# Measurability of precession and eccentricity for heavy binary-black-hole
mergers

Yumeng Xu  Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057
Zürich, Switzerland  Eleanor Hamilton  Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, 8057 Zürich, Switzerland

###### Abstract

Gravitational wave detections offer insights into the astrophysical
populations of black holes in the universe and their formation processes.
Detections of binaries consisting of black holes lying outside the bulk
distribution of the astrophysical population are particularly intriguing. In
this study, we perform an injection analysis within the intermediate-mass
black hole range, utilizing the NR surrogate model NRSur7dq4 and a selection
of NR waveforms from the SXS and RIT catalogues. Our investigation focuses on
the detectability of precession and its potential degeneracy with
eccentricity, especially for short signals with only a few cycles in band.
While total mass, mass ratio, and $\chi_{\mathrm{eff}}$ are generally well
recovered, the recovery of $\chi_{\mathrm{p}}$ is largely limited, and noise
significantly impacts the recovery of some parameters for short signals. We
also find that eccentricity lower than 0.2 is insufficient to mimic precession
in parameter estimation when assuming a quasi-circular signal. Our results
suggest that a certain degree of precession is necessary to produce evidence
of high precession in parameter estimation, but it remains challenging to
conclusively determine which effect is responsible for the high precession
observed in events like GW190521. We emphasize the importance of caution when
interpreting properties of a binary from short signals and highlight the
potential benefits of future third generation detectors and eccentric waveform
models for more exhaustive exploration of parameter space.

Precessing, Parameter estimation

##  I Introduction

Over the past few years, the LIGO-Virgo-KAGRA (LVK) network of gravitational
wave detectors has successfully completed three runs, resulting in the
detection of over 90 gravitational wave events [1]. With the upcoming fourth
run (O4), the network is expected to detect 3 times more events [2]. These
detections bring us to a new era of astrophysics and provide us with a new
possibility to answer questions about the astrophysical makeup of the
universe.

There are two particular questions of interest: the properties and population
of intermediate-mass black holes (IMBHs) and the existence of black holes
within the pair-instability supernova (PISN) mass gap. IMBHs have masses in
the range $10^{2}-10^{5}M_{\odot}$, bridging the gap between stellar BHs and
supermassive BHs. The understanding of the properties and population of IMBHs
might provide the missing link to explain the formation of supermassive black
holes [3]. The PISN mass gap refers to the range of black hole masses between
$65M_{\odot}$ and $120M_{\odot}$, which is disfavoured to be formed from
stellar origin. Within this gap, when a star with a helium core reaches the
end of central carbon burning and collapses inwards, there will be a complete
disruption of the core and no compact object will remain[4, 5, 6, 7, 8].
Alternative formation channels for BHs within this gap — including second-
generation BHs, stellar mergers in young star clusters, black hole mergers in
active galactic nucleus disks, primordial black holes, and gas accretion —
have been widely discussed [9, 10, 11, 12, 13, 14, 15].

Distinguishing the formation channels relies on accurate measurements of key
properties of the binary, in particular precession and eccentricity[16, 17]. A
quasi-circular binary black hole (BBH) is characterised by its intrinsic
parameters; the total mass $M_{\mathrm{tot}}$ of the binary system and mass
ratio $q=m_{2}/m_{1}$, where $m_{1}>m_{2}$ is the mass of the primary black
hole, and the spins $\vec{S}_{1}$ and $\vec{S}_{2}$. If we consider a binary
on an eccentric orbit then we add at least one additional intrinsic parameter;
the eccentricity $e$.

Spin-induced precession occurs when the spins are not aligned with the orbital
angular momentum. Precession introduces modulations of the GW amplitude and
phase [18, 19]. From post-Newtonian theory, we can construct an effective spin
$\chi_{\mathrm{eff}}=(\vec{S}_{1}/m_{1}+{\vec{S}_{2}}/{m_{2}})\cdot\hat{L}/M_{\mathrm{tot}}$
[20, 21] which characterizes the impact of the components of the spins aligned
with the orbital angular momentum of the binary $\hat{L}$. The effective spin
can be measured better than the individual spins [22, 23]. The components of
the spins which lie perpendicular to $\hat{L}$ cause $\hat{L}$ to precess
about the total angular momentum of the binary,
$\hat{J}=\hat{L}+\hat{S}_{1}+\hat{S}_{2}$. The dominant effects of precession
on the GW signal can be characterized by the effective precession spin
$\chi_{p}=\max(A_{1}S_{1\perp},A_{2}S_{2\perp})/(A_{1}m_{1}^{2})$ where
$A_{1}=2+3q/2$ and $A_{2}=2+3/(2q)$ [24]. $S_{1\perp}$ and $S_{2\perp}$ are
the magnitudes of the components of the spins that lie in the orbital plane.

We can use GW detections to build up a picture of the population of BHs in the
Universe. The LVK network has detected a number of events which fall into
either the population of IMBHs or the PISN mass gap [25]. Most of the
detection are marginal cases. Among these detections, GW190521 [26, 9] is
particularly interesting as it is the first-ever detection of a black hole in
PISN mass gap merging with another black hole to form an IMBH and it is also
the only one that’s not marginal. There have only been three signals reported
by the LVK so far which show any support for precession: GW190412, GW190521
and GW200129_065458 [27, 26, 1, 28, 29]. Eccentricity can also cause
modulations in the amplitude and phase of the signal [30] which we haven’t had
highly confident detection.

Difficulties in accurately determining the properties of an IMBH system arise
because this kind of heavy BBH event has a very short duration and only has a
few cycles in the frequency range to which aLIGO and aVirgo are most sensitive
[31, 32]. For instance, GW190521 only has approximately 0.1s in duration and
around 4 cycles in the frequency band 30-80Hz [26]. The early inspiral and
even part of the late inspiral were outside of the sensitivity band of the
detectors at the time at which the signal was detected. This kind of short
signal leaves opens the possibility that the inferred properties of the binary
may not be accurate, since such a short signal will have features that can be
produced by a number of degenerate parameters. Consequently, the
interpretation of the source as a quasi-circular compact binary coalescence
consisting of inspiral, merger and ringdown phases is not certain. In the case
of GW90521, it has been seen that the signal can be shown to be consistent
with that originating from a number of less likely sources [33, 34, 35, 36,
37]. The parameter estimation (PE) is also likely to be disproportionately
affected by noise in the detector. Thus, checking the measurability of
precession and eccentricity is crucial to be confident of any astrophysical
inferences drawn from gravitational wave events within the IMBH mass range.

In this paper, we will consider only signals which have come from a compact
binary coalescence (CBC) source, though not necessarily a quasi-circular one.
We will examine the measurability of precession by analyzing injected
waveforms with known properties similar to those of GW190521. The paper is
organized as follows. In Sec. II, we give an introduction to the methodology
we will use in this paper. In Sec. III, we investigate the inferences that can
be made about signals from quasi-circular binaries. In particular, we consider
the robustness of these inferences with respect to noise in the detector.
These investigations are extended to include eccentric systems in Sec. IV.
However, since we don’t have an eccentric waveform model for the merger, we
will only check the bias of precession from the eccentric injection. The
detectability from future detectors is also discussed in Sec. V. Finally, we
give our conclusions and discussions from these analyses in Sec. VI.

##  II Methodology

To study these kinds of events by injection, we first must choose the most
appropriate waveform model for our investigation. The motivation for our
choices is outlined in Sec. II.1. The methods employed to generate the
waveforms and perform the injections are described in Sec. II.2. We discuss
the setup employed for the PE in Sec. II.3. The techniques used to further
analyse the results of the PE are then outlined in Secs. II.4 and II.5.

###  II.1 Waveform models

A number of precessing waveform models have been developed to date [38, 39,
40, 41, 42, 43, 44]. In our analysis, we use one of these models, NRSur7dq4
[41]. NRSur7dq4 is a merger-ringdown model based on a catalogue of 1528 NR
simulations taken from the SXS Collaboration catalog. These simulations, and
thus the model, cover the 7-dimensional parameter space of quasi-circular
binaries with generic spins. However, since it is based solely on numerical
relativity (NR) waveforms, it is limited in application to signals containing
minimal inspiral– specifically, high mass systems. In our studying cases,
which only contains the merger-ringdown stage, NRSur7dq4 is the most accurate
waveform available and therefore the most appropriate for use with in-depth
studies of a given event, such as presented in this paper [39, 44].

When considering quasi-circular binaries, we use this model both for
generating our simulated signal and for the PE we perform on this signal.
Using the same waveform model for both parts of the analysis reduces the
possibility that any effects we see are due to waveform systematics.

When considering eccentric signals, there are several eccentric waveform
models available [45, 46, 30, 47, 48] which one could use to produce the
injected signal. However, these waveform models are insufficient for our study
since the majority of our signal is composed of the merger and ringdown while
these models are most accurate in the inspiral regime. Instead we use
waveforms from the SXS Collaboration catalog of binary black hole simulations
[49] and the RIT Catalog of Numerical Simulations [50] to simulate the signal
from an eccentric binary. We recover these injections using NRSur7dq4 as in
the quasi-circular case, since to date all PE performed by the LVK has used
quasi-circular models.

###  II.2 Generation of injected signals

We simulate a GW signal with known source parameters using either NRSur7dq4 or
an appropriate NR waveform. This signal is then used to create a GW
“injection”, consisting of the GW strain as seen by the detectors. Inspired by
the only known precessing IMBH detection, we choose the parameters which form
the basis of our injections the max-loglikelihood parameters from the
posterior by the LVK for GW190521 [51] and listed in Tab. 1. These are the
values for the waveform that were found to agree best with the event data in
the original LVK analysis. We vary the selection of these parameters
individually. In what follows, we will refer to the injection in zero noise
using exactly these parameters as the “maxL injection”.

Table 1:  maxL values of GW190521 from the data release [51] associated with Ref. [26].  Symbol | Name | maxL value
---|---|---
$m_{1}(M_{\odot})$ | mass of primary black hole | 147.767
| (detector frame) |
$m_{2}(M_{\odot})$ | mass of secondary black hole | 121.063
| (detector frame) |
$\vec{S}_{1}$ | spin of primary black hole | [0.008, -0.071, 0.054]
$\vec{S}_{2}$ | spin of secondary black hole | [0.808, -0.358, 0.143]
$\theta_{\mathrm{jn}}(\mathrm{rad})$ | angle between line of sight | 0.953
| and total angular momentum |
$\iota(\mathrm{rad})$ | inclination | 0.834
$\psi(\mathrm{rad})$ | polarisation | 2.382
$\alpha(\mathrm{rad})$ | right ascension | 0.164
$\delta(\mathrm{rad})$ | declination | -1.143
SNR | signal to noise ratio | 15.403
$\phi(\mathrm{rad})$ | coalescence phase | 0.004
$\chi_{\mathrm{eff}}$ | effective spin | 0.094
$\chi_{\mathrm{p}}$ | effective precessing spin | 0.704
$d_{L}(\mathrm{parsecs})$ | luminosity distance | 2941.027

The GW signal $h$ from a CBC can be written as a combination of two
polarisations $h_{+}$ and $h_{\times}$ as $h=h_{+}-ih_{\times}$. It can be
decomposed into modes $h_{\ell m}$ via [52]

$h\left(\iota,\phi_{0}\right)=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}h_{\ell
m}\,^{-2}Y_{\ell m}\left(\iota,\phi_{0}\right)$ (1)

where the ${}^{-2}Y_{\ell m}$ are a basis of spin-weighted spherical harmonics
and $\left(\iota,\phi_{0}\right)$ give the direction of the radiation in the
source frame. The signal itself can be considered as a function of time or
frequency and depends on the parameters of the binary.

Both the NRSur7dq4 model and the NR waveforms supply the GW modes $h_{\ell
m}$, which are recombined to produce the GW strain $h$ using get_td_waveform
from PyCBC [53] and SimInspiralChooseTDWaveform from LALSimulation [54] as
part of the NR Injection Infrastructure [55] respectively. The $h_{+}$ and
$h_{\times}$ polarisations are then projected onto a detector network using
the appropriate detector response functions $F_{+}(\alpha,\delta,\psi)$ and
$F_{\times}(\alpha,\delta,\psi)$ using Detector.project_wave from PyCBC [53].
$\alpha,\delta,\psi$ are as defined as in Tab. 1. This, combined with a given
noise realisation, gives us the signal as would be seen by the detectors. This
signal forms our “injection”, upon which we perform our analysis.

In what follows, if the choice of noise realisation is left unspecified the
injections will be zero-noise. The purpose of these injections is to see an
unbiased PE recovery, since a particular realisation of Gaussian noise might
cause a bias in the PE, especially for the kind of short signal under
investigation. A zero noise injection can be treated as the average of all
Gaussian noise realisations.

For injections where we wish to investigate the effect of noise on a signal of
this kind, we generate coloured Gaussian noise from the power spectral density
(PSD) for a given detector. The addition of Gaussian noise to the signal is
intended to allow for an assessment of the robustness of the PE performed on
these signals. We anticipate that this will give a conservative limit on the
effect of noise on the signal since much stronger effects are possible from
real detector noise.

###  II.3 Bayesian Parameter Estimation

Having produced a simulated GW signal of known source parameters, we perform
the same analysis on our simulated signal as is performed on detected signals
by the LVK. This analysis is intended to determine the parameters of the
source.

Given the data, the posterior distribution for a given black hole parameter is
given by [56]

$P(\mathrm{parameter}|\mathrm{data})\propto
P(\mathrm{data}|\mathrm{parameter})\times P(\mathrm{parameter})\,,$

where the first term on the right-hand side is the likelihood, which is the
probability of the data given the parameter. The second term is the
distribution we assume for a given parameter, known as the prior. The prior
should be astrophysically motivated. Our assumptions for the priors employed
in this study are identical to those used in the LVK analysis of GW190521 [9],
except the priors are uniform on chirp mass (70-150 $M_{\mathrm{tot}}$) and
mass ratio (0.17-1.0) to cover the parameter space of IMBH event.

We used the PE code Parallel-Bilby [57, 58, 59] with the nested sampler
Dynesty [60] in order to analyse each of our injected signals.

The PSDs of the characteristic sensitivity curves are used to simulate the
stationary Gaussian noise and compute the likelihood function. The various
characteristic sensitivity curves we used in our analysis are shown in Fig. 1.
The PSDs used in the bulk of the analysis come from the LVK results for
GW190512 (referred to as event PSDs) [51] which was calculated from the data
by the BayesLine algorithm [61]. We choose this PSD to make the analysis
closer to realistic. The curve for the detector H1 is shown, labelled _O3_H1_.
We also consider sensitivity curves for aLIGO and aVirgo [62, 63] (which will
be referred to as design PDSs) as well as future detectors Voyager, Cosmic
Explorer (CE) and Einstein Telescope (ET) [64] in the discussion surrounding
future detections in Sec. V.

Figure 1: Sensitivity curves of different interferometers. O3_H1 is the PSD of
LIGO Hanford when GW190521 was detected. aLigo and aVirgo curves are the
projected curves at design sensitivity. The other curves are proposed
sensitivity curves for future detectors.

###  II.4 Jensen–Shannon divergence

Having performed PE on each of our injected signals, we wish to be able to
perform a quantitative comparison of the results of our various analyses. We
quantify the deviation between two distributions using the Jensen-Shannon (JS)
divergence [65]. The JS-divergence is a variant of the Kullback-Leibler
divergence $D_{\mathrm{KL}}$ [66]. For discrete probability distributions P
and Q on the same probability space $\chi$

$D_{\mathrm{JS}}(P||Q)=\frac{1}{2}D_{\mathrm{KL}}(P||M)+\frac{1}{2}D_{\mathrm{KL}}(Q||M),$
(2)

where $M=(P+Q)/2$ and

$D_{\mathrm{KL}}(P||Q)=\sum_{x\in\chi}P(x)\log\left(\frac{P(x)}{Q(x)}\right).$
(3)

$D_{\mathrm{JS}}(P||Q)$ is a normalized quantity. The maximum value is
dependent on the base; under base 2, which we employ here, the Jensen–Shannon
divergence is bounded by 1. The lower the JS divergence, the closer the two
distributions. When the two distributions are identical, the divergence is
$D_{\mathrm{JS}}(P||P)=0$. A value of $D_{\mathrm{JS}}<0.05$ is generally
taken to mean that the two distributions are in good agreement [67].

###  II.5 Precessing SNR

One of the most interesting questions about IMBH binaries and those containing
BHs in the PISN mass gap is the degree to which it is possible to determine
whether the binary is precessing from the detected signal. In this analysis,
we therefore require a statistical tool to assist us in identifying whether we
can claim a detected GW to be unambiguously precessing. To do so, we use the
precessing SNR [68, 69], which is used to calculate whether the precession in
a binary can be confidently inferred from detection with a given PSD. The
precessing SNR is based on the two-harmonic approximation [68]. In this
decomposition, the precessing waveform strain is decomposed into five
harmonics. Each harmonic is a non-precessing signal. Therefore, if we want to
detect the precessing signal, there should be at least two dominant harmonics
$h^{0}$ and $h^{1}$. Thus the precessing SNR $\rho_{\textrm{p}}$ is defined by
the minimal SNR of $h^{0}$ and $h^{1}$

$\rho_{\textrm{p}}=\min(|\mathcal{A}_{0}h^{0}|,|\mathcal{A}_{1}h^{1}|),$ (4)

where $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ are the amplitude of each
harmonic.

A value of $\rho_{\textrm{p}}>2.1$ indicates the presence of the second
harmonic in the signal at a sufficient level that it is not likely to have
arisen from noise fluctuations [68, 69, 70]. $\rho_{\textrm{p}}=2.1$ was
therefore proposed to be a threshold above which a signal may be considered to
be unambiguously precessing. However, we note that in Ref. [71] it was found
that for a comparable mass ratio event, the threshold value of
$\rho_{\mathrm{p}}$ may have to be higher than 2.1 in order to confidently
claim a detection of precession. We therefore consider 2.1 to be a
conservative estimate of the threshold below which a detection cannot be
considered to be unambiguously precessing.

##  III Precession Detectability in IMBH Merger

To gain a deeper insight into the robustness of detecting strong precession in
IMBH binaries, we performed a wide-ranging injection study. We first validated
the existing understanding of the limitations in detecting precession for low
mass ratios and events with low to moderate signal-to-noise ratios. We then
looked at the impact of detector noise on such a short signal. Finally, we
considered the conditions under which one might be able to detect precession
in an IMBH event.

###  III.1 Exploring GW190521-like Injections and Varying Total Mass

First, we consider a system inspired by GW190521, the only confidently-
detected IMBH system to date. The properties of this system are listed in Tab.
1. We injected this signal as described in Sec. II.2.

The recovered posteriors for $M_{\mathrm{tot}}$, $q$ and $\chi_{\mathrm{eff}}$
are centred on the injected values, which suggests accurate recovery of these
parameters in zero noise. $\chi_{\mathrm{eff}}$ is particularly well-
recovered, with a narrow posterior.

Figure 2:  Results of maxL injection. The blue lines represent the posterior
samples from the LVK event data set. The orange lines are from the maxL
injection. The vertical dashed lines show the median (50%) and the bounds of
the 90% credible interval and the contours show the 50%, and 90% credible
interval. The contours of total mass $M_{\mathrm{tot}}$ and mass ratio $q$
show a good match. However, the effective spin and effective precession spin
show significant deviation from the LVK results.

However, the posterior for $\chi_{\textrm{p}}$ is not centred on the injected
value, although this value is included at 90% confidence. In fact, the JS
divergence for $\chi_{\textrm{p}}$ with respect to the prior is 0.03 bits and
0.06 bits when considering the injection using the design PSDs and event PSDs
respectively. We can therefore see that for the injection we are simply
recovering the prior for $\chi_{\textrm{p}}$ and we do not get any information
about the amount of precession in the signal. From this, we can conclude that
even when we have a highly precessing binary like the injected signal, we do
not expect to be able to measure the precession in the signal for such a low
mass ratio, low SNR system.

This conclusion is corroborated by the calculation of the precession SNR
$\rho_{\textrm{p}}$ for the injected signal. At 95% confidence, the precession
SNR of the injection is below 2.1, the minimum threshold required to claim an
unambiguous detection of precession. The low value of the precession SNR for
the event implies we would not expect to be able to make a confident detection
of precession here.

When comparing the maxL injections with the results for GW190521, we find that
the JS divergence between the posteriors of the LVK results and the maxL
injections is less than 0.02 bits for both $M_{\mathrm{tot}}$ and $q$. This
similarity indicates a good match between the injection and the values
measured for the event, meaning we have likely injected a signal similar in
total mass and mass ratio to our example IMBH binary.

However, our spin posteriors are not so similar. For example, we do not see
the support for the negative $\chi_{\mathrm{eff}}$ values seen in the event
posterior. This broadening of the posterior is likely due to detector noise.
The high value of the peak in the $\chi_{\textrm{p}}$ posterior distribution
and significant deviation from the prior (a JS divergence of 0.16 bits) seen
for GW190521 is not seen for our injection. It is therefore difficult to
confidently claim our injected signal has the same in-plane spin as our
example binary. Further, since we do not reproduce posteriors indicating a
high degree of precessing in the signal, the indications of this seen for
GW190521 are unlikely to be due to an equivalent or smaller degree of
precession than in our injected signal and more likely to be due to other
factors.

It is unlikely that the fiducial binary is exactly the same as the one that
produced GW190521, even though it was generated using the ”best matching
signal.” Therefore, we need to explore other parameters to see what the PE
from similar IMBH binaries looks like. Since the results obtained using the
event PSD and those obtained with the design PSD show negligible differences,
we will only use design PSD results as the maxL injection for comparison in
the following sections.

Figure 3:  Results of injections with different total mass. The lower total
mass injection has more cycles in the detectors. The injected total masses are
listed in the legend.

In addition, we also explore lower total-mass binaries, which will have more
cycles and duration in the aLIGO-aVirgo band. This means we can extract more
information from the signal, potentially increasing the confidence of
accurately recovering precession.

To achieve that, we inject signals varying the total mass while keeping other
parameters fixed. Since we are investigating black hole binaries in the PISN
mass gap and merging to an IMBH, we select 5 masses of primary black holes
from 65 $M_{\odot}$ to the maxL value 98 $M_{\odot}$ in the source frame. This
results in the injected total masses in detector frames being 177.76, 195.81,
213.86, 231.91, and 249.96 solar masses.

The results are shown in Fig. 3. It shows that the mass ratios recovered from
longer waveform injections are more accurate than the shorter ones. The
longest injected signal (blue line) has the best estimation, with the
narrowest posterior, of mass ratios, while the shortest injection is the
broadest. For the $\chi_{\mathrm{p}}$ recovery, even though the longest
injection is still not peaking close to the injected $\chi_{\mathrm{p}}$, the
max likelihood $\chi_{\mathrm{p}}=0.75$ is very close to the injected value
$\chi_{\mathrm{p}}=0.70$.

For lower mass systems that maintain a primary mass within the PISN gap and
merge into an IMBH, the results display a more robust indication of
precession. However, the precessing SNR remains below 2.1 at a 95% confidence
level, indicating that it is still insufficient to draw a definitive
conclusion.

In the following sections, we will continue our exploration of IMBH systems
and their detectability in the context of gravitational wave astronomy. By
considering GW190521 as a representative example and extending our analysis to
a broader range of scenarios and parameters, we aim to gain a more
comprehensive understanding of these systems and their implications for our
understanding of the universe.

###  III.2 Effect of Gaussian Noise on Short-Duration IMBH merger

In this section, we investigate the influence of Gaussian noise on the
parameter estimation of short signals, particularly focusing on the biases it
may introduce. We generated 38 different injections using our fiducial
waveform embedded in Gaussian noise (referred to as “noisy injections”), as
described in Sec. II.2. These noisy injections enable us to perform
preliminary investigations into the effects of noise on the recovered
parameters. We expect that detector noise will affect the results even more
strongly, so our investigation here provides a lower bound on these effects.

Figure 4:  Results of the noisy injections (green) compared with GW190521
(blue) and the zero-noise maxL injection (red). We highlight the injections
which give results closest to GW190521 (orange) and that give the greatest
deviation from the prior (lilac). Penultimate panel: maxL values from the
noisy injections (blue) compared to the 90% credible interval of the zero-
noise injection (orange). The values are all re-weighted by the range of the
priors. Final panel: JS-divergence between the posteriors and the prior.

It is clear from these results that noise fluctuations can significantly
impact the signal, especially for short signals. The impact is most pronounced
for the spin parameters, as evident from the maxL values for
$\chi_{\mathrm{p}}$ and $\chi_{\mathrm{eff}}$ in some injections that lie
outside the 90% credible interval of the zero-noise injection.

The total mass and mass ratio are also affected by noise, but to a lesser
extent. The addition of noise can shift the peak outside the 90% confidence
interval of the zero-noise injection for the total mass, although the overall
shift does not significantly alter the astrophysical conclusions about these
types of events.

The noise-induced changes in the posterior of $\chi_{\mathrm{p}}$ observed in
this analysis are consistent with the findings of Ref. [72]. However, some
differences might arise due to the choice of PSD employed in each analysis.

From Fig. 4, it is evident that the addition of noise results in deviations of
the recovered parameters compared to the zero-noise maxL injections. For the
parameters considered here, they do not show systematic biases, but are
instead distributed randomly around the zero-noise injection.

The addition of noise also reveals a second peak in the posteriors for
extrinsic parameters such as polarization $\psi$ and inclination $\iota$. This
second peak, which was present in the event posteriors but not seen in the
zero-noise injection, confirms that this feature arises due to the presence of
noise in the signal.

In light of these findings, caution should be exercised when inferring the
nature of events characterized by short signals based on the location of the
peak and the values of the maxL parameters. Real-world non-Gaussian noise
could introduce additional biases, emphasizing the importance of considering
the influence of noise when interpreting results.

###  III.3 Other affecting factors

####  III.3.1 Noise on Non-Spinning Systems

Figure 5:  Results of non-spinning noise injection. The injected signals are
generated with the same configuration as Fig. 4 except the spin components are
set to zero. The dots in the bottom panel represent the JS divergence from
posteriors with respect to the prior.

In the previous section, we found that a combination of Gaussian noise and a
highly precessing signal is sufficient to show indications of high in-plane
spins. This motivates us to investigate whether the evidence for high
precession can be mimicked purely by Gaussian noise from a non-spinning or
aligned-spin binary. To do so, we generated another set of injections using 30
random noise realizations. The injected signals have the same parameters as in
Tab. 1, but with spin components set to zero. We chose a non-spinning binary
as the limiting case.

The results of these injections are shown in Fig. 5. We find that the
intrinsic parameters are less affected by the addition of noise than in the
precessing case, as can be seen by comparison of the JS divergence. As
expected, the posteriors for $\chi_{\mathrm{eff}}$ are centred on zero (the
injected value) and do not appear to be noticeably impacted by the noise. This
is also clear for the posteriors of $\chi_{\mathrm{p}}$, where we essentially
recover the prior for each of our 30 injections. It is clearly unlikely that
evidence of high precession seen in the LVK results can be reproduced by pure
noise. The JS plot corroborates the fact that the noise cannot cause a
sufficient deviation from the prior to mimic a highly precessing system. It is
therefore not possible to produce evidence for precession in such a system
purely as a result of Gaussian noise

####  III.3.2 Effects of Varying Spin Magnitude

Since our results indicate that a non-spinning binary cannot give indications
of precession even in the presence of Gaussian noise, we aim to determine the
minimum in-plane spin magnitude required to obtain precession information for
our fiducal IMBH source.

The specific spin configuration significantly impacts the recovery of
$\chi_{\mathrm{p}}$ from a short, highly precessing signal [72]. For certain
configurations, there is a much higher probability of observing evidence of
precession in the signal. However, as demonstrated, this is not the sole
effect that can produce evidence of a highly precessing system. Since we do
not intend to investigate the effect of spin orientation here, we fixed the
orientation of the spin to the value given in Tab. 2.

Table 2:  Injected spins of varying magnitude. | $\chi_{\mathrm{p}}$ | $\chi_{\mathrm{eff}}$ | $a_{1}$ | $a_{2}$
---|---|---|---|---
$\chi_{p1}$ | 0.15 | 0.02 | 0.020 | 0.20
$\chi_{p2}$ | 0.30 | 0.04 | 0.040 | 0.39
$\chi_{p3}$ | 0.46 | 0.06 | 0.059 | 0.59
$\chi_{p4}$ | 0.62 | 0.08 | 0.079 | 0.79
$\chi_{p5}$ | 0.77 | 0.10 | 0.099 | 0.98
maxL | 0.70 | 0.09 | 0.090 | 0.90

As shown in Fig. 6, deviations from the prior in $\chi_{\mathrm{p}}$ start to
appear when $\chi_{\mathrm{p}}$ is around 0.7, and become significant when
$\chi_{\mathrm{p}}\sim 0.8$. The JS divergence between the posteriors and the
prior for $\chi_{\mathrm{p}}$ supports this conclusion, with only the
$\chi_{p5}$ injection showing a deviation of $\mathcal{O}\left(10^{-1}\right)$
bits. For a zero-noise injection, an in-plane spin of about $0.77$ and a total
spin greater than $0.9$ (beyond the calibration range of NRSur7dq4) is
necessary to produce a deviation from the prior that indicates high in-plane
spins.

This investigation can help inform our interpretation of the GW190521 event
and contribute to our astrophysical understanding of the IMBH binary
population. Notably, the JS divergence between $\chi_{p5}$ and the LVK results
is only 0.014 bits, which means the posteriors for $\chi_{\mathrm{p}}$ are
quite similar to those observed for the event. However, we have not yet
considered the effects of noise on the signal, potential degeneracy with
eccentricity, or the impact of extrinsic parameters. We will reevaluate any
conclusions drawn from this analysis in the following sections.

Figure 6:  Results of the injections with varying spin magnitude in zero
noise. The green lines are the injections with different spin magnitudes,
which are listed in Tab. 2. These are compared with the results from GW190521
(blue) and the zero-noise maxL injection (orange). The bottom panel give the
JS-divergence from posteriors to prior. Figure 7:  Results of the injections
varying spin magnitude into the $N_{\mathrm{max}}$ noise realisation. The
green lines are the injections with different spin magnitudes, which are
listed in Tab. 2. These are compared with the results from GW190521 (blue) and
the zero-noise maxL injection (orange).

To investigate the minimum spin magnitude required when accounting for
detector noise, we injected our signal into the $N_{\mathrm{max}}$ noise
realization, which produced the largest inferred value of $\chi_{\textrm{p}}$
as discussed in Section III.2. The results are presented in Fig. 7. We find
that in the presence of noise, a value of $\chi_{\mathrm{p}}$ greater than 0.4
is necessary to provide evidence of high in-plane spin values. A systematic
exploration using a range of noise realizations has not been conducted due to
computational resource limitations. However, combined with the results in
Section III.3.1, this analysis shows that a high inferred effective precession
spin for this type of event requires both a non-zero value of
$\chi_{\textrm{p}}$ (indicating some degree of precession in the source
signal) and noise. The presence of noise in the signal considerably reduces
the magnitude of the spin that can be claimed for this event.

####  III.3.3 Extrinsic Parameters and Their Impact

The measurability of precession is affected by the extrinsic parameters,
especially polarization $\psi$ and inclination $\iota$ [69]. These parameters
are poorly measured from the signal. We examine whether we could confidently
detect precession from a highly precessing, equal mass IMBH by choosing the
values of $\iota$ and $\psi$ that maximize the likelihood of detecting any
precession present in the signal.

Figure 8:  Calculation of $\rho_{\mathrm{p}}$ with varying inclination $\iota$
and polarization $\psi$ using maxL parameters. The color bar on the right
displays the value of $\rho_{\mathrm{p}}$. The blue circular dots in this
figure represent the maxL value, the white triangular dots are the minimum
values, and the red diamond dots are the maximum values.

We calculated $\iota$ and $\psi$ for which the precessing SNR
$\rho_{\mathrm{p}}$ is maximised for a binary with the properties listed in
Tab. 1. The dependency of $\rho_{\textrm{p}}$ on $\psi$ and $\iota$ is shown
in Fig. 8. From this, we can see that our choice of $\iota$ and $\psi$ for our
example system lie in a region where, compared to the maximum possible value
of $\rho_{\mathrm{p}}$ for this event, we are only moderately confident in our
ability to detect precession. The maximum value of $\rho_{\mathrm{p}}$
possible for such a system is around 1.43, which is still below the threshold
to unambiguously claim detection of precession, as discussed in Sec. II.5. We
therefore do not expect to be able to claim a detection of precession for any
set of extrinsic parameters for this event

Nonetheless, we aim to examine the influence of these extrinsic parameters on
our ability to interpret the signal. We generated and injected another
waveform based on the parameters in Table 1, altering the values of $\iota$
and $\psi$ to maximize $\rho_{\mathrm{p}}$ to investigate how optimizing these
extrinsic parameters impacts the likelihood of detecting precession for a
given system. We chose the first maximum in Fig. 8 ($\iota=1.759$ and
$\psi=1.005$) since the four maxima are equivalent. We also investigated the
effect of noise on these modified extrinsic parameters, using the
$N_{\mathrm{max}}$ noise realization.

Figure 9:  Results of injections using optimized extrinsic parameters. The
green line represents the zero-noise injection with $\iota=1.759$ and
$\psi=1.005$, the values that maximize the precessing SNR. The yellow line
represents a noisy injection with the same extrinsic parameters and the
$N_{\mathrm{max}}$ noise realization.

From the results (shown in Fig. 9), the peak of the posteriors for the
inclination $\iota$ and polarisation $\psi$ in the zero-noise max
$\rho_{\mathrm{p}}$ injection is centred on the injected value.

Similar to our original case, we observe only a single peak for each of these
parameters in the absence of noise. With the addition of noise (using the
$N_{\mathrm{max}}$ realization), the single-peak distributions disappear, and
we see posteriors that closely resemble the LVK results.

We also see that when injecting the parameters that maximize the precessing
SNR, the peak of the posterior for $\chi_{\textrm{p}}$ shifts to slightly
higher values. However, it is still insufficient to claim a detection of
precession. Interestingly, unlike the case with sub-optimal values of $\iota$
and $\psi$, the addition of noise does not significantly affect the
distribution for $\chi_{\textrm{p}}$, so even with noise, we cannot recreate
evidence for high in-plane spins for a binary at this sky location. This is
consistent with our expectation since $\rho_{\textrm{p}}$ remains below the
threshold at which we expect to measure precession. Regardless of the true
values of $\psi$ and $\iota$, we still would not be able to detect precession
for an equal mass, highly precessing binary. However, the apparent evidence of
precession in the signal may enable us to rule out certain sky positions.

##  IV Investigation of eccentricity

It has been suggested that for this kind of system, evidence of precession in
the signal may degenerate with indications of eccentricity [34, 35, 73]. To
examine this possibility, we investigate the likelihood of seeing evidence for
precession in the signal from a heavy eccentric binary when recovering such a
signal with a non-eccentric waveform model. We use a selection of NR waveforms
taken from the SXS and RIT catalogues as discussed in Sec. II.1.

We first performed a systematic investigation of low eccentricity ($e<0.2$)
systems. This is motivated in part by the public availability of NR waveforms
of sufficient length to perform this study. Nonetheless, we expect the results
of this study to be of general interest since most astrophysical models
predict the majority of binaries have $e<0.2$ by the time they enter the LIGO
band ($>20\mathrm{Hz}$) [74, 75, 76].

However, some astrophysical models [77, 78, 79] do suggest binaries with much
higher eccentricities are possible — that would make this a very rare event
and provide much information about the formation channel of the binary.
Therefore, we also performed a limited study of highly eccentric systems.

Before using any NR waveforms in our analysis, we need to have a consistent
method of estimating eccentricity from these waveforms where possible, as will
be outlined in Appendix A. We then perform an injection study with these
waveforms, following the procedure detailed in Sec. II.2. The results of this
study are given in Sec. IV.

Figure 10:  The results of the SXS injections. The posteriors from the
eccentric injections (green) are compared with the results from GW190521
(blue) and the zero-noise maxL injection (orange). The penultimate panel shows
the maxL values of the eccentric injections (blue) compared to the 90%
confidence interval of the zero-noise injection (orange). The bottom panel
shows the JS divergence between the posteriors and the prior.

We use the Numerical Relativity Injection Infrastructure [55] to generate a
waveform with the same total mass and extrinsic parameters shown in Tab. 1. We
then perform the injection as detailed in Sec. II.2.

Since the eccentricity decays with time, we need to choose a reference time
$t_{\mathrm{ref}}$ at which to report the eccentricity. Here we choose
$t_{\mathrm{ref}}=-2500M$ because this is the earliest consistent time among
all the waveforms taken from the SXS catalogue. The $t=0$ is defined as the
merger time, which is the peak of the strain. The estimated eccentricities are
given in Tab. 3.

For the cases taken from the RIT catalog, we note that the waveforms only
contain 0.5 cycles prior to the merger. All the methods proposed to estimate
the eccentricity from the waveform [80, 81, 82, 83] are not valid for such
short waveforms. Consequently, we have to rely on the eccentricity reported in
the metadata of each waveform [50] and cannot perform an independent estimate.
Thus, we cannot ensure the consistency of eccentricities with those we report
for the SXS waveforms. To distinguish the two estimates, we label the
eccentricity taken directly from the metadata as $e_{r}$.

###  IV.1 Results

The results of the systematic investigation into the impact of comparatively
low eccentricity on the quasi-circular PE of a high mass system are shown in
Fig. 10. The median and 90% distributions of the key parameters from the
posterior are also given in Tab. 3. From the $\chi_{\mathrm{p}}$ plot, we can
see that for a signal at SNR 15.4 from a high mass non-spinning binary with
$q\leq 3$ and $e\leq 0.2$, the effect of eccentricity on the signal does not
mimic the effect of precession. The JS divergence of the $\chi_{\mathrm{p}}$
posteriors from the prior is $\mathcal{O}\left(10^{-2}\right)$ bits, implying
that the PE is simply recovering the prior in these cases. From this, we can
infer that the evidence of high-precessing spins cannot be produced by
eccentricity for short signal, heavy mass binaries with $e<0.2$.

In Ref. [35], it was proposed that GW190521 originated from a highly eccentric
black hole merger with $e=0.69^{+0.17}_{-0.22}$. Here, we consider two
waveforms, the first a non-spinning waveform (RIT:eBBH:1318) and the second
with $\chi_{\mathrm{p}}=0.7$ (RIT:eBBH:1639), both with the initial
eccentricity $e_{r}=0.75$. The precessing waveform was found in [35] to be
that which most closely matched the posteriors from GW190521. We performed
both a zero-noise and noisy injection using the $N_{\textrm{max}}$ noise
realisation of each of these waveforms.

Figure 11:  Results from the high eccentricity injections. In zero-noise,
neither the non-spinning eccentric waveform (green) nor the precessing
eccentric waveform (yellow) shows evidence of high in-plane spins. This is
changed with the addition of noise, though the non-spinning waveform (orange)
does not indicate such high spins as the precessing waveform (purple). The
bottom panel shows the JS divergence between the posteriors and the prior.

The results are shown in Fig. 11 and the median and 90% distributions of the
key parameters from the posterior are given in Tab. 4. We can see that the
non-spinning eccentric waveform does not cause significant deviation from the
prior for $\chi_{\mathrm{p}}$, with only JS divergence $D_{\mathrm{JS}}=0.006$
bits. It shows no difference to the non-spinning non-eccentric case. The
injected signal with both a high degree of precession and eccentricity shows a
greater deviation from the prior — with a JS divergence of 0.034 bits.
However, this is similar to the amount of information about the degree of
precession in the system that we are able to infer from a high precession,
quasi-circular signal (which has a JS divergence of 0.029 bits with respect to
the prior). It is therefore difficult to identify whether the indication of
precession in these results is due to the high in-plane spins or the high
eccentricity. In addition, the combination of eccentricity and precession is
still insufficient to reproduce the very high degree of precession seen in the
posteriors for GW190521.

Adding noise to the signal causes both the zero-spin and the precessing cases
to show significant deviation from the prior (0.058 and 0.104 bits
respectively). As in the non-eccentric case, noise appears to be a key
requirement in creating apparent evidence for high in-plane spins. Both the
precessing and the non-spinning signals are now close to now show significant
deviations from the prior with a peak between 0.5 and 0.7 (with the non-
spinning case peaking at lower values). For the precessing signal, this is as
was seen in the quasi-circular case. For the non-spinning system, this
deviation from the prior is specific to the eccentric case. In the presence of
noise, it is therefore possible to produce evidence for a precessing system
from a non-spinning, eccentric binary. There is therefore no clear evidence
that this signal is any more likely to have come from an eccentric binary than
from a quasi-circular one.

We therefore do not see any degeneracy between eccentricity and precession for
this kind of high-mass system in the absence of noise. In the presence of
noise, either precession or high eccentricity (or a combination of both) can
give the appearance of a highly-precessing signal. It is not, however,
possible to identify the cause.

##  V Higher SNR and future detectors for heavy mass binaries

Figure 12:  Results of injections with varying SNR. The SNR for the maxL
injection is 15. We show an injection of SNR 30 (green) and SNR 45 (yellow).
Only the injection with SNR 45 has $\rho_{\mathrm{p}}$ at 90% confidence,
implying that only above this SNR can we unambiguously claim a detection of
precession.

Besides, we want to explore under what conditions we will be able to
confidently detect precession from a highly precessing system with high total
mass and a mass ratio close to 1. We examined two possibilities: our example
binary at a closer distance, and thus with a higher SNR for an aLIGO and
aVirgo network; and the same source at the same distance but using the PSD
from future third-generation (3G) detectors [64] which will be more sensitive
and thus the detection will have a higher SNR.

To explore how high an SNR is needed for an aLIGO and aVirgo network to detect
precession, we performed two injections with the maxL parameters but with
varying SNRs (SNR 30 and SNR 45). Fig. 12 shows that in order to be able to
confidently detect precession (i.e. for $\rho_{\mathrm{p}}>2.1$) for such a
high total mass, low mass ratio, highly precessing signal (at 95% confidence),
the SNR of the signal needs to be greater than 45, and possibly even higher,
if we increase our threshold value of $\rho_{\mathrm{p}}$ in accordance with
the findings of Ref. [71].

Figure 13:  Results of the injections with CE, ET, Voyager and a detector
network formed by CE and ET (labelled Network). The grey dashed vertical lines
are the injected values. The y-axes in the first three panels are plotted as
log scale to show the 90% credible interval more clearly.

We considered the increase in sensitivity provided by both individual 3G
detectors (using the PSDs shown in Fig. 1) and by a theoretical detector
network. This detector network is formed by “ET” (a single interferometer in
the position of V1) and “CE” (a single interferometer in the position of H1),
using the detector response of V1 and H1 respectively. The only change we
consider is an improvement in sensitivity and thus use the appropriate 3G PSD.
We assume the detectors to have the same configuration and orientation as
current ground-based detectors and ignore other properties of 3G detectors
such as the triangle configuration. The results from the PE are shown in Fig.
13.

With the increase in sensitivity of the detectors, the injected signals are
much louder and the posterior distributions are consequently narrower. For
each of the 3G detectors, all of the parameters of the injected signal are
well recovered, even for a signal of such a short duration. For a few
parameters, the distributions are now sufficiently narrow that the injected
value, while very close to the peak of the distribution, lies outside or on
the edge of the 90% credible interval. Two examples of this are the recovery
of $\chi_{\mathrm{eff}}$ with CE and $\chi_{\mathrm{p}}$ with ET. From this,
we can see that we will require more precise waveform models or PE for dealing
with such high SNR events.

##  VI Discussion

In this work, we have investigated the detectability of precession and
eccentricity for gravitational wave events within the IMBH mass range,
focusing on the challenges presented by short signals with only a few cycles
in band. To explore this, we performed an injection study using the NR
surrogate model NRSur7dq4 and a selection of NR waveforms taken from the SXS
and RIT catalogues. We assessed the degree to which we could measure
precession and its degeneracy with eccentricity.

We first injected a signal with the maxL parameters extracted from the
detection of our fiducial binary GW190521 and analyzed our ability to recover
various parameters. We then extended our analysis to different total masses
within the IMBH range. We found that the total mass, mass ratio, and
$\chi_{\mathrm{eff}}$ are generally well recovered. However, in the fiducial
parameter injection and most cases involving injections with varying total
masses, we found that we essentially just recover the prior for
$\chi_{\textrm{p}}$, giving us limited information about the degree of
precession in the injected signal. Since we injected a relatively highly-
precessing binary, we therefore investigated other possible causes of the high
degree of precession seen in the results for GW190521.

One possible cause we explored was the effect of noise on the signal. We found
no evidence of systematic bias in the recovery of the parameters due to
Gaussian noise. However, the maxL values of $\chi_{\mathrm{eff}}$ and
$\chi_{\mathrm{p}}$ of some of the injections lie outside of the 90% credible
interval of the zero-noise injection. Our results show that noise strongly
affects the recovery of some parameters for a signal of this length, and we
should be cautious of making strong claims about such a short signal. While a
highly precessing signal with noise could be responsible for the suggestion of
a high degree of precession in the signal, we found that noise alone appears
insufficient to mimic high precession. We require $\chi_{\mathrm{p}}>0.75$ for
a zero-noise injection, while only $\chi_{\mathrm{p}}>0.4$ is needed in a
given noise realization for replicate the high precession in posterior. This
implies that some degree of precession or eccentricity is required to
reproduce these results, assuming the signal to come from a BBH.

The extrinsic parameters of the binary do not significantly affect our
conclusions, although optimizing the values of $\iota$ and $\psi$ can increase
the chance of detecting precession. Interestingly, for these optimized values,
the addition of noise does not have a noticeable impact on the recovery of the
intrinsic parameters, unlike with generic values of $\iota$ and $\psi$.

Our investigation of the degeneracy of eccentricity with precession for this
event reveals no degeneracy between spin and eccentricity for a low
eccentricity ($e\leq 0.2$) system at such high total mass that we see only
merger-ringdown. This is also the case for the highly eccentric waveforms
taken from the RIT catalogue; for the non-spinning eccentric case we only
recover the prior for $\chi_{\mathrm{p}}$, while the highly precessing
eccentric injection is still insufficient to reproduce LVK the results seen
for GW190521 and it is impossible to determine whether the slight deviation
from the prior is due to the effect of precession or eccentricity on the
signal. Similar to the quasi-circular case, the addition of noise makes it
possible to reproduce evidence of highly precessing spins in the signal, for
both the non-spinning and the precessing injections. It is consequently
difficult to conclusively state which of these effects could be responsible
for indications of high precession, such as those seen for GW190521. Thus, our
analysis reveals no clear correlation between eccentricity and precession.

We find that the main requirement for reproducing evidence of high in-plane
spins is noise, but some degree of precession or eccentricity is also
required. Our study highlights the challenges of making conclusive statements
about the properties of the binary from which a short signal originates, and
we hope that future 3G detectors can help to explore this kind of event
further. In order to confidently detect precession for an equal mass, high
total mass, and highly precessing binary, we require an SNR above 45, which is
most likely to occur with 3G detectors. A network of two or more detectors can
improve the recovery of all parameters, especially in the case of high SNR
detections.

In conclusion, this study provides a semi-systematic analysis of the possible
bias and degeneracy of a short signal from an IMBH binary. We have highlighted
the need for caution when making conclusions about the properties of a binary
from a short signal, and the importance of exploring other parameters to
better understand the PE from similar IMBH binaries. However, due to the cost
of computational power and time, we cannot do a full systematic analysis for
all parameter space. We hope with the future machine learning PE tools, we can
do more exhaustive exploration.

###### Acknowledgements.

We would like to thank Philippe Jetzer, Shubhanshu Tiwari, Maria Haney,
Charlie Hoy and Rhys Green for helpful discussions. Y. Xu was supported by
China Scholarship Council. E. Hamilton was supported by grant IZCOZ0_189876
from the Swiss National Science Foundation (SNSF). The authors are grateful
for computational resources provided by Cardiff University and supported by
STFC grant ST/I006285/1. This research has made use of data, software and/or
web tools obtained from the Gravitational Wave Open Science Center
(https://www.gw-openscience.org/ ), a service of LIGO Laboratory, the LIGO
Scientific Collaboration and the Virgo Collaboration. LIGO Laboratory and
Advanced LIGO are funded by the United States National Science Foundation
(NSF) as well as the Science and Technology Facilities Council (STFC) of the
United Kingdom, the Max-Planck-Society (MPS), and the State of
Niedersachsen/Germany for support of the construction of Advanced LIGO and
construction and operation of the GEO600 detector. Additional support for
Advanced LIGO was provided by the Australian Research Council. Virgo is
funded, through the European Gravitational Observatory (EGO), by the French
Centre National de Recherche Scientifique (CNRS), the Italian Istituto
Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions
by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan,
Monaco, Poland, Portugal, Spain. All the code to perform the analysis in this
paper can be found on [84]

##  Appendix A Eccentricity estimator

In general relativity, eccentricity cannot be uniquely defined [85]. We must
therefore choose a consistent method of estimating the degree of eccentricity
present in these NR waveforms. There are a number of different methods to do
this [80, 81, 82, 83]. Here we use the estimator defined in [83] because it
does not rely on a quasi-circular fit of the orbital frequency. The
eccentricity at time $t$ is defined as

$e_{\omega}(t)=\frac{\sqrt{\omega_{p}^{NR}}-\sqrt{\omega_{a}^{NR}}}{\sqrt{\omega_{p}^{NR}}+\sqrt{\omega_{a}^{NR}}},$
(5)

where $\omega_{a,p}^{NR}$ are the GW frequency at apastron and periastron
respectively. $\omega_{a,p}^{NR}$ are estimated by fitting an ansatz of the
form

$\omega_{a,p}^{NR}=c_{a,p}\frac{1+n_{a,p}t}{1+d_{a,p}t},$ (6)

through the value of the GW frequency at the apastron or periastron
respectively. The quantities $c_{a,p}$, $n_{a,p}$ and $d_{a,p}$ are the fit
parameters.

##  Appendix B NR data injection results

Here we present the properties of the injections performed in Sec. IV. We also
give the median and 90% confidence interval of the results of the injections.
Tab. 3 is for the SXS injections while Tab. 4 is for the RIT injections.

Table 3: SXS eccentric injections | $q_{\mathrm{ref}}$ | $\chi_{\mathrm{eff}_{\mathrm{ref}}}$ | $\chi_{p_{\mathrm{ref}}}$ | $e_{t2500}$ | $q$ | $M_{\mathrm{tot}}$ | $M_{\mathrm{chirp}}$ | $\chi_{\mathrm{eff}}$ | $\chi_{\mathrm{p}}$ | $\rho_{\mathrm{p}}$ | $\theta_{jn}$
---|---|---|---|---|---|---|---|---|---|---|---
$BBH_{1355}$ | 1.00 | -0.00 | 0.00 | 0.06 | $0.85^{+0.13}_{-0.25}$ | $273.61^{+20.67}_{-19.58}$ | $118.35^{+9.10}_{-10.68}$ | $0.03^{+0.24}_{-0.25}$ | $0.44^{+0.37}_{-0.30}$ | $0.24^{+0.57}_{-0.20}$ | $0.53^{+2.19}_{-0.41}$
$BBH_{1356}$ | 1.00 | 0.00 | 0.00 | 0.08 | $0.86^{+0.12}_{-0.30}$ | $270.17^{+20.46}_{-17.83}$ | $116.74^{+9.40}_{-9.88}$ | $0.03^{+0.24}_{-0.29}$ | $0.39^{+0.45}_{-0.30}$ | $0.24^{+0.54}_{-0.21}$ | $0.57^{+2.25}_{-0.45}$
$BBH_{1357}$ | 1.00 | -0.00 | 0.00 | 0.10 | $0.84^{+0.14}_{-0.23}$ | $269.00^{+20.01}_{-17.77}$ | $116.18^{+9.10}_{-9.22}$ | $-0.01^{+0.22}_{-0.26}$ | $0.42^{+0.36}_{-0.30}$ | $0.24^{+0.56}_{-0.20}$ | $0.53^{+2.30}_{-0.40}$
$BBH_{1358}$ | 1.00 | -0.00 | 0.00 | 0.10 | $0.87^{+0.12}_{-0.23}$ | $272.75^{+23.40}_{-17.97}$ | $117.97^{+10.40}_{-8.27}$ | $0.02^{+0.26}_{-0.23}$ | $0.38^{+0.42}_{-0.27}$ | $0.23^{+0.55}_{-0.20}$ | $0.59^{+2.23}_{-0.44}$
$BBH_{1359}$ | 1.00 | -0.00 | 0.00 | 0.11 | $0.85^{+0.13}_{-0.25}$ | $269.71^{+17.28}_{-15.76}$ | $116.64^{+7.75}_{-8.75}$ | $0.00^{+0.21}_{-0.21}$ | $0.37^{+0.42}_{-0.27}$ | $0.22^{+0.48}_{-0.18}$ | $0.56^{+2.30}_{-0.42}$
$BBH_{1360}$ | 1.00 | -0.00 | 0.00 | 0.15 | $0.83^{+0.15}_{-0.21}$ | $271.16^{+18.60}_{-17.23}$ | $117.18^{+8.26}_{-8.60}$ | $0.04^{+0.21}_{-0.27}$ | $0.42^{+0.37}_{-0.30}$ | $0.26^{+0.57}_{-0.21}$ | $0.61^{+2.23}_{-0.44}$
$BBH_{1361}$ | 1.00 | -0.00 | 0.00 | 0.16 | $0.88^{+0.11}_{-0.24}$ | $273.05^{+19.98}_{-16.62}$ | $118.37^{+8.14}_{-8.22}$ | $0.03^{+0.20}_{-0.20}$ | $0.34^{+0.44}_{-0.24}$ | $0.20^{+0.47}_{-0.16}$ | $0.62^{+2.27}_{-0.45}$
$BBH_{1362}$ | 1.00 | 0.00 | 0.00 | 0.20 | $0.88^{+0.11}_{-0.23}$ | $266.89^{+20.56}_{-17.81}$ | $115.59^{+9.10}_{-8.36}$ | $-0.05^{+0.26}_{-0.23}$ | $0.39^{+0.40}_{-0.29}$ | $0.25^{+0.60}_{-0.21}$ | $0.67^{+2.20}_{-0.51}$
$BBH_{1363}$ | 1.00 | 0.00 | 0.00 | 0.21 | $0.86^{+0.13}_{-0.27}$ | $270.08^{+18.68}_{-17.90}$ | $116.81^{+8.29}_{-9.63}$ | $0.01^{+0.24}_{-0.27}$ | $0.41^{+0.43}_{-0.31}$ | $0.22^{+0.53}_{-0.18}$ | $0.64^{+2.29}_{-0.51}$
$BBH_{0832}$ | 2.00 | 0.01 | 0.80 | 0.02 | $0.63^{+0.21}_{-0.23}$ | $271.26^{+25.67}_{-23.46}$ | $114.17^{+12.31}_{-16.40}$ | $0.10^{+0.23}_{-0.21}$ | $0.32^{+0.36}_{-0.23}$ | $0.28^{+0.56}_{-0.21}$ | $0.69^{+2.09}_{-0.40}$
$BBH_{1364}$ | 2.00 | 0.00 | 0.00 | 0.05 | $0.54^{+0.24}_{-0.18}$ | $272.87^{+20.78}_{-20.62}$ | $112.02^{+12.24}_{-14.20}$ | $-0.03^{+0.22}_{-0.29}$ | $0.32^{+0.41}_{-0.23}$ | $0.43^{+0.93}_{-0.32}$ | $0.94^{+1.62}_{-0.44}$
$BBH_{1365}$ | 2.00 | 0.00 | 0.00 | 0.06 | $0.53^{+0.25}_{-0.21}$ | $269.98^{+23.55}_{-24.39}$ | $110.61^{+13.70}_{-18.08}$ | $-0.02^{+0.21}_{-0.29}$ | $0.34^{+0.37}_{-0.24}$ | $0.43^{+0.75}_{-0.31}$ | $0.90^{+1.76}_{-0.44}$
$BBH_{1366}$ | 2.00 | 0.00 | 0.00 | 0.10 | $0.57^{+0.23}_{-0.20}$ | $286.92^{+34.22}_{-28.49}$ | $119.02^{+17.36}_{-18.17}$ | $0.13^{+0.28}_{-0.26}$ | $0.32^{+0.45}_{-0.23}$ | $0.32^{+0.63}_{-0.24}$ | $0.88^{+1.78}_{-0.47}$
$BBH_{1367}$ | 2.00 | -0.00 | 0.00 | 0.10 | $0.56^{+0.22}_{-0.19}$ | $281.37^{+32.06}_{-25.60}$ | $116.24^{+17.18}_{-17.69}$ | $0.04^{+0.25}_{-0.24}$ | $0.35^{+0.49}_{-0.28}$ | $0.38^{+0.81}_{-0.31}$ | $0.87^{+1.74}_{-0.38}$
$BBH_{1368}$ | 2.00 | -0.00 | 0.00 | 0.10 | $0.55^{+0.20}_{-0.16}$ | $268.31^{+20.35}_{-18.58}$ | $110.73^{+10.85}_{-12.79}$ | $-0.05^{+0.19}_{-0.25}$ | $0.29^{+0.39}_{-0.22}$ | $0.39^{+0.79}_{-0.30}$ | $0.90^{+1.64}_{-0.41}$
$BBH_{1369}$ | 2.00 | 0.00 | 0.00 | 0.20 | $0.59^{+0.23}_{-0.20}$ | $279.26^{+27.85}_{-25.76}$ | $116.75^{+13.25}_{-16.94}$ | $0.06^{+0.24}_{-0.23}$ | $0.30^{+0.42}_{-0.22}$ | $0.30^{+0.70}_{-0.23}$ | $0.77^{+1.82}_{-0.37}$
$BBH_{1370}$ | 2.00 | 0.00 | 0.00 | 0.18 | $0.54^{+0.22}_{-0.19}$ | $273.75^{+29.60}_{-24.22}$ | $112.21^{+15.41}_{-17.10}$ | $0.03^{+0.24}_{-0.26}$ | $0.26^{+0.44}_{-0.20}$ | $0.31^{+0.77}_{-0.25}$ | $0.83^{+1.76}_{-0.41}$
$BBH_{1371}$ | 3.00 | 0.00 | 0.00 | 0.06 | $0.41^{+0.23}_{-0.17}$ | $274.96^{+28.10}_{-28.14}$ | $106.88^{+18.36}_{-22.79}$ | $-0.05^{+0.21}_{-0.29}$ | $0.26^{+0.41}_{-0.19}$ | $0.39^{+0.98}_{-0.30}$ | $1.16^{+1.36}_{-0.59}$
$BBH_{1372}$ | 3.00 | 0.00 | 0.00 | 0.09 | $0.49^{+0.25}_{-0.19}$ | $299.51^{+35.96}_{-35.12}$ | $120.89^{+19.60}_{-24.81}$ | $0.04^{+0.27}_{-0.27}$ | $0.32^{+0.50}_{-0.23}$ | $0.33^{+0.80}_{-0.25}$ | $2.02^{+0.59}_{-1.47}$
$BBH_{1373}$ | 3.00 | 0.00 | 0.00 | 0.09 | $0.43^{+0.23}_{-0.17}$ | $277.26^{+30.60}_{-29.48}$ | $108.48^{+18.42}_{-23.11}$ | $-0.02^{+0.23}_{-0.34}$ | $0.28^{+0.45}_{-0.21}$ | $0.41^{+1.10}_{-0.31}$ | $1.07^{+1.42}_{-0.51}$
$BBH_{1374}$ | 3.00 | -0.00 | 0.00 | 0.18 | $0.49^{+0.22}_{-0.21}$ | $305.11^{+41.77}_{-40.67}$ | $123.04^{+20.89}_{-28.62}$ | $0.10^{+0.36}_{-0.28}$ | $0.37^{+0.48}_{-0.27}$ | $0.36^{+0.79}_{-0.27}$ | $1.08^{+1.51}_{-0.53}$
Table 4: RIT eccentric injections | $q_{\mathrm{ref}}$ | $\chi_{\mathrm{eff}_{\mathrm{ref}}}$ | $\chi_{p_{\mathrm{ref}}}$ | $e_{r}$ | $q$ | $M_{\mathrm{tot}}$ | $M_{\mathrm{chirp}}$ | $\chi_{\mathrm{eff}}$ | $\chi_{\mathrm{p}}$
---|---|---|---|---|---|---|---|---|---
$eBBH_{1318}$ | 1.00 | 0.00 | 0.00 | 0.75 | $0.80^{+0.18}_{-0.31}$ | $275.50^{+28.25}_{-27.09}$ | $118.27^{+12.70}_{-14.77}$ | $-0.10^{+0.29}_{-0.33}$ | $0.47^{+0.39}_{-0.34}$
$eBBH_{1639}$ | 1.00 | 0.00 | 0.70 | 0.75 | $0.81^{+0.17}_{-0.27}$ | $259.57^{+24.54}_{-20.61}$ | $111.68^{+11.15}_{-10.50}$ | $-0.17^{+0.31}_{-0.33}$ | $0.55^{+0.33}_{-0.38}$
$eBBH_{1318}$ with noise | 1.00 | 0.00 | 0.00 | 0.75 | $0.79^{+0.19}_{-0.26}$ | $280.87^{+30.65}_{-26.39}$ | $120.62^{+13.15}_{-13.11}$ | $-0.00^{+0.37}_{-0.37}$ | $0.59^{+0.31}_{-0.41}$
$eBBH_{1639}$ with noise | 1.00 | 0.00 | 0.70 | 0.75 | $0.68^{+0.25}_{-0.22}$ | $272.61^{+27.67}_{-24.01}$ | $115.65^{+12.08}_{-12.77}$ | $-0.04^{+0.33}_{-0.31}$ | $0.64^{+0.28}_{-0.40}$

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