Adding eccentricity to quasicircular binary-black-hole waveform models
AAdding eccentricity to quasi-circular binary-black-hole waveform models
Yoshinta Setyawati
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Callinstr. 38, 30167 Hannover, GermanyLeibniz Universit¨at Hannover, 30167 Hannover, Germany andInstitute for Gravitational and Subatomic Physics (GRASP) Department ofPhysics, Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands
Frank Ohme
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Callinstr. 38, 30167 Hannover, Germany andLeibniz Universit¨at Hannover, 30167 Hannover, Germany (Dated: January 28, 2021)The detection of gravitational-wave signals from coalescing eccentric binary black holes would yield un-precedented information about the formation and evolution of compact binaries in specific scenarios, such asdynamical formation in dense stellar clusters and three-body interactions. The gravitational-wave searches bythe ground-based interferometers, LIGO and Virgo, rely on analytical waveform models for binaries on quasi-circular orbits. Eccentric merger waveform models are less developed, and only few numerical simulations ofeccentric mergers are publicly available, but several eccentric inspiral models have been developed from thePost-Newtonian expansion. Here we present a novel method to convert any circular analytical merger modelinto an eccentric model. First, using numerical simulations, we investigate the additional amplitude and fre-quency modulations of eccentric signals that are not present in their circular counterparts. Subsequently, weidentify suitable analytical descriptions of those modulations and interpolate key parameters from twelve nu-merical simulations designated as our training data set. This allows us to reconstruct the modulated amplitudeand phase of any waveform up to mass ratio three and eccentricity 0.2. We find that the minimum overlap of thenew model with numerical simulations is 0.984 over our test data set. A Python package pyrex easily carriesout the computation of this method.
I. INTRODUCTION
Coalescing stellar-mass black-hole binaries are one of theprimary sources of gravitational-wave (GW) signals detectedby the ground-based interferometers, Advanced Laser inter-ferometer Gravitational-wave Observatory (aLIGO) [1], Virgo[2], and KAGRA [3]. In the first three observing runs (O1-O3), detection pipelines assumed binary-black-hole (BBH)mergers to have negligible eccentricity when entering the or-bital frequencies to which aLIGO, Virgo, and KAGRA aresensitive [4–6]. BBHs formed in an isolated environmentthrough a massive stellar evolution are expected to circularizeand therefore have undetectable eccentricity by the time theyenter the LIGO band [7]. However, BBHs with a detectableeccentricity can form in a dense stellar cluster through dynam-ical capture [8, 9].A possible scenario is that the binary gains eccentricity dueto gravitational torques exchanged with a circumbinary disk[10]. Eccentric BBHs can also form from three-body inter-actions [9], where the BBH behaves as the inner binary. Inthis system, the Kozai-Lidov [11, 12] mechanism triggers theoscillation that boosts the eccentricity.Interactions of BBHs in a typical globular cluster suggesta significant eccentric BBH merger rate. As many as ∼
5% ofbinaries may enter the LIGO detector band ( f ≥
10 Hz) witheccentricities e > . / Virgo / KAGRA de-tection rate [14]. Besides, the detection of eccentric BBHmergers could capture e ff ects from the extreme gravity regimeand therefore can be used for testing the general theory of rel-ativity [17, 18].We highlight the significance of detecting GWs from eccen-tric BBHs. Constructing template models for eccentric wave-forms is challenging, and we aim to make progress towardsthis goal especially for the late inspiral and merger regimesthat are most accessible with today’s observations. One of themain di ffi culties in developing an eccentric waveform modelis that only a few numerical relativity (NR) simulations withhigher eccentricity are available. Thus, many studies focus ondeveloping eccentric models from the Post-Newtonian (PN)expansion. The development of full inspiral-merger-ringdown(IMR) eccentric waveform models is currently an actively re-searched area [19–21].Huerta et al. [19] construct a time-domain eccentric non-spinning waveform model ( e < .
2) up to mass ratio 5.5,where e is the eccentricity 10 cycles before the merger. Theirmodel is called ENIGMA , a hybrid waveform that has been cal-ibrated using a set of numerical simulations and trained us-ing Gaussian process regression (GPR). Ref. [20] presents alow eccentricity model ( e < .
2) called
SEOBNRE using theexpansion of the e ff ective-one-body (EOB) waveform family.This model has been used in the analysis of O1-O2 and some a r X i v : . [ g r- q c ] J a n events in the O3 data [24–26]. A more up to date EOB for-malism is demonstrated in [22, 23]. Hinder et al. [21] presenta time-domain, non-spinning eccentric waveform model up tomass ratio q = m / m = e ref ≤ .
08 starting at seven cycles before the merger.Like Ref. [19], the early inspiral of this model is hybridizedwith a PN expansion to produce a full IMR model in a
Math-ematica package [27]. Apart from the studies above, non-spinning, low-eccentric frequency domain models from thePN expansion are publicly available in the LIGO algorithmlibrary (LAL) [28–30].We present a promising method to add eccentricity to quasi-circular systems independent of the PN expansion. We ap-ply this method to non-spinning, time domain waveforms, al-though in principle it can be used in more general settings.Our technique focuses on a fast reconstruction of the nearmerger eccentric BBH waveform and can be applied to anyanalytical circular non-spinning model. We build our modelfrom 12 NR simulations and tested against further 8 NR sim-ulations from the open SXS catalog [31]. Our method is verysimple and can be applied to any circular time-domain modelobtained from, e.g., the Phenomenological [32–34] or EOB[35, 36] family.We model the deviation from circularity visible in the am-plitude and phase of eccentric GW signals. This deviationis modelled across the parameter space and can be simplyadded to any quasi-circular model, which elevates that modelto include eccentric e ff ects. This approach is inspired by the”twisting” technique that is applied for reconstructing pre-cessing spins from an aligned-spin model to build, e.g., the IMRPhenomP family [33, 37–40]. The dynamic calibration ofthe waveform model is motivated by our previous study [41]and the regression techniques tested in detail in [42].We calibrate our model for mass ratios q ≤ e ≤ .
2, and provide it as a Python package called pyrex [43]. Our model has been constructed for a fiducial 50 M (cid:12) BBH and can then be rescaled for other total masses M . Wefind that the overlap of all our test data against NR is above98%. Moreover, we expand the construction to earlier regimethan the calibrated time span. Although we do not calibratefor higher mass ratios, the early inspiral, and higher orbitaleccentricity, we allow the building of waveforms beyond theparameter boundaries used for calibration.The organization of this manuscript is as follows. In Sec-tion II, we present the methodology to construct this model.Section III discusses the primary outcome and the faithfulnessof our model. Finally, section IV summarizes and concludesthe prospect of our studies. Throughout this article, we usegeometric units in which G = c = II. METHOD
Using NR simulations, we investigate the frequency andamplitude modulations in eccentric BBHs signals and imple-ment them in analytical waveforms to develop our model. Asdescribed by Peters [7], the orbital eccentricity in binary sys- tems decreases over time due to energy loss through GW ra-diation. Pfei ff er et al. [44] investigated this in numerical sim-ulations of the SXS catalog. The authors point out that one ofthe main di ff erences in the evolution of low eccentric initialdata compared to quasi-circular binaries is an overall time andphase shift, where the quasi-circular data represent the binaryat a point close to merger. Following these studies, Hinderet al. [21] showed that the GW emissions from low-eccentricbinaries and circular binaries are indistinguishable near themerger stage. Specifically, Hinder et al. suggest that one onlyloses 4% of the signal when substituting the GW emissionfrom low-eccentric binaries with circular orbits 30 M beforethe peak of the amplitude ( t = t = − M . We then substitute theGW strain at t > − M with the circular model for the samebinary masses. A. Data preparation
We use 20 non-spinning NR simulations from the SXS cat-alog up to mass ratio 3 and eccentricity 0.2 to build our model(see Table I). We follow the definition of eccentricity e comm inRef. [21] as the eccentricity measured at the referenced fre-quency, x = ( M ω ) / = . + ” and ” × ” polarization using the spin-weighted spherical harmonics with the following expression[45]: h + − ih × = Mr ∞ (cid:88) (cid:96) = m = (cid:96) (cid:88) m = − (cid:96) h (cid:96) m ( t ) − Y (cid:96) m ( ι, φ ) , (1)where M and r are the total mass of the system and distancefrom the observer respectively, − Y (cid:96) m are the spin-weightedspherical harmonics that depend on the inclination angle ι andthe phase angle φ , and h (cid:96) m ( t ) can be extracted from the NRdata in the corresponding catalog. We construct our model for h ± , the leading contribution of spherical harmonic modeswith (cid:96) = m = ±
2. Ref. [21] suggests that other, subdom-inant modes are less significant for nearly equal mass sys-tems with low eccentricity. Here we consider only moder-ately small eccentricities, therefore we only model the dom-inant mode. For future studies, subdominant harmonics willbe important to model high eccentricity signals accurately.We prepare the data as follows. Firstly, we align all thewaveforms in the time-domain such that the peak amplitudeis at t =
0. Subsequently, we remove the first 250 M from thestart of the waveforms due to the junk radiation, and the last30 M before t = t > − M . We then decompose TABLE I. NR simulations from the SXS catalog used in this studywith mass ratio q = m / m , eccentricity at the reference fre-quency e comm , and the number of orbits before the maximum am-plitude of (cid:107) h (cid:107) . e comm is the eccentricity at the reference frequency( M ω ) / = e comm = − at the reference fre-quency.Case Simulations Training / test q e comm N orbs h ± into amplitude ( A ), phase ( Ψ ), and the phase derivative, ω = d Ψ dt , where the referenced frequency follows Ref [21].We model amplitude A and frequency ( ω ) as a sim-ple quasi-circular piece plus an oscillatory function. The fi-nal model then yields the phase ( Ψ ) by integrating the fre-quency. B. Eccentricity estimator
In numerical simulations, eccentricity is often discussed asa consequence of imperfections in the initial data [46]. It man-ifests itself as small oscillations on top of the gradual binaryevolution, where the oscillation’s amplitude is proportional tothe eccentricity (see A and ω plots in Fig. 2 and 3). Weuse this residual oscillation as a key to estimating the eccen-tricity evolution.Mrou´e et al. [47] compare various methods to estimate ec-centricity using e X ( t ). The orbital eccentricity is proportionalto the amplitude of a sinusoidal function, e X ( t ), expressed by e X ( t ) = X NR ( t ) − X c ( t )2 X c ( t ) , ⇔ e X ( X c ) = X NR ( X c ) − X c X c , (2)where X is either ω or A , and X c ( t ) is the X quantity incircular binaries instead of low-order polynomial fitting func-tions that are often used in the literature. We reverse this .
00 0 .
05 0 .
10 0 .
15 0 . comm . . . . . q trainingtest FIG. 1. The training and test data shown by the red circles and theblue plus signs, respectively, are located in the parameter space ofmass ratio and eccentricity. We use 20 NR simulations from the SXScatalog and divided them into 12 NR training data set and 8 test dataset. − − − − − . − . − . . . . . h SXS:BBH:1364 fullchopped
FIG. 2. The full and the chopped waveform, SXS:BBH:1364 simu-lation ( q = , e comm = . h mode, and the orange line presents the time range used in this study.We remove the first 250 M due to the junk radiation and modulate theresidual oscillation at − M ≤ t ≤ − M . relation to convert a circular model (with given X c ( t )) to aneccentric model using an analytical description of the oscilla-tory function e X ( X c ). We apply the Savitzky-Golay filter [48]to smooth the e X ( t ) curves from noises caused by numericalartefacts. Savitzky-Golay is a digital filter applied to smooththe selected data points without altering the signal directionby fitting the adjacent data with a low-degree polynomial fits. − − − −
500 0t/M0 . . . . . . . . A q=2 − − − −
500 0t/M0 . . . . . . . ω q=2 FIG. 3. Left: training data amplitudes for q = (cid:96) = , m = t > − M due to circularization. The leftpanel shows the amplitude of h , the right panel shows the timederivative of the phase, ω = d Ψ / dt . .
06 0 .
08 0 .
10 0 .
12 0 . A c − . − . − . − . . . . . . e X A .
06 0 .
08 0 .
10 0 .
12 0 . Mω c Mω FIG. 4. The eccentricity estimator from A plotted against the cir-cular amplitude A c (left), and the eccentricity estimator from ω plotted against the circular omega ω c (right) with the same mass ra-tio. Di ff erent colors show di ff erent cases of training data for massratio q =
2. We smooth the data from numerical artefacts using theSavitzky-Golay filter (see text).
As a first sanity check, we compute the orbital eccentricityusing the eccentricity estimator ( e X ) and find that the resultsagree with a maximum relative error of 8.9% against e comm quoted in the SXS catalog and given in Table. I. In Fig. 4, wepresent the eccentricity estimators e X ( X c ) as a function of itscircular amplitude and frequency, A c and ω c , respectively. C. Fitting e X Our main goal is to model an eccentric waveform by mod-ulating the amplitude and phase of a circular model. To con-struct the model, we interpolate the additional oscillation ofan eccentric waveform depending on its eccentricity and massratio, where the relationship between the circular and the ec-centric model is expressed in Eq. (2). Accordingly, we lookfor a fitting function to model e X ( X c ) that relies on the desiredparameters ( q , e ) and reverse Eq. (2) to obtain the eccentricamplitude and frequency. We then integrate the frequency toobtain the eccentric phase and construct the eccentric h .In a suitable parametrization, the eccentricity estimator e X is a decaying sinusoidal function (see Fig. 4) with its ampli-tude defined by the orbital eccentricity e [47]. To model e X for various eccentricities and mass ratios, we fit e X with a setof free parameters modifying a damped sinusoidal function.These parameters are two amplitude quantities ( A and B ), afrequency ( f ), and phase ( ϕ ) with the following relation: e X ( X c ) = Ae B X κ c sin( f X κ c + ϕ ) . (3) A , B , f , and ϕ are standard damped sinusoidal parameters ob-tained from the optimized curve fitting.We use a X κ c instead of X c to describe the evolution of theresidual oscillations of the amplitude and frequency mainlybecause of the following reasons. X c is a rapidly evolvingfunction. Therefore, it is more di ffi cult to model e X with astandard sinusoidal function with a constant frequency. Al-though it is in principle possible to use X c directly in themodel, we would have to slice the data into multiple smalltime windows that overlap. Thus, the results will be lesssmooth; one would have to blend all those individual func-tions defined on small intervals into one big function. Besides,we cannot guarantee our result beyond our calibration range,especially for the early inspiral. Using a power law allows usto fit the entire region with one set of free parameters. How-ever, we note that the power law of X c induces a twist result-ing in infinitely large eccentricities for the very early inspiralstage. That is a problem with assuming exponential decay, andthe fact that the power-law we use has a negative exponent.We fit our model e X from the starting frequency f low =
25 Hz for a circular BBH with a total mass M = M (cid:12) . Thepower law for ω c is κ = − /
24, and for A c is κ = − / ff erent values to cali-brate with higher eccentricity, higher mass ratio, or a di ff erentstarting frequency.By optimizing the curve fit between e X and Eq. 3, we obtainthe four quantities ( A , B , f , ϕ ) for all training data. The rela-tion between the mass ratio ( q ), eccentricity ( e ), and the threeparameters A , B , f are shown in Fig. 5. The amplitude compo-nents, A and B are strongly correlated to eccentricity, whereasthe mass ratio determines the frequency square. Hence, weperform one dimensional linear interpolation across eccentric-ity to obtain the value of A and B . Similar to that, we linearlyinterpolate the f across mass ratios. We choose f instead of f because the data is smoother for interpolation. The squareroot of f gives either positive or negative values. However,this ambiguity can be absorbed by the phase parameter ϕ .The phase parameter ϕ is an additional degree of freedomthat we cannot explore su ffi ciently with the available NR data.We expect it to correlate strongly with the mean anomaly. Be-cause the orientation of the ellipse is astrophysicly less inter-esting than the value of the eccentricity, we do not attempt tomodel the e ff ect of varying the mean anomaly other than in-troducing the phenomenological nuisance parameters ϕ . Weinterpolate the other parameters when generating a new wave-form model with di ff erent mass ratios and referenced eccen-tricities.We apply a one-dimensional interpolation for each keyquantity shown in Fig. 5. A and B are interpolated over di ff er-ent eccentricities e , f is interpolated over mass ratio q , andthe phase of the oscillation ϕ can be chosen arbitrarily. . . . . e A A . . . . . q A A exp( B A ) . . . . . . . . . . . . q . . . . e f A . . . . . . . . . . . e A ω . . . . . q A ω exp( B ω ) . . . . . . . . . . . . q . . . . e f ω . . . . . . . . FIG. 5. Key quantities of A (left) and ω (right) of a damped sinusoidal function obtained from the curve fitting (see Eq. 3). The amplitudeparameters ( A and B ) depend strongly on the eccentricity ( e ), whereas the square of the frequency ( f ) is correlated to mass ratio ( q ). We leave ϕ as a free nuisance parameter that we maximize over when comparing to the test data. The left colorbar corresponds to the bottom panel andthe right colorbar to the top panel, respectively Once we obtain the eccentricity estimators e X using the in-terpolated quantities, we substitute the results to reconstruct A and ω using Eq. 2. To construct Ψ , we integrate ω numerically using the trapezoidal rule. We truncate the wave-form at t = − M and join it with the non-spinning circularmodel. We then smooth the transition with the Savitzky-Golayfilter at − M < t < − M .We then build h ± as the combination of the amplitude andphase as follows: h (cid:96) m = A (cid:96) m e − i Ψ (cid:96) m . (4)To reconstruct the gravitational wave strain h = h + − h × , wecompute the spin-weighted spherical harmonics Y (cid:96) m ( ι, φ ) andemploy Eq. 1. III. RESULTS
We built a new non-spinning eccentric model by modulat-ing the residual amplitude and phase oscillations of the circu-lar analytical models,
IMRPhenomD [32] and
SEOBNRv4 [49].
IMRPhenomD is an aligned-spin IMR model that was origi-nally built in frequency domain and calibrated to numericalsimulations for mass ratios q ≤ SEOBNRv4 is an aligned-spin time-domain IMR model [49, 50] that has been calibratedto 140 NR waveforms produced with the SpEC code up tomass ratio 8 and extreme-mass-ratio signals.As described in Sec. II, we interpolate the residual ampli-tude and phase oscillations of the training data set for thegiven mass ratio and eccentricity. To construct a new, eccen-tric waveform for the intermediate to near merger regime, wethen use one of the non-spinning circular models with the de-sired mass ratio, compute the eccentricity estimators ( e X ) from the analytical description given in Eq. (3), and reconstructedthe desired eccentric waveform model for each test data.We evaluate the result by computing the overlap betweenthe new model against NR test data. The overlap is maximizedover a time and phase shift, as well as the free phase o ff setsof the residual oscillations. We define the unfaithfulness M between two waveforms ( h , h ) by the following equation: M = − max { t , Ψ ,ϕ A ,ϕ ω } (cid:20) (cid:90) f f ˜ h ( f ) ˜ h ∗ ( f )S n ( f ) d f (cid:107) h (cid:107)(cid:107) h (cid:107) (cid:21) , (5)where S n is the sensitivity curve of the corresponding GWinterferometer.We investigate three sensitivity curves for the future GWdetectors, aLIGO A + , Einstein Telescope (ET), and CosmicExplorer (CE). LIGO A + is the future GW interferometerwith 1.7 times better sensitivity than the current detector andexpected to start observing in mid-2024 at the earliest [51].ET is a 10 km GW observatory planned to be built in the bor-der between Belgium, Germany, and the Netherlands whichcould be operating in the mid 2030s [52]. ET is expected tohave higher sensitivity towards the low frequency range. CEis a 40 km third-generation GW detector which has highersensitivity towards low redshift ( z >
10) that planned to startobserving in the 2030s [53]. Since our model focuses on thelate inspiral case, and because the unfaithfulness is insensi-tive to a change in overall signal-to-noise ratio, the values ob-tained for the future third generation detectors show similarbehaviour [54]. Hence, we only show the overlap results forthe LIGO A + design sensitivity. A possible caveat is that ourmodel might not fill the LIGO A + band down to 10 Hz. Thus,there is a chunk of inspiral power missing in the signal.Fig. 6 visually compares the strain h ± of each NR test datawith the new eccentric non-spinning signal built from analyt-ical models, IMRPhenomD and
SEOBNRv4 for a 50 M (cid:12)
BBHwith inclination angle ι = φ c =
0. Using our method, we find that the minimum overlapbetween the new model against NR is 0.984 (log M = -1.8)over all of our test data set. The minimum overlap occurs atthe highest eccentricity in the test data set.Although we calibrated the new model for limited rangesin mass ratio, eccentricity and time, we let the production ofthe new model go beyond our calibration range. In Fig. 7, weshow the unfaithfulness of the new model against NR data testfor various total masses with the aLIGO A + design sensitiv-ity curve. The left panel shows the unfaithfulness within thecalibrated frequency range, between 25 Hz and the ISCO fre-quency scaled over the total mass. Similarly, the right panelpresents the unfaithfulness beyond the calibrated frequencyrange, between 20 Hz and the ringdown frequency. We usethe definition of the ISCO and ringdown frequency as follows: f ISCO = / (6 / π M ) , (6)and f RD = . / M . (7)Fig. 7 shows that the mismatches decrease toward highertotal mass systems. As the total mass increases, the over-lap computation covers a smaller waveform regime towardsmerger in the frequency space. Since the eccentricity de-creases over time, the near merger regime has lower eccen-tricities. Thus, the overlap between the model and the cor-responding NR simulation is better for the higher-mass sys-tems compared to the lower-mass ones. For comparison, wefind that mismatches between circular analytical models andthe eccentric NR test data are at least one order of magnitudeworse than the results we find for our eccentric model.Furthermore, we test how well one can extract the param-eters of an eccentric signal h ( q , e ) by comparing with variouswaveforms with di ff erent eccentricities e and mass ratios q .We generate a pyrex waveform ( q = e = . q , e ) using thesame analytical waveform model. The results are shown inFig. 8. We emphasize that in this study, we did not run a stan-dard parameter estimation (PE) pipeline that stochastically ex-plores a much greater parameter space. In particular, we donot consider varying the total mass or spin. Hence, our resultsare only a first indication of potential parameter ambiguities.Our results in Fig. 8 show that the overlap between the gener-ated waveform and other waveforms having similar mass ratiobut di ff erent eccentricities is high, suggesting that an accuratemeasurement of the eccentricity is challenging for high-massBBH systems where only the late inspiral and merger are ac-cessible through the GW detection. IV. CONCLUSION AND FUTURE PERSPECTIVES
The detection of GWs from an eccentric BBH mergerwould be crucial step towards understanding the physical evo-lution of compact binary coalescences and the nature of BBHin globular clusters. Due to limitations in waveform mod-eling, the current search and parameter estimation pipelines in the LIGO / Virgo data analysis rely on analytical waveformmodels for circular binaries. One of the limitations to developeccentric BBH models is the small number of eccentric NRsimulations. NR simulations that are publicly available havelow eccentricities ( e ≤ .
2) at M ω / = e X , using a damped sinusoidal fit,where the fitting function is built upon four key parameters.We then perform a one-dimensional interpolation for each keyparameter ( A , B , and f ) to build the eccentric waveform withthe desired mass ratio and eccentricity. One of our modelparameters, ϕ , shows no clear correlation with the physicalparameters we explore. However, the small number of NRsimulations used here did not allow us to model the e ff ect ofvarying the mean anomaly in detail, and we expect ϕ to repre-sent this degree of freedom. When quantifying the agreementbetween our model and the test data, we maximize over thisnuisance parameter.We then build a new model using the fitting values of e X andthe amplitude and frequency of the circular model which herewe take from IMRPhenomD and
SEOBNRv4 . Our new modelhas overlap between 0.984 and 0.9996 over all NR simula-tions in our test data set with the LIGO A + design sensitivitycurve. We hint that we need more training and test data setfor further development of this model beyond the current pa-rameter boundaries. The computation of our method can beperformed easily and fast in a Python package pyrex [43].Although we calibrate our model to a 50 M (cid:12) BBH ( q ≤ e ≤ .
2) starting at frequency f low =
25 Hz, we let the compu-tation go slightly beyond the calibrated range. The calibratedtime range of the waveform is between the late inspiral upto near merger phase, but we can extend the model throughmerger and ringdown by using the circular data. For the earlyinspiral, an analytical PN model could be used to completethe description of the entire coalescence. This way, our ap-proach can be adapted to develop a complete IMR eccentricmodel. This would be especially important for future gener-ations of GW interferometers as they have higher sensitivityespecially in the low frequency range. Careful studies of ec-centric search and parameter estimation are needed to to detecteccentric compact binary coalescences and its origin.
ACKNOWLEDGMENTS
The authors would like to thank David Yeeles, MariaHaney, and Sebastian Khan for useful discussions. Compu-tations were carried out on the Holodeck cluster of the MaxPlanck independent Research Group ’Binary Merger Obser-vations and Numerical Relativity’ and the LIGO Laboratory − − − h × − q= 1, e= 0.099 NRlog M = -2.4log M = -2.6 q= 1, e= 0.1 NRlog M = -2.5log M = -2.5 q= 1, e= 0.142 NRlog M = -1.8log M = -1.9 q= 1, e= 0.144 NRlog M = -1.8log M = -1.9 − . − . − . . t ( s ) − − h × − q= 2, e= 0.06 NRlog M = -3.1log M = -3.5 − . − . − . . t ( s ) q= 2, e= 0.095 NRlog M = -2.4log M = -2.5 − . − . − . . t ( s ) q= 2, e= 0.096 NRlog M = -2.1log M = -2.3 − . − . − . . t ( s ) q= 3, e= 0.092 NRlog M = -2.5log M = -2.6 FIG. 6.
IMRPhenomD (orange) and
SEOBNRv4 (green) circular waveforms twisted into eccentric models. log M is the log mismatch of IMRPhenomD against the NR waveform (shown in blue) and log M gives the log mismatch of SEOBNRv4 against NR with the same massratio and eccentricity, respectively. The total mass of the system is M = M (cid:12) , and the mass ratio ( q ) and eccentricity ( e ) are shown in the titleof each plot. We employ the A + design sensitivity curve starting at f =
35 Hz (see text) to compute the match. The black vertical lines markthe range in which we perform the interpolation and compute the match.
20 40 60 80 100 120 140
M/M (cid:12) − − − − M
25 Hz to ISCO
20 40 60 80 100 120 140
M/M (cid:12) − − M
20 Hz to Ringdown
FIG. 7. Mismatch results of eccentric variants of
IMRPhenomD and
SEOBNRv4 against the NR test data for di ff erent total masses assumingaLIGO A + design sensitivity. Left: 25 Hz to ISCO frequency (within the calibration range), and right: from 20 Hz to ringdown frequency(beyond the calibration range), where we define the ringdown frequency as f RD = . / M . computing cluster at California Institute of Technology, sup-ported by National Science Foundation Grants PHY-0757058 and PHY-0823459.. This work was supported by the MaxPlanck Society’s Research Group Grant. [1] Aasi J, et al. Advanced LIGO. Classical and Quantum Grav-ity. 2015 March;32:074001. Available from: http://stacks. iop.org/0264-9381/32/i=7/a=074001 . .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . e . . . . . . . . . q IMRPhenomD .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . eSEOBNRv4 − − − − − − − − FIG. 8. Comparison with the highest eccentricity in the test data set, e = . q =
2. We generated an eccentric waveform model derivedfrom the non-spinning circular model, IMRPhenomD and SEOBNRv4, and compare the signal with models for di ff erent mass ratios andeccentricities. Waveforms with higher parameter distance have lower overlap. The colorbar shows the log mismatch.[2] Acernese F, et al. Advanced Virgo: a second-generation inter-ferometric gravitational wave detector. Classical and QuantumGravity. 2014 December;32:024001. Available from: http://stacks.iop.org/0264-9381/32/i=2/a=024001 .[3] Akutsu T, Ando M, et al. KAGRA: 2.5 generation interfero-metric gravitational wave detector. Nat Astron. 2019 Jan;3:35–40. Available from: .[4] Abbott BP, et al. GWTC-1: A Gravitational-Wave TransientCatalog of Compact Binary Mergers Observed by LIGO andVirgo during the First and Second Observing Runs. Phys RevX. 2019 Sep;9:031040. Available from: https://link.aps.org/doi/10.1103/PhysRevX.9.031040 .[5] Abbott BP, et al.. 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