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Impact of noise on the measurability of precession and eccentricity for a GW190521-like event

Information from gravitational wave detectors can be used to inform our understanding of the astrophysical populations of black holes in the universe, as well as our understanding of their formation. Detections from binaries composed of black holes which lie outside the bulk of the distribution of the astrophysical population are of particular interest. In this paper, we perform an in-depth study of GW190521, the first detection of an intermediate mass black hole where the mass of the primary black holes lies in the pair-instability supernova mass gap. It has been suggested that there is evidence of both precession and eccentricity in the signal, which would have clear implications for the formation channel of the binary. However, since this short signal consists of only the merger-ringdown portion there is a high possibility that the parameter estimation might be biased. We find that for a signal of this length, the dominant effect on the estimation of the spin parameters is the noise in the detector. It is therefore not possible to make conclusive statements distinguishing whether this system is either precessing or eccentric.

Information from gravitational wave detectors can be used to inform our understanding of the astrophysical populations of black holes in the universe, as well as our understanding of their formation. Detections from binaries composed of black holes which lie outside the bulk of the distribution of the astrophysical population are of particular interest. In this paper, we perform an in-depth study of GW190521, the first detection of an intermediate mass black hole where the mass of the primary black holes lies in the pair-instability supernova mass gap. It has been suggested that there is evidence of both precession and eccentricity in the signal, which would have clear implications for the formation channel of the binary. However, since this short signal consists of only the mergerringdown portion there is a high possibility that the parameter estimation might be biased. We find that for a signal of this length, the dominant effect on the estimation of the spin parameters is the noise in the detector. It is therefore not possible to make conclusive statements distinguishing whether this system is either precessing or eccentric.

I. INTRODUCTION
On May 21, 2019 at 03:02:29 UTC the LIGO-Virgo-KAGRA (LVK) collaboration detected a gravitational wave (GW) event, designated GW190521 [1,2]. The most likely interpretation of this event, as reported by the LVK, is a quasi-circular compact binary coalescence (CBC). GW190521 is a very short event (approximately 0.1s in duration) and only has around 4 cycles in the frequency band 30-80Hz, which is the frequency range to which current generation ground-based detectors aLIGO and aVirgo are most sensitive [3,4]. The signal peaked at a frequency of 60Hz. The early inspiral and 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 open 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, and the signal can be shown to be consistent with that originating from a number of less likely sources [5][6][7][8][9]. The parameter estimation (PE) is also likely to be disproportionately affected by noise in the detector. In this paper, we will assume the signal to have come from a CBC source, though not necessarily a quasi-circular one.
The GW signal h from a CBC can be written as a combination of two polarisations h + and h × as h = h + − ih × . It can be decomposed into modes h m via where the −2 Y m are a basis of spin-weighted spherical harmonics and (ι, φ 0 ) 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. A quasi-circular black-hole-binary (BBH) is characterised by its intrinsic parameters; the total mass M 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 S 1 and S 2 . If we consider a binary on an eccentric orbit then we add at least one additional intrinsic parameter; the eccentricity e.
In addition, the signal as measured at the detectors will be further characterised by a set of extrinsic parameters, which include polarisation ψ, inclination ι, right ascension α, declination δ and luminosity distance d L . From post-Newtonian theory, we can construct an effective spin χ eff = ( S 1 /m 1 + S 2 /m 2 ) ·L/M tot [10,11] which characterizes the impact of the components of the spins aligned with the orbital angular momentum of the binaryL. The effective spin can be measured better than the individual spins [12,13]. The components of the spins which lie perpendicular toL causeL to precess about the total angular momentum of the binary, J =L +Ŝ 1 +Ŝ 2 . This effect is known as spin-induced precession. The dominant effects of precession on the GW signal can be characterized by the effective precession spin χ p = max(A 1 S 1⊥ , A 2 S 2⊥ )/(A 1 m 2 1 ) where A 1 = 2 + 3q/2 and A 2 = 2 + 3/(2q) [14]. S 1⊥ and S 2⊥ are the magnitudes of the components of the spins that lie in the orbital plane.
Assuming GW190521 is due to a CBC, this event is of particular interest because it is the first-ever detection of an intermediate-mass black hole (IMBH), which have masses in the range 10 2 − 10 5 M . The posterior for the final black hole mass M f = 142 +28 −16 M shows no support below 100 M . IMBHs bridge the gap between stellar BHs and supermassive BHs (SMBHs) and might be the missing link to explain the formation of SMBHs [15]. Additionally, the probability of the mass of the primary black hole being below 65 M is 0.32%. GW190521 is therefore also the first detection of a merging black hole in arXiv:2211.09561v1 [gr-qc] 17 Nov 2022 the range of the PISN (Pair-instability supernova) mass gap (65 − 120M ). When a star with a helium core in the range of the PISN mass gap 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 [16][17][18][19][20]. Thus, for a black hole in the PISN mass gap, stellar-origin formation is disfavoured. 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 [2,[21][22][23][24][25][26] These different formation channels all predict different properties of the black hole, such as the magnitude and orientation of its spin [27]. Accurate inference of the properties of a binary containing a black hole in the PISN mass gap can therefore be used to inform conclusions about possible formation channels. It is therefore particularly important to ascertain the degree to which PE for this signal is accurate, in order to be confident of any astrophysical inferences drawn from this signal [27,28].
There is some support in the original LVK analysis that this signal originated from a highly precessing binary, with χ p = 0.68 +0. 25 −0.37 [1]. There have only been three signals reported by the LVK so far which show any support for precession: GW190412, GW190521 and GW200129 065458 [1,[29][30][31][32]. As discussed, spin-induced precession occurs when the spins are not aligned with the orbital angular momentum. Precession introduces modulations of the GW amplitude and phase [33,34]. There will be bias in the PE if we neglect to consider precession when analysing signals from precessing systems.
There are also several studies which claim that this event might be an eccentric binary merger [6,7], with eccentricities up to 0.7 [7]. Eccentricity can also cause modulations in the amplitude and phase of the signal. These are distinct from precession since we expect eccentricity to be radiated away throughout the inspiral, so the effects will be strongest during the inspiral and weaker or non-existent through merger and ringdown, unless the binary merges very soon after formation. In contrast, precession effects continue right through merger. Additionally, unlike for precessing systems, eccentricity does not cause increased excitation of higher-order modes as compared with the circular case [35][36][37]. Due to the circularisation of eccentric binaries throughout their inspiral, it is expected that the vast majority of binaries seen by current ground-based detectors will have low, likely negligible, eccentricites [35,38,39].
In this paper, we investigated whether we could recreate the high precession seen in the PE for GW190521 by analysing injected waveforms with known properties. We also investigated the effects of noise and of varying the extrinsic parameters as well as the degeneracy of precession with eccentricity. 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 this signal assuming that it came from a quasi-circular binary. 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. Finally, we give our conclusions and discussions from these analyses in Sec. V.

II. METHODOLOGY
To study this event by injection, we first must choose the most appropriate waveform model for our investigation. The motivation for our choices is outlined in Sec. II A. The methods employed to generate the waveforms and perform the injections are described in Sec. II B. We discuss the setup employed for the PE in Sec. II C. The techniques used to further analyse the results of the PE are then outlined in Secs. II D and II E.

A. Waveform models
A number of precessing waveform models have been developed to date [40][41][42][43][44][45][46]. A subset of three [42,43,45] of these models were used in the original LVK analysis of GW190521 [1]. In our analysis, we use one of these models, NRSur7dq4 [43]. 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 these cases, such as GW190521 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 [41,46].
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 [47][48][49][50][51] 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 [52] and the RIT Catalog of Numerical Simulations [53] 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.

B. 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. For non-eccentric injections, we choose for our source parameters the max-loglikelihood parameters reported by the LVK and repeated in Tab. I for convenience. These are the values for the waveform that were found to agree best with the event data in the original LVK analysis. In what follows, we will refer to the injection in zero noise using exactly these parameters as the "maxL injection". Both the NRSur7dq4 model and the NR waveforms supply the GW modes h m , which are recombined to produce the GW strain h using get td waveform from PyCBC [54] and SimInspiralChooseTDWaveform from LALSimulation [55] as part of the NR Injection Infrastructure [56] respectively. The h + and h × polarisations are then projected onto a detector network using the appropriate detector response functions F + (α, δ, ψ) and F × (α, δ, ψ) using Detector.project wave from PyCBC [54]. α, δ, ψ are as defined as in Tab. I. 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 colored Gaussian noise from the power spectral density (PSD) for a given detector.

C. 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 [57] P (parameter|data) ∝ P (data|parameter)×P (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 [2], except the priors are uniform on chirp mass (70-150 M tot ) and mass ratio (0.17-1.0) instead of component masses as suggested in Ref. [58].
We used the PE code Parallel-Bilby [59][60][61] with the Dynesty sampler [62] 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) [63]. The curve for the detector H1 is shown, labelled O3 H1. We also consider sensitivity curves for aLIGO and aVirgo [64,65] (which will be referred to as design PDSs) as well as future detectors Voyager, Cosmic Explorer (CE) and Einstein Telescope (ET) [66] in the discussion surrounding future detections in Sec. III F.

D. 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 [67]. The JS-divergence is a variant of the Kullback-Leibler divergence D KL [68]. For discrete probability distributions P and Q on the same probability space χ where M = (P + Q)/2 and D 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 JS (P ||P ) = 0. A value of D JS < 0.05 is generally taken to mean that the two distributions are in good agreement [69].

E. Precessing SNR
One of the most interesting questions about GW190521 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 [70,71], 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 [70]. 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 ρ p is defined by the minimal SNR of h 0 and h 1 where A 0 and A 1 are the amplitude of each harmonic. A value of ρ 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 [70][71][72]. ρ 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. [73] it was found that for a comparable mass ratio event, the threshold value of ρ 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. INVESTIGATION OF PRECESSION
In order to better understand the indications of high in-plane spins in the LVK analysis of GW190521, we performed a wide-ranging injection study in order to investigate a range of possible causes. We first confirmed the current understanding of our inability to detect precession for a low mass ratio, low-moderate SNR event. 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 for a GW190521-like signal.

A. Max Likelihood injection
We first examine whether the maxL injection can produce the same results as seen in the LVK analysis of GW190521. To do this, we produced a signal with the parameters given in Tab. I and injected it as described in Sec. II B. We consider both design and event PSDs.
The posterior samples of key parameters from the maxL injection are plotted in Fig. 2, alongside the LVK results. From the LVK results, we can see that the total mass is around 270 M in the detector frame and the mass ratio is close to 1. The peak of the χ eff distribution is close to zero, which indicates that the aligned spin components are very small. In contrast, low in-plane spins (χ p < 0.31) are excluded at 90% confidence. The large deviation from prior and the shift in the peak of the χ p distribution towards higher values could imply large in-plane spins.
From the comparison between the maxL injections and the LVK results, we can see that the total mass M tot and mass ratio q of the maxL injection match the values measured for the event well; the JS divergence between the posteriors of the LVK results and the maxL injections is less than 0.02 bits for both the total mass and the mass ratio in both cases. The posteriors of M tot and q are also centred on the injected values. This implies that we expect to be able to accurately recover these parameters for this kind of system and, at least in zero noise, we can be confident in our detection of these properties of the system.
However, the posteriors for the spin parameters don't reproduce what we see for GW190521 for the injections with either set of PSDs. The effective spin χ eff has a narrower distribution implying that we are recovering χ eff better for the injected signal than for the event. The peak of the distribution is centred on the injected values, implying that we are recovering the parameter well. However, we do not see the support for the negative χ eff values seen in the event posterior, since this broadening of the posterior is likely due to detector noise. The peak and 90% confidence interval of the posterior of χ p are shifted relative to the event posterior. In fact, the JS divergence for χ p with respect to the prior is 0.03 bits and 0.06 bits for the maxL injection when considering the injection using the design PSDs and event PSDs respectively. We can therefore see that for the maxL injection we are simply recovering the prior for χ 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 implies that the high value of the peak in the χ p posterior distribution and significant deviation from the prior (a JS divergence of 0.16 bits) for the event is unlikely to be due to a high degree of precession in the signal and more likely to be due to other factors. This conclusion is corroborated by the calculation of the precession SNR ρ p for the detection and the injected signal. At 95% confidence, the precession SNR of both the event GW190521 and the maxL 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.
From these results, we can conclude that the apparent evidence for precession in GW190521 is more likely to be due to factors which may be able to mimic the appearance of precession in the signal, such as noise and eccentricity. We therefore investigated the effect of these factors on the signal. Since the event PSD results and design PSD results show negligible differences, we will only use design PSD results as the maxL injection for comparison in the following sections.

B. Effect of Gaussian noise
To explore any possible bias in the inferred parameters introduced by the presence of noise, we performed a series of injections consisting of the maxL waveform in Gaussian noise as described in Sec. II B. In order to comprehensively investigate the effect of noise, we produced 38 different injections, all with different Gaussian noise realisations. We will refer to any injections containing Gaussian noise generated in this fashion as "noisy injections". The posteriors of key parameters are plotted in Fig. 3.
From Fig. 3, we can see that the addition of noise causes deviations in the posteriors of the recovered parameters from 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.
It is clear from these results that some of the noisy injections have the potential to mimic the LVK results, especially the noise realisation highlighted in orange. We designate this noisy injection which produces posteriors which are, on average, closest to the LVK results as N c and the noise realisation which causes the maximum deviation (JS divergence) from the prior for χ p as N max . The posteriors from the N c injection show a good match with the LVK results for the parameters M , q, χ eff and χ p ; the JS divergence is O 10 −2 bits for each of these intrinsic parameters. This implies that the bias towards higher precessing spins seen in the PE for the event may be due to the effect of noise fluctuations on such a short signal.
From Fig. 3 we can clearly see that for such a short signal, different noise realisations can have a significant impact on the signal. The effect of noise on the signal is most pronounced for the spin parameters; from the penultimate panel in Fig. 3, we can see the maxL values for χ p and χ eff for some of the injections do not lie in the 90% credible interval of the zero noise injection (indicated by the orange markers). The total mass and mass ratio are also affected by the noise, albeit to a lesser degree. For the total mass, the 90% confidence interval is fairly narrow and, in some cases, the addition of noise is sufficient to shift the peak outside the 90% confidence interval of the zero-noise injection. However, the overall shift is not large enough to significantly alter any astrophysical conclusions drawn about this event. Conversely, for an event of this short duration, the shape of the distribution for the mass ratio q can be significantly altered by the addition of noise. In contrast with the spin parameters, however, almost all of the maxL values fall into the 90% credible interval of the zero-noise injection.
Caution should therefore be taken in inferring anything about the nature of an event characterised by such a short signal from the location of the peak and the values of the maxL parameters. Indeed, while the correct parameters lie within the 90% confidence interval for the majority of the noise realisations, there are a small minority of cases where they lie just outside it. It is therefore clear that the location of the peak and the maxL parameters may well not be close to the correct value.
The noise-induced changes in the posterior of χ p that we see here were also seen in Ref. [74]. In that analysis, the posteriors for the maxL parameters injected into zero noise show a stronger deviation from the prior than we see in Fig. 2. Further, the 10 different Gaussian noise realisations considered in that analysis also suggest a possible bias in the noise-induced shift in the prior, although this is not true for all cases. We anticipate that these differences are most likely due to the different choices of PSD employed in each of the analyses.
The fifth and sixth panels in Fig. 3 shows the effect of noise on extrinsic parameters polarization ψ and inclination ι. After adding the noise, a second peak is visible in the posteriors for these extrinsic parameters for some of the injections. The second peak was present in the event posteriors but was not seen for the zero noise injection.

C. Non-spinning systems
Having seen in the previous section that a combination of gaussian noise and a highly precessing signal is sufficient to reproduce the results seen in the detection of GW190521, we now want to investigate whether the evidence for precession in the LVK posterior can be produced purely by noise i.e. whether the signal from a non-spinning or aligned-spin binary could be distorted by noise in a way that could mimic a highly precessing signal. Thus we generated another set of injections using another 30 random noise realisations. The injected signal is now generated with the same maxL parameters as before, except for the spin components, which are all set to zero. We chose to investigate a non-spinning binary as the limiting case, since the LVK results indicate low or negligible aligned spins.
The results of these injections are shown in Fig. 4. We find that the intrisic 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 one would expect, the posteriors for χ eff are centred on zero (the injected value). Furthermore, they do not appear to be noticeably impacted by the noise. This is also clear for the posteriors of χ p , where we essentially recover the prior for each of our 30 injections. It is clearly unlikely that the apparent 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 what we see in the LVK results. It is therefore not possible to produce evidence for precession in such a system purely as a result of detector noise.

D. Effects of varying spin magnitude
Since our results show that the signal is unlikely to come from a non-spinning binary, we want to know the minimum in-plane spin magnitude required to reproduce the χ p posterior seen in the LVK event for a source with the same total mass, mass ratio and extrinsic parameters.
The specific spin configuration significantly affects the recovery of χ p from a short, highly precessing signal [74]. For certain configurations, there is a much higher chance of seeing evidence of precession in the signal. However, as we have shown, this is not the only effect which can produce evidence of a highly precessing system. Since we do not wish to investigate the effect of the spin orienta- tion here, we fixed the orientation of the spin to the value inferred from the LVK result and varied the overall spin magnitude. This keeps the ratio of the value of χ p to the value of χ eff constant. The values of χ p and χ eff in each of these injections are listed in Tab. II. We first considered the zero noise case to see whether a precessing binary alone (i.e. without the addition of noise) is capable of producing the results seen in the detection of GW190521, and, if so, the spin magnitude required to do this. We then consider the effect of noise on these conclusions.
From Fig. 5, we can see that only when the value of χ p is around 0.7, do we start to see the deviation from the prior in χ p and only for χ p ∼ 0.8 is this deviation sufficient to reproduce the LVK results. This conclusion is also supported by the JS divergence between the posteriors and the prior for χ p ; only the χ p5 injection shows a deviation of O 10 −1 bits. Additionally, the JS divergence between χ p5 and the LVK results is only 0.014 bits, which means the posteriors for χ p are very close to those seen for the event. For a zero noise injection, it requires an in-plane spin of around 0.77, and a total spin of greater than 0.9 (which is beyond the edge of the calibration range of NRSur7dq4) to produce evidence of high in-plane spins as seen in GW190521.
To explore the minimum spin magnitude required if we consider noise in the detector, we injected our signal into the N max noise realisation, which produced the largest inferred value of χ p as discussed in Sec. III B. These results are shown in Fig. 6. We find that in the presence of noise, a value of χ p greater than 0.4 is required to hint at the evidence of high in-plane spin values. Here we have not done a systematic exploration using a range of noise realisations due to limits on computational resources. Nevertheless, combined with the results in section III C, this shows to have a high inferred effective precession spin for this kind of event, we require both a non-zero value of χ p (and thus some degree of precession in our source signal) and noise. However, the presence of noise in the signal significantly lowers the magnitude of the spin that can be claimed for this event.

E. Extrinsic parameters
The measurability of precession is affected by the extrinsic parameters, especially polarization ψ and inclina- tion ι [71]. These parameters are poorly measured by the detection, meaning the maxL values measured for GW190521 used in the preceding analysis may not be even close to the correct ones. We investigate whether we might be able to confidently detect precession for a GW190521-like signal, if instead we choose the values of ι and ψ that maximise the likelihood of detecting any precession present in the signal. We calculated ι and ψ for which the precessing SNR ρ p is maximised for this event. The dependency of ρ p on ψ and ι is shown in Fig. 7. From this, we can see the maxL value lies in a region where, compared to the maximum possible value of ρ p for this event, we are only moderately confident in our ability to detect precession. The maximum value of ρ p possible for a GW190521-like signal is around 1.43, which is below the threshold to unambiguously claim detection of precession, as discussed in Sec. II E. We therefore do not expect to be able to claim a detection of precession for any set of extrinsic parameters for this event Nevertheless, we wish to explore the effect of these extrinsic parameters on our ability to interpret the signal. We therefore generated and injected another waveform based on the maxL parameters but changed the values of ι and ψ to maximize ρ p , to investigate how optimizing these extrinsic parameters affects the likelihood of detecting precession for a given system. The four maxima in Fig. 7 are equivalent, so we chose the first one ι = 1.759 and ψ = 1.005. We also investigated the effect of noise on these altered extrinsic parameters, using the N max noise realisation.
From the results (shown in Fig. 8), the peak of the posteriors for the inclination ι and polarisation ψ in the zero-noise max ρ p injection is centred on the injected value. As in the maxL case, we see only a single peak for each of these parameters in the absence of noise. With the addition of noise (using the N max realisation), the single-peak distributions once again disappear and we see posteriors that are much more similar to the LVK results.
We can also see that when injecting the parameters which maximise the precessing SNR, the peak of posterior for χ p is shifted to marginally higher values. However, it is still insufficient to reproduce the LVK result. Interestingly, the addition of noise does not seem to affect the distribution much, unlike in the maxL case, 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 ρ p is still below the threshold at which we expect to be able to measure precession. Regardless of the true value of ψ and ι, we still wouldn't be able to detect precession for a GW190521-like event. However, the apparent evidence of precession in the signal may enable us to rule out certain sky positions.

F. Consdering SNR and future detectors
Finally, 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: an equivalent binary to GW190521 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 [66] which will be more sensitive and thus the detection will have a has ρp at 90% confidence, implying that only for above this SNR can we unambiguously claim a detection of precession.
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 SNR (SNR 30 and SNR 45). Fig. 9 shows that in order to be able to confidently detect precession (i.e. for ρ 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 ρ p in accordance with the findings of Ref. [73].
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. 10.
With the increase in sensitivity of the detectors, the injected signals are much louder and the posterior distributions consequently narrower. For each of the 3G detctors, all of the parameters of the injected signal are well recovered, even for a signal of such 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 χ eff with CE and χ p with ET. From this we can see that we will require more preceise waveform models or PE for dealing with such high SNR events.

IV. INVESTIGATION OF ECCENTRICITY
It has been suggested that for this event, evidence of precession in the signal may be degenerate with indications of eccentricity [6,7]. 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 A.
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 majority of binaries have e < 0.2 by the time they enter the LIGO band (> 20Hz) [35,75,76].
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 Sec. IV A. We then perform an injection study with these waveforms, following the procedure detailed in Sec. II B. The results of this study are given in Sec. IV B.

A. Eccentricity estimator
In general relativity, eccentricity cannot be uniquely defined [80]. 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 [81][82][83][84]. Here we use the estimator defined in [84] because it does not rely on a quasi-circular fit of the orbital frequency. The eccentricity at time t is defined as where ω N R a,p are the GW frequency at apastron and periastron respectively. ω N R a,p are estimated by fitting an ansatz of the form ω N R a,p = c a,p 1 + n a,p t 1 + d a,p t , 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.
Since the eccentricity decays with time, we need to choose a reference time t ref at which to report the eccentricity. Here we choose t 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 eccentricity are given in Tab. III. For the cases taken from the RIT catalog, we note that the waveforms only contain 0.5 cycles prior to merger. All the methods proposed to estimate the eccentricity from the waveform [81][82][83][84] are not valid for such short waveforms. Consequently, we have to rely on the eccentricity reported in the metadata of each waveform [53] 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 .

B. Results
We use the Numerical Relativity Injection Infrastructure [56] to generate a waveform with the same total mass and extrinsic parameters shown in Tab. I. We then perform the injection as detailed in Sec. II B.
The results of the systematic investigation into the impact of comparatively low eccentricity on the quasicircular PE of a high mass system are show in Fig. 11. The median and 90% distributions of the key parameters from the posterior are also given in Tab. III. From the χ p plot, we can see that for a signal at SNR 15.4 from a high mass non-spinning binary with q ≤ 3 and e ≤ 0.2, the effect of eccentricity on the signal does not mimic the effect of precession. The JS divergence of the χ p posteriors from the prior is O 10 −2 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 seen in GW190521 is not due to eccentricity for binaries with e < 0.2.
In Ref. [7], 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 χ p = 0.7 (RIT:eBBH:1639), both with the initial eccentricity e r = 0.75. The precessing waveform was found in [7] to be that which most closely matched the posteriors from GW190521. We performed both a zero-noise and noisy injection using the N max noise realisation of each of these waveforms.
The results are shown in Fig. 12 and the median and 90% distributions of the key parameters from the posterior are given in Tab. IV. We can see that the nonspinning eccentric waveform does not cause significant deviation from the prior for χ p , with only JS divergence D JS = 0.006 bits. It shows no difference to the nonspinning 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 highly precession, quasicircular 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 reproducing the LVK results. Both the precessing and the non-spinning signals are now close to reproducing the LVK results (although it should be noted that for the non-spinning case, the peak remains lower and we do not exclude low spins at such high confidence). 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 reproduce the LVK results 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. Results from the high eccentricity injections. In zero-noise, neither the non-spinning eccentric waveform (green) nor the precessing eccentric waveform (yellow) show 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.

V. DISCUSSION
In this work, we have investigated our ability to make conclusive statements about the properties of a GW190521-like signal; that is a signal with only a few cycles in band, assumed to have come from a high total mass, equal mass ratio system. To achieve this end 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 investigated the effect of noise on the signal, the degree to which we could measure precession and its degeneracy with eccentricity.
We first injected a signal with the maxL parameters reported for the detection of GW190521. The total mass, mass ratio and χ eff are well recovered and centred on the injected values. While the injected in-plane spin values are within the 90% confidence interval for χ p , we find that we essentially just recover the prior giving us no information about the degree of precession in the injected signal. The precessing SNR for this system also implies we expect to be unable to detect precession from such a signal. The indications of a high degree of precession in the PE of GW190521 therefore cannot be unambiguously attributed to the presence of a high degree of precession in the signal We therefore investigated other possible causes of the high degree of precession seen in the PE of GW190521.
The first of these possible causes that we investigated was the effect of noise on the signal. To do this we performed over 30 injections, each with a different Gaussian noise realisation. We found no evidence of systematic bias in the recovery of the parameters due to noise. However, the maxL values of χ eff and χ p of some of the injections lie outside of the 90% credible interval of the zero-noise injection. Noise therefore very 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. One noise realisation N c potentially mimics the LVK results for GW190521, implying that a highly precessing signal with noise could be responsible for the suggestion of a high degree of precession in the signal.
We therefore wished to investigate whether a highly precessing signal was required to reproduce evidence of a highly precessing system in the PE or whether noise was the primary requirement. We found a non-spinning system gives no evidence of precession, meaning it is not possible to have hints of a highly precessing system from noise alone. However, while we require χ p > 0.75 to see any evidence of precession in an otherwise GW190521like signal in zero noise, we require only χ p > 0.4 in a given noise realisation, thus decreasing the magnitude of the spin required. This further implies that GW190521 may not be from a highly precessing system, although it is unlikely to have come from a quasi-circular nonspinning system.
These conclusions are not significantly affected by altering the extrinsic parameters of the binary, although it is possible to optimise the values of ι and ψ to increase the chance that precession will be measureable. Interestingly, for these optimised values, the addition of noise does not have a noticeable impact on the recovery of the intrinsic paramters, unlike with generic values of ι and ψ.
In order to confidently detect precession for an equal mass, high total mass, highly precessing binary, we require an SNR above 45. High SNR detections are most likely to occur with 3G detectors. In this case we are able to confidently recover all parameters, expecially when considering a network of two or more detectors.
Finally, we investigated the degeneracy of eccentricity with precession for this event. We see no degener-acy between spin and eccentricity for a low eccentricity (e ≤ 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 χ 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 is responsible in the posteriors seen for GW190521.
From the above studies, we find that the main requirement for reproducing the results seen in the LVK analysis of GW190521 is noise. Noise has a very strong impact on a signal of this length, which makes it difficult to make conclusive statements about the signal and the properties of the binary from which it originates. Nevertheless, noise alone appears insufficient to reproduce LVK results. This implies that some degree of precession or eccentricity is required to reproduce these results, assuming the signal to come from a BBH. However, we cannot conclusively state whether the signal is from either a precessing or an eccentric binary.
In this paper, we have semi-systematically analyzed the possible bias and degeneracy of this short signal. 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.

ACKNOWLEDGMENTS
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 [85]

Appendix: NR data injection results
Here we present the properties of the injections performed in Sec. IV B. We also give the median and 90% confidence interval of the results of the injections. Tab. III is for the SXS injections while Tab. IV is for the RIT injections.