# A Rosetta Stone for eccentric gravitational waveform models

Alan M. Knee Department of Physics & Astronomy, University of British
Columbia, Vancouver, BC V6T 1Z1, Canada  Isobel M. Romero-Shaw School of
Physics & Astronomy, Monash University, Clayton, VIC 3800, Australia  OzGrav:
The ARC Centre of Excellence for Gravitational Wave Discovery, Clayton, VIC
3800, Australia  Department of Applied Mathematics and Theoretical Physics,
Cambridge CB3 0WA, United Kingdom  Paul D. Lasky School of Physics &
Astronomy, Monash University, Clayton, VIC 3800, Australia  OzGrav: The ARC
Centre of Excellence for Gravitational Wave Discovery, Clayton, VIC 3800,
Australia  Jess McIver Department of Physics & Astronomy, University of
British Columbia, Vancouver, BC V6T 1Z1, Canada  Eric Thrane School of Physics
& Astronomy, Monash University, Clayton, VIC 3800, Australia  OzGrav: The ARC
Centre of Excellence for Gravitational Wave Discovery, Clayton, VIC 3800,
Australia

###### Abstract

Orbital eccentricity is a key signature of dynamical binary black hole
formation. The gravitational waves from a coalescing binary contain
information about its orbital eccentricity, which may be measured if the
binary retains sufficient eccentricity near merger. Dedicated waveforms are
required to measure eccentricity. Several models have been put forward, and
show good agreement with numerical relativity at the level of a few percent or
better. However, there are multiple ways to define eccentricity for
inspiralling systems, and different models internally use different
definitions of eccentricity, making it difficult to directly compare
eccentricity measurements. In this work, we systematically compare two
eccentric waveform models, SEOBNRE and TEOBResumS, by developing a framework
to translate between different definitions of eccentricity. This mapping is
constructed by minimizing the relative mismatch between the two models over
eccentricity and reference frequency, before evolving the eccentricity of one
model to the same reference frequency as the other model. We show that for a
given value of eccentricity passed to SEOBNRE, one must input a $20$-$50\%$
smaller value of eccentricity to TEOBResumS in order to obtain a waveform with
the same empirical eccentricity. We verify this mapping by repeating our
analysis for eccentric numerical relativity simulations, demonstrating that
TEOBResumS reports a correspondingly smaller value of eccentricity than
SEOBNRE.

††software: PyCBC (Biwer et al., 2019), LALSuite (LIGO Scientific
Collaboration, 2018), SciPy (Virtanen et al., 2020), Bilby (Ashton et al.,
2019; Romero-Shaw et al., 2020), Matplotlib (Hunter, 2007)

##  1 Introduction

The LIGO-Virgo-KAGRA (LVK) collaboration has announced a total of 90 likely
detections111This is the total number of candidate signals with at least a 50%
probability of being astrophysical. Different thresholds can yield different
numbers of detections. of gravitational waves (GWs) from the first three
observing runs of the Advanced LIGO (Aasi et al., 2015), Advanced Virgo
(Acernese et al., 2015), and KAGRA (Akutsu et al., 2019) interferometers.
These detections are consistent with the inspiral and subsequent merger and
ringdown of coalescing compact binaries, including binary black hole (BBH),
binary neutron star, and neutron star-black hole systems (Abbott et al.,
2019a, 2021a, 2021b, 2021a). Analyses of the publicly available strain data
have yielded additional likely detections (Nitz et al., 2021; Olsen et al.,
2022; Venumadhav et al., 2020). BBH mergers constitute the majority of the
detections thus far. By studying their GW emissions, the source properties of
BBH systems can be inferred (e.g., Abbott et al., 2016, 2020), including the
masses of the two black hole companions, their spins, and their orbital
eccentricities.

Precise measurements of the intrinsic parameters describing BBH systems can
reveal important clues about how such systems are formed (Abbott et al.,
2019b, 2021b, 2021). In general, coalescing BBH systems can be formed via one
of two proposed formation channels: isolated binary evolution, facilitated by,
e.g., common envelope (Bethe & Brown, 1998; Belczynski et al., 2002; Voss &
Tauris, 2003) or chemically homogeneous evolution (Mandel & de Mink, 2016; de
Mink & Mandel, 2016), and dynamical assembly within dense stellar
environments, such as in the cores of globular (Sigurdsson & Hernquist, 1993;
Portegies Zwart & McMillan, 2000; Zwart & McMillan, 2002; Rodriguez et al.,
2016) and nuclear (Belczynski & Banerjee, 2020; Gerosa & Fishbach, 2021) star
clusters.

Orbital eccentricity is a promising signature of dynamical assembly.
Gravitational radiation efficiently circularizes binary systems (Peters &
Mathews, 1963; Peters, 1964), and any initial eccentricity present at the
epoch of isolated binary formation is expected to be almost completely damped
away when the GW signal reaches the observing band of ground-based detectors
($\sim 10$ Hz). Dynamical binaries, however, can sometimes form with
sufficiently small separation such that there is insufficient time for the
system to fully circularize before its radiation becomes observable, allowing
these sources to be distinguished from isolated binaries based on their
eccentricity (Rodriguez et al., 2018a, b; Lower et al., 2018; Samsing, 2018;
Zevin et al., 2019, 2021). Possible hints of eccentricity have already been
detected in a few BBH candidates observed by the LVK network (Romero-Shaw et
al., 2019, 2021, 2022), with GW190521 being a notable case study (Romero-Shaw
et al., 2020; Gayathri et al., 2022). As the number of detections increases,
measurements of eccentricity will place tighter constraints on the fraction of
BBH mergers which are of dynamical origin.

Currently, the literature features two eccentric inspiral-merger-ringdown
waveform models that have been used for some form of GW parameter estimation:
TEOBResumS (Damour & Nagar, 2014; Nagar et al., 2016, 2018, 2020a, 2020b;
Riemenschneider et al., 2021; Chiaramello & Nagar, 2020; Nagar et al., 2021)
and SEOBNRE (Cao & Han, 2017; Liu et al., 2020, 2022).222ENIGMA (Huerta et
al., 2018) is another eccentric model, but has so far only been used for
creating injections that were later recovered with a quasi-circular
approximant. There are also the inspiral-only models EccentricFD (Huerta et
al., 2014) and TaylorF2e (Moore & Yunes, 2019), which were used by Wu et al.
(2020) and Lenon et al. (2020) to analyze BBH and binary neutron star events,
respectively. For high-mass eccentric candidates like GW190521, it is
essential to use waveforms that include merger physics. These are time-domain,
aligned-spin models which employ the effective one-body (EOB) formalism
(Buonanno & Damour, 1999, 2000) to solve the general-relativistic two-body
problem. Both waveforms have been validated against numerical relativity
simulations of eccentric BBH systems (Hinder et al., 2018) with low-to-
moderate eccentricity ($\lesssim 0.3$), giving mismatch factors (defined later
in Equation 9) no larger than 2-3% (Cao & Han, 2017; Chiaramello & Nagar,
2020). In addition to TEOBResumS and SEOBNRE, several other eccentric waveform
models are under development (e.g., Ramos-Buades et al., 2022; Islam et al.,
2021; Chen et al., 2021; Yun et al., 2021; Setyawati & Ohme, 2021; Hinder et
al., 2018).

In parameter estimation, the waveform model typically needs to be evaluated on
the order of $10^{5}$ times or more. Neither of the waveforms introduced above
are fast enough for direct use in parameter estimation in their current “out-
of-the-box” states, and certain workarounds are required. Romero-Shaw et al.
(2019, 2020, 2021, 2022) constrained the eccentricities of 62 BBH candidates
with SEOBNRE by importance-sampling the parameter space with a computationally
cheaper quasi-circular waveform, and subsequently reweighting to the eccentric
posterior (e.g., Payne et al., 2019). Another study by O’Shea & Kumar (2021)
reanalyzed two BBH candidates with TEOBResumS by loosening the error tolerance
of the ODE integrator to speed up evaluation times. More recently, Iglesias et
al. (in prep.) reanalyzed five BBH candidates with TEOBResumS and the rapid
parameter estimation algorithm RIFT (Lange et al., 2018). Though they did not
conduct parameter estimation, Zevin et al. (2021) also used TEOBResumS to
evaluate the impact of selection effects on the detection of eccentric BBH
mergers.

By analyzing individual GW events with different waveform models, we can learn
more about the physics of the source system, as well as build confidence in
our inferences if the models yield similar results. However, the notion of
eccentricity is inherently ambiguous in general relativity, where particles in
a two-body system are not restricted to perfectly elliptical orbits. In
practice, different eccentric waveform models apply different definitions of
eccentricity, and so it is currently not straightforward to reconcile
measurements obtained with different eccentric waveforms.

As interest in eccentric binaries grows, it is becoming increasingly important
for our analyses to implement consistent definitions of eccentricity. In this
work, we address this problem by constructing a mapping between the
eccentricity definitions of TEOBResumS and SEOBNRE. Our approach is to
minimize the relative mismatch between these two models over the parameters of
interest, namely their eccentricity and reference frequency. We then evolve
the eccentricities to the same frequency. Using this mapping, a measurement of
eccentricity obtained with one model can be converted into an equivalent
eccentricity measured by the other model, allowing for a more direct
comparison.

The rest of this paper is organized as follows: In Section 2, we discuss the
challenges of defining eccentricity in general relativity and how this leads
to discrepancies between waveform models; in Section 3, we outline our method
for mapping the eccentricities defined by TEOBResumS and SEOBNRE; in Section 4
we discuss the results of our eccentricity mapping analysis; finally, in
Section 5, we offer concluding remarks.

##  2 Background

###  2.1 Eccentric compact binaries

Eccentricity is most intuitively understood in the Newtonian regime, where it
simply describes the amount by which an orbit deviates from a circle. In the
Keplerian parameterization, the eccentricity, $e$, is defined through the
orbital equation of motion

$r(t)=\frac{a(1-e^{2})}{1+e\cos\phi(t)}\,,$ (1)

where $r$ is the radial distance from the focus, $a$ is the semimajor axis,
and $\phi$ is the true anomaly. An eccentricity of $e=0$ is a circular orbit,
$0<e<1$ is an eccentric orbit, and $e\geq 1$ is a parabolic (if equal) or
hyperbolic unbound orbit. For a binary system with component masses $m_{1,2}$,
each with semimajor axes $a_{1,2}$, the orbital (angular) frequency obeys
Kepler’s third law,

$\bar{\omega}^{2}a^{3}=GM\,,$ (2)

where $M=m_{1}+m_{2}$ is the total mass, and $a=a_{1}+a_{2}$ is the sum of the
semimajor axes of each object. The quantity $\bar{\omega}$ represents an
average orbital frequency, which we refer to as the “Keplerian” frequency.
Ignoring the inspiral for a moment and considering just the motion of the two
objects along unchanging, closed orbits, the instantaneous orbital frequency
is given by (e.g., Poisson & Will, 2014)

$\omega(t)=\frac{\bar{\omega}\sqrt{1-e^{2}}}{[1-e\cos
E(t)]^{2}}\approx\bar{\omega}[1+2e\cos(\bar{\omega}t)]\,,$ (3)

where $E(t)$ is the eccentric anomaly, and the second expression follows in
the low-eccentricity limit. Thus, in eccentric binaries the frequency consists
of a Keplerian component, $\bar{\omega}$, which is constant in the absence of
radiation-reaction, plus an oscillating component with amplitude
$2e\bar{\omega}$ and period equal to that of the orbit, which vanishes in the
quasi-circular limit. The same is true of the GW signal, which in the case of
eccentric systems receives contributions from several harmonics of the orbital
frequency, instead of the power being concentrated in the second harmonic as
in quasi-circular systems (Peters & Mathews, 1963). The emission of GWs is
also asymmetric, with greater power radiated during a periastron passage
compared to apastron.

This picture becomes more complicated when we properly account for the energy
lost in GWs, introducing a dissipative radiation-reaction force which causes
the system to inspiral. At leading order, the semimajor axis and eccentricity
of the binary will decay according to Peters’ equations (Peters, 1964):

$\bigg{\langle}\frac{{\rm d}a}{{\rm
d}t}\bigg{\rangle}=-\frac{64}{5}\frac{G^{3}m_{1}m_{2}M}{c^{5}a^{3}(1-e^{2})^{7/2}}\bigg{(}1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\bigg{)}\,;$
(4) $\bigg{\langle}\frac{{\rm d}e}{{\rm
d}t}\bigg{\rangle}=-\frac{304}{15}\frac{G^{3}m_{1}m_{2}M}{c^{5}a^{4}(1-e^{2})^{5/2}}\bigg{(}1+\frac{121}{304}e^{2}\bigg{)}\,,$
(5)

where angle brackets indicate that the derivatives are orbit-averaged
quantities. It is important to note that the Keplerian interpretation of
eccentricity is only meaningful in an adiabatic sense, when the system is
evolving slowly enough such that the orbits remain nearly closed on orbital
timescales. The adiabatic approximation is thus accurate during the early
inspiral, but inevitably breaks down closer to merger. In the highly
relativistic regime, alternative ways of measuring eccentricity which better
reflect the true dynamics of the system are needed.

Eccentricity is challenging to define rigorously in general relativity. At the
core of this problem is the fact that, while its physical effect on GW
emission is observable, eccentricity is nonetheless gauge-dependent. As a
result, there is no single definition of eccentricity in general relativity
that can be applied universally, and several definitions have been conceived
within different contexts (see e.g., Loutrel et al., 2019a, for a review). In
the post-Newtonian formalism, eccentricity is frequently expressed in terms of
the orbit-averaged quantities $e_{r}$, $e_{t}$, and $e_{\phi}$, which is known
as the quasi-Keplerian parameterization (Damour & Deruelle, 1985, 1986;
Blanchet, 2014). In contrast, the so-called osculating method for measuring
the eccentricity (Damour et al., 2004; Pound, 2010) does not use orbit-
averaged quantities, and yields behaviour that apparently contradicts post-
Newtonian theory (Loutrel et al., 2019b). In the field of numerical
relativity, eccentricity is usually measured in terms of the coordinate
separation between the binary components (Boyle et al., 2007; Pfeiffer et al.,
2007; Tichy & Marronetti, 2011), or the orbital or GW frequency of the system
(Mroue et al., 2010; Buonanno et al., 2011; Ramos-Buades et al., 2019),
neither of which provide exactly the same measurement since these methods
assume a coordinate system and are therefore gauge-dependent.

###  2.2 Eccentric waveforms

In waveform modelling, it is typical to define time-dependent quantities such
as eccentricity, inclination angle and spin tilts in terms of a reference
frequency.333Recent work, e.g., by Mould & Gerosa (2022), seeks to define
time-dependent quantities at past time infinity. Eccentric models therefore
utilize a reference eccentricity, $e_{0}$, defined at some GW reference
frequency, $f_{\rm ref}$, which are both taken as inputs by the models, in
addition to the masses and spins. Note that these models do not employ any
cosmology, and therefore the parameters are implicitly defined in the detector
frame. For TEOBResumS and SEOBNRE, the input eccentricity and reference
frequency are used to determine a set of adiabatic initial conditions from
which the trajectories of the binary components are evolved. It is important
to emphasize that the eccentricities supplied to the waveform models do not
correspond to systems with the same physical eccentricity, and therefore these
“waveform eccentricities” carry different meaning depending on the waveform
model.

Figure 1: Comparison of the two waveform models, TEOBResumS and SEOBNRE, for
an equal mass, non-spinning BBH with total mass $M_{\rm det}=100\,M_{\odot}$
in the detector frame, showing the $(\ell=2,|m|=2)$ waveform frequency as a
function of time. The waveforms are given identical input parameters, with
waveform eccentricity $e_{0}=0.1$ at a reference frequency of $10$ Hz
(indicated by a horizontal line). For the same input value of eccentricity,
the frequency oscillations in TEOBResumS are noticeably larger, implying a
higher simulated eccentricity. TEOBResumS also interprets the reference
frequency differently (described in Section 2.2), causing it to start at a
lower frequency than SEOBNRE. Lastly, the waveforms start from different
points in the orbit: TEOBResumS starts from apastron (a trough), and SEOBNRE
starts from periastron (a peak).

The instantaneous GW frequency calculated with TEOBResumS and SEOBNRE (using
identical parameters) are compared in Fig. 1. For non-precessing binaries, the
$(\ell=2,|m|=2)$ waveform can be simply decomposed into an amplitude and
complex phase,

$h_{+}(t)-ih_{\times}(t)=A_{22}(t)\exp[-i\Phi_{22}(t)]\,,$ (6)

where $h_{+,\times}$ are the two GW polarizations, and the GW frequency is
given by the time derivative of the phase,

$2\pi f_{22}(t)=\frac{{\rm d}\Phi_{22}}{{\rm d}t}\,.$ (7)

The shapes of the orbits are encoded in the frequency evolution. As explained
in Section 2.1, the frequency consists of a monotonically increasing Keplerian
component, and an eccentricity-induced oscillating component. The amplitude of
this oscillating component provides a direct measure of the eccentricity of
the system, and can be used to compare the eccentricity of two different
waveform families. Furthermore, we can identify local maxima in the frequency
oscillations with periastron passages, and local minima with apastron
passages. The oscillations become suppressed as the inspiral progresses due to
the circularization of the system.

The eccentric EOB models we examine in this work, TEOBResumS and SEOBNRE,
ostensibly use the Keplerian definition of eccentricity, yet produce waveform
templates with apparently different eccentricities as determined by the
amplitude of the frequency oscillations, from which one can derive an
empirical measure of eccentricity (Mora & Will, 2002)

$e_{\omega}=\frac{\omega_{\rm p}^{1/2}-\omega_{\rm a}^{1/2}}{\omega_{\rm
p}^{1/2}+\omega_{\rm a}^{1/2}}\,,$ (8)

where $\omega_{\rm p,a}$ are the orbital or GW frequencies measured at
periastron and apastron passage, respectively. There are multiple reasons for
this discrepant behaviour:

  1. 1.

There are a number of differences between the two waveform models related to
their treatment of the conservative dynamics of the system and the radiation-
reaction under eccentric conditions. Furthermore, the models employ a
different logic for setting the initial conditions of the system, which
couples with the above factors to render an effectively model-dependent
definition of eccentricity. Given the same input value of eccentricity, the
waveforms end up simulating different levels of eccentricity, as judged by
Equation 8.

  2. 2.

The models are calibrated differently to numerical relativity simulations,
which can yield additional mismatch at merger.

  3. 3.

The models use different conventions for the reference frequency, $f_{\rm
ref}$, which the waveforms also use as the starting frequency of the system.
SEOBNRE defines $f_{\rm ref}$ with respect to the Keplerian frequency of the
GW radiation. For TEOBResumS, the user is able to define $f_{\rm ref}$ as
either the GW frequency at periastron, apastron, or the mean of the two. The
last option is closest to what SEOBNRE uses, but it is still not equivalent.
For the same value of $f_{\rm ref}$, TEOBResumS in general begins from a lower
initial frequency than SEOBNRE, since its definition corresponds to a slightly
smaller Keplerian frequency than SEOBNRE. For waveform eccentricity
$e_{0}=0.2$, the difference in starting frequency reaches about 10% which, at
lower masses, translates into substantial difference in waveform length.444For
example, changing the initial frequency of a $250\,M_{\odot}$ system from $10$
Hz to $9$ Hz increases the time to merger by $0.15$ seconds, but increases it
by $2.7$ seconds for a $50\,M_{\odot}$ system. This means that in addition to
the models using intrinsically different eccentricity definitions, TEOBResumS
parameterizes eccentricity at an earlier reference point than SEOBNRE.

  4. 4.

Neither model admits the mean anomaly as an input parameter. Instead, the
models initialize the system from opposite orbital configurations: TEOBResumS
always starts from apastron, SEOBNRE always starts from periastron. Because of
this, the arguments of periastra are often misaligned unless one carefully
tunes the reference frequency to avoid a persistent phase offset in the
frequency oscillations. Work is underway to incorporate a variable mean
anomaly in parameter estimation with these models (Islam et al., 2021);
however, at current detector sensitivity, neglecting this parameter is not
expected to bias results (Clarke et al., 2022).

Some or all of these factors may be present when comparing other eccentric
waveform models.

The amount of agreement between two complex waveforms, $h_{1}$ and $h_{2}$, is
quantified with their match (or overlap), defined through a noise-weighted
inner product in some frequency band maximized over a time and phase of
coalescence (Flanagan & Hughes, 1998; Lindblom et al., 2008),

$\mathcal{O}(h_{1},h_{2})=\max_{t_{0},\Phi_{0}}\frac{\langle
h_{1}|h_{2}\rangle}{\sqrt{\langle h_{1}|h_{1}\rangle\langle
h_{2}|h_{2}\rangle}}\,,$ (9)

where

$\langle h_{1}|h_{2}\rangle=4\Re\int_{f_{\rm low}}^{f_{\rm
high}}\frac{\tilde{h}_{1}(f)\tilde{h}_{2}^{*}(f)}{S_{\rm n}(f)}\,{\rm d}f\,.$
(10)

Here, tildes denote a Fourier transform, and $S_{\rm n}(f)$ is the noise power
spectral density (PSD) in the detector. The match is normalized such that a
value of $\mathcal{O}=1$ means the waveforms are identical (within the
specified band), and $\mathcal{O}=0$ means the waveforms are maximally
different. The difference between the waveforms is thus given by their
mismatch, $1-\mathcal{O}(h_{1},h_{2})$. In Figure 2, we show the relative
mismatch between TEOBResumS and SEOBNRE as a function of waveform
eccentricity. To make the comparison clearer, we use Equation 3 to adjust the
reference frequency555This formula is only an approximation because the
eccentricities are not interchangeable. However, we find that it works well
within the eccentricity range explored by this work. supplied to TEOBResumS,
ensuring that it is initialized with the same initial Keplerian frequency as
SEOBNRE. If we do not do this, the mismatch oscillates as the eccentricity
changes due to the arguments of periastra going in and out of phase. With
consistent reference frequencies, we can more clearly observe that the
mismatch between the models increases as the eccentricity increases, reaching
$1\%$ for $e_{0}\sim 0.05$ and exceeding $10\%$ for $e_{0}\sim 0.2$ (with
white noise). The models also differ in their handling of the transition from
eccentric to quasi-circular dynamics, which we show in Figure 3 by computing
the mismatch between the eccentric ($e_{0}>0$) and quasi-circular ($e_{0}=0$)
configurations of each model as a function of waveform eccentricity. Figure 3
shows that SEOBNRE approaches quasi-circular behaviour as $e_{0}\rightarrow
0$. In contrast, the eccentric configuration of TEOBResumS, known as
TEOBResumS-DALI, displays a minimum level of mismatch with its quasi-circular
counterpart, TEOBResumS-GIOTTO, for arbitrarily small (but non-zero)
eccentricities, reflecting the systematic differences between the
models.666Specifically, SEOBNRE utilizes quasi-circular initial conditions
(Buonanno et al., 2006) adjusted by an eccentric factor (Cao & Han, 2017), and
so it obtains exactly quasi-circular initial conditions in the limit of zero
eccentricity. TEOBResumS-DALI implements post-adiabatic initial conditions
(Chiaramello & Nagar, 2020; Damour et al., 2013; Hinderer & Babak, 2017), but
is discontinuous with TEOBResumS-GIOTTO, in part due to a difference in the
description of radiation-reaction which does not vanish in the limit of
$e_{0}\rightarrow 0$.

Figure 2: Mismatch between TEOBResumS and SEOBNRE as a function of waveform
eccentricity at $10$ Hz, showing how the mismatch increases with higher
eccentricity. Curves are shown for equal mass systems with different total
masses as indicated. The reference frequencies supplied to TEOBResumS are
adjusted so that the waveform starts with a Keplerian frequency of $10$ Hz,
ensuring consistency with the way SEOBNRE defines the reference frequency. We
show mismatches calculated for both white noise (solid curves) and LVK
sensitivity (dashed curves).

##  3 Method

In order to construct a map between different definitions of eccentricity, we
find the optimal values of the waveform eccentricity and reference frequency
which minimize the mismatch (Equation 9), given fixed component masses and
aligned spin. This will tell us what input eccentricity we need to pass to
TEOBResumS to obtain a waveform template that is as similar as possible to a
given SEOBNRE template, and vice versa.

Figure 3: Mismatch between the eccentric and quasi-circular version of
TEOBResumS and SEOBNRE ($50\,M_{\odot}+50\,M_{\odot}$, non-spinning), plotted
as a function of waveform eccentricity at $10$ Hz. As the eccentricity of
SEOBNRE decreases, there is a corresponding decrease in mismatch with its
quasi-circular counterpart. In contrast, the mismatch between the eccentric
model TEOBResumS-DALI and the quasi-circular model TEOBResumS-GIOTTO levels
off at a minimum value, reflecting a discontinuity between the behaviours of
the two models. As in Figure 2, the mismatches are shown for white noise
(solid curves) and LVK sensitivity (dashed curves), and we use the same
frequency adjustment for TEOBResumS.

Our strategy is as follows. Since SEOBNRE takes much longer to evaluate than
TEOBResumS, we generate a fiducial set of 550 SEOBNRE waveforms with detector-
frame total masses ranging from $50\,M_{\odot}$ to $300\,M_{\odot}$, and
eccentricities from $e_{0}^{\rm SEOB}=0$ to $0.4$ (uniformly) at $10$ Hz. We
then minimize the mismatch with TEOBResumS by varying its waveform
eccentricity and reference frequency,777We have to minimize over $f_{\rm ref}$
because the models are initialized from opposite starting points in the orbit.
We adopt the Keplerian convention for the reference frequency, as explained in
Section 2.2, meaning we need to use Equation 3 to adjust the reference
frequency passed to TEOBResumS so that it starts from the desired Keplerian
frequency. keeping all other parameters identical to SEOBNRE. To examine how
the eccentricity conversion varies with the mass ratio and spins, we also
generate SEOBNRE waveforms with mass ratios from $q=m_{1}/m_{2}=1$ to $4$, and
with component spins888Note that SEOBNRE is not valid for large spins
$|\chi_{1,2}|\geq 0.6$. between $\chi_{1}=\chi_{2}=-0.5$ and $0.5$. We
consider only the case where the spins are aligned with the orbital angular
momentum of the system (i.e. non-precessing), as neither model allows for
misaligned spins. For our mismatch calculations, we integrate over a frequency
band starting from $10$ Hz and ending at the Nyquist frequency of $2048$ Hz.
As the SEOBNRE model is initialized from periastron, the initial
(instantaneous) frequency of the waveform will generally be above the
requested frequency of $10$ Hz. We avoid any issues that may arise from having
the waveforms begin in-band by generating them from a lower frequency than
$10$ Hz. We use Peters’ equations (Equations 4–5) to back-evolve the
eccentricity as needed, relating the semimajor axis to the GW frequency with
Kepler’s third law, $a=[GM/(\pi f_{\rm GW})^{2}]^{1/3}$. We back-evolve to $5$
Hz in all cases except for the $50\,M_{\odot}$ waveforms, for which we use $7$
Hz. Similarly, we ensure that the TEOBResumS waveforms do not start in-band by
varying the reference frequency between $5$–$10$ Hz (or $7$–$10$ Hz in the
low-mass case). The start of the inspirals are tapered with a half-Tukey
window rolled over 0.2 seconds to prevent spectral leakage from contaminating
the Fourier transforms. Only the (2,2)-mode waveforms are considered in our
analysis, since the publicly available version of SEOBNRE does not currently
include higher-order modes.

In order to combine information from both GW polarizations, we calculate the
mismatch by substituting the detector response (Thorne, 1987),

$h(t)=F_{+}(\alpha,\delta,\psi)h_{+}(t)+F_{\times}(\alpha,\delta,\psi)h_{\times}(t)\,,$
(11)

directly into Equation 9, where $F_{+,\times}$ are the antenna patterns, which
are functions of the sky location of the source (right ascension, $\alpha$,
and declination, $\delta$), and the polarization angle, $\psi$. We set the sky
location by choosing the optimal orientation relative to the Livingston
detector at some arbitrary reference time, setting $\psi=0$ and assuming face-
on inclination.

Figure 4: Left panel: Mismatch between TEOBResumS waveforms and a fiducial
SEOBNRE waveform for an equal-mass non-spinning BBH with total mass
$200\,M_{\odot}$ in the detector frame, assuming a white noise PSD. The
TEOBResumS waveforms are distributed over a grid in waveform eccentricity and
(Keplerian) reference frequency. The SEOBNRE waveform has $e_{0}^{\rm
SEOB}=0.1$ at $10$ Hz. The colour scale shows the log-scaled mismatch between
the waveforms at each grid point, where darker shading indicates a smaller
mismatch. The regions of lower mismatch coincide with successive alignments of
the arguments of periastron. The red cross marks the grid point with the
lowest mismatch, which sits at the centre of a bounding box that is used as
the search range for the optimization algorithm. The true minimum found by the
optimizer is given by the cyan star, which is at a mismatch of
$1-\mathcal{O}=0.0071$. For reference, the mismatch found when supplying
identical parameters to both waveforms is $0.1567$. The cyan curve shows the
eccentricity evolution track calculated with Peters’ equations, which is used
to forward-evolve the eccentricity to $10$ Hz. Right panels: Comparison of the
fiducial SEOBNRE waveform (as observed in a detector; red curve) with two
TEOBResumS waveforms, in which we supply either identical parameters as
SEOBNRE (blue curve) or the optimal parameters (dashed black curve) from the
parameter search that give the minimum mismatch. The top panel shows the time-
domain strain, and the bottom panel shows the frequency evolution. The plotted
waveforms are each aligned in coalescence phase and time.

Due to the highly oscillatory behaviour of the mismatch as a function of
eccentricity and reference frequency, we determine the minimum mismatch point
using a two-step procedure. For each SEOBNRE template, we first perform a
coarse grid search over the waveform eccentricity of the TEOBResumS model,
$e_{0}^{\rm TEOB}$, and in the reference frequency. Then, we obtain a more
precise answer by using a Nelder-Mead optimizer to search over a small region
centered on the lowest-mismatch grid point. An example mismatch grid is shown
in the left panel of Figure 4. By construction, the minimum mismatch point
corresponds to an eccentricity measured at a lower frequency than $10$ Hz,
where $e_{0}^{\rm SEOB}$ is defined. We correct for this by again using
Peters’ equations, this time to forward-evolve $e_{0}^{\rm TEOB}$ to $10$
Hz.999This introduces a small error on the evolved eccentricity, as Peters’
equations are less accurate in this frequency range at higher masses.

We construct maps assuming both white noise (uniform) and LVK design-noise
sensitivity PSDs. Using white noise tells us fundamentally how the
eccentricity definitions differ, whereas using detector PSDs tells us what
eccentricity one would actually measure with the two waveform models. These
methods need not give the same answer. The detector curves weigh frequency
bins differently, tending to weigh higher frequencies near merger more
strongly at the expense of lower inspiral frequencies, where the eccentricity
is larger. In contrast, white noise applies equal weighting at all
frequencies. The results presented in Section 4 mainly use white noise, but we
include examples of eccentricity maps computed with LVK noise in Appendix A.

##  4 Results

###  4.1 Waveform eccentricity maps

Figure 5: The mapping between the eccentricity definitions used by SEOBNRE and
TEOBResumS obtained with our mismatch-minimization procedure, for various
choices of total (detector-frame) mass, equal-masses and zero spins. The
mismatch calculations assume white noise. All eccentricities are quoted at a
reference GW frequency of $10$ Hz. The data points are coloured by the minimum
mismatch between the TEOBResumS and SEOBNRE waveforms before evolving to $10$
Hz. The vertical lines indicate where the minimum mismatch crosses certain
thresholds: waveforms to the right of the dashed line have
$1-\mathcal{O}>2\%$, and to the right of the solid line have
$1-\mathcal{O}>5\%$. The absence of any of these lines means the minimum
mismatch never passes that threshold within the explored parameter ranges. The
insets show the ratio of the two eccentricities as a function of $e_{0}^{\rm
SEOB}$. To avoid unphysical model behaviour, we have to limit the maximum
SEOBNRE eccentricity for heavier systems.  Figure 6: Effect of varying the
mass ratio and component spins on the the eccentricity mapping between
TEOBResumS and SEOBNRE (at 10 Hz). The total mass here is fixed at
$150\,M_{\odot}$. The top left panel is repeated from Figure 5. Again, the
insets show the ratio of the mapped eccentricities, and the vertical lines
denote the same mismatch thresholds from Figure 5.

In this section, we discuss the results of our analysis of TEOBResumS and
SEOBNRE, using the mismatch-minimization procedure described in Section 3. The
mapping between the two eccentricity definitions are shown in Figure 5, in
which all eccentricities are defined at $10$ Hz. Across the parameter space we
explore, we find that a smaller value for eccentricity must be passed to
TEOBResumS compared to SEOBNRE in order to produce the best-matching waveform.
If we represent the mapping from SEOBNRE eccentricity to TEOBResumS
eccentricity by $e_{0}^{\rm TEOB}=ke_{0}^{\rm SEOB}$, then the conversion
factor, $k$, typically fluctuates between $0.5$-$0.8$. In other words, for a
given numerical value of SEOBNRE eccentricity, we must pass a $20$-$50\%$
smaller value of eccentricity to TEOBResumS to obtain an equivalently
eccentric waveform. This is true provided the reference eccentricity passed to
the waveforms is also made to be consistent, as noted in Section 2.2.

The conversion factor is sensitive to the intrinsic parameters of the system,
including both the eccentricity parameter itself as well as the masses and
spins. As shown in the insets of Figure 5, the conversion factor is stable at
lower masses and high eccentricities, but as we go to lower eccentricities
this factor changes, requiring larger adjustments to match the waveforms. This
is likely a consequence of the models’ diverging behaviour in the small-
eccentricity regime (see Figure 3). Relatedly, the eccentric versions of the
two models cannot be reconciled at low but non-zero eccentricities. When the
SEOBNRE eccentricity is sufficiently small, we find that the quasi-circular
TEOBResumS-GIOTTO waveform yields a lower mismatch than any eccentric
TEOBResumS-DALI waveform. Thus, Figure 5 shows that small enough SEOBNRE
eccentricities are mapped onto zero TEOBResumS eccentricity. Figure 6 shows
how the conversion varies slightly with the masses and spins, but remains
between $20$-$50\%$ except in the case of larger aligned-spins, where the
needed adjustment reaches as low as $10\%$.

Figure 5 shows that the minimum mismatch between the waveforms increases with
larger eccentricity. We have drawn vertical lines to denote when the mismatch
passes $2\%$ and $5\%$, giving a sense of the eccentricity and mass range for
which our eccentricity maps yield roughly equivalent waveforms. As expected,
lighter systems exceed these thresholds at lower eccentricities than heavier
systems, as these system retain a larger fraction of their inspiral in-band.
Since it is during the inspirals where the models are maximally different,
this contributes to higher overall mismatch. For example, at detector-frame
mass $100\,M_{\odot}$ the lowest possible mismatch between SEOBNRE and
TEOBResumS is greater than $5\%$ for eccentricities $e_{0}^{\rm SEOB}\gtrsim
0.2$ at $10$ Hz. However, for a $250\,M_{\odot}$ system that mismatch never
exceeds $2\%$ for $e_{0}^{\rm SEOB}<0.2$.

We also observe that the relation between $e_{0}^{\rm SEOB}$ and $e_{0}^{\rm
TEOB}$ appears to oscillate as a function of eccentricity for higher-mass
systems, becoming more noticeable at higher masses. It is not clear yet why
this occurs, but is probably related to systematic differences between the two
models when measuring the eccentricity closer to merger.

###  4.2 Numerical relativity analysis

To complement our eccentricity definition study, we repeat our mismatch-
minimization procedure to measure the eccentricities of numerical relativity
simulations, cross-checking the results with our eccentricity maps. We analyze
a set of eccentric BBH simulations provided in the SXS catalog (Boyle et al.,
2019; Hinder et al., 2018), which includes non-spinning and aligned-spin
configurations, and mass ratios ranging from $q=1$ to 3. The waveforms are
scaled to a total mass of $150\,M_{\odot}$, which ensures each simulation has
frequency content starting below $10$ Hz, which we again integrate from in our
mismatch calculations. We use only the $(\ell=2,|m|=2)$ waveforms, and we
remove the initial junk radiation from the simulations.

Figure 7: Eccentricities in our chosen set of numerical relativity simulations
measured with TEOBResumS and SEOBNRE using our mismatch-minimization
procedure. The eccentricities are evolved to a frequency of $10$ Hz. The
simulations were scaled to a total mass of $150\,M_{\odot}$ in the detector
frame, and the mismatches are computed with respect to either white (circles)
or LVK (crosses) noise, integrating from $10$ Hz. Note that three simulations
(SXS:BBH:1136, SXS:BBH:1149, SXS:BBH:1169) lack an SEOBNRE measurement due to
having spins outside the allowed range of this model, $|\chi_{1,2}|<0.6$.

The measured eccentricities are shown in Figure 7, where we report results
assuming both white and LVK noise. These results are consistent with our
eccentricity maps, with TEOBResumS consistently measuring a smaller
eccentricity than SEOBNRE to compensate for its larger simulated eccentricity.
At LVK sensitivity, we find that TEOBResumS-DALI cannot yield as low of a
mismatch as TEOBResumS-GIOTTO for a subset of the simulations, indicated by a
measured eccentricity of zero. This is due to a combination of factors, namely
the differing behaviours of the two models at small eccentricities, and
because the detector PSDs down-weight the inspiral frequencies (where the
eccentricity is largest) relative to the merger frequencies.

###  4.3 Limitations

TEOBResumS and SEOBNRE possess multiple differences aside from their different
definitions of eccentricity. Consequently, there will be potentially
measurable differences between the waveforms that cannot be completely
eliminated by varying their parameters. Furthermore, our procedure does not
uniquely fit for the different eccentricity definitions. Systematic
differences between the models related to their treatment of radiation-
reaction, the conservative dynamics (via the definition of the Hamiltonian),
and their numerical relativity calibration will inevitably affect the best-
fitting parameters to some degree. We examine the utility of our eccentricity
maps by comparing the optimal signal-to-noise ratio (SNR) of the residuals
between the mapped TEOBResumS and SEOBNRE waveforms with their minimum
mismatch. This tells us whether these difference are detectable at LVK
sensitivity, which would degrade the usefulness of our eccentricity maps.

For a given PSD, the optimal SNR is defined as the noise-weighted inner
product of the waveform with itself,

$\rho_{0}=\sqrt{\langle\delta h|\delta h\rangle}\,,$ (12)

where $\delta h=h_{\rm TEOB}-h_{\rm SEOB}$ is the residual strain after
aligning the waveforms in coalescence time and phase. As a general rule, this
difference is potentially detectable if (Flanagan & Hughes, 1998; Lower et
al., 2018)

$1-\mathcal{O}\gtrsim\rho_{0}^{-2}\,,$ (13)

where we average over sky location, polarization angle, and orbital
inclination. Since the mismatch calculations used to construct the
eccentricity maps assume white noise PSDs, we recompute these mismatches under
LVK sensitivity to facilitate comparison with the SNRs. If the above condition
is not met, i.e. that $1-\mathcal{O}<\rho_{0}^{-2}$, then no differences
between the waveforms can be detected (Pürrer & Haster, 2020).101010Note that
Chatziioannou et al. (2017) and Pürrer & Haster (2020) argue that the right-
hand-side of Equation 13 should contain a factor of $D-1$, where $D$ is the
number of waveform parameters. Including this factor would increase the
mismatch threshold required to distinguish the waveforms, expanding the region
of parameter space where the waveforms cannot be distinguished. Equation 13
thus gives a more conservative range for where our eccentricity maps are most
useful.

Figure 8: Detectability of the residual differences between SEOBNRE and
TEOBResumS after minimizing their mismatch. The dark blue (red) curve shows
the minimum mismatch between the waveforms, assuming LVK design sensitivity,
as a function of SEOBNRE eccentricity at 10 Hz for a $M=100\,M_{\odot}$
($M=250\,M_{\odot}$) BBH at a distance of 1 Gpc (5 Gpc). The light red/blue
curves show the inverse squared optimal SNR of the difference, $\delta
h=h_{\rm TEOB}-h_{\rm SEOB}$, between the mismatch-minimized waveforms, again
computed with LVK sensitivity. For reference, we have drawn horizontal dashed
lines for certain values of optimal SNR. The differences are detectable
wherever the darker-shaded curve crosses above the lighter curve of the same
base colour. The differences between the waveforms for $100\,M_{\odot}$ at 1
Gpc become detectable above roughly $e_{0}^{\rm SEOB}=0.25$, whereas they are
never detectable for the $M=250\,M_{\odot}$ system at 5 Gpc in the parameter
range we explored.

Figure 8 shows the results of our mismatch-SNR comparison. We consider two
classes of sources in our analysis: a $100\,M_{\odot}$ BBH (detector frame)
located at $1$ Gpc, and a $250\,M_{\odot}$ BBH located at 5 Gpc. We find it is
unlikely that we would be able to detect the residual difference across much
of the parameter space, especially in distant, high-mass sources with only a
handful of cycles in-band such as GW190521. For such systems, the SNR of the
residuals registers between $1$-$2$ at LVK sensitivity. Our method will be
less effective for less massive, highly eccentric systems, where we see a
greater build-up in SNR from $\delta h$ due to the substantially longer
observable inspiral. For our specific example of a $100\,M_{\odot}$ BBH shown
in Figure 8, the residual SNR reaches the $5$-$10$ range for eccentricities
$e_{0}^{\rm SEOB}\gtrsim 0.25$ at $10$ Hz, where it starts to be comparable to
the mismatch. We note, however, that this eccentricity range is above what was
considered by Romero-Shaw et al. (2019, 2020, 2021, 2022), which probed
SEOBNRE eccentricities up to only $e_{0}^{\rm SEOB}=0.2$. If the system was
located further away, then we would be even less likely to detect any
difference between the mapped waveforms.

##  5 Summary

The importance of orbital eccentricity as a probe of dynamical BBH formation
underscores the need to have an in-depth understanding of how eccentricity is
defined by different waveform models. These models can also differ in the
definition of the reference frequency and initial mean anomaly, further
complicating the interpretation of our measurements.

In this work, we perform various studies aimed at reconciling the eccentricity
definitions used by two eccentric EOB waveform models, TEOBREsumS and SEOBNRE.
We find the optimal values of the waveform eccentricities and reference
frequencies which minimize the mismatch between the models, and then evolve
the eccentricities to a common frequency of $10$ Hz. Our analysis shows that
TEOBResumS simulates a larger eccentricity for the same numerical input at
SEOBNRE, which must be adjusted downwards to match the eccentricity of
SEOBNRE. However, we also find that our method does not work to resolve very
small eccentricities due to the behaviours of the two models at small
eccentricities. In addition, we highlight other waveform differences that must
be accounted for when comparing parameter estimation results, mainly that the
waveforms initialize the system from opposite orbital positions, and the use
of inconsistent reference frequencies.

We supplement this work with a comparison to a set of eccentric numerical
relativity simulations, showing that the eccentricities measured by TEOBResumS
and SEOBNRE via our mismatch procedure are consistent with our established
maps. Additional work, however, is still required to fully understand why
these two models yield different eccentricity measurements, which we discuss
briefly in Appendix B.

Lastly, we examine the robustness of our method in terms of producing a
faithful conversion between the two models. For low masses and high
eccentricities, our mismatch minimization returns waveforms with residual
mismatches that are potentially detectable at LVK sensitivity. For heavier and
more distant systems like GW190521, we find that these differences are likely
not detectable. The definition of eccentricity could potentially be
disentangled from other features of the waveform by measuring the eccentricity
directly from its frequency evolution. The exploration of a standardized
eccentricity parameter that could be used to compare measurements across
different models is currently underway (Shaikh et al., in prep.; Bonino et
al., in prep.).

We thank Alessandro Nagar, Rossella Gamba, Piero Rettegno, and Zhoujian Cao
for answering our queries about their respective waveform models. We thank
Vijay Varma and Harald Pfeiffer for helpful discussions concerning
eccentricity definitions. We also thank Harald Pfeiffer for providing comments
on this manuscript. This material is based upon work supported by NSF’s LIGO
Laboratory which is a major facility fully funded by the National Science
Foundation. We are grateful for computational resources provided by the LIGO
Laboratory and supported by National Science Foundation Grants PHY-0757058 and
PHY-0823459. A.M.K. and J.M. acknowledge funding support from the Natural
Sciences and Engineering Research Council of Canada (NSERC) through the
Discovery Grants program. A.M.K. was also supported by the John I. Watters
Research Fellowship. I.M.R.-S, P.D.L, and E.T. acknowledge support from the
Australian Research Council (ARC) Centre of Excellence CE170100004. I.M.R.-S.
also acknowledges support received from the Herchel Smith Postdoctoral
Fellowship Fund.

##  Appendix A Eccentricity maps assuming LVK sensitivity

Figure 9 shows how our eccentricity maps appear when using LVK sensitivity to
calculate the waveform mismatch. Specifically, we used the Advanced LIGO zero-
detuned high-power noise curve provided in LALSimulation (LIGO Scientific
Collaboration, 2018). The relation between the two eccentricities is
noticeably less smooth, as shown in Figure 9. Compared to our white noise
results, the eccentricity of SEOBNRE must be much higher before the lowest-
mismatch TEOBResumS waveform is not quasi-circular. This occurs because the
LVK sensitivity curve weighs higher frequencies near merger more strongly than
deeper in the inspiral, and so the system must have larger eccentricity closer
to merger before it can be detected with TEOBResumS.

Figure 9: Eccentricity maps calculated with LVK sensitivity instead of white
noise. This plot shows that the same information as in Figure 5, except we
replace the flat PSDs with detector PSDs in the calculation of the mismatch.

##  Appendix B Eccentricity constraints on GW events

In Figure 10, we show eccentricity constraints obtained with SEOBNRE and
TEOBResumS for ten selected GW events detected by the LVK. These events are
chosen because, of all events in GWTC-2, they have the highest support for the
eccentric-waveform hypothesis when analysed with SEOBNRE (Romero-Shaw et al.,
2021). We employ the Bayesian inference package Bilby (Ashton et al., 2019;
Romero-Shaw et al., 2020) to perform parameter estimation on these events. We
reuse posterior samples for SEOBNRE that were first presented in previous work
(Romero-Shaw et al., 2021), which were obtained by first sampling the
posterior with a quasi-circular waveform model, IMRPhenomD, and then
reweighting with SEOBNRE (see Romero-Shaw et al. (2019); Payne et al. (2019)
for a full description of the likelihood reweighting process). Posteriors for
TEOBResumS are obtained by using this model directly in parameter estimation,
without any reweighting, but with a loosened error tolerance as demonstrated
in O’Shea & Kumar (2021). For TEOBResumS, we use a log-uniform prior on
eccentricity in the range $10^{-4}\leq e_{10}\leq 0.3$.

###  B.1 Discussion of GW190521

Figure 10 shows that we infer different eccentricity posterior distributions
between SEOBNRE and TEOBResumS for these ten events, with the latter yielding
either an uninformative (uniform) distribution, or a distribution that is
peaked slightly above the SEOBNRE posterior. Despite being peaked away from
zero eccentricity, most of the events appear consistent with quasi-
circularity, as their posteriors have support over the full prior range. The
exception to this is GW190521, which shows more evidence for the eccentric
hypothesis when analyzed with SEOBNRE (Romero-Shaw et al., 2020), with little
support for $e_{0}^{\rm SEOB}<0.1$ at 10 Hz. However, when analyzed with
TEOBResumS, the posterior is uninformative. This discrepancy is curious.

Figure 10: Marginal posterior probability distributions for eccentricity on
ten selected GW events, obtained with either SEOBNRE (red) or TEOBResumS
(blue). Note that GW190909 has since been downgraded to a non-astrophysical
trigger (Abbott et al., 2021b). In many cases the peaks in the posterior
distributions rail against the upper prior limit. The upper prior limit used
for TEOBResumS is at the right edges of the plots. The upper prior limit of
SEOBNRE is shown by the dashed vertical lines. The upper prior limit of
TEOBResumS is $e_{0}^{\rm TEOB}=0.3$ at 10 Hz.

We suggest a few possibilities as to why we infer different posteriors for
GW190521. One possibility is that the bulk of the TEOBResumS posterior support
is located above the upper limit of the prior, causing us to miss the true
peak in the eccentricity posterior. This hypothesis is supported by the fact
that the peak of the SEOBNRE posterior is very close to the prior limit. Note,
however, that our eccentricity mapping study shows that a given SEOBNRE
eccentricity will translate into a smaller numerical value of TEOBResumS
eccentricity after evolving to consistent reference frequencies. Thus, if the
TEOBResumS distribution is indeed peaked at a higher eccentricity, then it is
not measuring the same physical eccentricity as SEOBNRE. An alternative
explanation is that the TEOBResumS posterior is consistent with quasi-
circularity, as Figure 9 shows that systems with $e_{0}^{\rm SEOB}\lesssim
0.15$ at 10 Hz can have a best-matching TEOBResumS waveform with $e_{0}^{\rm
TEOB}=0$ assuming LVK noise. This can occur because the sensitivity curves
down-weight the lower inspiral frequencies relative to the merger, making it
harder to detect any eccentricity. A third possibility is that the event is
not eccentric at all, but it is precessing, and the precessing waveform is
best fit by SEOBNRE with $e_{0}^{\rm SEOB}>0$ while the TEOBResumS waveform
can be fit using $e_{0}^{\rm TEOB}\approx 0$.

The inconsistent results between SEOBNRE and TEOBResumS are not yet
understood. Our eccentricity maps are a step forward in this regard, as it
partially disentangles the definition difference from other systematics.
However, further work is needed to determine how one waveform could measure
significant eccentricity while the other measures none. This will likely
require injection studies in which one adds an eccentric (and/or precessing)
waveform to realistic detector noise and attempts to recover the signal with a
different eccentric model by running full parameter estimation, instead of
calculating mismatches as done in this work.

Comparing our results to those of Iglesias et al. (in prep.), we find
qualitatively consistent eccentricity posterior distributions, when
differences in prior shape, prior volume, reference frequency, analysis
technique, and waveform model are considered. In particular, we note that
their use of a higher reference frequency both reduces the eccentricity and
reduces the evidence for eccentricity, thereby disfavouring the eccentric
hypothesis.

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