Biases in parameter estimation from overlapping gravitational-wave signals in the third generation detector era
Anuradha Samajdar, Justin Janquart, Chris Van Den Broeck, Tim Dietrich
BBiases in parameter estimation from overlapping gravitational-wave signals in thethird generation detector era
Anuradha Samajdar , , , Justin Janquart , , Chris Van Den Broeck , , and Tim Dietrich , Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands Institute for Gravitational and Subatomic Physics (GRASP),Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands Department of Physics, University of Milano – Bicocca, Piazza della Scienza 3, 20126 Milano, Italy Institut für Physik und Astronomie, Universität Potsdam, Haus 28,Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany and Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany (Dated: February 16, 2021)In the past few years, the detection of gravitational waves from compact binary coalescences withthe Advanced LIGO and Advanced Virgo detectors has become routine. Future observatories willdetect even larger numbers of gravitational-wave signals, which will also spend a longer time in thedetectors’ sensitive band. This will eventually lead to overlapping signals, especially in the case ofEinstein Telescope (ET) and Cosmic Explorer (CE). Using realistic distributions for the merger rateas a function of redshift as well as for component masses in binary neutron star and binary blackhole coalescences, we map out how often signal overlaps of various types will occur in an ET-CEnetwork over the course of a year. We find that a binary neutron star signal will typically have tensof overlapping binary black hole and binary neutron star signals. Moreover, it will happen up to tensof thousands of times per year that two signals will have their end times within seconds of each other.In order to understand to what extent this would lead to measurement biases with current parameterestimation methodology, we perform injection studies with overlapping signals from binary blackhole and/or binary neutron star coalescences. Varying the signal-to-noise ratios, the durations ofoverlap, and the kinds of overlapping signals, we find that in most scenarios the intrinsic parameterscan be recovered with negligible bias. However, biases do occur for a short binary black hole or aquieter binary neutron star signal overlapping with a long and louder binary neutron star event whenthe merger times are sufficiently close. Hence our studies show where improvements are requiredto ensure reliable estimation of source parameters for all detected compact binary signals as we gofrom second-generation to third-generation detectors.
I. INTRODUCTION
The direct observation of gravitational waves (GWs) [1]has had a tremendous impact in fundamental physics[2–5], astrophysics [6–15], and cosmology [16–18], andstarting from the observation of the binary neutron star(BNS) signal GW170817 [9] has opened a new era inmulti-messenger astronomy with GWs [19–22]. The thirdobserving run (O3) of Advanced LIGO [23] and AdvancedVirgo [24] ended in March 2020, and together these inter-ferometers have found more than 50 GW candidates [25],with 39 candidates observed during the first half of O3 [8].The detector sensitivities will be improved further, andthe frequency with which signals are observed is expectedto keep increasing in coming years. In particular, in thetransition to the envisaged third generation (3G) era,with Einstein Telescope (ET) [26, 27] and Cosmic Ex-plorer (CE) [28–30], the detection rate will go up steeply,and signals will also spend much longer times in the de-tectors’ sensitive band [31]. As first pointed out in [32]and studied further in this paper, the probability of over-lapping signals will then become signficant.In view of this, it will be important to assess to whatextent the science goals of 3G detectors (see e.g. [31, 33–42]) may be affected by signals overlapping with eachother. Apart from science with signals from compactbinary coalescences (CBCs), this includes searches for primordial backgrounds, since the subtraction of “fore-ground” CBC sources [30, 43–47] will rely on our abilityto characterize them individually. As shown in [48, 49],even using current data analysis techniques, the detec-tion rates of individual CBC sources would likely not besignificantly impacted by the occurrence of overlappingsignals. However, a study of the effect on parameter es-timation had not yet been performed.Earlier works [50–52] have studied parameter estima-tion for single sources in the 3G era. Here we take thefirst step in assessing possible biases in the recovery of pa-rameters characterizing a GW signal when signals fromdifferent sources are simultaneously present in the de-tectors’ sensitive band. Before doing this, we map outhow often signal overlaps of various types will occur ina network of two CEs and one ET over the course of ayear, assuming realistic distributions for merger rate as afunction of redshift and for component masses in binaryneutron star and binary black hole (BBH) coalescences.We find that a typical BNS signal will be overlapped bytens of BBH signals. Moreover, BBH or BNS signalswhose mergers occur within seconds from each other willbe quite common. Since these are the cases for which wecan expect the largest parameter estimation biases to oc-cur, we focus on them in setting up simulations wherebysignals are added to synthetic data from the ET-CE net-work, and analyzed using current state-of-the-art param- a r X i v : . [ g r- q c ] F e b eter estimation techniques. We explore various scenariosof signals from different kinds of sources overlapping: twoBBH signals, two BNS signals, and a BBH signal witha BNS. For our simulations we choose signal parametersconsistent with what has been observed as being repre-sentative of each kind of source: parameter values sim-ilar to the ones of GW170817 [53] to represent a BNS,similar to the ones of GW150914 [54] to represent a high-mass BBH, and similar to the ones of GW151226 [55] fora lower-mass BBH. We find that in most cases, the in-trinsic parameters can be recovered with negligible bias.However, if the merger times of the two signals are suffi-ciently close, considerable biases can occur when a shortBBH signal or a quieter BNS signal overlaps with a louderBNS signal. Though our study should be considered ex-ploratory, it already points to where improvements overcurrent parameter estimation pipelines will be needed themost.This paper is structured as follows. In Sec. II we ob-tain detection rate estimates for signals in the 3G era,from which we calculate overlap rates. In Sec. III welay out the settings and methods we use for parameterestimation. Parameter estimation results for various sce-narios are shown in Sec. IV. A summary and conclusionsare presented in Sec. V, where we also give recommen-dations for future improvements of parameter estimationtechniques. II. OVERLAP RATE ESTIMATESA. Methodology
Before looking at the impact of overlapping signals onparameter estimation for the individual ones, we want toaddress the question of how frequently such overlaps willoccur, depending on the type. Previous characterizationsof the overlap probabilities for 3G detectors were basedon the duty cycle , which is defined as the ratio of thetypical duration of a particular type of event (BNS orBBH) to the average time interval between two succes-sive events of that type, assuming some fixed canonicalvalues of the component masses for each type [32]. How-ever, here we also want to allow for overlaps of mixedtypes, and for a range of component masses (and hencesignal durations) within a given type, so as to arrive at adetailed assessment of overlap rates. Therefore, what wewill do is to assume particular merger rates as functionof redshift for BBH and BNS, as well as component massdistributions, and on the basis of these create simulated“catalogs” of signals in the detectors. This will allow usto make quantitative statements regarding BNS signalsoverlapping with other BNS signals and with BBHs, andthe same for overlaps of BBH with BBH events, in a much more detailed and realistic fashion. We start by estimating the number of individualBBH and BNS coalescences that happen in a given vol-ume, up to a maximum redshift which is chosen to be z max = 30 for BBH events and z max = 6 for BNS events[31, 32, 48, 57]. For this we need the intrinsic merger ratedensity for the events as a function of redshift. We willassume that the compact binaries originate from stellarpopulations, and adopt the merger rate estimates of Bel-cynski et al. [58] with Oguri’s analytical fit [59] , whosegeneral expression is R GW ( z ) = a e a z e a z + a . (1)Here R GW is expressed in Gpc − yr − , and the coeffi-cients a i , i = 1 , . . . , depend on the star populationsthat are considered; see Fig. 1. For our purposes, weconsider the combination of population I and II stars forBNS, and populations I, II, and III for BBH, as the con-tribution of the latter type of stars is important only atredshifts of (cid:38) . However, these relations are rescaled tomatch the local merger rate estimates obtained observa-tionally by LIGO and Virgo so far; see [60]. In this work,we focus on the lowest, the median, and the highest localrate for each type of event. For BNS, the lowest, median,and highest local rates are, respectively,
80 Gpc − yr − ,
320 Gpc − yr − , and
810 Gpc − yr − , which are obtainedby changing the value of a to 2480, 9920, and 25110, re-spectively. On the other hand, for the BBH events, weapply a multiplicative constant to the sum of the popula-tion I and II and the population III rates, equal to 0.0709,0.112, and 0.178 for the lowest, median, and highest localrates, which are . − yr − , . − yr − , and . − yr − , respectively.An intrinsic merger rate density R GW ( z ) is then con-verted to an observed merger rate density as function ofredshift by multiplying by the differential comoving vol-ume [32]: R obsGW ( z ) = R GW ( z ) dV c dz ( z ) . (2)To obtain dV c /dz , we assume the Planck13 cosmologicalmodel [61] of
Astropy [62, 63]. Since neutron star-black hole (NS-BH) rates are less certain (seee.g. [13, 56]), we will not consider them here, but we expect gen-eral conclusions regarding parameter estimation to largely carryover when signal durations are similar. Strictly speaking this merger rate distribution refers to BBHmergers. However, when computing the merger rate density(see e.g. [48, 57, 58]), one assumes a time delay distribution(e.g. P ( t d ) ∝ /t d ), with a minimum time delay that is higherfor BBH than for BNS. Using the distribution of [58] for bothBNS and BBH (with some overall rescaling) then implies that wewill underestimate the BNS merger rate density [57] and hencethe frequency of overlaps involving BNS signals. z R G W ( z ) ( G p c y r ) Total BBH merger rate for stellar progenitors
Pop I and II starsPop III starsTotal rate for stellar progenitors
FIG. 1. The BBH merger rate density according to Oguri’sfit [59] for population I, II, and III stars, as well as the totalrate, when all the star populations are accounted for.
As a next step, we simulate the population of sys-tems by constructing a “catalog”, and determine whichevents are actually detected. For the BBH popula-tion, we assume that the masses follow the “ power law+ peak ” distribution presented in Ref. [60] for the pri-mary component mass, and the corresponding power lawdistribution for the mass ratio, through which we sam-ple the secondary mass [60]. For BNS events we dis-tribute component masses uniformly, where for the pri-mary mass m ∈ [1 , . M (cid:12) , and for the secondary mass m ∈ [1 M (cid:12) , m ] . Events are distributed over comovingdistance D according to R GW ( z ) , converting between D and z using the above mentioned cosmology and cuttingoff at the maximum redshifts z max stated above. Sky po-sitions and unit normals to the orbital plane are taken tobe uniform on the sphere.In this work we assume a network of two CEs locatedat the LIGO Hanford and Livingston sites, and one ETlocated at the Virgo site. For each event we calculate theoptimal signal-to-noise ratios (SNRs) in the three obser-vatories, which are added in quadrature to obtain a net-work SNR. In computing SNRs we only consider the in-spiral part of binary coalescence, so that in the stationaryphase approximation [64] and for a single interferometer[65] SNR = 12 (cid:114)
56 1 π / cD (1 + z ) / (cid:18) G M c (cid:19) / × g ( θ, φ, ψ, ι ) (cid:112) I ( M ) . (3)Here M = ( m m ) / / ( m + m ) / is the chirp mass in the source frame. The geometric factor is given by g ( θ, φ, ψ, ι ) = (cid:18) F ( θ, φ, ψ )(1 + cos( ι ) ) + 4 F × ( θ, φ, ψ ) cos( ι ) (cid:19) / , (4)where F + , × are the beam pattern functions in terms ofsky position ( θ, φ ) and polarization angle ψ , while ι is theinclination angle. We take Einstein Telescope to consistof three detectors with ◦ opening angle, arranged in atriangle with sides of 10 km [66], and add the correspond-ing SNRs in quadrature; for Cosmic Explorer we assumea single L-shaped detector of 40 km arm length [28, 29].Finally, I ( M ) = (cid:90) f high f low f − / S h ( f ) df. (5)Here f low is a low-frequency cut-off that depends on theobservatory; we set f low = 5 Hz for both ET and CE,though lower values may be achieved in the case of ET[67, 68]. For f high we use the frequency of the innermoststable circular orbit: f high ( m , m , z ) = 11 + z π √ c GM , (6)where M = m + m is the total mass. We take the noisepower spectral density (PSD) S h ( f ) to be ET-D in thecase of Einstein Telescope [26, 27]; for the projected PSDof Cosmic Explorer, see [28, 29].The network SNR, denoted SNR net , is defined asSNR = (cid:88) i =1 SNR i , (7)where the sum is over the two CE and the one (triangu-lar) ET observatories. We consider an event as detectableif the network SNR is above 13.85 ( = √ × ), with-out imposing SNR thresholds in individual observatories.For the BNS and BBH mass ranges considered here, thismeans that detection rates will mainly be driven by theCEs, but we note that ET will have an advantage athigher masses [69].Finally, signals will be present in a detector for a du-ration given by τ = 2 . (cid:18) . M (cid:12) M (cid:19) / (cid:20)(cid:18) Hz f low (cid:19) / − (cid:18) Hz f high (cid:19) / (cid:21) s . (8)Simulated catalogs of events happening over the courseof a year are constructed as follows. The year is split intoa grid in which each cell corresponds to one second, andmerger times are drawn from a uniform distribution overthese cells. For a given type of event (BNS or BBH), oneassociates to each merger time a mass pair, redshift, skyposition, and orientation of the orbital plane drawn fromthe corresponding distributions, as well as a signal du-ration computed from Eq. (8). Doing this for the threechoices of local merger rate, and in each case puttingtogether the BNSs and BBHs, catalogs of events are ob-tained. Finally, within each catalog, it is assessed whichevents will be detectable with the ET-CE network ac-cording to the criteria spelled out above, leading to anoverview of what we may expect to be contained in oneyear’s worth of data. In particular, we can check howoften and in what way events tend to overlap, dependingon their types. B. Overlap estimates
The three different local merger rates give the followingtypical numbers of events happening over one year, priorto imposing detectability thresholds: ∼ , , BBH events, and , , BNSevents for the low, median, and high local rate, respec-tively. The network of two CEs and one ET will detect93% of BBHs and 35% of BNSs. The number of detectedsignals is shown in Table I for the three local rates, alongwith median and 90% spreads on SNRs, and a breakdownof detections according to their loudness.Within our simulated catalogs of events, we can lookat the numbers of detected signals that overlap depend-ing on the types. We focus on two quantities: (i) thenumber of seconds in a year where at least two detectedsignals have their merger, and (ii) the typical number ofmergers that happen during the time a given signal is ina detector’s sensitivity band.The numbers of seconds in a year that have at leasttwo mergers taking place is given in Table II; clearly thiswill happen frequently over the course of a year. In-deed, we find that even more than two mergers can oc-cur within the same second. The proportion of detectedsignals merging together with at least one other goes upwith increasing local merger rate, potentially reachingthousands per year.In addition to the scenario where different compact bi-nary mergers happen in the same second, we investigatethe typical number of mergers that will happen over theentire duration of a BNS event while it is in band, de-pending on their type; see Table III and Fig. 2. BecauseBNS events are in the detector band for a long time (sev-eral hours for f low = 5 Hz), quite a number of such over-laps will indeed occur. If one does the same for BBHs,one finds that either zero or one BBH or BNS merger (at90% confidence) will happen in its duration; this is dueto BBH events being shorter (the median duration being ∼ seconds).Before moving on to parameter estimation issues, letus briefly look at other future GW observatories that arebeing planned or considered. Constructing simulated cat-alogs of detectable sources in the same way as above, andfocusing on the high local merger rate, we find that over F r a c t i o n o f t h e B N S s i g n a l s Fraction of BNS with a given number of overlaps
Overlapping compact binary mergersOverlapping BBH mergersOverlapping BNS mergers
FIG. 2. Fraction of detected BNS mergers with a given num-ber of compact binary mergers (blue), BBH mergers (red),and BNS mergers (green) taking place while the BNS signalis in band. the course of a year, Advanced LIGO+ [70] will typicallyhave no events merging within the same second, and onlya few occurrences of a BBH merging in the duration of aBNS (assuming f low = 15 Hz). For Voyager [71] we find O (1) instances of two events merging within the samesecond, and BNS signals will typically have at most oneother signal’s merger in their duration (for f low = 10 Hz).These numbers refer to signals detectable with a singleinterferometer (with SNR threshold 8) rather than witha network of them, but it will be clear that overlappingsignals are going to become an important considerationmainly in the 3G era.
III. PARAMETER ESTIMATION SETUP
Having established that third-generation detectors willsee a considerable number of overlapping signals whosemergers occur very close in time, we want to find outwhat this will imply for parameter estimation. To thisend, we simulate BBH and BNS signals in a networkconsisting of one ET and two CE observatories as in theprevious section, assuming stationary, Gaussian noise fol-lowing the PSDs used above.Since we expect parameter estimation biases to bemore pronounced when SNRs of overlapping signals aresimilar to each other, and on the low side, we focus onnetwork SNRs roughly between 15 and 30. We consideroverlapping events whose merger times either coincide(as a proxy for merger within the same second), or areseparated by 2 seconds, again because these are the typesof scenarios where biases will likely be the largest. Thenumber of overlaps from the previous section that sat-isfy these criteria is given in Table IV, for different localmerger rates; we see that they will be fairly common. net net > net > net > net > BBH
Low rate 53756 . +94 . − . . +93 . − . . +94 . − . BNS
Low rate 98898 . +22 . − .
17 (0.017%) 298 (0.30%) 2712 (2.7%) 44350 (48%)Median rate 396793 . +22 . − .
73 (0.018%) 1257 (0.32%) 10659 (2.7%) 177296 (45%)High rate 1004525 . +22 . − .
196 (0.020%) 3255 (0.32%) 27135 (2.7%) 448610 (45%)
TABLE I. The number of events detected by a network of two CEs and one ET in one year of simulated data, the mediannetwork SNRs and their 90% spreads, and the detection numbers and percentages (in brackets) for different choices of minimumnetwork SNR.
Rate BBH mergers > 1 BNS mergers > 1 Any mergers > 1Low rate 48 310 750Median rate 127 2412 7347High rate 303 15581 20149
TABLE II. The number of seconds in a year with at leasttwo mergers occurring, depending on their types.Rate Number of Number of Number ofBBH mergers BNS mergers any typeLow rate +10 − +16 − +23 − Median rate +14 − +58 − +77 − High rate +21 − +144 − +164 − TABLE III. Typical numbers of compact binary mergers hap-pening during the time a BNS signal is in band.
In our parameter estimation studies, for definitenesswe take the BBH events to have masses similar to thoseof either GW150914 (a higher-mass, shorter-duration sig-nal) or GW151226 (a lower-mass, longer-duration event),while for BNSs we take the masses to be similar to thoseof GW170817. Overlapping signals are given different in-jected sky locations. All analyses are done with 3 differ-ent noise realizations. For each example of overlappingsignals, parameter estimation is also done on the indi-vidual signals, for the same noise realizations, in order toassess what biases occur. Fig. 3 provides an overview ofthe various overlap scenarios that will be considered inthe rest of this paper, in terms of masses and SNRs.To reduce computational cost, we focus on non-spinning sources. A BBH signal is then characterizedby parameters (cid:126)θ = { m , m , α, δ, ι, ψ, D L , t c , ϕ c } , where m , m are the component masses, ( α, δ ) specifies thesky position in terms of right ascension and declination, ι and ψ are respectively the inclination and polarizationangles which specify the orientation of the orbital planewith respect to the line of sight, D L is the luminosity dis-tance, and t c and ϕ c are respectively the time and phaseat coalescence. BNS signals have two additional parame-ters (Λ , Λ ) , corresponding to the (dimensionless) tidaldeformabilities [72–76].In this work we focus specifically on potential biases in Run BBH-BBH BBH-BNS BNS-BNSLow rate
Median rate
11 304 6752
High rate
15 1594 41306
TABLE IV. Number of pairs of binary coalescence events withboth SNRs between 15 and 30, and such that their mergersoccur within 2 seconds or less from each other. intrinsic parameters. For BBHs, results will be shown forthe total mass M = m + m and mass ratio q = m /m (with the convention m ≤ m ). For BNSs, we showchirp mass M instead of total mass, since that param-eter is usually the best-determined one for long signals.As the individual tidal deformabilities tend to be poorlymeasurable for the SNRs considered here, we will beshowing results for a parameter ˜Λ defined as [77] ˜Λ = 1613 (cid:88) i =1 , Λ i m i M (cid:16) − m i M (cid:17) , (9)since this is how tidal deformabilities enter the waveformphase to leading (5PN) order [72].In the Bayesian framework, all information about theparameters of interest is encoded in the posterior prob-ability density function (PDF), given by Bayes’ theorem[78]: p ( (cid:126)θ |H s , d ) = p ( d | (cid:126)θ, H s ) p ( (cid:126)θ |H s ) p ( d |H s ) , (10)where (cid:126)θ is the set of parameter values and H s is the hy-pothesis that a GW signal depending on the parameters (cid:126)θ is present in the data d . For parameter estimationpurposes, the factor p ( d |H s ) , called the evidence for thehypothesis H s , is effectively set by the requirement thatPDFs are normalized. Assuming the noise to be Gaus-sian, the likelihood p ( d | (cid:126)θ, H s ) of obtaining data d ( t ) giventhe presence of a signal h ( t ) is determined by the propor-tionality p ( d | (cid:126)θ, H s ) ∝ exp (cid:20) −
12 ( d − h ( (cid:126)θ ) | d − h ( (cid:126)θ )) (cid:21) , (11) s t r a i n
1e 24 tc-2 tc m =1.68, m =1.13SNR=30 SNR=20 SNR=15101 s t r a i n
1e 23 m =41, m =33SNR=3052 54 56 58 60 62 time [sec.] +1.1262595e9202 s t r a i n
1e 24 m =15, m =8SNR=13.5 s t r a i n
1e 24 m =15, m =8SNR=15 tc-2 tc
52 54 56 58 60 62 time [sec.] +1.1262595e9101 s t r a i n
1e 23 m =41, m =33SNR=30 s t r a i n
1e 24 m =1.68, m =1.13SNR=30 tc-2 tc
52 54 56 58 60 62 time [sec.] +1.1262595e9202 s t r a i n
1e 24 m =1.38, m =1.37SNR=20 FIG. 3. Individual waveforms and the overlap scenarios con-sidered in our simulations. All signals are injected in 3 differ-ent simulated noise realizations for a third-generation detec-tor network. Signals are either overlapped using the same endtime (blue waveforms), or 2 seconds earlier than the “primary”signal’s end-time (orange waveforms).
Top three panels:
BNSsignals (top) with an SNR of 30, 20, or 15 being overlappedwith either a high-mass BBH signal (middle; GW150914-like)or a low-mass BBH signal (bottom; GW151226-like).
Mid-dle panels:
Overlapping waveforms in the case of two BBHsignals. The higher-mass BBH signal (bottom; GW150914-like) is overlapped with the lower-mass BBH signal (top;GW151226-like).
Bottom panels:
Overlapping waveforms inthe case of two BNS signals. where the noise-weighted inner product ( · | · ) is defined as ( a | b ) = 4 (cid:60) (cid:90) f high f low ˜ a ∗ ( f ) ˜ b ( f ) S h ( f ) df. (12)Here a tilde refers to the Fourier transform, and S h ( f ) is the PSD, as in the previous section. Due to computa-tional limitations, in our parameter estimation studies weuse a lower frequency cut-off of f low = 23 Hz. Since bothET and CE will be sensitive down to lower frequenciesthan that, we expect that our choice will lead to conser-vative estimates of parameter estimation biases, as thesame signal will in reality accumulate more SNR when itis visible in the detector already from a lower frequency.Our choices for the prior probability density p ( (cid:126)θ |H s ) inEq. (10) are similar to what has been used for the anal-yses of real data when BBH or BNS signals were presentwith masses similar to the ones specified in Fig. 3. In allcases we sample uniformly in component masses. For theGW150914-like signals, we do this in the range m , m ∈ [10 , M (cid:12) . For analyzing the GW151226-like signals,the component mass range is m , m ∈ [3 , . M (cid:12) , andin addition we restrict chirp mass to M ∈ [5 , M (cid:12) andmass ratio q to the range [0 . , . For BNSs we samplecomponent masses in the range m , m ∈ [1 , M (cid:12) , re-stricting M ∈ [0 . , M (cid:12) , while tidal deformabilities aresampled uniformly in the range Λ , Λ ∈ [0 , . Whenwe show PDFs for the derived quantity ˜Λ , they will havebeen reweighted with the prior probability distribution ofthis parameter induced by the flat priors on componentmasses and Λ , Λ , such as to effectively have a uniformprior on ˜Λ .To sample the likelihood function in Eq. (11),we use the LALInference library [79], and specifi-cally the lalinference_mcmc algorithm. The wave-forms we use for the BNS and BBH signals are IMRPhenomD_NRTidalv2 [80–82] and
IMRPhenomD [83, 84]respectively, both computed with the waveform libraryLALSimulation. To inject the signals and add noise tothem, we use standard tools available within the LAL-Simulation package. All these codes are openly accessiblein LALSuite [85].Before performing parameter estimation, we verify thedetectability of the individual signals in the overlap sce-narios of Fig. 3 using the PyCBC software package [86].We inject overlapping signals in noise generated from thePSD and check that the individual signals show up astriggers with masses that are consistent between detec-tors, at a network SNR above a threshold of 8. Thisturns out to be true for all the cases considered, exceptfor two BBH signals merging at the same time. In thelatter case we still have triggers in individual detectors,but with masses differing by up to ∼ M (cid:12) . Using theSNRs in single detectors as detection statistics, detec-tion is still achieved. For all scenarios, the end times ofindividual signals tend to be identified with a precisionof a few milliseconds [87]; when subsequently performingparameter estimation, we use a prior range for end timethat is centered on the true end time, leaving an intervalof 0.1 s on either side.For parameter estimation, all simulations are donewith three different noise realizations. In the next sec-tion, results are shown for one of those; for the other twonoise realizations, see Appendix A.As usual, the one-dimensional PDF p ( λ |H s , d ) for aparticular parameter λ is obtained from the joint PDF p ( (cid:126)θ |H s , d ) by integrating out all other parameters. Inassessing the effect on parameter estimation of signalsthat overlap in various ways, we will frequently be com-paring one-dimensional PDFs for the same parameter indifferent situations. A convenient way of quantifying thedifference between two distributions p ( λ ) and p ( λ ) is bymeans of the Kolmogorov-Smirnov (KS) statistic [88, 89].Let P ( λ ) , P ( λ ) be the associated cumulative distribu-tions; then the KS statistic is just the largest distancebetween these two:KS = sup λ | P ( λ ) − P ( λ ) | . (13)By construction, this yields a number between 0 and 1;if the KS statistic is close to zero, then the distributions p ( λ ) and p ( λ ) will be considered close to each other. IV. RESULTSA. Overlap of a BNS signal with a BBH signal
First we look at the results of parameter estimation forthe overlap of a BNS signal with a BBH, either ending atthe same time or with the BBH signal ending 2 secondsearlier than the BNS. This is the scenario shown in thetop panels of Fig. 3. We perform parameter estimationfirst on the BNS and then on the BBH, with priors asspecified in the previous section.
1. BNS recovery
Fig. 4 shows posterior probability distributions for in-trinsic parameters characterizing the BNS signal, for 3different SNRs of the BNS, and the different overlap sce-narios. The PDFs tend to widen with decreasing SNR,as expected. We see that estimation of the mass param-eters are essentially unaffected, regardless of the type ofoverlapping BBH signal (GW150914-like or GW151226-like) or of its merger time (identical to that of the BNS,or 2 seconds earlier). For a given SNR of the BNS, thePDFs for the tidal parameter ˜Λ differ slightly more be-tween the overlap scenarios. However, we note that mostof the information on tides enters the signal at high fre-quencies, where the detectors are less sensitive; and infact, as shown in Appendix A (Fig. 8), differences in theunderlying noise realization tend to have a larger effecton the measurement of ˜Λ than overlapping signals. P D F SNR=30SNR=30SNR=30SNR=30SNR=30 P D F SNR=20SNR=20SNR=20SNR=20SNR=20
500 1500 25001.1946 1.1948 1.1950 P D F SNR=15SNR=15SNR=15SNR=15SNR=15 q GW150914-tcGW150914-tc-2 GW151226-tcGW151226-tc-2 BNS
FIG. 4. Posterior PDFs showing estimation of intrinsic pa-rameters when the BNS signal has SNR = 30 (top row), SNR= 20 (middle row), and SNR = 15 (bottom row). Results areshown for the cases when the GW150914-like signal ends atthe same time as the BNS signal (
GW150914-tc ), when it ends2 seconds earlier (
GW150914-tc-2 ), when the GW151226-likesignal ends at the same time as the BNS (
GW151226-tc ), whenit ends 2 seconds earlier (
GW151226-tc-2 ), and finally whenthe injected signal is only the BNS (
BNS ). The true values ofthe parameters are indicated by vertical, black lines.
22 23 24 25M P D F GW151226GW151226GW151226GW151226GW151226GW151226GW151226 P D F GW150914GW150914GW150914GW150914GW150914GW150914GW150914
FIG. 5. Posterior PDFs for total mass and mass ratio, forthe GW150914-like signal (top panel) and the GW151226-like signal (bottom panel) when they are respectively beingoverlapped with a BNS signal of SNR = 30 (solid lines), SNR= 20 (dashed lines), and SNR = 15 (dotted lines). The over-laps are being done when the BBH and the BNS end at thesame time ( tc ), and when the BBH ends 2 seconds before theBNS ( tc-2 ). Finally, posterior PDFs for the two BBH signalsby themselves are shown as green, dashed-dotted lines ( BBH ).The injected parameter values are indicated by black, verticallines.
BBH overlapped BNS (SNR = 30) BNS (SNR = 20) BNS (SNR = 15) M q ˜Λ M q ˜Λ M q ˜Λ GW150914-tc
GW150914-tc-2
GW151226-tc
GW151226-tc-2
GW150914-tc GW150914-tc-2 GW151226-tc GW151226-tc-2
M q M q M q M q
BNS (SNR = 15) – – 0.0504 0.0807 0.00933 0.0117 0.0687 0.0657BNS (SNR = 20) – – 0.0427 0.0698 0.0107 0.0106 0.0727 0.0700BNS (SNR = 30) – – 0.0379 0.0673 0.0187 0.183 0.0819 0.0793TABLE VI. Values of the KS statistic comparing PDFs for BBH parameters (columns) in the BNS+BBH overlap scenarios(rows) with the corresponding PDFs when there is no overlapping BNS signal. In the case of a GW150914-like signal mergingat the same time as a BNS, the sampler fails to find the signal, but other scenarios are not so problematic. For GW151226, theslightly higher values for the tc-2 case compared to the tc case are likely due to the signals being placed in a slightly differentpart of the noise stream (two seconds earlier) from the BBH-only cases that are used for comparison. The numbers shown herecorrespond to the PDFs in Fig. 5. We conclude that an overlapping BBH signal does nothave much impact on the estimation of the BNS param-eters, even if the BBH merger time is arbitrarily close tothat of the BNS. This is corroborated by the KS statisticsin Table V, which compare PDFs for the various overlapscenarios with the corresponding PDFs in the absence ofoverlapping signals. It is reasonable to assume that plac-ing a BBH signal even earlier in the BNS would also havehad little impact.
2. BBH recovery
Figure 5 shows parameter estimation on the BBHswhen the SNR of the BNS signal is varied from 30, to20, to 15. Table VI has the corresponding KS statisticscomparing with PDFs obtained in the absence of over-lap. Again results are shown for a particular noise re-alization; see Fig. 9 in Appendix A for two other noiserealizations. We see that when the BBH signal has a timeof coalescence 2 seconds earlier than the BNS ( tc-2 inthe figure), the signal is well recovered. However, whenthe BBH signal and the BNS signal end at the same in-stant of time, the BBH recovery deteriorates, and in thecase of the GW150914-like signal, the sampling processin fact fails to find the signal. For the GW151226-likesignal, while the estimates are offset from their true val-ues, there is some measurability of the signal when thetimes of coalescence of the BBH and BNS are the same.The different outcomes between the GW150914-like andGW151226-like injection are likely due to the short dura-tion of the GW150914-like signal, effectively leading to adistortion of the entire signal when the merger happensat the same instant as the BNS merger. By contrast, themuch longer inspiral of the GW151226-like signal implies many more wave cycles for the parameter estimation al-gorithm to latch on to. Finally, as the SNR over theunderlying BNS signal is varied (keeping the SNR of theBBH signal the same), the PDFs for the BBH show es-sentially no change. Placing a BBH signal only 2 secondsbefore the BNS merger causes the BBH to be recoveredwithout appreciable biases, so it is reasonable to assumethat placing a BBH signal still earlier in the BNS inspiralwould also have little effect on its recovery.
B. Overlap of 2 BBH signals
The scenario being analyzed here is the one in the mid-dle panels of Fig. 3. Fig. 6 shows the posterior PDFs ontotal mass M and mass ratio q when two BBH signalsof different masses are being overlapped, compared withparameter estimation on the same signals in situationswhere there is no overlap ( BBH ). The corresponding KSstatistic values are given in Table VII. We find the resultsto be consistent within statistical fluctuations. Here too,the signals are overlapped once with the same coalescencetimes ( tc ), and once with one of the signals, GW150914,ending 2 seconds earlier ( tc-2 ). The SNRs of the two sig-nals, GW150914-like, and GW151226-like, are 30 and 15,respectively. As can be seen in the Figure, the two BBHsignals’ parameters can be extracted without any biaseseven when they end simultaneously. Again see AppendixA for other noise realizations, with the same conclusion. GW150914-tc GW150914-tc-2 GW151226-tc GW151226-tc-2
M q M q M q M q tc-2 cases are likely due to the signalsbeing in a slightly different part of the noise stream (two seconds earlier) from the BBH-only cases used for comparison.However, in all cases there is no significant bias. The numbers shown here correspond to the PDFs in Fig. 6.BNS1 ( tc ) BNS1 ( tc-2 ) BNS2 ( tc ) BNS2 ( tc-2 ) M q ˜Λ M q ˜Λ M q ˜Λ M q ˜Λ
72 74 76 78 P D F GW150914GW150914GW150914 P D F GW151226GW151226GW151226
FIG. 6. Posterior PDFs for total mass and mass ratio whena GW150914-like signal and a GW151226-like signal are be-ing overlapped at the same trigger times ( tc ) and when thetrigger time of the GW150914-like BBH ends 2 seconds ear-lier ( tc-2 ), compared with parameter estimation in the ab-sence of overlap ( BBH ). The top panel shows the recovery ofthe GW150914-like signal and the bottom one that of theGW151226-like signal. Black vertical lines indicate the truevalues of the parameters.
C. Overlap of 2 BNS signals
Finally, we analyze the simulations in the bottom pan-els of Fig. 3. Figure 7 shows the recovery of BNS pa-rameters for each BNS signal when two BNS signals arebeing overlapped, again with either the same coalescencetimes and when one of the BNSs (henceforth BNS2) ends2 seconds earlier than the other BNS signal (henceforthBNS1). For KS statistic values comparing PDFs withthe corresponding non-overlapping cases, see Table VIII.BNS1 and BNS2 respectively have SNRs of 30 and 20,and component masses ( m , m ) = (1 . , . M (cid:12) and ( m , m ) = (1 . , . M (cid:12) . These particular choicescause both signals to have very similar chirp masses. P D F BNS1BNS1BNS1 P D F BNS2BNS2BNS2
FIG. 7. Posterior PDFs showing recovery on chirp mass, massratio and tidal deformability ˜Λ when two BNSs, referred toas BNS1 and BNS2, are being overlapped at the same timeof coalescence ( tc ) and when BNS2 ends 2 seconds earlierthan BNS1 ( tc-2 ). These are compared with results in theabsence of overlap ( BNS ). The top panel is for the recovery ofBNS1 and the bottom one for the recovery of BNS2. The solidblack vertical lines indicate the injected values of the sourcebeing recovered each time. We note that when the timesof coalescence of the two BNSs are the same, the parameterestimates recovered are those of BNS1, whose injected valuesare also shown in the bottom panel, as dashed vertical blacklines.
Given these masses, their tidal deformabilities, ˜Λ = 303 for BNS1 and ˜Λ = 292 for BNS2, follow the equationof state APR4; these were the simulated signals usedfor investigating systematics in the measurements onGW170817 in Ref. [53].In Fig. 7, the top panel shows the posterior PDFs onchirp mass, mass ratio, and tidal deformability for BNS1when BNS2 ends at the same time ( tc ) and when BNS2ends 2 seconds earlier ( tc-2 ), together with the casewhere only BNS1 is present in the data ( BNS ). The bot-tom panels show the same, but for the recovery of BNS2.When the two signals end at the same time, the pa-rameters characterizing BNS1 are being recovered, which0likely happens because of the higher SNR of BNS1. Asthe tidal deformabilities of the two sources are so close,the PDFs for ˜Λ look similar in all cases. However, alsolooking at the mass parameters, parameter estimation israther robust when the signals end 2 seconds apart. V. CONCLUSIONS
Given regular improvements in the sensitivity ofgravitational-wave detectors and especially the plannedconstruction of the next generation of interferometers,it will become increasingly likely that individually de-tectable gravitational-wave signals will end up overlap-ping in the data. In this paper we (i) assessed how oftendifferent types of overlap will happen in ET and CE, and(ii) tried to quantify the impact this would have on pa-rameter estimation with current data analysis techniques.To address the question of the nature and frequency ofdifferent overlap scenarios, for each of three possible localmerger rates, we constructed a “catalog” of signals in ETand CE, enabling a more in-depth study of overlaps thanin previous works. We showed that there will be a sig-nificant number of signals for which the merger happenswithin the same second, varying from tens to thousandsdepending on the local merger rate. Additionally, thesubstantial increase in the duration of BNS events dueto the improved low frequency sensitivity of third gener-ation observatories will lead to the occurrence of up totens of other signals overlapping with a given BNS.Motivated by these results, we performed the first de-tailed Bayesian analysis study on possible biases thatmay arise in future as detection rates become higher andoverlapping signals start to occur. We focused on over-lapping signals for which the end times were close to eachother, so that in particular there is overlap at times whereboth signal amplitudes are high; it is this type of situ-ation where we expect parameter estimation biases tobe the most pronounced. Specifically, merger times weretaken to be either the same (as a proxy for being arbi-trarily close to each other), or separated by 2 seconds.Our preliminary conclusions (based on a limited numberof investigations) are as follows:• When BBH signals are overlapping with a BNS sig-nal of similar SNR, parameter estimation on theBNS is hardly affected, even with the merger timeof the BBH arbitrarily close to that of the BNS.Presumably this is due to the much larger numberof BNS wave cycles in band compared to the BBH.• However, in the same scenario, parameter estima-tion on the BBH can be subject to significant biasesif the BBH is high-mass, so that its signal is short.That said, the problem largely disappears when theBNS and BBH merger times are separated by 2 sec-onds, or when the BBH is low-mass.• When two BBHs with sufficiently dissimilar masses overlap with close-by merger times, parameter es-timation on either of the signals will not be muchaffected.• When two BNS signals overlap with close-bymerger times, parameter estimation will recover thelouder signal reasonably well. With a 2 second sep-aration of merger times, good-quality parameter es-timation can already be done on the two signalsseparately.These results suggest that current parameter estima-tion techniques will, in several types of situations of in-terest, already perform reasonably well in the 3G erawhen applied to overlapping signals, even when the in-dividual signals have similar SNRs, and even when theSNRs are on the low side given the projected distributionfor these observatories. Nevertheless, a number of ques-tions remain. What happens when SNRs are graduallyincreased? Related to this is the choice of lower cut-offfrequency; to what extent will parameter estimation im-prove as one goes to f low = 5 Hz or even lower, so thatsignals have a much larger number of wave cycles in thedetector’s sensitive band? Though not the focus here, athigher SNRs the use of currently available waveform ap-proximants to analyze BNS signals in 3G detectors wouldlead to biases in the estimation of ˜Λ even in the absenceof overlap [90], also motivating further research in wave-form modeling. Spins were not included in our study,but it would be of interest to see their effect: large pre-cessing spins will complicate parameter estimation in thecase of BBHs, while for BNSs, having access to the spin-induced quadrupole moment can aid in determining tidaldeformabilities [91]. Finally, what happens when over-laps involve (much) more than two signals, e.g. a longBNS signal overlapping with a large number of BBH sig-nals? These questions are left for future work.In order to make optimal scientific use of the capabil-ities of 3G detectors, it will be appropriate to developBayesian parameter estimation techniques for which thelikelihood function assumes multiple signals to be presentin a given stretch of data, e.g. replacing Eq. (11) by p ( d |{ (cid:126)θ i } , H s ) ∝ exp (cid:34) − (cid:18) d − N (cid:88) i =1 h ( (cid:126)θ i ) (cid:12)(cid:12)(cid:12)(cid:12) d − N (cid:88) i =1 h ( (cid:126)θ i ) (cid:19)(cid:35) , (14)with N the number of signals found by a detectionpipeline, and (cid:126)θ i , i = 1 , . . . , N the associated parameters.Additionally, one could let N itself be a parameter tobe sampled over, thus allowing for an a priori unknownnumber of signals in the given stretch of data. In all this,it may be possible to borrow from techniques developedin the context of somewhat related problems in GW dataanalysis, such as the characterization of the large numberof (in this case near-monochromatic) signals from galac-tic white dwarf binaries in the space-based LISA [92–98],BNSs in BBO [43], or supermassive black hole binariesin pulsar timing searches [99].1 ACKNOWLEDGMENTS
We are grateful to Elia Pizzati, Surabhi Sachdev, Anu-radha Gupta, and Bangalore Sathyaprakash for sharingand discussing their results on a related study. A.S., J.J,and C.V.D.B are supported by the research programmeof the Netherlands Organisation for Scientific Research (NWO). The authors are grateful for computational re-sources provided by the LIGO Laboratory and supportedby the National Science Foundation Grants No. PHY-0757058 and No. PHY-0823459. We are grateful for com-putational resources provided by Cardiff University, andfunded by an STFC grant supporting UK Involvement inthe Operation of Advanced LIGO. [1] B. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J.Lett. , L22 (2016), arXiv:1602.03846 [astro-ph.HE].[2] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.Lett. , 221101 (2016), [Erratum: Phys.Rev.Lett. 121,129902 (2018)], arXiv:1602.03841 [gr-qc].[3] B. P. Abbott et al. (LIGO Scientific, Virgo), Astrophys.J. 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We have performed all our simulations in three different noise realizations. To avoid plots getting too busy, inSec. IV we only showed results for one of these; here we also give them for the other two noise realizations.In the case of a BNS overlapping with a BBH, the measurements on the BNS are shown in Fig. 8 and those on theBBH in Fig. 9. The corresponding KS values are given in Tables IX and X, respectively. For measurements on themass parameters of the BNS, we find that the results are consistent between noise realizations. For the tidal parameter ˜Λ , the PDFs differ somewhat more; compare the right columns in the two panels of Fig. 8. This is likely because mostof the information on tides enters the signal at higher frequencies, where the variance of the noise is larger; hence themeasurement of ˜Λ will be more affected by the noise realization than the mass measurements, especially when SNRsare not high. Indeed, though not shown here explicitly, KS statistics for ˜Λ between different noise realizations, butfor the same overlap situation, tend to be significantly larger than within the same noise realization but for differentoverlaps. For parameter estimation on the BBH, there are differences in the PDFs for the masses when the BBHmerger time coincides with that of the BNS, but not so much if it occurs 2 seconds earlier.In the case of two overlapping BBH signals, parameter estimation results are shown in Fig. 10, and KS statistics inTable XI. The results are quite robust under a change of noise realization.Finally, the case of two overlapping BNSs with different noise realizations is shown in Fig. 11, and KS statisticsin Table XII. As in the case of a BNS overlapping with a BBH, the PDFs for the masses are not much affected bydifferences in noise, but the ones for ˜Λ are more susceptible. P D F SNR=30SNR=30SNR=30SNR=30SNR=30 P D F SNR=20SNR=20SNR=20SNR=20SNR=20
500 1500 25001.1944 1.1948 1.1952 P D F SNR=15SNR=15SNR=15SNR=15SNR=15 q GW150914-tcGW150914-tc-2 GW151226-tcGW151226-tc-2 BNS P D F SNR=30SNR=30SNR=30SNR=30SNR=30 P D F SNR=20SNR=20SNR=20SNR=20SNR=20
500 1500 25001.1946 1.1948 1.1950 P D F SNR=15SNR=15SNR=15SNR=15SNR=15 q GW150914-tcGW150914-tc-2 GW151226-tcGW151226-tc-2 BNS
FIG. 8. Posterior PDFs for BNS parameters when a BNS and BBH signal are being overlapped; same as Fig. 4 when injectionsare done in two other noise realizations (left and right panels). BBH overlapped BNS (SNR = 30) BNS (SNR = 20) BNS (SNR = 15)
Noise realization 2 M q ˜Λ M q ˜Λ M q ˜Λ GW150914-tc
GW150914-tc-2
GW151226-tc
GW151226-tc-2
Noise realization 3 M q ˜Λ M q ˜Λ M q ˜Λ GW150914-tc
GW150914-tc-2
GW151226-tc
GW151226-tc-2
72 74 76 78 P D F GW150914GW150914GW150914GW150914GW150914GW150914GW150914 P D F GW151226GW151226GW151226GW151226GW151226GW151226GW151226 P D F GW150914GW150914GW150914GW150914GW150914GW150914GW150914 P D F GW151226GW151226GW151226GW151226GW151226GW151226GW151226
FIG. 9. Posterior PDFs for BBH parameters when a BNS and BBH signal are being overlapped; same as Fig. 5 when injectionsare done in two other noise realizations.BNS overlapped
GW150914-tc GW150914-tc-2 GW151226-tc GW151226-tc-2
Noise realization 2
M q M q M q M q
BNS (SNR = 15) – – 0.0134 0.011 0.00832 0.00890 0.411 0.398BNS (SNR = 20) – – 0.0104 0.0109 0.0169 0.0172 0.390 0.377BNS (SNR = 30) – – 0.0100 0.0113 0.0140 0.0146 0.367 0.357
Noise realization 3
M q M q M q M q
BNS (SNR = 15) – – 0.0168 0.0100 0.0140 0.0142 0.318 0.131BNS (SNR = 20) – – 0.0189 0.0131 0.0132 0.0137 0.322 0.315BNS (SNR = 30) – – 0.0287 0.295 0.0136 0.0130 0.334 0.327TABLE X. Values of the KS statistic comparing PDFs for BBH parameters (columns) in the BNS+BBH overlap scenarios(rows) with the corresponding PDFs when there is no overlapping BNS signal, when injections are done in two other noiserealizations. As before, when the GW150914-like signal ends at the same time as a BNS, it is not found by the samplingalgorithm, but other scenarios are less problematic. The numbers shown correspond to the PDFs in Fig. 9, noise realisation 2corresponding to the left panel and noise realisation 3 to the right panel.
GW150914-tc GW150914-tc-2 GW151226-tc GW151226-tc-2
M q M q M q M q
M q M q M q M q
72 74 76 78 P D F GW150914GW150914GW150914 P D F GW151226GW151226GW151226 P D F GW150914GW150914GW150914 P D F GW151226GW151226GW151226
FIG. 10. Posterior PDFs for BBH parameters when a BNS and BBH signal are being overlapped; same as Fig. 6 when injectionsare done in two other noise realizations. P D F BNS1BNS1BNS1 P D F BNS2BNS2BNS2 P D F BNS1BNS1BNS1 P D F BNS2BNS2BNS2
FIG. 11. Posterior PDFs when two BNS signals are being overlapped; same as Fig. 7 when injections are done in two othernoise realizations. BNS1 ( tc ) BNS1 ( tc-2 ) BNS2 ( tc ) BNS2 ( tc-2 ) M q ˜Λ M q ˜Λ M q ˜Λ M q ˜Λ M q ˜Λ M q ˜Λ M q ˜Λ M q ˜Λ˜Λ