Resolving the eccentricity of stellar mass binary black holes with next generation ground-based gravitational wave detectors

Next generation ground-based gravitational wave (GW) detectors are expected to detect $\sim 10^4 \mbox{-} 10^5$ binary black holes (BBHs) per year. Understanding the formation pathways of these binaries is an open question. Orbital eccentricity can be used to distinguish between the formation channels of compact binaries, as different formation channels are expected to yield distinct eccentricity distributions. Due to the rapid decay of eccentricity caused by the emission of GWs, measuring smaller values of eccentricity poses a challenge for current GW detectors due to their limited sensitivity. In this study, we explore the potential of next generation GW detectors such as Voyager, Cosmic Explorer (CE), and Einstein Telescope (ET) to resolve the eccentricity of BBH systems. Considering a GWTC-3 like population of BBHs and assuming some fiducial eccentricity distributions as well as an astrophysically motivated eccentricity distribution (Zevin et $al.$ (2021)), we calculate the fraction of detected binaries that can be confidently distinguished as eccentric. We find that for Zevin eccentricity distribution, Voyager, CE, and ET can confidently measure the non-zero eccentricity for $\sim 3\%$, $9\%$, and $13\%$ of the detected BBHs, respectively. In addition to the fraction of resolvable eccentric binaries, our findings indicate that Voyager, CE, and ET require typical minimum eccentricities $\gtrsim 0.02$, $5\times 10^{-3}$, and $10^{-3}$ at $10$ Hz GW frequency, respectively, to identify a BBH system as eccentric. The better low-frequency sensitivity of ET significantly enhances its capacity to accurately measure eccentricity.


INTRODUCTION
The LIGO (Aasi et al. 2015) and Virgo (Acernese et al. 2015) have detected O (100) compact binary mergers (Abbott et al. 2016(Abbott et al. , 2017c(Abbott et al. , 2019(Abbott et al. , 2020;;Venumadhav et al. 2020;Abbott et al. 2021a,b;Nitz et al. 2021;Olsen et al. 2022;Nitz et al. 2023).One of the most interesting questions these observations are capable of answering is related to the formation scenarios of compact binaries, especially of binary black holes (BBHs) which will be observed in large numbers with the present and next generation GW detectors.There are broadly two formation channels of compact binaries: isolated formation channel and dynamical formation channel (Mapelli 2020(Mapelli , 2021;;Mandel & Farmer 2022).In isolated formation channel, two massive stars in a binary system co-evolve in the galactic field, undergo supernova explosions and form BBHs. In the dynamical formation, a BBH is formed due to gravitational interaction in dense stellar environments such as globular and nuclear star clusters.The current observations of BBH mergers by LIGO and Virgo pose a challenge in terms of being explained solely through a single formation channel.A mixture of formation channels is preferred over a single formation channel (Abbott et al. 2021c;Bouffanais et al. 2021;Wong et al. 2021;Zevin et al. 2021a).
Future gravitational wave (GW) detectors with improved sensitivity will detect a large number of BBH mergers.For example, a LIGO ★ E-mail: pankajsaini@cmi.ac.in detector with Voyager technology is expected to detect ∼ 10 4 BBH mergers per year (Baibhav et al. 2019).Moreover, third-generation (3G) detectors such as Cosmic Explorer (CE) (Abbott et al. 2017a;Reitze et al. 2019) and Einstein Telescope (ET) (Punturo et al. 2010;Sathyaprakash et al. 2011;Maggiore et al. 2020) are expected to detect ∼ 10 5 BBH mergers per year (Baibhav et al. 2019;Evans et al. 2023;Gupta et al. 2023).Different formation channels leave unique imprints on the properties of BBH systems, providing valuable clues about their formation history.
Orbital eccentricity is one such unique feature that can give us clues about the formation channels of the binaries as different formation channels are expected to follow different eccentricity distributions.The ellipticity of a binary's orbit at a given time depends on how these binaries formed.Since the emission of GWs carries away the energy and angular momentum from the binary system, with time, the elliptical orbits tend to become circular (Peters 1964).In the small eccentricity limit, the eccentricity approximately decreases with GW frequency as   / 0 ≈ (  0 /  ) 19/18  (Peters 1964).Here   is the 'time-eccentricity' in the quasi-Keplerian representation of the binary orbit (Damour et al. 2004) when binary emits the GWs at dominant mode frequency  , and  0 is the initial value of   at a reference frequency  0 . 1 For example, a binary with an initial orbital eccentricity  0 = 0.6 when emitting GWs at 0.1 Hz reduces to one 1 The definition of eccentricity is model dependent.One should ideally use a model-agnostic definition of eccentricity when comparing eccentricity mea-with eccentricity ∼ 0.1 when it enters the ET band at 1 Hz.The eccentricity reduces to ∼ 0.02 when it starts emitting GWs in the frequency band of CE and Voyager at 5 Hz.The eccentricity further reduces to ∼ 5 × 10 −3 when it emits GWs at 20 Hz (the lower cut-off frequency of LIGO and Virgo).
In current GW detectors, the measurement of eccentricity for GW150914-like BBHs requires  0 ≳ 0.05 (at 10 Hz GW frequency) (Lower et al. 2018;Favata et al. 2022).Next generation GW detectors with improved sensitivity will enable us to constrain smaller eccentricities.Apart from the overall improvement in sensitivity, 3G detectors will have excellent low-frequency sensitivity where binaries can retain a larger eccentricity.LIGO-Virgo-KAGRA currently employ quasi-circular templates for the detection of BBHs.Template-based matched-filter searches for eccentric binaries require accurate eccentric waveform model.Including eccentricity in parameter space increases its dimensionality and hence the number of templates and computational cost.Various alternate search methods have been proposed for the detection of BBHs (Tiwari et al. 2016;Cheeseboro & Baker 2021;Lenon et al. 2021;Pal & Nayak 2023).
Reference (Lenon et al. 2020) reanalysed the detected binary neutron star (BNS) signals GW170817 (Abbott et al. 2017c) and GW190425 (Abbott et al. 2020) with eccentric waveform models and found that these systems have  0 ≤ 0.024 and  0 ≤ 0.048, respectively.Reference (Chen et al. 2021) studied the signal-to-noise ratio (SNR) sky distributions of eccentric and quasi-circular BBHs for second and third-generation GW detectors.The potential of next generation GW detectors to measure orbital eccentricity remains unexplored.In this paper, we study the ability of next generation GW detectors to constrain the eccentricity of BBH systems once a BBH system has been confidently detected.
Typically, binaries formed through isolated formation channels are expected to be in quasi-circular orbits by the time they enter the frequency band of ground-based detectors (Peters 1964;Hinder et al. 2008).Isolated binaries can have a non-negligible eccentricity due to the large natal kick imparted due to the second supernova.However, it happens well before the merger that the binary is likely to shed away all its eccentricity by the time it is observed in the frequency band of the ground-based detector.N-body simulations suggest that the formation of highly eccentric binaries is inevitable in dense stellar environments (Antonini & Perets 2012;Antonini et al. 2014Antonini et al. , 2016;;Rodriguez et al. 2018b).Dynamically formed binaries in dense star clusters such as globular star clusters, and nuclear star clusters can retain moderate to high values of eccentricity when observed in the frequency band of ground-based detectors (Wen 2003;O'Leary et al. 2009;Bae et al. 2014;Samsing et al. 2014;Zevin et al. 2017;Samsing 2018;Gondán et al. 2018;Rodriguez et al. 2018b;Rasskazov & Kocsis 2019;Zevin et al. 2019;Gröbner et al. 2020;Gondán & Kocsis 2021;Tagawa et al. 2021;Zevin et al. 2021b;Dall'Amico et al. 2023).
Close to the centre of the supermassive black hole (SMBH), stellar mass binaries can be bound to the SMBH and form a triple system.Highly eccentric binaries can be formed through this secular GW evolution (Antonini et al. 2010;Antonini & Perets 2012;Naoz et al. 2013;Michaely & Perets 2014;Prodan et al. 2015;Antonini et al. 2017a;Hoang et al. 2018;Hamers et al. 2022).The gravitational interaction of a binary system with the third body can also increase its eccentricity (Silsbee & Tremaine 2017;Liu & Lai 2018;Rodriguez & Antonini 2018).The potential mechanisms in triple star surements from different analyses (Bonino et al. 2023;Arif Shaikh et al. 2023).
systems that can lead to the formation of high eccentricity include isolated stellar flyby interaction of wide stellar mass BBHs in the field (Michaely & Perets 2019;Raveh et al. 2022), wide binaries in the galactic centre (Michaely & Naoz 2023;Rozner & Perets 2023), evolution of wide hierarchical triple systems under the influence of galactic tidal field (Grishin & Perets 2022), mergers in gaseous environment (McKernan et al. 2012;Tagawa et al. 2018;Rozner & Perets 2022), dynamical instability induced due to mass-loss or mass transfer between stars (Perets & Kratter 2012).Around 5% of all dynamical mergers in globular clusters can give rise to BBHs with eccentricities ≳ 0.1 at 10 Hz GW frequency (Samsing & Ramirez-Ruiz 2017;Samsing & D'Orazio 2018;Samsing 2018;Rodriguez et al. 2018a;Rodriguez et al. 2018b). 2ext generation ground-based detectors would observe compact binaries at frequencies ≳ 1 Hz.Since binary is expected to be more eccentric in the past, detectors observing in the low-frequency regime would be able to constrain eccentricity with high accuracy.The Laser Interferometer Space Antenna (LISA) (Babak et al. 2017) is a planned space-based mission that will be observing in the frequency range 10 −4 − 0.1 Hz.Mergers of SMBHs will be the main targets of LISA.LISA can measure eccentricities  0 > 10 −2.5 (at one year before the merger) for supermassive BBHs (Garg et al. 2023).However, there is a possibility of observing heavy stellar mass BBHs by LISA during their early inspiral before merging in the frequency band of groundbased detectors (Sesana 2016;Vitale 2016;Moore et al. 2019;Wong et al. 2018;Ewing et al. 2021).The eccentricity measurement of stellar mass BBHs in the LISA band can be used to distinguish between formation channel of these binaries (Nishizawa et al. 2016;Randall & Xianyu 2019).Combining the information from LISA and high-frequency observations of next generation ground-based detectors such as CE and ET can improve the parameter estimation of compact binaries.The multiband observation of stellar-mass black hole binaries can help in better measurement of eccentricity (Klein et al. 2022).
There are already claims for the presence of eccentricity in the detected GW events (Romero-Shaw et al. 2020, 2021;O'Shea & Kumar 2021;Gayathri et al. 2022;Romero-Shaw et al. 2022).However, the eccentricity can be mimicked by spin-precession effects.Due to the lack of an inspiral-merger-ringdown waveform model that includes both eccentricity and spin precession, it is hard to distinguish between the two effects, especially for the short signals such as GW190521 (Romero-Shaw et al. 2023).However, eccentricity is not mimicked by the spin-induced precession for long-duration signals (Divyajyoti et al. 2023).
Apart from the eccentricity, the spin orientation of the component BHs in a binary can also be used to discern between different formation channels (Rodriguez et al. 2016a;Stevenson et al. 2017;Talbot & Thrane 2017a;Vitale et al. 2017;Farr et al. 2017;O'Shaughnessy et al. 2017;Farr et al. 2018;Gerosa et al. 2018a;Gerosa et al. 2018b).Due to tidal interactions, isolated binaries are expected to have their spins aligned with the orbital angular momentum of the binary.However, dynamically formed BBHs are likely to have their spins misaligned.Hence these BBHs are expected to have isotropic spin distribution (Rodriguez et al. 2016b).Redshift evolution of merger rate can also be used to constrain formation scenarios (Rodriguez & Loeb 2018;Fishbach et al. 2018;Ng et al. 2022).
Here, we focus on the measurement of eccentricity to distinguish circular and eccentric binary systems.If the eccentricity of a BBH system is measured with relative error Δ 0 / 0 < 1, we will call it a resolved binary.This study examines the ability of Voyager, CE, and ET to measure the eccentricity of BBHs, drawn from a particular eccentricity distribution.More precisely, considering few fiducial eccentricity distributions as well as an astrophysically motivated eccentricity distribution given in Zevin et al. (2021b), we quantify the fraction of detected binaries that can be confidently identified as eccentric.We also find that it requires typical eccentricities ≳ 0.02, 5 × 10 −3 , 10 −3 at 10 Hz GW frequency for Voyager, CE, and ET, respectively, to distinguish between eccentric and circular binary systems.In Section 2.1, we discuss the GW detectors considered in the study.Section 2.2 discusses the simulated binary black hole population.Section 2.3 explains the method of calculating statistical errors using Fisher information matrix and the eccentric waveform model used for the analysis.Section 3 describes the results of the paper.Section 4 provides the summary and outlook of the study.Throughout the paper, we use units  =  = 1.

Detectors
We consider three ground-based GW detector configurations: (i) A LIGO detector operating at the Voyager sensitivity (Adhikari et al. 2020) with lower cut-off frequency  low = 5 Hz.
(ii) Cosmic Explorer (Reitze et al. 2019) in USA with 40 km arm length at its design sensitivity.The lower cut-off frequency is chosen to be  low = 5 Hz.
(iii) Einstein Telescope (Punturo et al. 2010) in Europe with 10 km arm length at its design sensitivity.This will be an underground facility with a triangular shape.The lower cut-off frequency is taken to be  low = 1 Hz.

Binary black hole population
We consider a GWTC-3 like population of stellar mass BBHs with the following distributions: • Primary mass: The primary mass ( 1 ) of BBH population follows PowerLaw+Peak model (Abbott et al. 2023) which is a mixture of a power law and a Gaussian distribution (Talbot & Thrane 2017b).The values of parameters defining the PowerLaw+Peak model are taken to be:  peak = 0.04,  = 3.4,  min = 5.08,  max = 86.85,  = 33.73,  = 3.56,   = 4.83.
• Mass ratio: The mass ratio of BBH population follows a power law distribution () =   with  = 1.08.
• Spins: We assume that the spin angular momentum vectors of binary components are aligned with the orbital angular momentum vector of the binary (non-precessing).The magnitudes of dimensionless spins for both black holes are drawn from a Beta distribution (Talbot & Thrane 2017b) where   and   are the shape parameters that determine the mean and variance of the distribution.The values are   = 1.6,   = 4.11.
• Eccentricity distributions: Table 1 summarizes the eccentricity distributions considered in this study.We consider two fiducial eccentricity distributions: Loguniform and Powerlaw with two eccentricity ranges:  0 ∈ (10 −4 , 0.2) and  0 ∈ (10 −7 , 0.2).In Powerlaw distribution, the powerlaw index 1.2 is chosen such that it does not deviate significantly from the Loguniform distribution.The upper limit of distributions is fixed to be  0 = 0.2, beyond which the waveform model employed in this study becomes less accurate.We also consider an expected astrophysical eccentricity distribution based on cluster simulations given in Fig. 1 of Zevin et al. (2021b).This will be referred to as Zevin eccentricity distribution from here onwards.The Fig. 1 of Zevin et al. (2021b) contains two eccentricity distributions plotted by dashed and solid lines.We use the eccentricity distribution shown in dashed lines which represents the intrinsic eccentricity distribution of the considered population.Similar to the Loguniform and Powerlaw eccentricity distributions, we restrict Zevin eccentricity distribution to the range  0 ∈ (10 −7 , 0.2).Note that Zevin eccentricity distribution is based on cluster simulations which are based on many astrophysical assumptions.We draw eccentricity samples from the one-dimensional marginalized distribution and do not account for the correlations of eccentricity with other binary parameters, assuming them to be small.
• Redshift: Sources are distributed uniformly in comoving volume according to the following redshift distribution where    is the comoving volume element at redshift z.The factor (1 + ) −1 converts the detector frame time to the source frame time.We choose maximum redshift  max = 2.5 which corresponds to a luminosity distance   = 20.9Gpc.The corresponding   is calculated assuming Planck18 (Aghanim et al. 2020) cosmology using Astropy (Astropy Collaboration et al. 2018).
Note that there might be correlations of eccentricity distribution with other binary parameters such as mass, mass-ratio, and redshift distribution in an astrophysical N-body dynamical model (Zevin distribution) and one should in principle use multiparameter distribution for drawing population.The measurement errors in eccentricity however strongly depend on the eccentricity itself and weakly dependent on other binary parameters.This can be seen from the approximate leading-order scaling in Eq. (4.18b) in Favata et al. (2022).The measurement errors in eccentricity for a given detector scale as the inverse square of eccentricity.Therefore small correlations of eccentricity with other binary parameters are less likely to affect our results.

Methodology
Using the BBH population discussed in Sec.2.2, we calculate measurement uncertainties on binary parameters.The statistical errors on binary parameters can be forecasted using the Fisher information matrix framework (Finn 1992;Cutler & Flanagan 1994;Poisson & Will 1995).This framework gives the 1 width around the injected values of the binary parameters.The inner product between two frequency-domain signals h1 (  ) and h2 (  ) is defined as where   (  ) is the one-sided noise power spectral density (PSD) of the detector and * represents the complex conjugation.The limits of integration in Eq. ( 3) are fixed by the sensitivity of the detector and the properties of the source.The SNR  is defined as the norm of the signal, The Fisher information matrix Γ  is defined as where   is the set of waveform parameters.The covariance matrix is obtained by taking the inverse of the Fisher matrix 1 errors on binary parameters   are calculated by taking the square root of the diagonal elements of the covariance matrix We use TaylorF2Ecc waveform model (Moore et al. 2016) which is an inspiral waveform model and accounts for the leading order [O ( 2 0 )] corrections due to eccentricity in the phasing expression.The TaylorF2Ecc waveform model is valid for small eccentricities  0 ≲ 0.2.The waveform model does not account for the eccentricity corrections to the amplitude.Since GW detectors are more sensitive to the GW phase than to the amplitude, small eccentricity corrections to the amplitude will be less important than eccentricity corrections to the phase.The full expression for GW phasing can be found in Eq. (6.26) of Moore et al. (2016).The circular part of Eq. (6.26) in (Moore et al. 2016) is for non-spinning binaries.We add alignedspin terms to the circular part of Eq. (6.26) (Arun et al. 2005(Arun et al. , 2009;;Buonanno et al. 2009;Mishra et al. 2016).
The lower limit of integration (  low ) in Eq. ( 3) is set by the lower cut-off frequency of the detector.For Voyager3 and CE,  low is set to be 5 Hz.For ET,  low is taken to be 1 Hz.The upper cut-off frequency (  high ) is the redshifted frequency corresponding to the innermost stable circular orbit (  isco ) of the remnant Kerr BH (Bardeen et al. 1972;Husa et al. 2016;Hofmann et al. 2016).The  isco is a function of two-component masses and spins.The full expression can be found in appendix C of Favata et al. (2022).For CE, we use the noise PSD from Abbott et al. (2017b).The analytical fit for CE noise PSD is given in Eq. (3.7) of Ref. Kastha et al. (2018).For ET, we use design sensitivity ET-D (Hild et al. 2011).The ET noise PSD is scaled by a factor of (sin 2 60 • ) −1 to convert it to the effective sensitivity of ET's triangular design.

RESULTS
In this section, we examine the capability of Voyager, CE, and ET to measure the orbital eccentricity of a binary system.First, we calculate the fraction of total sources that are detected in each detector above a certain threshold SNR.In Sec.3.1, we quantify the fraction of detected sources that can be resolved.To recall, a binary with Δ 0 / 0 < 1 is called a resolved binary.We show the cumulative distribution of resolved binaries for each eccentricity distribution in Sec.3.2.In Sec.3.3, we investigate the minimum values of eccentricity that can be measured with Voyager, CE, and ET.

Fraction of detected binaries with resolved eccentricity
We draw 20000 BBH samples from mass, spin, eccentricity, and redshift distributions discussed in Sec.2.2.For Voyager, CE, and ET, we calculate the SNR for all these sources.The detection threshold is set to be  th ≥ 8 for all the detectors.Table 2 summarizes the fraction of sources that are detected in Voyager, CE, and ET with  th ≥ 8.For each detector, we perform five independent realisations of the simulation and quote the median value of the fraction of detected binaries.Voyager detects 33% of the total sources, while CE and ET detect 100% and 97% of the total sources with  th ≥ 8, respectively.The sources that do not pass the detection threshold are discarded from further analysis.
For detected binaries, we calculate 1 errors on binary parameters using the Fisher information matrix.Table 3 shows the fraction of detected binaries for which eccentricity can be resolved.A source with Δ 0 / 0 < 1 can be called eccentric with ≳ 68% confidence.The value quoted is the median value of the five independent realisations.Among all eccentricity distributions, Loguniform(10 −4 , 0.2) represents the maximum number of resolved sources for all three detectors.This is because Loguniform(10 −4 , 0.2) has the maximum number of sources with higher eccentricity.For this eccentricity distribution, Voyager can confidently measure non-zero eccentricity for 14% of the detected sources.CE can resolve 36% of the sources, while ET can resolve eccentricity for 55% of the sources.The number of resolved binaries in ET is ∼ 1.5 times larger than CE and ∼ 4 times larger than Voyager.This is due to the longer inspiral of the GW signal and better low-frequency sensitivity of ET.The effect of the lower cut-off frequency of ET on the fraction of resolved binaries is discussed in Appendix A.
For Zevin(10 −7 , 0.2), Voyager can confidently distinguish 3% of the detected sources from circular binaries.CE can distinguish 9% of the sources as eccentric, while ET can resolve 13% of the detected sources.In Sec.3.3, we investigate which part of the eccentricity distributions dominates the resolved binaries.
Eccentricity distributions based on N-body simulations (including Zevin eccentricity distribution) predict that a reasonable fraction of binaries can retain  0 > 0.2 at 10 Hz GW frequency.A binary with a higher value of eccentricity is easier to be distinguished as eccentric.We do not include sources with  0 > 0.2 (at 10 Hz) in our analysis.Hence the fraction of resolved binaries quoted in Table 3 represents a conservative lower limit.

Fraction of resolved binaries with excellent eccentricity measurement
Having discussed the ability of GW detectors to constrain the fraction of different eccentricity distributions by resolving the eccentricity of binaries in Sec.3.1, in this section, we discuss the cumulative distribution of Δ 0 / 0 for resolved binaries to get insight into the number of resolved binaries and their corresponding measurement precision.
For the Zevin(10 −7 , 0.2) distribution, Voyager can measure the eccentricity of ∼ 15% of the resolved sources with accuracy better than 10%.CE measures ∼ 52% of the resolved sources with Δ 0 / 0 < 0.1, whereas ET measures eccentricity for ∼ 75% of the resolved sources (13%) with better than 10% precision.ET shows remarkable capabilities for measuring the eccentricity of binary black holes.The dependence of eccentricity measurement on the binary's eccentricity is explained in the next Section.

Minimum resolvable eccentricity for Voyager, Cosmic Explorer, and Einstein Telescope
To gain insights into the measured values of eccentricities and their corresponding precision, we plot Δ 0 / 0 as a function of  0 for resolved binaries in Fig. 2 for one of the (randomly chosen) realisations.Figure 2 shows the dependence of eccentricity measurement on binary's eccentricity.Five panels represent different eccentricity distributions.In each panel, distinct colours represent different detectors.For all eccentricity distributions, each detector takes a different region in ( 0 , Δ 0 / 0 ) plane with ET measuring more number of sources with better precision followed by CE and Voyager.As expected, the most precise measurement of eccentricity comes from the larger eccentricities.
It can be observed from the figure that it requires  0 ≳ 10 −3 at 10 Hz GW frequency for ET to confidently distinguish a binary system as eccentric.For CE and Voyager, it requires  0 ≳ 5 × 10 −3 and  0 ≳ 0.02 (at 10 Hz GW frequency), respectively, to confidently measure a non-zero eccentricity.The better measurement of eccentricity by ET compared to CE is mainly due to two effects.The first is due to the better sensitivity of ET at low frequencies (below 8 Hz) compared to CE and the second is because the binary system has a relatively larger value of eccentricity when it enters the frequency band of ET (1 Hz).For example, a binary with  0 = 0.05 at 10 Hz GW frequency had an eccentricity of ∼ 0.1 when it entered the CE and Voyager band at 5 Hz.The binary had an eccentricity of ∼ 0.4 when entering the ET band at 1 Hz4 .Hence ET provides better measurements of eccentricity.
It is clear that Voyager, CE, and ET can resolve only the high eccentricity part of the eccentricity distributions.None of the detectors is able to resolve eccentricity ≲ 10 −3 at 10 Hz GW frequency.In Zevin eccentricity distribution, the BBHs formed via GW capture have  0 ≳ 10 −3 at 10 Hz GW frequency.Hence next generation GW detectors such as Voyager, CE, and ET will be able to confidently distinguish a BBH system as eccentric that is likely formed via GW capture in globular clusters.However, there are other channels that can produce BBHs with  0 ≳ 10 −3 at 10 Hz.These include gravitational interaction of field triples (Silsbee & Tremaine 2017;Antonini et al. 2017b;Rodriguez & Antonini 2018;Michaely & Perets 2019;Grishin & Perets 2022;Raveh et al. 2022), three-body mergers in globular clusters (Samsing 2018)  The initial orbital eccentricity  0 on the x-axis is defined at a reference gravitational-wave (dominant mode) frequency of 10 Hz.Hoang et al. 2018).Though next generation GW detectors would be able to identify a BBH system as eccentric formed through these channels, distinguishing these formation scenarios from each other may be more challenging.

SUMMARY AND OUTLOOK
In this work, we have studied the ability of next generation GW detectors such as Voyager, CE, and ET to measure the eccentricity of BBHs.The key goal was to investigate the fraction of detected binaries that can be confidently distinguished as eccentric by measuring a non-zero value of eccentricity.The precise measurement of eccentricity can be used to distinguish different formation channels as the eccentricity of a binary system depends on its formation history.
We find that for Zevin eccentricity distribution, Voyager can resolve the eccentricity for ∼ 3% of the detected BBHs.CE can resolve eccentricities for ∼ 9% of the detected BBHs, whereas ET can resolve the eccentricity of ∼ 13% of the detected BBHs.Moreover, a subpopulation of resolvable eccentric binary systems will have an extremely precise measurement of eccentricity.For Voyager, (15-20)% of the resolved binaries will have eccentricity measurement better than 10%, while for CE and ET, (50-60)% and (65-75)% of the resolved binaries will have Δ 0 / 0 ≲ 10%, respectively.ET's low-frequency sensitivity plays a crucial role in the measurement of eccentricity.
Another important aspect of this study is the determination of the minimum value of eccentricity that is required to confidently identify a binary system as eccentric.We find that it requires eccentricities ≳ 0.02 (at 10 Hz GW frequency) for Voyager to identify a binary as eccentric.For CE and ET, the minimum eccentricities that can be measured to distinguish a binary as eccentric are ≳ 5 × 10 −3 and ≳ 10 −3 at 10 Hz GW frequency, respectively.In summary, next generation GW detectors would be able to confidently distinguish BBH systems with  0 ≳ 10 −3 (at 10 Hz GW frequency) from circular binaries.
Dynamically formed binaries are expected to have isotropic spin distribution.Our waveform model does not account for the precessional effect.Moreover, TaylorF2Ecc waveform model does not take into account the higher modes due to eccentricity.It would be interesting to study the effect of precession and higher modes (due to eccentricity) on eccentricity measurement.The ability of GW detectors to reconstruct the actual eccentricity distribution can be explored in a future study.Table A1.The impact of ET's lower cut-off frequency (  low ) on the fraction of resolved binaries.The simulation is repeated five independent times and the median value is quoted.The  low plays a crucial role in the eccentricity measurement.sources.ET at  low = 5 Hz can resolve a larger fraction of eccentricity distribution compared to CE.This is because of the better low-frequency sensitivity of ET (below 8 Hz) compared to CE.For Loguniform(10 −7 , 0.2), there is a ∼ 10% and ∼ 26% reduction in the fraction of resolved sources when  low is increased from 1 Hz to 3 Hz and 5 Hz, respectively.Similar kind of trends can be seen for Powerlaw(10 −4 , 0.2) and Powerlaw(10 −7 , 0.2) distributions.For Zevin eccentricity distribution, there is a 7% and 23% reduction in the fraction of resolved sources, when  low is increased from 1 Hz to 3 Hz and 5 Hz, respectively.
Figure A1 shows the effect of  low on the cumulative probability distribution of Δ 0 / 0 < 1.It can be seen that the number of sources with smaller errors reduces as the  low increases from 1 Hz to 3 Hz and 5 Hz.Hence, the lower cut-off frequency of ET plays a critical role in the measurement of low-frequency effects such as eccentricity.
This paper has been typeset from a T E X/L A T E X file prepared by the author.

Figure 1 .
Figure 1.The cumulative probability distribution of Δ 0 / 0 for resolved binaries.Different panels show different eccentricity distributions.Three colours in each panel represent different GW detectors.For each detector, thin lines with the same colour represent five different realisations of the population.ET measures a larger fraction of binaries with smaller errors for all eccentricity distributions.The lower frequency cut-off frequency (  low ) for Voyager and CE is 5 Hz.For ET,  low = 1 Hz.

Figure 2 .
Figure2.Scatter plots of Δ 0 / 0 with  0 for those sources that have Δ 0 / 0 < 1.Each panel shows different eccentricity distributions.The distinct colours represent different detectors.The initial orbital eccentricity  0 on the x-axis is defined at a reference gravitational-wave (dominant mode) frequency of 10 Hz.

Figure A1 .
Figure A1.The effect of ET's lower cut-off frequency on the cumulative probability distribution.The number of sources with smaller errors reduces as the  low increases from 1 Hz to 3 Hz and 5 Hz.

Table 1 .
Eccentricity distributions considered in this study.The initial eccentricity  0 is defined at 10 Hz GW frequency.

Table 2 .
Fraction of binaries that are detected with SNR ≥ 8.The median value of five independent realisations is quoted.The median value is rounded to its nearest integer value.

Table 3 .
Fraction (in %) of detected binaries with Δ 0 / 0 < 1 The fraction of detected binaries that are measured with Δ 0 / 0 < 1 for Voyager, CE, and ET.To account for statistical fluctuations of the results, the exercise is carried out five times and the median values are reported.