Lidov
mechanism [22, 23, 24], which will be detected when entering the detection
frequency band. Gravitational waves of BBHs mergers in globular clusters
entering the LIGO sensitive band, 10% of them still maintain an eccentricity
more than 0.1 according to Refs. [15, 16]. GW190521 [25] is considered to be
currently the only BBH merger with high mass and high eccentricity
$e=0.69_{-0.22}^{+0.17}$ through 611 numerical relativity simulations [26].
With the improvement of detector sensitivity, more and more eccentric BBHs
mergers will be detected by next-generation ground-based gravitational-wave
detectors Einstein Telescope(ET) [27] or Cosmic Explorer(CE) [28].

Errors may exist or signal-to-noise ratio will be reduced when using circular
orbital waveforms for parameter estimation [29, 30]. There are some numerical
relativistic simulations of eccentric BBHs mergers [31, 32, 33, 34]. Parameter
estimation of gravitational waves requires millions of waveform templates. In
general, complete numerical relativity (NR) simulation yields the most
accurate gravitational waveform, but each NR simulation takes several weeks
and months and is computationally expensive. Until now several studies
developed many analytical gravitational waveforms based on post-Newtonian (PN)
approximation in Refs. [35, 36, 37, 38, 39, 40, 41, 42, 43, 36, 44, 45, 46,
47, 48, 49] or effective-one-body (EOB) in Refs. [50, 51, 52, 53, 54]. Islam
_et al._ [55] developed a waveform based on the calibration of complete NR.
Some state of art surrogate models of full inspiral-merger-ringdown (IMR)
eccentric gravitational waveform have been developed based on hybrid of PN and
NR [56, 57, 58, 59] or EOBNR [53, 60, 54, 61, 62, 63, 52]. There are also some
special cases such as eccentric extreme-mass-ratio inspirals (EMRI) [64, 65,
66, 67, 68, 69] and gravitational wave bursts with high eccentricity [70]. It
is not so accurate to replace the part of the waveform that is close to the
merger with the circular orbit waveform of NR. But the complete NR BBHs
simulation with eccentricity is rare and not publicly available, which makes
it necessary to develop phenomenological model which can generate fast and
accurate numerically relativistic eccentric gravitational waveforms.

The main purpose of this paper is to do an extended study on the
phenomenological model that converts a circular orbital waveform into an
eccentric orbital waveform proposed by Setyawati and Ohme [1]. The model can
quickly and simply produce a complete numerical relativistic eccentric orbital
waveform based on corresponding circular orbital waveform. In addition to the
waveform from Simulating eXtreme Spacetime(SXS) catalog, we add some waveforms
from the Rochester Institute of Technology (RIT) catalog, making the range of
mass ratio extended to $q\in[1,7]$, and the eccentricity range extended to
$e\in[0,1]$, and the time range of the waveform extended from the fixed time
range $t\in[-1500M,-29M]$ to any time range even at $t=12000M$ before merging.
We extend the model can not only to dominant mode, but to higher order modes
including 3-3, 2-1, 4-4, 5-5, 3-2,4-3 modes. We also apply the model to the
spin-aligned case, but for this case the model needs to be adjusted in some
ways. When we apply this model to the most complex spin precession situation,
it is time to build a more complete model to convert the waveform without spin
and eccentricity into the waveform with eccentricity and spin precession. This
implies a phenomenological and universal relationship between the eccentric
waveform and the circular orbit waveform, which can help us understand the
relationship between eccentricity, spin and precession in BBH mergers.

This article is organized as follows: In Sec. II, we firstly introduce the
numerical relativity waveform data we used and some basic notions about
gravitational waves in Sec. II.1. We make a detailed description to the
eccentricity estimators in Sec II.2. Next, we introduce a method to measure
eccentricity from eccentric waveform in Sec. II.3. Finally, We describe the
fitting process to the dominant mode in Sec. II.4, and the extended research
on higher-order mode in Sec. II.5, spin-aligned in Sec. II.6 and spin-
precessed waveforms in Sec. II.7, respectively. In section III, we give the
fitting results and overlap for every case, and analyze them. In section IV,
we give conclusions and outlook. Throughout this article, we use geometric
units that $G=c=1$. Component masses of BBHs are $m_{1}$ and $m_{2}$. The
total mass is $M$, which we set to unity for simplifying expression. The mass
ratio $q$ is defined as $q=m_{1}/m_{2}$ and $m_{1}>m_{2}$. We denote the black
hole’s dimensionless spin vectors by $\vec{\chi}_{i}=\vec{S}_{i}/m_{i}^{2}$
for $i=1,2$.

##  II Method

After combining some new waveforms , the range of parameters we studied such
as mass ratio, eccentricity, time and spin had been expanded, which allow us
to comprehensively investigate the model and its applicable range. Here, we
want to present enough details in the article, and thus add much content to
present the essence of the model, it is different in some ways with the Ref.
[1].

###  II.1 Numerical relativity data

There are many numerical relativity collaborations that have performed lots of
numerical simulations of BBHs mergers, but there are few publicly available
simulations with eccentricity or high mass ratios. The data we used come from
two collaborations. The first is Simulating eXtreme Spacetimes (SXS)
Collaboration which evolves the initial data using a multi-domain spectral
method [71, 72, 73, 74, 75] with a first-order version of the generalized
harmonic formulation [76, 77, 78, 75] of Einstein’s equations with constraint
damping with Spectral Einstein Code (SpEC) [79]. SXS catalog has published 23
sets of non-spinning eccentric waveforms with mass ratios $q\in[1,3]$ and
eccentricity range $e\in[0,0.2]$. The other set of waveforms we used was from
the Rochester Institute of Technology (RIT) [80]. The simulations in the RIT
Catalog were evolved using the LazEv code [81] implementation of the moving
puncture approach [82] and the BSSNOK formalism of evolution systems [83, 81,
84]. The LazEv code uses the Cactus [85] /Carpet [86] /EinsteinToolkit [87]
infrastructure. The 4th release of RIT catalog published 824 eccentric BBHs NR
simulations including spinning, spin-aligned and spin-precessing cases with
eccentricities from 0 to 1 [34].

Waveforms are obtained by computing the Newman-Penrose scalar $\Psi_{4}$ at a
finite radius and extrapolating to null infinity. $\Psi_{4}$ can be expanded
by the spin-weighted $s=-2$ spherical harmonic function
${}_{-2}Y_{\ell,m}(\theta,\phi)$ as

$r\Psi_{4}=\sum_{\ell,m}{r\Psi_{4}}_{(\ell m)}{}_{-2}Y_{\ell,m}(\theta,\phi),$
(1)

where $r$ is the extraction radius. Gravitational wave strain $h$ and its
relationship with $\Psi_{4}$ can be expressed as

$rh=r\left(h_{+}-ih_{\times}\right)=\sum_{\ell,m}rh_{\ell
m}{}_{-2}Y_{\ell,m}(\theta,\phi)$ (2)

and as r goes to infinity

$\Psi_{4}(t)=\frac{\partial^{2}}{\partial t^{2}}h(t),$ (3)

where $h_{+}$ and $h_{\times}$ represent the two polarizations of
gravitational waves, respectively. We can decompose $\Psi_{4}$ and $h$ into a
combination of amplitude and phase:

$\Psi_{4(lm)}=A_{lm}(t)\exp\left[-i\varphi_{lm}(t)\right]$ (4)
$h_{lm}=\mathcal{A}_{lm}(t)\exp\left[-i\Phi_{lm}(t)\right]$ (5)

and amplitude and phase of $h_{lm}$ can be obtained by

$\mathcal{A}_{lm}=|h_{lm}|$ (6) $\omega_{lm}=\frac{d\Phi_{lm}}{dt}.$ (7)

Define effective spin in the z direction:

$\chi_{\mathrm{eff}}=\frac{m_{1}\chi_{1,z}+m_{2}\chi_{2,z}}{m_{1}+m_{2}},$ (8)

where $\chi_{1,z}$ and $\chi_{2,z}$ are dimensionless spins in the $z$
direction for two black holes. We get the gravitational wave strain $rh$
instead of $r\Psi_{4}$ from the SXS catalog and the RIT catalog, but in fact
the model is applicable for both according to analysis in Sec. II.2. In the
processing of waveforms, for the waveforms of SXS and RIT, we cut off the
first 300 and the first 100 respectively due to junk radiation. Other
processing for waveforms is similar to the Ref. [1]. The waveforms we used and
associated parameters in the RIT and SXS catalogs are given in Appendix A for
TABLE 1 and TABLE 2. Because SXS waveform catalog does not give initial
eccentricity, we use the method in Sec. II.3 to measure the eccentricity of
the waveform at the first 300. For the RIT waveform, we directly use the
reference eccentricity given in the catalog. All waveform parameter
distributions for mass ratio $q$, effective spin $\chi_{eff}$ and initial
eccentricity $e_{0}$ are given in FIG. 1. We only show the range of
eccentricity $e\in[0,0.45]$ here because the model is dependent on the length
of the waveform, so a sufficiently long waveform is required, and many shorter
waveforms are excluded. Relevant analysis is in Sec. II.3.

Figure 1:  Waveform parameter distribution of the data for mass ratio $q$,
effective spin $\chi_{eff}$ and initial eccentricity $e_{0}$ come from SXS and
RIT catalog.

###  II.2 Eccentricity Estimator

It is necessary to introduce the eccentricity estimator in detail. According
to Ref. [88], eccentricity estimator is derived from Newton’s formula
containing distance, orbital phase or orbital frequency that can be used to
estimate eccentricity. In the Ref. [88], an eccentricity estimator is defined
as an oscillatory function

$e_{X}(t)=e\cos\left(\Omega{t}+\phi\right),$ (9)

where $X$ represents the object used to measure eccentricity, and $e$,
$\Omega$ and $\phi$ is eccentricity, frequency and phase respectively. $X$ can
be separation, frequency, amplitude, phase, or derivative of frequency from
orbital dynamics or waveform. Based on separation

$d(t)=d_{0}\left[1+e\cos\left(\Omega
t+\phi_{0}\right)\right]+O\left(e^{2}\right),$ (10)

where $d_{0}$ and $\phi_{0}$ are the average distance and initial phase in
Newtonian orbit. We get separation eccentricity estimator:

$e_{d}(t)=\left(1\right)\left(\frac{d(t)-\bar{d}(t)}{\bar{d}(t)}\right),$ (11)

where $\bar{d}(t)$ represents the secular average to $d(t)$, and $1$ is its
coefficient. Based on orbital phase:

$\Phi(t)=\Phi_{0}+\Omega_{0}t+2e\sin\left(\Omega
t\right)+O\left(e^{2}\right),$ (12)

where $\Phi_{0}$ and $\Omega_{0}$ are phase offset and average frequency. We
get orbital phase eccentricity estimator:

$e_{\Phi}(t)=(1)\left(\Phi(t)-\bar{\Phi}(t)+\Phi_{1}\right),$ (13)

where $\bar{\Phi}(t)$ is the secular average to $\Phi(t)$ and $\Phi_{1}$
represents a phase offset. Taking the first time derivative with respect to
the phase, we get orbital frequency eccentricity estimator:

$e_{\Omega}(t)=\left(\frac{1}{2}\right)\left(\frac{\Omega(t)-\bar{\Omega}(t)}{\bar{\Omega}(t)}\right),$
(14)

where $\bar{\Omega}(t))$ is the secular average to $\Omega(t)$. We can not
only get the eccentricity estimator based on the orbital dynamical quantities,
but we can get it from the waveform. According to Ref. [89], based on the
frequency, amplitude and phase of the Weyl scalar $\Psi_{4}$, we can also
define the associated eccentricity estimators. In the Ref. [89], according to
Eq.(4), Weyl scalar $\Psi_{4}$ of the 2-2 mode and its frequency can be
expressed as

$\Psi_{4(22)}=A_{22}(t)\exp\left[i\varphi_{22}(t)\right]$ (15)
$\varpi_{22}={\mathrm{d}}\varphi_{22}/{\mathrm{d}}t,$ (16)

where

$\displaystyle A_{22}(t)$ $\displaystyle=K_{1}\left(1+\frac{39}{8}e\cos\Omega
t\right)+\mathcal{O}\left(e^{2}\right)$ (17) $\displaystyle\varphi_{22}(t)$
$\displaystyle=-2\Omega t-\frac{21}{4}e\sin\Omega
t+\mathcal{O}\left(e^{2}\right)$ $\displaystyle\varpi_{22}(t)$
$\displaystyle=-2\Omega\left(1+\frac{21}{8}e\cos\Omega
t\right)+\mathcal{O}\left(e^{2}\right),$

which are first-order approximation of the eccentricity, where $K_{1}$ is a
constant. The associated eccentricity estimators are:

$\displaystyle e_{A_{22}}(t)$
$\displaystyle=\left(\frac{8}{39}\right)\left(\frac{A_{22}(t)-\bar{A}_{22}(t)}{A_{22}(t)}\right)$
(18) $\displaystyle e_{\varphi_{22}}(t)$
$\displaystyle=\left(\frac{4}{21}\right)\left[\varphi_{22}(t)-\bar{\varphi}_{22}(t)+\varphi_{220}\right]$
$\displaystyle e_{\varpi_{22}}(t)$
$\displaystyle=\left(\frac{8}{21}\right)\left(\frac{\varpi_{22}(t)-\bar{\varpi}_{22}(t)}{\varpi_{22}(t)}\right),$

where $\varphi_{220}$ represents a phase offset to $\varphi_{22}$. According
to Eq.(5), the 2-2 mode of gravitational waves strain $h$ and its frequency
can be expressed as

$h_{22}=\mathcal{A}_{22}(t)e^{i\Phi_{22}(t)}$ (19)
$\omega_{22}={\mathrm{d}}\Phi_{22}/{\mathrm{d}}t,$ (20)

where

$\displaystyle\mathcal{A}_{22}(t)=K_{2}\left(1+\frac{3}{2}e\cos\Omega
t\right)+\mathcal{O}\left(e^{2}\right)$ (21)
$\displaystyle\Phi_{22}(t)=-2\Omega t-3e\sin\Omega
t+\mathcal{O}\left(e^{2}\right)$
$\displaystyle\omega_{22}(t)=-2\Omega\left(1+\frac{3}{2}e\cos\Omega
t\right)+\mathcal{O}\left(e^{2}\right),$

which are first-order approximation of the eccentricity, where $K_{2}$ is a
constant. The associated eccentricity estimators are:

$\displaystyle e_{\mathcal{A}_{22}}(t)$
$\displaystyle=\left(\frac{2}{3}\right)\left(\frac{\mathcal{A}_{22}(t)-\bar{\mathcal{A}}_{22}(t)}{\mathcal{A}_{22}(t)}\right)$
(22) $\displaystyle e_{\Phi_{22}}(t)$
$\displaystyle=\left(\frac{1}{3}\right)\left[\Phi_{22}(t)-\bar{\Phi}_{22}(t)+\Phi_{220}\right]$
$\displaystyle e_{\varpi_{22}}(t)$
$\displaystyle=\left(\frac{2}{3}\right)\left(\frac{\omega_{22}(t)-\bar{\omega}_{22}(t)}{\omega_{22}(t)}\right).$

Although some aspects are different in form, the ref. [88] puts the average on
the denominator, but the ref. [89] not. What they express is the difference
between a certain quantity of the eccentric waveform and its average value,
and the meaning is the same. In fact, no matter it is based on the orbital
dynamics, or the gravitational waveform, whether it is based on the 2-2 mode
or the high-order modes, it turns out that we can obtain the associated
eccentricity estimator, and they can be summed up in the form:

$e_{X_{1}}(t)=\left(k_{1}\right)\left(\frac{{X_{1}}(t)-{\bar{X}_{1}}(t)}{{\bar{X}_{1}}(t)}\right)$
(23)
$e_{X_{2}}(t)=\left(k_{2}\right)\left[{X_{2}}(t)-{\bar{X}_{2}}(t)+X_{20}\right],$
(24)

where $X_{1}$ represents the quantity relative to orbital distance, frequency
or amplitude, $X_{2}$ represents the quantity relative to phase, and $X_{20}$
represents a phase offset, and $k_{1}$, $k_{2}$ can be some constant. Then we
will find that $\bar{X}$ is the quantity $X_{c}$ come from its corresponding
circular orbit waveform. We need to emphasize that there is a difference
between waveform and orbital dynamics, which is reflected in the different
constant coefficients of the eccentricity estimator derived. However, the
difference in the constant coefficients $k_{1}$ only affects the magnitude of
the Eq.(23), not the overall behavior of the eccentricity estimator. So the
phenomenological fitting model which we will introduce later is still
applicable for them.

###  II.3 Measuring eccentricity of waveform

There are many definitions of eccentricity, and many ways to measure
eccentricity [88, 89, 90]. Each of them have their own scope of application.
In general, estimating eccentricity is mainly from the perspectives of
waveform and orbital dynamics. The eccentricity estimators Eq.(11), Eq.(13),
Eq.(14) to estimate the eccentricity is based on some orbital dynamical
quantities. The eccentricity estimators Eq.(19), Eq.(23) to estimate the
eccentricity is based on some quantities of waveform. In the Ref. [34], RIT
catalog estimates the eccentricity by the dynamical coordinate distance $d$:

$e_{d}=d^{2}\ddot{d}/M.$ (25)

We can obtain the eccentricity at each moment by interpolating the amplitude
of the eccentricity estimator. But the larger the eccentricity, the more the
eccentricity estimator deviates from the behavior of the cosine or sine
function, so that it is not suitable for measuring high eccentricity. When we
study high eccentricity, we have to introduce a new method capable of
measuring high eccentricity according to the Ref. [91]:

$e_{\Omega}(t)=\frac{\Omega_{p}^{1/2}-\Omega_{a}^{1/2}}{\Omega_{p}^{1/2}+\Omega_{a}^{1/2}},$
(26)

where $\Omega$ is orbital frequency. And $\Omega_{a}$ and $\Omega_{p}$ are
orbital frequency at apastron $\Omega_{a}$ and perisatron $\Omega_{p}$,
respectively. Although it is orbital frequency $\Omega$ here, it can apply to
frequency $\omega$ of waveforms. Our approach is similar to Ref. [58]. First
We use the python function find_peaks to find the apastron and perisatron of
the waveform. We then measure the eccentricity of the waveform at the apastron
and perisatron. Next we use the python function cubic_spline interpolation to
obtain a continuous evolution of the eccentricity. FIG. 2 shows the evolution
of frequency, apastron, perisatron and eccentricity of the waveform
SXS:BBH:1360. According to error propagation formula:

$\delta
e_{\omega}=\frac{\delta\omega}{\left(\omega_{a}^{1/2}+\omega_{p}^{1/2}\right)^{2}}\left[\frac{\omega_{a}^{1/2}}{\omega_{p}^{1/2}}+\frac{\omega_{p}^{1/2}}{\omega_{a}^{1/2}}\right].$
(27)

The conservatively estimated error of frequency in the Ref. [58] is about
$\delta\omega_{a}=\delta\omega_{p}=\delta\omega=0.0001$ caused by the
different resolutions of the numerical simulations. The statistical error
introduced by it in the measurement of eccentricity is $\delta
e_{\omega}\approx 0.001$, which is not the main error. The main error is
caused by interpolation, but it is not easy to quantify it. When the
eccentricity is about 0.4, the interpolation error can be as high as 0.1. The
fewer points that can be interpolated, the higher the error. For the research
in this paper, if a waveform has more than 6 cycles and the eccentricity is
lower than 0.3, the measurement error for eccentricity is not very large, but
as the eccentricity increases, the measurement error becomes larger and
larger. In FIG. 3, we show the frequency of the waveforms separately, which
have 5 cycles, 4 cycles, 3 cycles, 2 cycles, 1 cycle, 0 cycle. For a waveform,
when the eccentricity is small or the distance between the BBH is large, we
can get more cycles, such as SXS:BBH:1360 in the FIG. 2. On the contrary, we
can only get a few cycles. If the eccentricity of the waveform is very large
and close to 0.55, BBH will merge directly without any cycle such as 0 cycle
in the FIG. 3. The way we measure eccentricity is by interpolating with
cubic_splines, which requires at least four points to be interpolated. At the
same time, fitting of the eccentricity estimator in Sec. II.4 also requires
enough waveform cycles to get enough information about mass ratio and initial
eccentricity, so that we can obtain a sufficiently accurate waveform, which
also determines the length of the waveform that we used is at least
$[-1500,-300]$ to be reliable. Therefore, for cases such as 0 cycle, 1 cycle,
2 cycles and 3 cycles in the FIG. 3, we cannot obtain the eccentricity of the
waveforms, and for cases such as 4 cycles and 5 cycles, the error of the
obtained eccentricity is large. For these reasons, the waveforms we can use
are limited, and the range of their initial eccentricity is limited, so we
only list limited waveforms in the Sec. II.1.

Figure 2: Time evolution of waveform frequency $\omega$, apastron
$\omega_{a}$, perisatron $\omega_{p}$, and eccentricity $e$ for numerical
simulation SXS:BBH:1360. Figure 3: Frequency of waveforms which have 5 cycles,
4 cycles, 3 cycles, 2 cycles, 1 cycle, 0 cycle. For 0 cycle, 1 cycle, 2 cycles
and 3 cycles, we cannot obtain the eccentricity of the waveforms, and for 4
cycles and 5 cycles, the error of the obtained eccentricity is large.

###  II.4 Fitting process

Due to the circularization of the gravitational radiation, eccentric waveform
has the same properties as the circular orbit waveform near merger [9]. So we
only study the waveform before merger. We use the same notation as Ref. [1]
for eccentricity estimator equivalent to taking $k_{1}=\frac{1}{2}$ in the
Eq.(23):

$e_{X}(t)=\frac{X_{\mathrm{e}}(t)-X_{\mathrm{c}}(t)}{2X_{\mathrm{c}}(t)},$
(28)

where $X$ represents the amplitude $\mathcal{A}$ or frequency $\omega$, and
the subscripts $e$ and $c$ represent the eccentric and the circular orbital
waveform, respectively. Due to the monotonic correspondence between frequency,
amplitude and time for circular orbital waveform, there is an assumption
implicit in the Ref. [1] for the fitting model:

$\displaystyle e_{X}\left(t\right)$
$\displaystyle=e_{X}\left(t(X_{c})\right)=\frac{X_{\mathrm{e}}\left(t(X_{c})\right)-X_{\mathrm{c}}}{2X_{\mathrm{c}}}$
(29)
$\displaystyle=\frac{X_{\mathrm{e}}\left(X_{c}\right)-X_{\mathrm{c}}}{2X_{\mathrm{c}}}=Ae^{BX_{c}^{\kappa}}\sin\left(fX_{c}^{\kappa}+\varphi\right),$

where $A$, $B$, $f$, $\varphi$ and $\kappa$ are fitting parameters. We
directly write the functional relationship between the quantities, because it
is not obvious that there is a functional relationship between the eccentric
quantities $X_{e}$ and the circular quantities $X_{c}$. Instead of taking
$\kappa$ as a fixed value -59/24 for amplitude or -83/24 for frequency as in
the Ref. [1], we regard $\kappa$ as a free parameter which will allow us to
generalize the fitting to arbitrary time ranges, arbitrary mass ratio and
higher-order modes. It is some coincident that the Ref. [1] uses $\kappa$ as a
fixed constant to fit and obtain good results, because they only use the 2-2
mode with time range $t\in[-1500,-29]$ and mass ratio $q\in[1,3]$. According
to the analysis in Sec. II.2, the constant $k_{1}=\frac{1}{2}$ in the formula
can be any other constant, which does not affect the fitting result of the
Eq.(29) to the waveform. We still use it for convenience. Unlike any other
fitting model to get a local fitting to waveform, the model is a global
fitting to the waveform, which reflects global nature of waveform. The fitting
uses the python function op.curve_fit, which uses the non-linear least square
fitting method to obtain the best fit.

We can fit the quantities such as amplitude $\mathcal{A}$ and frequency
$\omega$ of waveform well through the Eq.(29), and then look for the
relationship between initial eccentricity $e$, mass ratio $q$ and fitting
parameters $A$, $B$, $f$, $\varphi$ and $\kappa$. When the data of the
waveforms is not so much, it is difficult for us to discover the relationship
between them. After adding the waveforms form RIT catalog, the data is more
and the relationship between the parameters is more obvious. After obtaining
the relationship between the fitting parameters and the waveform parameters,
we can cover the entire parameter range by interpolation or polynomial
fitting. By inverting Eq.(29), we get the amplitude or frequency of the
eccentric waveform we want as follows:

$\displaystyle X_{e}(t)$ $\displaystyle=X_{e}(t(X_{c}))=X_{e}(X_{c})$ (30)
$\displaystyle=2X_{c}e_{X}\left(t(X_{c})\right)+X_{c}$
$\displaystyle=2X_{c}Ae^{BX_{c}^{\kappa}}\sin\left(fX_{c}^{\kappa}+\varphi\right)+X_{c}.$

In order to improve the fitting effect, we can also set $\kappa$ to different
$\kappa_{1}$ and $\kappa_{2}$:

$e_{X}\left(t\right)=Ae^{BX_{c}^{\kappa_{1}}}\sin\left(fX_{c}^{\kappa_{2}}+\varphi\right).$
(31)

But the results of the fitting show that it does not work, because

(i) The larger parameter space makes it difficult to find relationships
between fitting parameters and waveform parameters. The error is larger when
we obtain eccentric waveforms through the Eq.(30).

(ii) The existence of the free parameter $\kappa_{1}$ makes
$e^{BX_{c}^{\kappa_{1}}}\approx 1$ unstable, thus destroying the magnitude
relationship $A\approx Ae^{BX_{c}^{\kappa}}$, thereby destroying the
proportional relationship between $A$ and eccentricity $e$, and introducing a
larger error. It is very important that the eccentricity $e$ is proportional
to $A$, because it makes the fitting more accurate. So we must maintain the
relationship.

Unlike the Ref. [1], we do not want to construct a complete inspiral-merger-
ringdown waveform, but study the waveform of 300 prior to merger. Near
merging, eccentricity is very small and negligible. The part close to merging
cannot be well fitted, and forced fitting will only bring errors. We give the
fitting results of several different cases as follows:

(i) The same mass ratio $q=1$, the same time range $t\in[-2000,-300]$, and
different eccentricity (see FIG. 4).

(ii) Different mass ratios $q\in[1,7]$, the same time range
$t\in[-2000,-300]$, and similar initial eccentricity (see FIG. 4 (b) and FIG.
5)

(iii) The same mass ratio $q=1$, different time range, and different
eccentricity for numerical simulation RIT:eBBH:1422 (see FIG. 6).

Figure 4: Fitting results for amplitude $\mathcal{A}$ of waveforms of 2-2 mode
of the same $q=1$, time range $t\in[-2000,-300]$ but different initial
eccentricities. Figure 5: Fitting results for frequency $\omega$ of waveforms
of 2-2 mode of the similar initial eccentricity, time range $t\in[-2000,-300]$
but different mass ratio $q\in[2,7]$. Figure 6: Fitting results for frequency
$\omega$ of waveforms of 2-2 mode of RIT:eBBH:1422 for different time range.

From fitting results of each case, we can draw some conclusions:

(i) From FIG. 4, we find the larger the eccentricity, the worse the fitting
effect. When the eccentricity is very small, for $e=0.0522$, we can almost
achieve complete fitting, but when $e=0.3592$, we cannot achieve complete
fitting. For the case with larger eccentricity, it is not easy to obtain a
waveform of time range $t\in[-2000,-300]$, because it requires BBH to have a
larger initial separation.

(ii) From FIG. 4 (b) and FIG. 5, we find the fitting model can be applied to
mass ratios $q\in[1,7]$. We find that a poor fitting result is obtained in
FIG. 5 (e), because there are some problems with the numerical simulation
RIT:eBBH:1357 itself, not with the fitting model.

(iii) From FIG. 6, we find the fitting model can be applied to a very long
time range. Even for $t\in[-12000,-300]$, the morphology of the waveform can
be roughly grasped.

###  II.5 Extend to higher-order modes

In fact, this model can be applied not only to 2-2 mode, but also to higher-
order modes. Exactly the same as 2-2 mode, we must maintain the one-to-one
correspondence between the eccentric waveform and the circular waveform. As an
example, when we study the eccentric nonspinning 2-1 mode, we have to use the
associated circular nonspinning 2-1 mode. Here, we only list the 3-3, 2-1,
4-4, 5-5, 3-2, 4-3 modes of some waveforms given in the SXS catalog, because
some other modes are not given in the catalog, and there are not many high-
order modes of waveforms in the RIT catalog. In FIG. 7, we can see that there
is a one-to-one correspondence between the eccentric nonspinning high-order
modes and circular ones, which makes the same fitting model applicable to
them. Here, we only take numerical simulation SXS:BBH:1368 with mass ratio
$q=1$, time range $t\in[-2000,-300]$ and initial eccentricity $e_{0}=0.0929$
as an example to give the fitting result for higher-order modes. In FIG. 8, we
present the fitting results for each modes. But in fact, the variation
behavior of all the higher-order modes is exactly the same as the 2-2 mode,
which means that the model can also be applied to higher-order modes with
other mass ratios, other eccentricities, and other time range.

Figure 7: Amplitude $\mathcal{A}$ of high-order modes including 3-3, 2-1, 4-4,
5-5, 3-2, 4-3 modes of waveform SXS:BBH:1368 expressed as $e$ and its
corresponding circular orbit waveform SXS:BBH:1165 expressed as $c$. Figure 8:
Fitting results for amplitude $\mathcal{A}$ of 3-3,4-4,5-5,3-2,4-3,2-1 modes
of waveform SXS:BBH:1368 for the same time range $t\in[-2000,-300]$ and
initial eccentricity $e=0.0929$.

###  II.6 Extend to spin-aligned

Spin is an important parameter of gravitational waves, which is very necessary
to contain. From the numerical simulations RIT:eBBH:1282, RIT:eBBH:1740,
RIT:eBBH:1763 and RIT:eBBH:1899, we can see that the spin has the same effect
as eccentricity to speed up BBHs merger. We discover that we can establish a
relationship between eccentric spin-aligned waveforms and circular orbit
waveforms from two perspectives.

(i) As in the previous case, we can establish a relationship as Eq.(29)
between eccentric spin-aligned waveforms and circular orbit spin-aligned
waveforms. We must maintain the consistency of spins for eccentric waveforms
and circular waveforms. For example, if initial dimensionless spins of BBH
${\chi_{z}}_{1}=-0.5$, ${\chi_{z}}_{2}=-0.5$ for one of the waveforms, it must
be the same for another. Here, we take the eccentric waveform as
RIT:eBBH:1899, whose $q=1$, two dimensionless spins are ${\chi_{z}}_{1}=-0.5$,
${\chi_{z}}_{2}=-0.5$, time range is $t\in[-3000,-300]$ and initial
eccentricity $e=0.1110$, and the circular orbit waveform as SXS:BBH:0325,
which has the same mass ratio, time range and dimensionless spin. In FIG. 9
(a), we present the fitting result.

(ii) We try another way. We keep the waveform RIT:eBBH:1899, but we take the
nonspinning circular orbital waveform SXS:BBH:0180 not the spin-aligned
circular orbital SXS:BBH:0325 as seed circular orbital waveform. We find that
if we want to fit the waveform, we have to introduce a shift parameter $g$ in
the Eq. (29) as follows:

$e_{X}\left(t\right)=Ae^{BX_{c}^{\kappa}}\sin\left(fX_{c}^{\kappa}+\varphi\right)+g.$
(32)

The parameter $g$ reflects the influence of spin on the waveform and straight
line $e_{\mathcal{A}}=g$ is the approximate axis of symmetry of the waveform.
In FIG. 9 (b), we show the fitting result and the position of the parameter
$g$. We select the waveforms RIT:eBBH:1282, RIT:eBBH:1740, RIT:eBBH:1763,
RIT:eBBH:1899 to study the influence of spin on the fitting effect. We find
that, the spin makes the entire waveform translate $g$ to the negative half
axis of the $e_{\mathcal{A}}$-axis. The larger the spin, the greater the
translation effect(see FIG. 9 (c)). We discover a strictly proportional
relationship between $g$ and the absolute value of effective spin $\chi_{eff}$
of BBH(see FIG. 10). It can be expressed as

$g=a\left|\chi_{\rm eff}\right|,$ (33)

where a=-0.04355, obtained by linear fitting. What we need to emphasize is
that the waveforms in the FIG. 9 (c) do not have the same initial eccentricity
at time $t=-3000$, because both eccentricity and spin have an impact on the
evolution of the waveform. If we want to obtain a counterpart of RIT:eBBH:1899
with the same eccentricity from RIT:eBBH:1282, we can use the method
introduced in Sec. III.1.

Figure 9: From left to right. Figure (a) is the fitting result of the
amplitude eccentricity estimator of waveform 1899, which comes from a circular
orbit waveform with the same spin as it. Figure (b) is the fitting result of
the amplitude eccentricity estimator of waveform 1899, which comes from a
circular orbit waveform without spin, the line $g$ shows the effect of spin on
it. Figure (c) are the amplitude eccentricity estimators of waveform
RIT:eBBH:1282, RIT:eBBH:1740, RIT:eBBH:1763, RIT:eBBH:1899, which come from a
circular orbit waveform without spin, the corresponding colors lines $g$ shows
the effect of spin on them. Figure 10: The red ’x’s are the relationship
between $g$ from the FIG. 8 (c) and effective spin
$\left|\chi_{\mathrm{eff}}\right|$, and the blue line represents a linear fit
to them.

###  II.7 Extend to spin-precession

When the spin direction of BBHs is inconsistent with the direction of orbital
angular momentum, it will cause orbital precession. The larger the effective
spin, the stronger the precession effect. The spin-precessing waveform with
eccentricity is extremely rare. In principle, we can generate the eccentric
waveform from two perspectives of spin and non-spinning as in Sec. II.6, but
since the corresponding waveform of the circular orbit with similar spin is
lacking, here we only study non-spinning case. We take the numerical
simulation BBH:eBBH:1631 as study object because of its large effective spin,
long time range $t\in[-12000,0]$, and low initial eccentricity $e_{0}=0.19$.
In FIG. 11 (a), we present the waveform BBH:eBBH:1631 which has been shifted
$|g|=0.03$ towards the positive semi-axis of $e_{\mathcal{A}}$-axis. This
value is obtained through the Eq.(33). This operation removes the spin effect
of the waveform, leaving only the eccentricity and precession effects. It is
not easy to obtain the precession effect of the waveform. But when the
eccentricity of the waveform is relatively low, we can interpolate the peaks
of the waveform, and obtain the midpoint of the upper and lower peaks. The
reason for this operation is that the symmetry axis of the nonspinning
waveform is approximately located at the midpoint of peaks of the waveform. We
can accurately model the precession effect by a polynomial fit, but any other
analytical method is also possible. We can analytically express the precession
effect as follows:

$f_{p}=\sum_{i=0}^{n}a_{i}{{\mathcal{A}}_{c}}^{i},$ (34)

where $a_{i}$ is the polynomial fit coefficient. In FIG. 11 (a), for the sake
of accuracy, we fit the precession effect by a 10 order polynomial fit. If we
express the spin effect as $f_{s}=g$, then we can get a nonspinning and
nonprecessing amplitude eccentricity estimator by

$\displaystyle{e_{\mathcal{A}}}_{nons,nonp}={e_{\mathcal{A}}}_{s,p}-f_{s}-f_{p},$
(35)

where the subscripts $nons$ and $nonp$ stand for nonspinning and
nonprecessing, respectively, and $s$ and $p$ stand for spin and precession. If
we want to obtain a precessing and spining amplitude eccentricity estimator,
we just invert Eq.(35). In the FIG. 11 (b), we subtract the precession effect
and then obtain an $e_{\mathcal{A}}$ without spin or precession. However, what
we want to emphasize is that the waveform in FIG. 11 (b) are not the same as
RIT:eBBH:1282, due to the influence of spin and eccentricity on the evolution
of the waveform, whose imprints have been left in it.

In summary, incorprating Eq.(29), Eq.(33) and Eq.(34), if we denote the
eccentricity effects as $f_{e}$, we can obtain an eccentric spin-precessing
$e_{\mathcal{A}}$ by gradually adding these effects to a nonspinning,
nonprecessing circular orbit waveform, which can be described as follows:

$\displaystyle{e_{\mathcal{A}}}_{e,s,p}$
$\displaystyle=\mathcal{A}_{c}+f_{e}+f_{s}+f_{p}$ (36)
$\displaystyle=Ae^{B\mathcal{A}_{c}^{\kappa}}\sin\left(f\mathcal{A}_{c}^{\kappa}+\varphi\right)+a\left|\chi_{eff}\right|$
$\displaystyle+\sum_{i=0}^{n}a_{i}{{\mathcal{A}}_{c}}^{i}.$

Then, we obtain the corresponding amplitude by the Eq.(30)

$\displaystyle{\mathcal{A}}_{e,s,p}$
$\displaystyle=2{\mathcal{A}}_{c}\left(\mathcal{A}_{c}+f_{e}+f_{s}+f_{p}\right)+{\mathcal{A}}_{c}$
(37)
$\displaystyle=2{\mathcal{A}}_{c}[Ae^{B\mathcal{A}_{c}^{\kappa}}\sin\left(f\mathcal{A}_{c}^{\kappa}+\varphi\right)$
$\displaystyle+a\left|\chi_{eff}\right|+\sum_{i=0}^{n}a_{i}{{\mathcal{A}}_{c}}^{i}]+{\mathcal{A}}_{c}.$

We can also get the frequency $\omega_{e,s,p}$ by the same process. Both of
them are fundamental components of gravitational waves.

Figure 11: Figure (a) is amplitude eccentricity estimator of waveform
BBH:eBBH:1631 which has been shifted $|g|=0.03$ towards the positive semi-axis
of $e_{\mathcal{A}}$-axis. $f_{p}$ represents the precession effect. Figure
(b) is a non-spinning, non-precessing amplitude eccentricity estimator of
waveform after subtracting the precession effect.

##  III results

###  III.1 2-2 mode

####  III.1.1 Fitting results

Some waveforms with large errors need to be discarded. The $q=6$ waveforms are
not shown due to their own problems from numerical simulation, because they
deviate far from the behavior shown by the parameters of other waveforms. We
only show the behavior here for amplitude and time range $t\in[-2000,-300]$,
but the fitting parameters for frequency or other time range share the same
behavior. The fitting results of parameters $A_{\mathcal{A}}$,
$B_{\mathcal{A}}$, $f_{\mathcal{A}}$, $\kappa_{\mathcal{A}}$ for different
mass ratio $q$ are shown in FIG. 12, where the subscript $\mathcal{A}$
represents that it is the fitting of amplitude. We take the values of
$A_{\mathcal{A}}$ and $f_{\mathcal{A}}$ as positive, and $B_{\mathcal{A}}$ and
$\kappa_{\mathcal{A}}$ as positive and negative. In fact, the values of
$A_{\mathcal{A}}$ and $f_{\mathcal{A}}$ can be positive or negative, depending
on the parity of the sine function, while $B_{\mathcal{A}}$ and
$\kappa_{\mathcal{A}}$ must be positive and negative. The sign of
$A_{\mathcal{A}}$ and $f_{\mathcal{A}}$ does not affect the relationship
between them and eccentricity $e$. The parameter $\varphi_{\mathcal{A}}$ has
no effect on the morphological properties of the amplitude and frequency of
the waveform, but only translates the frequency and amplitude on the time
coordinate. That is, it does not affect eccentricity $e$ and mass ratio $q$ of
the waveform. So $\varphi_{\mathcal{A}}$ is a free parameter. We also discover
it has a certain periodicity.

Figure 12:  Fitting results of parameters $A_{\mathcal{A}}$,
$B_{\mathcal{A}}$, $f_{\mathcal{A}}$, $\kappa_{\mathcal{A}}$ for different
mass ratio $q$ eccentricity $e$.

As we see in FIG. 12, the parameters $A_{\mathcal{A}}$, $B_{\mathcal{A}}$,
$f_{\mathcal{A}}$, and $\kappa_{\mathcal{A}}$ are related to the mass ratio
$q$ and can be judged by the hierarchical phenomenon of the curves.

(i) There is a strict proportional relationship between $A_{\mathcal{A}}$ and
eccentricity $e$. The larger the eccentricity is, the larger $A_{\mathcal{A}}$
is, which comes from the relationship between the amplitude eccentricity
estimator $e_{\mathcal{A}}$ and the eccentricity $e$. In FIG. 12 (a), we see
that $A_{\mathcal{A}}$ is not only related to eccentricity, but also to mass
ratio $q$.

(ii) The correspondences between $B_{\mathcal{A}}$ and $f_{\mathcal{A}}$ and
eccentricity are very similar, but their magnitudes are not the same. They are
distinctly stratified at different mass ratios (see in FIG. 12 (b) and (c)).
And there is a slightly monotonic relationship between them and eccentricity.
However, we cannot ignore this monotonic relationship, because the accuracy of
the waveform is extremely demanding on these two parameters.

(iii) The parameter $\kappa_{\mathcal{A}}$ is also related to both mass ratio
and eccentricity. When we obtain the fitting result of the higher-order modes,
we will find that the function of $\kappa_{\mathcal{A}}$, $f_{\mathcal{A}}$
and $B_{\mathcal{A}}$ is to adjust the Eq.(28), making it suitable for
different mass ratios and higher-order modes.

(iv) The larger the eccentricity, the more spread out the points are, which
means larger errors due to interpolation and fitting, as analyzed in the Sec
II.3.

These parameters have a strong dependence on the eccentricity $e$ and mass
ratio $q$. When the amount of data is relatively small, it is difficult for us
to judge it as in Ref. [1]. When the amount of data is large, we can see it
clearly. Since the parameters have a complex relationship not only with the
mass ratio but also with the eccentricity, which makes it very difficult to
cover the parameter space of $q$ and $e$ at the same time. We can only fix the
mass ratio $q$ and then cover the eccentricity $e$ space. As shown in the FIG.
12, since there are too many data points and very scattered, it is difficult
to interpolate them. So we obtain a relationship between parameters
$A_{\mathcal{A}}$, $B_{\mathcal{A}}$ by polynomial fitting, $f_{\mathcal{A}}$,
$\kappa_{\mathcal{A}}$ and $e$. We use linear fit for $A_{\mathcal{A}}$. For
$B_{\mathcal{A}}$, $f_{\mathcal{A}}$ and $\kappa_{\mathcal{A}}$, the most
accurate results are obtained by a second order polynomial fitting, and others
are possible. It is worth emphasizing that polynomial fitting does not mean
that there must be a continuous monotonic relationship between these four
parameters and eccentricity, which is only an approximation. If we want to get
a more accurate relationship between them, we need more eccentric waveform
data.

####  III.1.2 Mismatch

Since our purpose is to reproduce the complete numerically relativistic
waveforms, they are naturally our comparison target. We use the leave-one-out
method, setting one waveform of all data as test data and the other waveforms
as training data to generate fitting parameters $A_{\mathcal{A}}$,
$B_{\mathcal{A}}$, $f_{\mathcal{A}}$ and $\kappa_{\mathcal{A}}$. Next we
obtain the corresponding fitting parameters through the eccentricity of the
left test waveform. After we get the fitting parameters, we can obtain the
corresponding amplitude $\mathcal{A}$ and frequency $\omega$ through Eq.(29).
After this, we have to integrate the frequency to get phase by

$\Phi=\int_{t_{1}}^{t_{2}}\omega dt,$ (38)

where ${t_{1}}$ and ${t_{2}}$ are the integral lower and upper limits. Then we
can reconstruct the test waveform by Eq.(19). In order to evaluate the
similarity between the test waveform and the newly reconstructed waveform, we
need to calculate overlap as in Ref. [31]:

$\mathcal{O}=\mathop{\max}\limits_{{t_{0},\Phi_{0},\varphi_{\mathcal{A}},\varphi_{\omega}}}\frac{\left\langle
h_{1},h_{2}\right\rangle}{\sqrt{\left\langle
h_{1},h_{1}\right\rangle\left\langle h_{2},h_{2}\right\rangle}},$ (39)

where $\left\langle h_{1},h_{2}\right\rangle$ is the inner product of waveform
$h_{1}$ and $h_{2}$ defined as

$\left\langle
h_{1},h_{2}\right\rangle=\left|\int_{t_{\min}}^{t_{\max}}h_{1}(t)h_{2}^{*}(t)dt\right|,$
(40)

where $h_{2}^{*}(t)$ is complex conjugate of $h_{2}(t)$. $t_{0}$ and
$\Phi_{0}$ are given time and phase $\varphi_{\mathcal{A}}$ and
$\varphi_{\omega}$ are free parameters inherited from the construction of
waveform. We calculate the overlap in time domain because the waveform we
constructed is a time domain waveform and each fitting parameter related to
eccentricity is equivalent to related to time. At the same time, we choose a
uniform Power Spectral Density (PSD) set to unity instead of the noise PSD of
LIGO or other gravitational wave detectors in calculation, in order to to
reflect the fitting effect in the entire time domain because we do not care
about the application on detection of gravitational waves, but only care about
the waveform itself. For convenience, we can also define mismatch or
unfaithfulness as

$\mathcal{M}=1-\mathcal{O}.$ (41)

We do not calculate $\mathcal{M}$ all the time range, but every 250 from
$t\in[-3000,-300]$ to $t\in[-1500,-300]$. However, we think it is common for
any other continuous time range. Here, we need not to consider the total mass
$M$, because all units have been cancelled in Eq.(39). When calculating
$\mathcal{M}$, we divide all eccentricities into four ranges, and use
different colors to represent them. Blue means $e\in[0,0.1]$. Green means
$e\in[0.1,0.2]$. Red means $e\in[0.2,0.3]$. Black means $e\in[0.3,0.4]$. The
purpose of this is to study the effect of eccentricity on $\mathcal{M}$. In
FIG. 13, we show the $\mathcal{M}$ between the waveform obtained by leave-one-
out method and the test waveform in different eccentricities $e\in[0,0.4]$,
different mass ratios $q\in[1,3]$ and different time ranges. For mass ratio
$q\in[4,7]$, since there are few waveforms, we can only use the fit of the
test waveform to calculate $\mathcal{M}$, which is also meaningful to reflect
the fitting effect of the model.

Figure 13: $\mathcal{M}$ between the waveform obtained by leave-one-out method
and the test waveform in different eccentricities $e\in[0,0.4]$, different
mass ratios $q\in[1,3]$ and different time ranges. For mass ratio $q\in[4,7]$,
since there are few waveforms, we can only use the fit of the test waveform to
calculate $\mathcal{M}$. In figure, blue means $e\in[0,0.1]$, green means
$e\in[0.1,0.2]$, red means $e\in[0.2,0.3]$ and black means $e\in[0.3,0.4]$.

####  III.1.3  Analysis

$\mathcal{M}$ reflects the similarity of the waveforms. From the FIG. 13, we
can see that

(i) For waveforms with different time ranges, we get different mismatch. When
the time range is very long, such as $t\in[-3000,-300]$ or longer, or the time
range is short, such as $t\in[-1500,-300]$ or shorter, the fitting effect we
obtained are not so good as time range $t\in[-2000,-300]$ and
$t\in[-2500,-300]$. The reason for it is that when the waveform is very long,
the model cannot fully capture all the overall properties of it, and when the
waveform is short, the model gives a large error due to too little information
given by the waveform.

(ii) For the waveforms with mass ratio $q\in[1,3]$, we obtain a relatively
small $\mathcal{M}$, but when the mass ratio is in $q\in[4,7]$, we cannot give
a small $\mathcal{M}$ even with the fitting of the waveform itself. It is not
because the model is not suitable for mass ratio $q\in[4,7]$, but the errors
in the waveforms themselves in the RIT catalog, which is particularly obvious
in the amplitude eccentricity estimator $e_{\mathcal{A}}$. So we do not show
it in FIG. 5 but present frequency eccentricity estimator $e_{\omega}$. Some
$e_{\mathcal{A}}$s come from RIT catalog has a strong local ups and downs,
which is obviously caused by errors of the waveforms themselves.

(iii) We stratify different eccentricities in order to show that we obtain
different $\mathcal{M}$ under different eccentricities. We find that if we
don’t consider $\mathcal{M}$ with a mass ratio $q\in[4,7]$ that has a large
error, when the eccentricity $e\in[0,0.1]$, we obtain an $\mathcal{M}$ less
than $10^{-4}$, and when the eccentricity $e\in[0.1,0.2]$, we obtain an
$\mathcal{M}$ less than $10^{-3}$, and when the eccentricity $e\in[0.2,0.3]$,
we obtain an $\mathcal{M}$ less than $10^{-2}$, and when the eccentricity
$e\in[0.3,0.4]$, we obtain an $\mathcal{M}$ less than $10^{-1}$. This implies
that as the eccentricity becomes larger, the model fits the waveform worse,
which is consistent with the conclusion we have drawn in the Sec.II.4.

####  III.1.4 Morphology of eccentric waveform

There is a clear difference in morphology between the eccentric waveform and
the circular orbit waveform. The separation of BBHs is relatively close and
far away at the perisatron and apastron. So the amplitude and frequency of the
waveform is relatively large at the perisatron and relatively small at the
apastron, which is the same for dominant mode in FIG. 2 or higher order modes
in FIG. 7. However, not only the eccentric waveform and the circular orbit
waveform are morphologically different, but also the low eccentricity waveform
and the high eccentricity waveform are very different in morphology. It is
difficult for us to see this morphological difference only through the
comparison between the eccentric waveforms, but the circular orbit waveform
and the eccentricity estimator provide us with a new perspective. The
eccentricity estimator is defined as a cosine function by Eq.(9). That is,
when the eccentricity estimator deviates from the behavior of the cosine
function, measuring eccentricity by the eccentricity estimator will introduce
errors [88]. In FIG. 14, we show the amplitudes of the waveforms with mass
ratio $q=1$, time range $t\in[-2000,-300]$ and initial eccentricities of
$e_{0}=0$ (circular), $e_{0}=0.0522$ (1355), $e_{0}=0.2014$ (1362) and
$e_{0}=0.3653$ (1286), respectively. $T_{p1}$ and $T_{a1}$ represent the time
of the perisatron passage and apastron passage of the first and second half
cycle of the waveform 1355 based on the circular orbit waveform. $T_{p2}$,
$T_{a2}$ and $T_{p3}$, $T_{a3}$ are for 1362 and 1286. From FIG. 14 (a), we
can find that, the greater the eccentricity of the waveform, the stronger the
oscillation in amplitude. As the eccentricity increases, the perisatron
passage and apastron passage of the waveform gradually show different
behaviors. The former becomes sharper and the latter becomes smoother. We can
get the ratio $T_{a3}/T_{p3}>T_{a2}/T_{p2}>T_{a2}/T_{p2}$, which means that
the time of the apastron passage is getting longer and longer than the time of
the perisatron passage. Behavior in amplitude is passed to the associated
amplitude eccentricity estimator. In order to show this effect, in FIG. 14
(b), we take the absolute value of the eccentricity estimator, so that
perisatron and apastron are at the same level, and then we connect all the
points to draw a trend line, where $a1$, $p1$, etc. are the corresponding
perisatron and apastron in the FIG. 14 (a). The ups and downs of these trend
lines indicate how far the eccentricity estimator deviates from the cosine
function. The trend line of waveform 1355 is roughly a straight line, implying
that it does not deviate from cosine behavior, but the trend line of waveform
1286 is an obvious broken line, implying that it deviates from cosine
behavior. Due to the monotonic relationship between the amplitude of the
circular orbit and time, the behavior of the amplitude eccentricity estimator
with respect to time will be passed to the behavior of the amplitude
eccentricity estimator with respect to the amplitude of the circular orbit(see
in FIG. 14 (c)). We can derive the behavior of the Eq.(28) by magnitude
analysis. From the FIG. 12, we get $B\sim 10^{-3}$, ${X_{c}}^{\kappa}\sim
10^{3}$ and $e^{BX_{c}^{\kappa}}\sim 1$. So the overall behavior of Eq.(29) is
a sine or cosine function which is similar to FIG. 4 (a). The trend lines in
FIG. 14 (c) show that 1355 does not deviate from sinusoidal behavior, while
1286 does. All in all, the degree of deviation of the eccentricity estimator
from the sinusoidal function determines the scope of application of the
phenomenological fitting model. The greater the eccentricity, the greater the
deviation of the eccentricity estimator from the sinusoidal behavior, which
also determines the model cannot be used for high eccentricity. As we can see
in the FIG. 13, different eccentricities give different fitting effects, so
when we want to obtain higher accuracy, we generally need to keep the
eccentricity at different intervals in $[0,0.4]$.

Figure 14: Amplitudes of the waveforms with mass ratio $q=1$, time range
$t\in[-2000,-300]$ and initial eccentricities of $e_{0}=0$ (circular),
$e_{0}=0.0522$ (1355), $e_{0}=0.2014$ (1362) and $e_{0}=0.3653$ (1286),
respectively. $T_{p1}$ and $T_{a1}$ represent the time of the perisatron
passage and apastron passage of the first and second half cycle of the
waveform 1355 based on the circular orbit waveform. $T_{p2}$, $T_{a2}$ and
$T_{p3}$, $T_{a3}$ are for 1362 and 1286. From the figure, we can find that
the eccentric behavior of the cosine function is passed from Figure (a) to
Figure (b) and then passed to Figure (c).

###  III.2 Other situations

####  III.2.1 Higher-order modes

Compared with the 2-2 mode, the high-order modes have a large difference in
amplitude and frequency, which leads to different magnitudes of parameters,
but their related behavior with eccentricity and mass ratio is the same as the
2-2 mode. Similarly, we can also obtain it according to the method in the Sec.
III.1. Here we will not go into details.

The calculation of mismatch $\mathcal{M}$ can be obtained through the Eq.(39).
We take fitting of the high-order modes 3-3, 2-1, 4-4, 5-5, 3-2, 4-3 mode with
mass ratio $q=2$, eccentricity $e\in[0,0.1]$ and time range $t\in[-2000,-300]$
as an example to show the mismatch $\mathcal{M}$ we get (see in FIG. 15). The
situation is similar for other parameters. Here we only use fitting to
calculate $\mathcal{M}$ because there is too little data for the high-order
modes of the eccentric waveforms. When we get enough high-order modes data of
the eccentric waveforms, we can also do the same procedure as the Sec. III.1.
From FIG. 15, we can see that the mismatch $\mathcal{M}$ of high-order modes
and the 2-2 mode share roughly the same behavior with the eccentricity.
Therefore, this model is able to fit the higher order modes very well.

Figure 15:  Mismatch $\mathcal{M}$ for 3-3, 2-1, 4-4, 5-5, 3-2, 4-3 modes with
mass ratio $q=2$, eccentricity $e\in[0,0.1]$ and time range
$t\in[-2000,-300]$.

####  III.2.2 Spin-aligned

As stated in the Sec. II.6, the waveform of spin aligned can be obtained in
two ways, one is the circular orbit waveform with spin, and the other is the
circular orbit waveform without spin. We can fit waveforms by Eq.(29) or
Eq.(32). Since the number of eccentric spinning waveforms is few, and it is
not easy to find an eccentric waveform whose magnitude and direction of
component spins are exactly the same as a circular orbital waveform, we can
only present the fitting effect for two cases here. We show mismatch
$\mathcal{M}$ of different time range with mass ratio $q=1$ in the FIG. 16 in
which RIT:eBBH:1740, RIT:eBBH:1763 and RIT:eBBH:1899 are for circular orbit
waveforms without spin, and the other two RIT:eBBH:1763 and RIT:eBBH:1899 are
for circular orbit waveforms with spin. For both cases, the mismatch
$\mathcal{M}$ we obtained is less than about $10^{-4}$ , indicating that the
model can fit them well.

Figure 16: Mismatch $\mathcal{M}$ of different time range with mass ratio
$q=1$. RIT:eBBH:1740, RIT:eBBH:1763 and RIT:eBBH:1899 are come from circular
orbit waveforms without spin, and the other two RIT:eBBH:1763 and
RIT:eBBH:1899 are come from circular orbit waveforms with spin.

####  III.2.3 Spin-precession

The eccentric spin-precessing BBHs merger is one of the most complicated cases
in numerical relativity, and there are few studies on it. Their waveforms are
interspersed with multiple complex effects and are difficult to analyze. In
the Sec. II.7, we propose a method for separating the eccentric spin-
precessing waveform into different effects as in Eq.(36). Now, we select the
part of the waveform RIT:eBBH:1361 in the time range $t\in[-3000,-300]$. Let
us take the amplitude as an example. First, we measure the eccentricity of the
waveform given by the FIG. 11 (b) at time $t=-3000$ and get $e=0.090$. Then we
obtain the values of each parameter $A_{\mathcal{A}}=0.037593098$
,$B_{\mathcal{A}}=0.000556893$, $f_{\mathcal{A}}=0.034178913$,
$\kappa_{\mathcal{A}}=-3.277506968$ corresponding to the eccentricity
$e=0.090$ through the FIG. 12. Next we obtain an eccentricity estimator with
mass ratio $q=1$, no spin and eccentricity $e=0.090$ via the Eq.(29). The
results are shown in FIG. 17 (a), where the blue solid line is named
“removed”, because we removed the spin and precession effects inside it, and
the dark orange dashed line is what we “got”. Then we add the spin effect
$-g=0.3$ and the precession effect $f_{p}$ to the ”got” waveform to obtain an
eccentricity estimator with spin-precession through the Eq.(36), which
precisely recovers the characteristics of the eccentricity estimator of
original waveform 1631(see in FIG. 17(b)). Finally, we get the amplitude of a
waveform with eccentricity and spin precession by the Eq.(37), which has a
good overlap with the amplitude of waveform 1631(see in FIG. 17 (c)). For
frequency $\omega$, the manipulations are the same. Combining with it, we get
a mismatch $\mathcal{M}=0.00056$, which means that we accurately reproduce the
original waveform 1631.

Figure 17: In figure (a), the blue solid line represents the part of the
waveform RIT:eBBH:1361 in the time range $t\in[-3000,-300]$, which is named
“removed”, because we removed the spin and precession effects inside it. The
dark orange dashed line represents a new waveform we “got” from the parameters
corresponding to the eccentricity $e=0.090$ through the FIG.12. In figure (b),
the blue solid line represents the “got” amplitude eccentricity estimator of
waveform that we add the spin effect $-g=0.3$ and the precession effect
$f_{p}$. The dark orange dashed line represents the 1631 amplitude
eccentricity estimator. In figure (c), The dark orange dashed line is the
“got” the amplitude of a waveform with eccentricity and spin precession by the
Eq.(37). The blue solid line represents the amplitude of waveform 1631.

Not all eccentric spin-precessing waveforms have a simple precession effect
like RIT:eBBH:1361. Waveform RIT:eBBH:1701 has an effective spin of zero due
to the spin anti-alignment of BBHs, leading to $g=0$. If we assume that the
precession effect and the eccentricity effect are independent of each other,
then the parameter $f$ only depends on the eccentricity. So we can force fit
its eccentricity estimator to try to obtain its precession effect(see in FIG.
18). As in the previous case, subtracting the fit, we get the precession
effect.

Figure 18: The blue solid line represents amplitude eccentricity estimator of
waveform 1701, and the dark orange dashed line represents a forced fit for it.

##  IV Conclusion and Outlook

The eccentricity and spin of gravitational waves can reflect the dynamics of
BBHs merger. However, there are very few public numerical relativity
simulations with eccentricity. Setyawati and Ohme [1] propose a novel method
to convert quasi-circular orbit waveforms into eccentric ones, but their
method is very limited because of the small range of parameters. Based on the
eccentric waveforms of the SXS catalog, we add some eccentric waveforms of the
RIT catalog, including the waveforms of mass ratio $q\in[1,7]$ , eccentricity
$e\in[0,0.4]$ , spin alignment, and spin precession, greatly expanding the
parameter space. We find that after setting the fixed constant parameter to
the variable parameter $\kappa$ as in Eq.(29), the applicability of the model
becomes wider, and can be applied to the case of mass ratio $q\in[1,7]$,
eccentricity up to $e=0.4$, time range up to $t\in[-12000,-300]$, high-order
modes and spin alignment. We use the leave-one-out method to verify this
model. For $e\in[0,0.1]$, it gives an overlap of more than 99.99%. For
$e\in[0.1,0.2]$, it gives an overlap of more than 99.9%. For $e\in[0.2,0.3]$,
it gives an overlap of more than 99%. For $e\in[0.3,0.4]$, it gives an overlap
of more than 90%. The reason for these phenomenons is that the larger the
eccentricity, the larger the deviation of the eccentricity estimator from the
cosine function due to the large change in the morphology of the eccentric
waveform, and the worse the fitting effect of the model. After adding a shift
parameter as in Eq.(32), the model can convert the waveform without spin and
eccentricity into a waveform with spin and eccentricity. That is, the spin
effect can be added to nonspinning waveform. For some waveforms with
relatively simple and obvious precession effects, we can separate the
precession effects. Finally, the spin precession waveform with eccentricity
can be regarded as the superposition of various effects including
eccentricity, spin and precession, which is very novel and simple. We can also
obtain models of complex precession phenomena by the model. We believe that
this phenomenological and universal relationship can not only help us to
generate fast and accurate gravitational waveforms with eccentricity and spin
precession, but also put up with a new perspective for understanding
eccentricity, spin and precession effects of the waveform BBHs dynamics.

Due to the small amount of waveform data used in this work and the lack of
generality in the cases of spin alignment and spin precession, we only study
some special cases for them. We hope that there will be more and more
numerical relativity simulations with eccentricity in the future, which will
make this phenomenological model richer and more accurate.

Once a large coverage on the parameters including mass ratio, eccentricity,
and spins can be considered in the phenomenological relationship, we will be
able to construct a large amount of waveforms easily. This could be used as
the waveform template in searching GWs by LIGO/Virgo [92] or by the upcoming
next generation gravitational wave detectors, such as the Einstein Telescope
[93]. As the waveforms are scalable to any mass, it can also be applied to the
merging of galactic center binary black holes, which happens in the early
universe, and could be detectable by the space borne gravitational wave
missions, such as Laser Interferometer Space Antenna (LISA) [94], and Tianqin
[95].

###### Acknowledgements.

The authors are very grateful to the SXS collaboration and the RIT
collaboration for the numerical simulation of eccentric BBHs mergers, and
thanks to Yun-Gui Gong, Xiao-Lin Liu, Ying-Yan Li, Chao Zhang, Qingwen Wu, and
Shi-Yan Tian for their helpful discussions. This work is in part supported by
the National Key R&D Program of China (2022SKA0130103, 2021YFC2203100), and by
the National Natural Science Foundation of China (Grant Nos. 12041306 and
U1931203). We also acknowledge the science research grants from the China
Manned Space Project with No. CMS-CSST-2021-B11. The computation is partly
completed in the HPC Platform of Huazhong University of Science and
Technology.

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##  Appendix A RIT and SXS waveforms

Table 1: Eccentric and circular orbit waveforms from the RIT catalog, which contains simulation number, type, mass ratio $q\in[1,7]$, initial separation $\operatorname{sep}_{ini}$, dimensionless spin component, effective spin, reference eccentricity $e_{ref}$ and merge time $t_{merge}$. case | simulation | type | $sep_{ini}$ | $q$ | $\chi_{1x}$ | $\chi_{1x}$ | $\chi_{1x}$ | $\chi_{1x}$ | $\chi_{1x}$ | $\chi_{1x}$ | $\chi_{eff}$ | $e_{ref}$ | $t_{merge}/M$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | RIT:eBBH:1282 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.1900 | 11764
2 | RIT:eBBH:1283 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.2775 | 6524
3 | RIT:eBBH:1284 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3276 | 4430
4 | RIT:eBBH:1285 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3600 | 3368
5 | RIT:eBBH:1287 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3916 | 2508
6 | RIT:eBBH:1286 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3994 | 2330
7 | RIT:eBBH:1289 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4071 | 2141
8 | RIT:eBBH:1288 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4148 | 1993
9 | RIT:eBBH:1291 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4224 | 1822
10 | RIT:eBBH:1290 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4300 | 1681
11 | RIT:eBBH:1293 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4375 | 1558
12 | RIT:eBBH:1292 | Nonspinning | 24.64 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4450 | 1410
13 | RIT:eBBH:1740 | Spin-aligned | 24.71 | 1 | 0.0 | 0.0 | -0.5 | 0.0 | 0.0 | -0.5 | -0.500 | 0.1900 | 10094
14 | RIT:eBBH:1763 | Spin-aligned | 24.75 | 1 | 0.0 | 0.0 | -0.8 | 0.0 | 0.0 | -0.8 | -0.800 | 0.1900 | 9054
15 | RIT:eBBH:1899 | Spin-aligned | 24.69 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.8 | -0.400 | 0.1900 | 10588
16 | RIT:eBBH:1741 | Spin-aligned | 24.71 | 1 | 0.0 | 0.0 | -0.5 | 0.0 | 0.0 | -0.5 | -0.500 | 0.3600 | 2286
17 | RIT:eBBH:1764 | Spin-aligned | 24.75 | 1 | 0.0 | 0.0 | -0.8 | 0.0 | 0.0 | -0.8 | -0.800 | 0.3600 | 1620
18 | RIT:eBBH:1900 | Spin-aligned | 24.69 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.8 | -0.400 | 0.3600 | 2483
19 | RIT:eBBH:1786 | Spin-aligned | 24.59 | 1 | 0.0 | 0.0 | 0.5 | 0.0 | 0.0 | 0.5 | 0.500 | 0.4375 | 2448
20 | RIT:eBBH:1807 | Spin-aligned | 24.56 | 1 | 0.0 | 0.0 | 0.8 | 0.0 | 0.0 | 0.8 | 0.800 | 0.4375 | 2985
21 | RIT:eBBH:1828 | Spin-aligned | 24.60 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.8 | 0.400 | 0.4375 | 2275
22 | RIT:eBBH:1787 | Spin-aligned | 24.59 | 1 | 0.0 | 0.0 | 0.5 | 0.0 | 0.0 | 0.5 | 0.500 | 0.4671 | 1866
23 | RIT:eBBH:1808 | Spin-aligned | 24.56 | 1 | 0.0 | 0.0 | 0.8 | 0.0 | 0.0 | 0.8 | 0.800 | 0.4671 | 2349
24 | RIT:eBBH:1829 | Spin-aligned | 24.60 | 1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.8 | 0.400 | 0.4671 | 1720
25 | RIT:eBBH:1631 | Spin-precessing | 24.62 | 1 | 0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.700 | 0.1900 | 12012
26 | RIT:eBBH:1701 | Spin-precessing | 24.64 | 1 | -0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.000 | 0.1900 | 11759
27 | RIT:eBBH:1632 | Spin-precessing | 24.62 | 1 | 0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.700 | 0.2775 | 6726
28 | RIT:eBBH:1702 | Spin-precessing | 24.64 | 1 | -0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.000 | 0.2775 | 6554
29 | RIT:eBBH:1633 | Spin-precessing | 24.62 | 1 | 0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.700 | 0.3600 | 3488
30 | RIT:eBBH:1703 | Spin-precessing | 24.64 | 1 | -0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.000 | 0.3600 | 3382
31 | RIT:eBBH:1634 | Spin-precessing | 24.62 | 1 | 0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.700 | 0.4375 | 1633
32 | RIT:eBBH:1704 | Spin-precessing | 24.64 | 1 | -0.7 | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 | 0.000 | 0.4375 | 1562
33 | RIT:eBBH:1883 | Spin-aligned | 24.70 | 2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.8 | -0.533 | 0.3600 | 2257
34 | RIT:eBBH:1422 | Nonspinning | 24.63 | 2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.1900 | 13074
35 | RIT:eBBH:1423 | Nonspinning | 24.63 | 2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3600 | 3714
36 | RIT:eBBH:1424 | Nonspinning | 24.63 | 2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4375 | 1667
37 | RIT:eBBH:1862 | Spin-aligned | 24.70 | 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.8 | -0.600 | 0.3600 | 2209
38 | RIT:eBBH:1468 | Nonspinning | 24.62 | 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.1900 | 15114
39 | RIT:eBBH:1469 | Nonspinning | 24.62 | 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3600 | 4249
40 | RIT:eBBH:1470 | Nonspinning | 24.62 | 3 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4375 | 1854
41 | RIT:eBBH:1491 | Nonspinning | 24.61 | 4 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.1900 | 17835
42 | RIT:eBBH:1492 | Nonspinning | 24.61 | 4 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3600 | 4878
43 | RIT:eBBH:1493 | Nonspinning | 24.61 | 4 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4375 | 2102
44 | RIT:eBBH:1514 | Nonspinning | 24.60 | 5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.1900 | 15781
45 | RIT:eBBH:1515 | Nonspinning | 24.60 | 5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3600 | 5539
46 | RIT:eBBH:1516 | Nonspinning | 24.60 | 5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4375 | 2496
47 | RIT:eBBH:1517 | Nonspinning | 24.60 | 5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4671 | 1601
48 | RIT:eBBH:1537 | Nonspinning | 24.59 | 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.1900 | 6268
49 | RIT:eBBH:1538 | Nonspinning | 24.59 | 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3600 | 3200
50 | RIT:eBBH:1539 | Nonspinning | 24.59 | 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4375 | 2213
51 | RIT:eBBH:1540 | Nonspinning | 24.59 | 6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4671 | 1544
52 | RIT:eBBH:1560 | Nonspinning | 24.59 | 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.1900 | 22937
53 | RIT:eBBH:1561 | Nonspinning | 24.59 | 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.3600 | 6491
54 | RIT:eBBH:1562 | Nonspinning | 24.59 | 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4375 | 2680
55 | RIT:eBBH:1563 | Nonspinning | 24.59 | 7 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.000 | 0.4671 | 1733
Table 2: Eccentric and circular orbit waveforms from the SXS catalog, which contains simulation number, type, mass ratio $q\in[1,7]$, initial separation $\operatorname{sep}_{ini}$, dimensionless spin component, effective spin, initial eccentricity $e_{0}$ after removing junk radiation, merge time $t_{merger}$ and orbit number $N_{orbits}$. case | simulations | type | q | $sep_{ini}$ | $\chi_{1x}$ | $\chi_{1y}$ | $\chi_{1z}$ | $\chi_{2x}$ | $\chi_{2y}$ | $\chi_{2z}$ | $\chi_{eff}$ | $e_{0}$ | $t_{merge}/M$ | $N_{orbits}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
2 | SXS:BBH:1355 | Nonspinning | 1 | 12.98 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0613 | 2552 | 13.9
3 | SXS:BBH:1356 | Nonspinning | 1 | 18.87 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1010 | 6001 | 22.34
4 | SXS:BBH:1357 | Nonspinning | 1 | 16.24 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1123 | 2889 | 14.76
5 | SXS:BBH:1358 | Nonspinning | 1 | 15.90 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1097 | 2656 | 14.08
6 | SXS:BBH:1359 | Nonspinning | 1 | 15.74 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1103 | 2530 | 13.75
7 | SXS:BBH:1360 | Nonspinning | 1 | 16.70 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1643 | 2373 | 13.14
8 | SXS:BBH:1361 | Nonspinning | 1 | 16.70 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1691 | 2325 | 12.98
9 | SXS:BBH:1362 | Nonspinning | 1 | 17.70 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.2195 | 2147 | 12.28
10 | SXS:BBH:1363 | Nonspinning | 1 | 17.70 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.2208 | 2109 | 12.18
11 | SXS:BBH:0154 | Spin-aligned | 1 | 15.37 | 0.0 | 0.0 | -0.8 | 0.0 | 0.0 | -0.8 | -0.8 | 0.0000 | 3805 | 13.24
12 | SXS:BBH:0325 | Spin-aligned | 1 | 15.87 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.8 | -0.4 | 0.0000 | 5104 | 17.34
13 | SXS:BBH:1165 | Nonspinning | 2 | 19.92 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0000 | 15792 | 40.34
14 | SXS:BBH:1364 | Nonspinning | 2 | 14.54 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0522 | 3200 | 16.14
15 | SXS:BBH:1365 | Nonspinning | 2 | 15.03 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0677 | 3181 | 16.06
16 | SXS:BBH:1366 | Nonspinning | 2 | 16.03 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1110 | 3073 | 15.62
17 | SXS:BBH:1367 | Nonspinning | 2 | 15.88 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1091 | 2955 | 15.31
18 | SXS:BBH:1368 | Nonspinning | 2 | 15.73 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1066 | 2850 | 15.03
19 | SXS:BBH:1369 | Nonspinning | 2 | 18.18 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.2220 | 2616 | 13.92
20 | SXS:BBH:1370 | Nonspinning | 2 | 17.70 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.2167 | 2377 | 13.23
21 | SXS:BBH:2265 | Nonspinning | 3 | 22.56 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0000 | 30923 | 65.56
22 | SXS:BBH:1371 | Nonspinning | 3 | 14.97 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0627 | 3707 | 18.17
23 | SXS:BBH:1372 | Nonspinning | 3 | 15.97 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1064 | 3565 | 17.66
24 | SXS:BBH:1373 | Nonspinning | 3 | 15.84 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1058 | 3451 | 17.31
25 | SXS:BBH:1374 | Nonspinning | 3 | 18.10 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.2264 | 3015 | 15.59
26 | SXS:BBH:1220 | Nonspinning | 4 | 15.20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0000 | 7301 | 26.26
27 | SXS:BBH:0056 | Nonspinning | 5 | 15.00 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0000 | 7980 | 28.81
28 | SXS:BBH:0181 | Nonspinning | 6 | 14.00 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0000 | 6695 | 26.47
29 | SXS:BBH:0298 | Nonspinning | 7 | 12.20 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0000 | 4251 | 19.68