Toward low-latency coincident precessing and coherent aligned-spin gravitational-wave searches of compact binary coalescences with particle swarm optimization
TToward low-latency coincident precessing and coherent aligned-spin gravitational-wavesearches of compact binary coalescences with particle swarm optimization
Varun Srivastava,
1, 2
K Rajesh Nayak, and Sukanta Bose
4, 5 Department of Physics, Syracuse University, Syracuse, New York 13244, USA Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India Indian Institute of Science Education and Research Kolkata, Mohanpur, West Bengal 741252, India Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India Department of Physics & Astronomy, Washington State University, 1245 Webster, Pullman, WA 99164-2814, U.S.A (Dated: November 16, 2018)We investigate the use of particle swarm optimization (PSO) algorithm for detection ofgravitational-wave signals from compact binary coalescences. We show that the PSO is fastand effective in searching for gravitational wave signals. The PSO-based aligned-spin coincidentmulti-detector search recovers appreciably more gravitational-wave signals, for a signal-to-noiseratio (SNR) of 10, the PSO based aligned-spin search recovers approximately 26 % more eventsas compared to the template bank searches. The PSO-based aligned-spin coincident search uses 48kmatched-filtering operations, and provides a better parameter estimation accuracy at the detectionstage, as compared to the PyCBC template-bank search in LIGO’s second observation run (O2) with400k template points. We demonstrate an effective PSO-based precessing coincident search with320k match-filtering operations per detector. We present results of an all-sky aligned-spin coherentsearch with 576k match-filtering operations per detector, for some examples of two-, three-, andfour-detector networks constituting of the LIGO detectors in Hanford and Livingston, Virgo andKAGRA. Techniques for background estimation that are applicable to real data for PSO-basedcoincident and coherent searches are also presented.
PACS numbers: 95.85.Sz, 04.30.Db, 97.60.Jd
I. INTRODUCTION
The Advanced LIGO and Advanced Virgo detectorshave now reached a sensitivity where they have alreadymade successful observations of gravitational waves(GWs) from multiple compact binary coalescences(CBCs), such as several binary black holes (BBHs) [2–7]and a binary neutron star (BNS) [8]. GWs fromastrophysical source can provide great insight intofundamental physics ranging from testing the generaltheory of relativity [16], validating no-hair theorem forblack holes [25], constraining the NS equation of state(EOS) [8, 18, 23]. In the upcoming years as the sensitivityof each detector improves one can probe deeper into theUniverse. The ground-based detectors are capable ofdetecting GWs primarily from BBH, NSBH and BNSbinaries, where the black holes masses are in the stellarto the intermediate mass range. We will study the use ofparticle swarm optimization (PSO) algorithm [29] detectthese sources.The present GW searches, with PyCBC [37, 44]and GstLAL [33], rely on a template-bank approachto detect GWs from CBCs [38, 39]. The PyCBCtemplate-bank [21] for LIGO’s second observation run(O2) comprises approximately 400k templates. Thus,for a given analysis segment of data from a detector,there are a total of 400k matched-filtering operations(MFOs). An early detection of a GW signal facilitatesthe follow-up of prospective sources in various regionsof the electromagnetic (EM) spectrum promptly. Thecombined knowledge of EM emission spectrum and the gravitational wave signal can be used to put constraintson the NS equation of state [11, 23, 32]. Theelectromagnetic counterparts are GRBs, kilonovae andvarious other transient and longer-lived signals arisingfrom BNS and in some cases NS-BH mergers. Thekilonovae are expected to evolve over a timescale from afew hours to a few days [34]. These light curve predictionswere verified by the electromagnetic observations ofGW170817 [35]. This time constraint demands forprompt GW alerts for EM follow-up. The PSO algorithm[29] provides a fast way to search for optimal solutions ina given parameter space. This property can be exploitedto speed up the process of detection of GWs.In our work, we demonstrate the effectiveness of thePSO algorithm in GW data-analysis by successfullyexecuting an all-sky blind coherent search that uses atotal of 576k MFOs per detector. We also show thatthe PSO can effectively search for precessing binariesusing coincident statistics with a total of 320k MFOsper detector. The high-speed and effectiveness of thePSO algorithm to provide optimal solutions in a givenparameter space makes the above two searches feasible.PSO can also be used to speed up the online aligned-spinCBC search for GW signal in data.The template-bank approach [39] for detectingthe GWs from CBCs in LIGO-VIRGO detectors isdiscretized (analogous to setting up grid-points inparameter space). The template bank is generated a r X i v : . [ a s t r o - ph . I M ] N ov by choosing a minimal overlap/match of M = 0 . The match here is a measure of the cross-correlation between thedata and a known template, and is defined more precisely in Sec. § III. parameter space is automatically manifested by thealgorithm. The points for template generation areproduced dynamically. The search parameters, thetemplate approximants can be actively changed andthe swarm intelligent algorithm will evolve accordinglyto generate the search template points, giving greaterflexibility. We will discuss these in greater detail insections § II and § VI. We will first describe briefly theaspects to gravitational wave data analysis. We will useboth the coherent search and coincident search statisticsto detect gravitational waves from CBCs, we will brieflyreview the idea behind these searches in section § III. ThePSO algorithm and the different variants explored in ourstudy are discussed in section § II. We will present theresults of our study in section § VI.
II. PARTICLE SWARM OPTIMIZATION
Particle swarm optimization (PSO) [29] originatedwith the aim to model the social behavior of animalsliving in a colony like swarms. However, simulationsreveal that the eventually swarms converged to optimalsolutions of a given function ( fitness function ) in theparameter space.Being social gives that individuals in the swarm theability to communicate with each other. This grants eachindividual in a given swarm or colony to be aware of theinformation gathered by all the other individuals in theswarm. The trajectory taken by an individual particlecan then be modeled by three independent behavioraltraits. These factors influence the path taken up byan individual during the period when it explores theparameter space. The first factor is inertia , implyingthat if an individual is moving in a given direction it willcontinue to move in that direction. The second trait isrelated to nostalgia , at any given instance of time, eachindividual has the tendency to give up the search andmove towards the best location explored by it, referredto as personal best (pBest). The last factor is basedon the collective ( social ) pool of knowledge. As eachindividual is aware of the best location explored by theentire swarm, the global best (gBest), each individualhas a tendency to give up the energy exhaustive searchprocess and move towards the global best location. Thesethree factors influencing the velocity of an individual canbe combined as v i,d ( t + 1) = ω · v i,d ( t ) + γ p · r p · ( p i,d ( t ) − x i,d ( t ))+ γ g · r g · ( g d ( t ) − x i,d ( t )) (1)where the subscript i marks the individual of the swarm, d represents the dimension. The factors r p and r g arestochastic factors which determine at any given instanceof time the instinct to move towards the pBest and gBestrespectively. The coefficients ω , γ p and γ g determinethe strength of the inertial, nostalgic and social factorsrespectively. The force of attraction towards the globaland personal best is assumed to be due to a harmonicpotential. The above PSO notations are consistentthroughout the paper. The position evolution equationis simply given by x i,d ( t + 1) = x i,d ( t ) + v i,d ( t ) (2)Thus, using PSO one can obtain optimal solutions of agiven fitness function in a parameter space. The steps ofthe PSO algorithm are briefly defined below. • Let S be the given parameter space in D -dimension.A swarm of N p particles is initialized with theparticle’s position vector x d having a uniform priordistribution in each dimension. Each particle is alsoassigned with a random velocity vector, v d witha norm uniformly distributed in the range [ − , f ( x d ) for the initial location of each particle, andthen the maximum value of f ( x d ) achieved is usedto initialize the gBest location. • The swarm is evolved with the equations 1 and 2,in discrete time steps of one. The pBest and gBestlocations are updated by checking if the fitnessfunction evaluated at the new position vectors ishigher than the previous step. • The evolution is terminated with a stop condition.Usually, a maximum number of steps N s arepredefined and used as a flag to stop the simulation.The value of gBest location at the end of thesimulation is considered as one of the optimalsolutions to the fitness function f ( x d ).We use the PSO algorithm to maximize the likelihoodfunction and find optimal solutions in the givenparameter space of compact coalescence binaries(CBCs).We extended the PSO scheme to search using multipleswarms over the parameter space. There are twoways this can be achieved. One in which all theswarms evolve independently of each other and thus,can be evolved simultaneously. The second way is toevolve subsequent swarms in a way that they use theinformation acquired by previous swarms ( relayed ). Thelatter can be used to give priors on search variables(instead of a subsequent blind search) and reduce theextent of parameter space. The subsequent swarms,in this case, can use the information acquired by thesearches carried out by previous swarms. The formersearch method has the advantage that it can be easilyparallelized across machine nodes.We use the information from the multiple swarmvariants of the relayed searches in two ways. The firstway is using the gBest information of previous swarmsand using the hostile swarm algorithm discussed insection II B. The second method is using the informationabout the parameters estimated. In GW data analysis,chirp mass ( M chirp ) is one of the parameters that is Coincident SNR (HL Network) N u m b e r o f e v e n t s Informed: M Chirp constrainedIndependent
FIG. 1. The figure compares the distribution of the SNR ofthe two sub-swarms in the case where – the second sub-swarmevolves independently of the first (pink) and the case wherethe second swarm uses the ±
30 % error on M Chirp obtainedby the first swarm. For 20k injections, we see that the laterscheme performs better as the sub-swarms achieve an overallhigher signal to noise ratio (SNR). recovered with good accuracy. We can use the M chirp estimated from previous swarms to set up a uniformprior on the subsequent swarms. Figure 1 compares theperformance between a search where multiple swarmsare independent to the one with the M Chirp constraint.We see that having the M Chirp prior significantlyimproves performance. Next, we describe some of thealternatives to the standard PSO algorithms to enhancethe performance specific to our problem.
A. Multiple Independent Swarms
The simplest extension to the standard PSO algorithmis the use of multiple swarms independent of each other toexplore the parameter space. The use of multiple swarmsis advantageous in the situation when the parameterspace has multiple optimum solutions. The evolutionequations of velocity and position are the same as 1 and 2for any given sub-swarm. At the end of the simulations,one obtains an optimal solution corresponding to eachsub-swarm. From this set of global best location exploredby each sub-swarm, we choose the one which gives thehighest value of the fitness function – i.e. matched filteror likelihood as the case may be.
B. Hostile Swarm Algorithm
Many species in nature compete amongst themselvesand exhibit territorial behaviors. We aim to extendand modify the PSO algorithm to model the territorialbehavior. In nature animal species maintain territoriesbecause of limitation of resources. Individual within acertain group interact amongst themselves but have ahostile attitude towards any intruder. Thus, the multiplesub-swarms are no longer independent but have a hostileattitude towards each other. The velocity equation for agiven sub-swarm is thus modified to v mi,d ( t + 1) = ω · v mi,d ( t ) + γ p · r p · ( p mi,d ( t ) − x mi,d ( t ))+ γ g · r g · ( g md ( t ) − x mi,d ( t )) − (cid:88) n ; n (cid:54) = m γ r · r r · F ( g nd ( t ) − x mi,d ( t )) (3)where the superscripts m (or n) represent the m-thsub-swarm. The function F represents the repulsionforce by which the hostile sub-swarms interact with eachother. The functional chosen for the repulsion can bearbitrary. We use an inverse square repulsion potential.However, when deciding the functional form of repulsionit is important to consider the size of the parameter spaceand the strength of repulsion. If the repulsion force isset to be very high then particle’s in a swarm will leakout of the parameter space and the boundary conditionwill randomly re-inject the particle back in the parameterspace. If there are a large number of particles leaving theparameter space at each step, then the swarm intelligencewill not evolve and optimal solutions will not be rendered.One way to step around this problem is re-normalizingthe particle’s velocity to v norm if the particle’s velocityis greater than some v o .The particles in a given swarm are repelled at anyinstant of time from the gBest location discovered byany other sub-swarms allowing an effective explorationin the parameters space as not only the convergence isdelayed but the repulsion leads the sub-swarms to searchunexplored locations in parameter space. However, thereare some more caveats to be addressed for the methodto function effectively. One major problem arises ifmultiple hostile swarms are evolved simultaneously. Theproblem arises that by chance if any of individuals intwo or more distinct swarm end up by chance exploringsome optimal (global or local) solution during the courseof their exploration. In an extreme case, consider twoswarms end up with the same gBest location then theirconvergence is drastically affected due to the repulsion.Multiple hostile swarms search can be relayed to solvethese issues.Sometimes, in a relayed search as well, the secondswarm despite the repulsion may end up close to theoptimal solution previously explored and from where itis repelled from again causing convergence issues. Tofurther reduce such the probability of this to happen onecan further add a distance constrained. To elaborate, ifthem ( m = 1, 2, ... ) evolved swarms have exploredgBest locations (cid:126)g , (cid:126)g , ... . Then the m+1 swarm isevolved in a way that it is constrained that none of theindividuals in the swarm can have their pBest or gBestcloser than the distances d , d , ... gBest of them previous swarms. Lastly, because of the repulsion potential, theconvergence of the swarms may be adversely affected.When the repelled swarm has evolved then to allowswarm convergence we turn off the repulsion and evolvethe repelled swarm with the standard PSO evolutionequation for five steps. This ensures that the swarmsmoothly converges. The velocity of each individual isalso reinitialized to very small values so that they don’tfly away from the converged location. All hostile swarmsearches end with five iterations using the standard PSOevolution equations. C. Remove PSO
In GW detection problem, we wish to generate possibletriggers of detected astrophysical events as soon aspossible. The PSO algorithm provides very quick swarmconvergence if there is a very sharp outlying global peak.This property could be exploited in GW data analysis.The following trick helps reduce the computational cost,specially in cases of GW triggers with higher SNRs. Ingeneral, the following modification is specially relevantin cases where the cost of computation of the fitnessfunction is large. In the last few iterations of the PSOsearch, the particles wander around the gBest with slightor no improvement in the estimate of gBest. For thepurpose of detecting the gBest location, sampling thepeak and calculating the posterior is not important, thusthe late iterations of PSO could be avoided if the swarmhas converged reducing the computational cost. Weallow the swarms to evolve for 40 iterations, however,if the SNR of the trigger is considerably high the swarmconverges early. Consider N iterations were allowed forthe swarm to evolve, we choose to remove the particlesthat have converged after M ( < N) iterations. We canremove the particles based on the following criteria andways. First, consider a hypersphere of radius r in thedimensional space around the global best location, thenremoving a fraction f of the particles within this volumeat every iteration following M. One could also considera hypercube around the global best location and removeparticles within it in the same way. The second wayto eliminate the grouped particles is by dividing thedimensions d , d , ... into n , n , ... bins respectively. Ifthere are multiple particles in any hypercuboid in theparameter space after M iterations, we remove a fractionf of the particles in that hypercuboid, each step followingM. D. PSO Variants Summary
The performance of different PSO variants wascompared. We use eight sub-swarms in our maximizationand find that multiple-independent swarms and thedifferent variant of hostile swarms all performedcomparably. We use multiple-independent swarm for therest of the study. In our subsequent sections, all theresults are presented for this variant.
III. GRAVITATIONAL WAVE DATA ANALYSIS
The matched filter is the technique prominently usedto meaningfully extract gravitational wave signals buriedin detector noise [39]. Given some data imprintedwith signal from an astrophysical source along withnoise . The matched-filter function is defined asthe cross-correlation of templates h I ( t, ξ ) (of knownparameters) with detector ( I ) data. The matched filteris weighted by the power spectral density of noise infrequency domain which is a representative of detectorsensitivity across the frequency band. The detector noise n I ( t ) is related to the power spectral density of noise S Ih ( f ) by the fourier space auto-correlation function ofnoise ˜ n I ( t ) given by (cid:104) ˜ n I ( f ) | ˜ n I ( f (cid:48) ) (cid:105) = δ ( f − f (cid:48) ) S Ih ( f ) (4)For a single detector the matched filter is given byequation 5.( s I | h I ) = 4 R e (cid:90) ∞ df ˜ h I ∗ ( f )˜ s I ( f ) S Ih ( f ) . (5)For a network of detectors, the matched filter functionis simply the sum of matched filter computed for agiven template in all the detectors in that network.In the absence of an astrophysical signal, the detectoroutput is a time series of noise. Detecting gravitationalwaves implies distinguishing between the former caseagainst the presence of astrophysical signals in data.To distinguish between the two one can either useBayesian inference approach or a frequentist approach.In Bayesian inference, the likelihood ratio is defined asthe ratio of the probability that data has signal present(Test Hypothesis) to the probability of no signal beingpresent (Null Hypothesis), mathematically expressed as: λ ( h ) = P ( s | h ) P ( s |
0) = e − ( s I − h I | s I − h I ) · . e − ( s I | s I ) · . (6)The above expression translates to the followinglog-likelihood measure for a given detectorln λ = ( s | h ) −
12 ( h | h ) (7)Based on the parameterization of the templates usedfor matched filtering, one can describe two different Data can be decomposed as s I ( t ) = n I ( t ) (noise) + h IT ( t, ξ )(astrophysical signal) h I ( t, ξ ) represents a time-domain template generated withparameters ξ and data is represented by s I ( t ). The fourier transform of a time-series a(t) is denoted by ˜ a ( f ). statistical approaches – coincident and coherent. We willaddress the coherent and coincident GW search methodsin detail for multi-detector scenarios in section § IV and § V respectively. We briefly describe the dimensionalityand extent of the parameter space of CBCs in section § III A. Next, we will describe the injection parametersused in our study along with the parameters of PSOsearch in section § III B
A. Parameter Space of Compact Binaries
The GW waveform from CBCs depends on seventeenparameters, when including generically spinningcomponents and eccentric orbits. Here we will considernon-spinning as well as spinning components in varioussimulation studies, but never eccentric orbits. While along inspiral into LIGO-like sensitivity band is expectedto render the orbit devoid of any eccentricity, certainevolutionary scenarios allow for in-band (late-time)residual eccentricities that cannot be ignored. Herewe choose to set aside the study of eccentric orbitsfor a future work. Thus, the signals studied here willrequire atmost 15 parameters. These parameters canbe divided into two categories – intrinsic and extrinsic.The intrinsic parameters are inherent to the source,such as the masses of the stellar object M and M in a binary and their respective spin vectors S and S . On the other hand, the extrinsic parameters arethe source luminosity distance D , the inclination ofthe orbital plane of the binary ι with respect to theline of sight, the source sky-position angles ( θ, φ ), thepolarization angle ψ , the coalescence phase φ o , andthe time of arrival t a of the signal at a given detectorlocation. The matched-filter function (eq. 5) involvescross-correlation of tens to a thousand of seconds longdata segment, making a search in a fifteen-dimensionalparameter space computationally expensive.The search statistics employed to detect GWs inthe network of detectors can be divided into two –the coincident and the coherent search. Each ofthese searches is tuned such that the search spaceis mathematically reduced and there is a subset ofparameters p s over which the search is carried. Someof the other parameters can be estimated given theestimates of individual parameter p s . Further in manyGW searches, the search space is reduced to lowerdimensions in order to reduce this computational burdenand to allow the possibility to make prompt GWdetections, crucial to EM follow-up. Typically the effectsof spins is reduced by considering the component spins ofthe individual objects in the binaries to be either alignedor anti-aligned with the orbital angular momentum. Asa result, the six-dimensional parameter extent of spinis reduced to two ( S z , S z ). For LIGO-O2 the bankincorporates for aligned spins S z and S z and componentmass parameters [21]. The total number of templatepoints in the PyCBC-O2 template bank were 400k. B. Injections and PSO Parameters
The aLIGO detectors are sensitive enough to detectGWs from CBCs in a frequency range from close to 20Hz to a few thousand Hertz. For effectively utilizing thecomputational time, we divide the parameter space ofCBCs into different groups. • BBH Injections: The masses of individualcomponents in this group range from 12 M (cid:12) to 80 M (cid:12) . We divide the BBH injectionsinto sets – high and low spin BBH injections.The spins on component masses are aligned oranti-aligned with the CBCs angular momentum.Each subset consists of 10k different injections. Inthe sub-group of low spin BBH CBCs, we constrainthe spins of individual objects to be in the rangefrom -0.5 to 0.5. For the high-spin BBH injectionset one of the component masses is forced to havethe spin in range (0.5 to 0.85), whereas the secondobject can either be aligned or anti-aligned with theformer with a spin up to 0.85. The injections aredistributed uniformly in SNR over the volume withminimum coincident SNR of 5.5 and the minimumSNR in any given detector of 4. • BNS Injections: The masses of individual NS is inthe range from 1 M (cid:12) to 2 M (cid:12) . We also assume thatthe BNS systems will have low effective spins whenthey are close to the merger [22]. Thus, the spinsof each individual object go up to a maximum of0.05. We allow the possibility of individual spins tobe aligned or anti-aligned. We generate 500 BNSinjections distributed uniformly in SNR over thevolume with minimum coincident SNR of 7.5 andthe minimum SNR in any given detector of 5. Theincreased minimum is to allow the recovery of agreater number of events and overcome the smallnumber statistics of recovered events. • Non-precessing NS-BH Injections: The massesare for this injection set are so chosen thatthe injections are more likely to give rise toelectromagnetic counter parts [9]. The black holemass is restricted to range from 5 M (cid:12) to 14 M (cid:12) .The NS mass and spins are varied in the same rangeas before. We restrict the spin on black holes to amaximum of 0.4. We have assumed that the spinof binaries is aligned. For higher black-hole spinsthe assumption breaks, the coupling of componentspins with the orbital angular momentum will causeprecession [14]. We generate 2k injections smeareduniformly in SNR over the volume with minimumcoincident SNR of 7.5 and a minimum SNR in anygiven detector of 5. • Precessing NS-BH Injections: The masses and SNRdistribution range is same as that of non-precessingNS-BH injection set. The difference is we allow the possibility of precessing spins in NS-BH binaries.The total BH spin is smeared uniformly in the rangeof 0.5 to 0.85. We generate 2k injections in this set.We use the theoretical design power spectral density(PSD) of aLIGO detectors [42], VIRGO and KAGRAto generate noise in each detector using PyCBC [36].The lowermost sensitive frequency of each detector isassumed to be 20 Hz for the generation of templates.The injected signals and the templates generated aresampled at 4096 Hz. We use IMRPhenomD 3.5PN [30]templates for PSO based aligned-spin coincident andcoherent searches. For precessing injections and templatepoints, we use IMRPhenomPv2 waveform model [26, 40].For the template bank search similarly, the injections andthe search uses SEOBNRv4-ROM-DoubleSpin waveformmodel [17] for a total mass greater than 4 M (cid:12) , which isin accordance with the template bank [21]. We ensurethat the injected signal and the waveform model usedin the search are the same. The approached opted forin our study is the following. The GW signal from theastrophysical parameters discussed above is simulated.Then, we add noise to the simulated GW signal, the noiseis whitened by the PSD of the corresponding detector.Then, the swarms are initialized and are used to optimizefor the coherent SNR or the coincident SNR.In aligned-spin coincident search, we use 40 iterationsof PSO and eight multiple swarms with 150 particles eachto explore the parameter space, a total of 48k MFOs perdetector. In precessing-spin coincident search, we use40 iterations of PSO and 10 multiple swarms with 800particles each to explore the parameter space, a totalof 320k MFOs per detector. The PSO parameters inequation 1 are set to ω = 0.5, γ p = 2 and γ g = 2. We usean electrostatic repulsion potential and a linear repulsionpotential to repel the particles of subsequent swarms fromthe gBest of previous in hostile swarm algorithm. Theparameters in 3 are the same values along with γ r ( lin ) =1.05 γ g and γ r ( ES ) = 25. The different variants of PSOhad similar performance. All our results are presentedfor multiple-independent swarm variant of PSO. IV. COINCIDENT SEARCH
The ground-based GW detectors have an antenna-likeall-sky sensitivity, lacking the ability to locate theGW source in the sky. Using multiple detectorsand triangulation techniques the source location isdetermined. Additionally, environmental or instrumentaldisturbances give rise to glitches in the detector. Someof these glitches mimic the GW signal from CBCs [1, 10].One fundamental discriminator to veto these glitches isthe time of arrival in a network of GW detectors withsimilar sensitivity operating at the same time. Theastrophysical signals in the detectors cannot be separatedin time by a time greater than the light travel time t c between any two corresponding detectors in the network.The glitches are uncorrelated across detectors and thisapproach drastically reduces the false positives arisingdue to the glitches. All our injections are into gaussiannoise. Another discriminator of noise glitches and signalis the chi-squared discriminator [13].In a network of detectors the coincident SNR iscomputed by match filtering a given template ˜ h I ( ξ, f )with data in each detector ˜ s I ( f ). However, thematch-filter output from detectors has to be combinedkeeping the light travel time distance constraint acrossany two detectors. To compute the coincident SNR,one of the detectors is taken as a reference detector andthe matched filter series is computed. The maximum ofthe match-filter gives the time of arrival at the referencedetector t refa . Next, the match-filter output for the sametemplate is computed across all other detectors. Givena light travel time between the reference detector andsome detector j in the network t ref ; jc . In coincidentsearch, for j detectors the maximum of match-filteroutput in the time window t refa ± . t ref ; jc , gives thetime of arrival t ja in the j detector. The value of themaximum of match-filter at t ja is added in quadratureto give the coincident SNR for the reference detector.The reference detector is then changed and the coincidentSNR is recomputed. During the search, the template’sparameters ξ are varied over the search space. We definethe best template which maximizes the coincident SNRwith template parameters ξ = ξ max . If the coincidentSNR for template ξ max is greater than some thresholdsdefined to discriminate against noise and signals ofastrophysical origin, we flag the event of astrophysicalimportance. The parameters ξ max are the first-handestimates of source parameters. Using algorithms likenested sampling, one can calculate the posterior andestimate parameters with greater accuracy and bettersampling in an offline search which is not constrainedby time.We use the PSO algorithm to maximize over thecoincident SNR described above. The templateparameters we maximize our search over are M chirp , η , s z , s z and ι (inclination of the orbit), in thegeneral case. Thus, the dimensionality of the searchspace is five. We include the orbital inclination asan independent parameter as in the upcoming yearswith multiple GW detectors the distance-inclinationdegeneracy is expected to be broken [45]. To show thatthe PSO search is not drastically affected by the change issearch parameters, we will compare the search with m , m , s z , s z and ι as search parameters in section § IV B.For consistency check we will compare the performance ofPSO search over high-spin and low-spin BBH injectionsin section § IV A. We also summarize the results of PSObased aligned-spin coincident search on BNS and NSBHinjections, in section § IV A. Next, we will use PSO toset up a precessing coincident search (dimensionality ofnine). The results of the precessing coincident searchover precessing NS-BH injections and the correspondingcomparison with PSO based aligned-spin coincidentsearch over the same set of injections is summarized in section § IV C. In section § IV D we will estimatethe background of PSO based coincident and coherentsearches using 100k gaussian noise realization and 30time-slide over each of them. Lastly in section § IV E wewill compare the performance of PSO based coincidentsearch with the O2-template bank. We will then varythe number of detectors in the network and present theperformance of PSO in section § V C.To summarize the errors in the estimation of differentCBC parameters in different searches we will usebox-and-whisker plots throughout the paper. The boxand whiskers plot is a projection of a histogram. Thelimits of the colored box extend from the lower quartile toupper quartile. “Whiskers” plotted on either side of thebox extend to 1.5 × inter-quartile range (IQR) . Outliersare points outside the whiskers and are marked as ‘+’signs. A. PSO based aligned-spin coincident searchesover BBH, BNS and NSBH injections
To estimate the performance of PSO algorithm andto test the effectiveness of dynamic generation of pointsin parameter space, we test our algorithm by comparingagainst the different set of injections defined in section § III B. First, we test the performance of PSO to recoverBBH injections. The BBH injections are divided intotwo categories – high and low spinning. Each of the twoinjection sets contains 10k injections, smeared uniformlycomponent masses, but the total spin of the two systemvaries as per the corresponding definitions in section § III B. We use the PSO algorithm to search over theparameter space and maximize the coincident SNR foran HL and HLVK detector network. The observationsare summarized in Fig. 4. Placing a threshold of 10 and14.25 on the HL and HLVK detector networks (explainedin section § IV D), we see the fraction of events recoveredfrom the two different parts of the parameter space ofBBH are almost the same and the error in estimation ofparameters also has a similar distribution.We will now extend the PSO method to a searchover aligned spin BNS and NSBH injections. Thiscombined with the above two searches completes theparameter space of the CBCs that can be detected byground-based GW detectors. We do a similar exercise asdiscussed before and maximize the coincident SNR forthese injections in HL detector network. The results aresummarized in Fig. 5. We get a precise measurementof the M chirp value which is a trademark characteristicfeature of binaries with NS. The error in other parametersis also lesser compared to BBH search. Lastly, theFig. 2 summaries the error in the time of arrival in If the histograms were Gaussian, the ends of the whiskers wouldbe at 4.7 σ on either side of the mean. Individual Detector Threshold: 8
NS-NS NS-BH BH-BH432101234 T a ( m s ) Individual Detector Threshold: 10
FIG. 2. For the different injection sets – BBH, BNS, andNSBH (defined in section § III B) used in our study the aboveplot summarizes the error in the time of arrival in a network oftwo detectors (HL) for the injections that were recovered. Thethresholds on individual detectors used to flag an injection asrecovered are shown above the corresponding subplot. Fromthe figure above it is evident that the time of arrival iseffectively measured in an aligned-spin coincident search. milliseconds for the different injection sets. Thus, wehave demonstrated that the dynamic template placingin PSO algorithm is effective to recover signals, almostindependent of the component spin of objects and thenature of CBC.
B. Flexibility of Search
The functional form of the ambiguity function changeswith the choice of parameters used in the search [24].Specifically, how the match of a given template with itsneighboring templates falls off with increasing differencein their parameter values varies with the parameterchoice, such as ( m , m ) as opposed to ( M chirp , η ). Theperformance of a template-bank based search dependson such choices [38]. The question that arises is whethersuch choices affect PSO-based searches as well. We usethe same 20k BBH injections described in the sectionabove to look for any such effects.We set up two different searches over the 20k BBHinjections simulated in the HL detector network. Onewith parameters ( m , m , s z , s z , ι ) while the other withparameters ( M chirp , η, s z , s z , ι ). Figure 3 compares theresult of the two searches. We see that the results, Individual Detector Threshold: 8 m m m m M Chirp M Chirp R e l a t i v e e rr o r ( % ) Individual Detector Threshold: 10
FIG. 3. For 20k BBH injections, the plot abovesummarizes the results when the PSO based coincident searchwith two detectors(HL). A search with mass parameters( m , m ) (blue) – the masses of individual objects in binariesis compared against a search done with mass parameters( M chirp , η ) (orange) – the chirp mass and the symmetric massratio of the binary. The plot shows that the performance isalmost identical in the two search methods. The number ofevents passing the thresholds (labeled above each subplot) ineach detector is also equal. in terms of detection efficiency and error estimates arealmost similar. The rate of convergence to the optimalsolutions might be different amongst the different familiesof search parameters but for reasonable iteration steps,PSO search indicates weak dependence on the searchparameters allowing more flexibility. C. Precessing NSBH search
In astrophysical scenarios, in NSBH binaries the BHare expected to have high spins due to the accretion ofthe NS matter onto the BH, this makes it more likelyfor the orbit to precess [14]. We aim to extend the CBCsearch from aligned spin to precessing search using PSOto search for precessing NSBH.We first define the parameters of the injected signalin this sub-domain of precessing NSBH signals. Wegenerate 2k injections where the mass of BHs range from5 M (cid:12) to 14 M (cid:12) and their total spin range from 0 to 0.85.For the NSs the mass ranges between 1 M (cid:12) to 2 M (cid:12) whereas their total spin is restricted in the range from0 to 0.05. These values are chosen from astrophysicalknowledge of these systems such that their GW emissionis detectable by the ground-based detectors.To recover these signals we set up a nine-dimensionalsearch over parameters ( m , m , S , S , ι ). UsingPSO we maximize the coincident SNR. However, toensure that the search is effective in this high-dimensionalparameter space we increase the number of particles in R e l a t i v e e rr o r ( % ) HL Network: High and Low Spin BBH m m m m M ch M ch η η R e l a t i v e e rr o r ( % ) HLVK Network: High and Low Spin BBH
HL Network X eff Error N u m b e r o f E v e n t s HS-BBH: . LS-BBH: . HLVK Network X eff Error N u m b e r o f E v e n t s HS-BBH: . LS-BBH: . FIG. 4. The plot above summarizes the errors on the different parameters of binary black-hole injections. For a total of 10000injections in each set, the errors shown above are for events that pass the corresponding network thresholds. We see for boththe detector network configurations, the coincident SNR maximization using PSO is effective to recover signals from both theinjection sets – high-spin (Blue) and low-spin (yellow). The error distribution is almost comparable for the HL and HLVKnetwork as a majority of the events in HLVK are recovered using two detectors in the network, this is also the reason for theslight under-performance. m m M ch m m M ch R e l a t i v e e rr o r ( % ) HL network: Threshold 10 η Error N u m b e r o f E v e n t s BNS: 63.6 0.15 0.10 0.05 0.00 0.05 0.10 0.15 η Error X eff Error N u m b e r o f E v e n t s BNS: 63.6 0.6 0.4 0.2 0.0 0.2 0.4 0.6 X eff Error
FIG. 5. The plot summarizes the relative errors in the estimation of parameters for 500 BNS (blue) and 2k NSBH (yellow)injections. From the plots above we see that the M Chirp is estimated with a great accuracy, a trademark characteristic of BNSand NSBH binaries. The estimated errors on other parameters are also consistent with the expected values from the statisticalapproach described in [12]. m m m m M Chirp M Chirp η η R e l a t i v e e rr o r ( % ) Non-Pres vs Pres Search over precessing NSBH: HL threshold 12
HL Network ( X eff Error) N u m b e r o f E v e n t s Non-Pres Search: 21.2Pres Search: 25.1
FIG. 6. For the 2k precessing NSBH injection, the plot compares the performance of a precessing PSO search (yellow) with analigned-spin PSO search (blue). The background is similar for the two, but for the same false alarm probability, the detectionefficiency of precessing search is higher. We see for the same set of injections with identical noise realization, precessing searchrecovers approximately 18 % more events compared to an aligned-spin PSO search with the same number of MFOs. The errorin mass parameters has similar distribution for the two searches but the estimation of χ eff is very accurate in a precessingPSO search. the PSO algorithm to 800. The number of independentswarms used is also increased to 10. The total numberof 320k MFO are performed in the process. We aim tocheck the capabilities of PSO to perform a precessingsearch to recover precessing injections. To look for anyperformance improvements or advantages obtained froma precessing search, we compare the results of the searchover precessing NSBH injections with precessing PSOsearch against aligned-spin search ( m , m , S z , S z , ι ),with identical PSO parameters, swarm-size, and number.Thus, the total number of MFOs are same and equal to320k in both the searches.The results of the two searches are summarized in Fig. 6. Using the same threshold on the two searches,we see out of the 2k precessing NSBH injections,the precessing search recovers 25.1 % compared to arecovery of 21.2% from aligned-spin PSO search. It hasbeen demonstrated that a non-precessing search wouldunderperform with respect to a precessing search whensearching over precessing injections [28]. By extendingour search to account for precession we improve therecovery of injections by almost 18 % with respect toPSO aligned-spin coincident search. The second strikingfeature of the precessing search is the accuracy in theestimation of the χ eff parameter of the precessing binary.In summary, PSO based precessing search is promising2and with parallelization techniques could be used in anonline search with 320k MFOs. D. Background Estimation
We use 100k gaussian noise injections to estimate thebackground of the coherent and coincident PSO basedsearches. To estimate the background of a given gaussiannoise stream, we use PSO to maximize the correspondingSNR – coincident or coherent. The background eventsare not correlated. For a coincident background trigger,the time of arrival between the two detectors of thenetwork must be greater than the light travel distancebetween the two detectors – no astrophysical signal willbe separated in two detectors greater than the light traveltime. To compute the background we use time-slideson the dynamic template points which generate anSNR greater than 5 in one detector. To computethe background coincident SNR we take a 200ms timewindow in the stream of other data ensuring that thistime window doesn’t overlap with the correspondingtime of arrival window for astrophysical signals in theformer detector. The maximum SNR in the 200ms timewindow of the second detector in added in quadraturewith the SNR of the first detector. The backgroundestimation of the coherent search in done in the sameway, only the time window of the coherent astrophysicalsignal is removed from the time-slide window. Wedo 30 time-slides for each noise realization. TheFig. 7 summarizes the estimated background for the twosearches in a network of two detectors. From the figure,we estimate the false alarm probability of each searchstatistics.We use the thresholds corresponding to the respectivefalse alarm probability over the 20k BBH injections usedin our study. The fraction of these injections which crossthe thresholds is flagged as recovered events. The fractionof events recovered from the 20k BBH injections, eachwith a different but unique noise realization consistentwith different search methods, are summarized in thetable I. The coherent search is computationally moreexpensive than the coincident searches. However, thefraction of events recovered by the coherent search ishigher than the fraction of events recovered by coincidentsearch. The PSO based coincident search also recoversmore events with higher SNRs compared to the templatebank used in out study, for the same set of injections dueto the dynamic placing of template points and its abilityto find optimal solutions of a function in parameter space.
E. Comparison with O2-Template Bank
We compare the performance of the PSO basedcoincident search with the O2-template bank over the20k BBH injections. The noise realization of eachinjection in each detector is unique and identical for F a l s e A l a r m P r o b a b ili t y Coherent Non-SpinCoherent SpinCoincident
FIG. 7. We take 100k noise realizations and estimatethe background by time sliding on each of the noiserealizations. We estimate the background for coincidentsearch, non-spinning coherent search and spinning coherentsearch using the method described in section § IV D. Theplot above summarizes the background for each search in anetwork of two detectors (HL). For a false alert probabilityof the order O (10 − ), the network coincident SNR is close to8.5, for the same false alarm probability the coherent searcheshave an SNR close to 8. The sharp drop in coherent SNRarises due to and is a trend expected for a non-degeneratedetector network [27]. We point out that the three statisticsare different and their corresponding SNR values alone for anyinjection are not the true measure of their effectiveness. For agiven injection, our coherent SNR will be substantially higherthan the coincident SNR. Thus, for performance comparison,we note that while for a false alarm of unity, the coincidentSNR is ∼ . ∼ both the searches. The template bank searches rely onthe discretization of the parameter space of CBCs in away that the points in the bank have an overlap of 97%with the neighboring templates in the bank. To detectGW signals the strain data is match-filtered with all thepoints in the template bank. For multiple detectors,the data is combined using coincidence statistics. Forthe template bank search, the threshold value of SNRfor a given injection in a single detector is 4.5. Thenew-SNR threshold for the template bank search has asingle detector threshold of 4. In the template bank andPSO based searches, we use the same sampling rate of4kHz and the noise realization for any given injectionis identical in the corresponding detectors. We combinethe triggers from the template bank search and get thetemplate that maximizes the coincident SNR for thegiven injections.Figure 8 compares the template bank and PSO basedaligned-spin searches in an HL detector network. ThePSO based search uses 48k MFO only. We see PSOperforms better except for low coincident SNR range of7-8 where template bank has fewer number of events.Implying that points in the template bank yielded higherSNRs whereas PSO based search resulted in lower SNRs.3 m m m m M Chirp M Chirp M T M T R e l a t i v e e rr o r ( % ) HL Network: Coincident SNR threshold 10
HL Network ( X eff Error) N u m b e r o f E v e n t s PSOTB
10 15 20 25 30 35 40 45 50
SNR distribution N u m b e r o f E v e n t s PSOTB
Coincident SNR N u m b e r o f E v e n t s PSO: Injections > SNR 10 =26.91%TB: Injections > SNR 10 =21.02%
FIG. 8. Comparison between template bank (TB, shown in Yellow) and PSO (Blue) to search over 20k BBH injections. PSOuses a total of 48k matched-filter operations whereas the O2-template bank has 400k template points. The first two plots fromthe top compare the error of coincident signal recovered with an SNR greater than 10. We see that PSO estimates parametersof the binaries with less error as compared to the TB search. The third plot shows the recovered SNR distribution of both thesearches for all the injections. The last plot shows the SNR distribution for events recovered with an SNR greater than 10. Thelast plot shows that PSO performs better for high SNR injections with approximately 28% more events than template bankrecovered with SNR greater than 10 – the same injection set and with an identical noise in each injection set in each detectorin the two searches. For an SNR threshold of 9, we find that the PSO based aligned-spin spin search recovers approximately4.5% more events as compared to the template bank. This improvement obtained by PSO is due to the algorithm’s capabilityto place template points dynamically and find the optimal solution in the parameter space to maximize the coincident SNR.The O2 template bank has higher SNR loss for anti-aligned spins [21]. A majority of the 28 % improvement in SNR with PSOarises in the SNR range from 10 to 14. However, for lower SNR events PSO under-performs in the SNR window from 7-9. Thisis partly due to the drifting of events to higher SNRs using PSO, as the total number of events in both the searches is thesame. The third plot shows PSO recovers fewer events in SNR range of 7-8 than TB. This effect is mitigated by increasing thematched-filter operations in PSO to 96k. Threshold Corresponding Coincident Coincident Coherent Coherent(Coincident, Coherent) False Alarm Probability TB (PSO based) No-spin Spin8.5, 8 5 × − < − < − χ eff is much moreaccurately measured in a PSO based search. Theerrors on parameters estimated are consistent with thestatistical errors predicted in [12]. If we consider thefraction of events from the 20k injections that arerecovered with an SNR greater than 10, we see PSObased search has a higher fraction. PSO outperformstemplate bank search by recovering approximately 28 %more events with SNR greater than 10. For an SNRthreshold of 9, the PSO based coincident search recoversapproximately 4.5% more events than the templatebank search. Higher recovered SNRs in the detectionprocess also help reduce the error in sky-localizationusing BAYESTAR [43]. V. COHERENT SEARCH
In the coincident search, described in the previoussection, the constraint on data to discriminate fromastrophysical signal was that of time of arrival differencebetween corresponding detectors. However, by virtueof its origin, gravitational waves are emitted coherently.Thus, GWs have an additional property of being coherentacross each detector. If we put this constraint that thesignals are coherently emitted from a source at location( θ, φ ). Thus, if the data has any signal of astrophysicalorigin it too would be coherent across multiple detectorswhich noise would not. By choosing a source at location( θ, φ ), one can find the time delay between the detectorsas GWs travel at the speed of light. If the source locationis known then the search is a targeted coherent search. Ifthe source location is not known then the search is blindcoherent search. In blind coherent search, the sourcelocation is a variable. By choosing different locationsacross the sky as a parameter along with other intrinsicparameters of CBCs, the coherent matched functioncan be maximized using PSO. We use the dominantpolarization basis to get the coherent wave statisticsin our search using PSO. For a detailed discussion oncoherent search refer to [19, 27]. We will present a briefsummary of coherent statistics.Using the same definitions of variables and functionsdefined in section III. The GW signal can be brokendown into two polarizations h + and h × . Each of thepolarization can be expressed in phase and amplitudeterms, dependent on the response of the detector[27]. For a given template parameter the gravitationalwaveform in the I th detector is given by h I ( t ) = (cid:88) µ =1 A µ h Iµ ( t ) . (8)For multi-detector in the dominant polarization basis, the coherent SNR is defined as ρ coh = ( s | F + h ) + ( s | F + h π ) ( F + h | F + h ) + ( s | F × h ) + ( s | F × h π ) ( F × h | F × h ) . (9)In our work, we maximize the coherent SNR of CBCsignals in a network of detectors, with and withoutspinning components. The coherent search filtersout the signal buried in noise by maximizing it overthe phase as astrophysical signals are coherent acrossmultiple detectors. The coalescing binaries with spinningindividual components causes the modulation in theprofile of the overall phase of the GW waveform [15, 20].We divide the injection into three classes – no-spinBBH injections, aligned low-spin BBH injections andaligned high-spin BBH injections. The parameters ofeach set are consistent with the definition in section § III B. On each of the injection sets, we use PSO tomaximize the coherent SNR over the signal parameters.We set up a coherent search in a four-dimensionalparameter space – component masses and source location( m , m , θ, φ ), referred to as non-spinning coherentsearch. We will extend the parameter space toincorporate aligned-spins extending the parameter spaceto six ( m , m , s z , s z , θ, φ ), referred to as spinningcoherent search. A. Effect of Spinning Injections on Coherent search
Coherent search maximizes over the phase overlapof the signal with the templates. The individualspin components modulate the phase of the GWwaveforms, giving rise to a degeneracy between thesource sky-position and component spins [15, 20], whichaffects the performance of any search, but especially thecoherent search. We aim to study the effect of componentspins of CBCs on the coherent search. To study the abovewe take three injection sets – non-spinning (componentmasses have no spin), low-spinning and high-spinningBBH injections, each set are defined by the same setof parameters described in section § III B. We maximizethe coherent SNR for each of the injections in differentsets in a network of two detectors. The search is setup in a 4 dimensional parameter space (without spin m , m , θ, φ ). We use 8 swarms with 300 particleseach, totaling to 576k match filtering operations (300 × × ×
6) during the search. The results of thesearch are summarized in Fig. 9. From the plot, itis evident that spinning binaries negatively impact theperformance of the coherent search. We see the numberof injections above the threshold (corresponding to thesame FAP incoherent search) are higher when injectionsare non-spinning or have aligned low-spins compared toaligned high-spins injections as summarized in table II.The errors in estimated parameters from the coherentsearch are also lower for aligned low-spin injections ascompared to aligned high-spin injections, evident from6
10 20 30 40 5010 N u m b e r o f E v e n t s SNR distribution
No-Spin-BBHLS-BBHHS-BBH R e l a t i v e e rr o r ( % ) Coherent SNR: 9.5 m m m m m m M Chirp M Chirp M Chirp R e l a t i v e e rr o r ( % ) Coherent SNR: 12
FIG. 9. The plot summarizes the result of a non-spinning coherent search over three different injection set parameters –Non-spinning BBH injections (blue), low-spin BBH injections (yellow) and high-spin BBH injections (pink). We find thatcoherent search is significantly affected if the injection parameters have spin. The parameters of PSO search are same for thesearch over each injection set. Amongst different injection sets, the component masses are the same but the spins of eachcomponent mass vary to satisfy the parameters of each class of injection set. We find the coherent search has a low error andhigher detection efficiency when the injections have no spin. As the CBC spin increases the detection efficiency and estimatesof CBC parameters drops. This spin-induced discrepancy is persistent with higher SNR thresholds (corresponding coherentSNR thresholds are labeled above each subplot). R e l a t i v e e rr o r ( % ) Low Spin BBH - Coincident SNR: 10, Coherent SNR: 9.5, FAP <10 M Chirp M Chirp M Chirp M T M T M T R e l a t i v e e rr o r ( % ) High Spin BBH - Coincident SNR: 10, Coherent SNR: 9.5, FAP <10 FIG. 10. We compare the performance of the spinningcoherent search with the template bank to standardizethe comparison of the results. The coherent search iscomputationally more expensive than the template banksearch. However, the advantage that the former offers are theability to recover a higher number of events as summarizedin table I. The plot shows for a network of two detectors HL,the error in the recovered injections (above the correspondingFAP thresholds). The two plots show the distribution forlow-spin (top) and high-spin (bottom) BBH injections. Wesee for low-spin BBH injections the non-spinning coherent(yellow) and spinning coherent (pink) searches performreasonably well in estimation the parameters. The errorsover recovered injections are lesser when compared to theO2-template bank (green). However, for high-spin BBHinjections, we see that coherent search under-performs inestimating the parameters in comparison to template bank.However, the number of injections that are recovered usingcoherent statistics for high-spin searches are higher than thetemplate bank. figure 9. One possible way to resolve this issue wouldbe to increase the number of particles and swarms used,but we don’t do so as the computational cost wouldincrease making it less likely for the coherent search tobe developed as an online search tool.
B. Extending the Parameter Space of CoherentSearch: Aligned-Spin Coherent Search
We try to resolve the problems with high-spinninginjections that coherent search runs into by extendingthe search to an aligned-spin search – six-dimensionalspace ( m , m , s z , s z , θ, φ ). We aim for detection ofCBCs using coherent search, thus, we do not increase thenumber of points or swarms to keep the computationalcost almost the same. The difference in cost of coherentsearch with aligned-spin and without spin is due to thegeneration of corresponding GW waveform templates.We use the template bank based PSO search to compare R e l a t i v e e rr o r ( % ) Low Spin BBH: 9.5 (HL), 11.7 (HLV), 13.4 (HLVK), FAP <10 M Chirp M Chirp M Chirp M T M T M T R e l a t i v e e rr o r ( % ) High Spin BBH: 9.5 (HL), 11.7 (HLV), 13.4 (HLVK), FAP <10 FIG. 11. The plot summarizes the performance of PSO basedspinning coherent searches for HL (green), HLV (yellow)and HLVK (pink) network. The top subplot shows theerror distribution for low-spin BBH injections while the lowerpanel shows the distribution for high-spin BBH injections.The trend that coherent search performs well for CBCinjections with low inherent spin as compared to high-spinCBC injections is consistent for a higher number of detectorsin the network. The errors distribution spreads for a highernumber of detectors because there are a higher number ofevents above the threshold. the performance of the coherent search. The results ofdifferent searches are summarized in Fig. 10. From thebackground estimation plots in section IV D we observethat the background of spinning and non-spinningcoherent search is almost identical. Putting the sameSNR threshold on 10k BBH injections of low and highspinning system, we observe that the error estimatesare similar for the two searches. However, out of the20k BBH injections in total, the aligned-spin coherentsearch recovers more signals as compared to aligned-spincoincident search, as summarized in the table II.
C. Network of detectors
We can extend PSO based search to higher number ofdetectors in the network. We aim to recover injectionswith HLV and HLVK network to do consistency checksfor coincident and spinning coherent searches. We seeby increasing the number of detectors the number ofevents recovered increase, as summarized in table II.From the table, it is evident that coherent search recoversa higher number of events than coincident search andthe trend is consistent with having multiple detectorsin the network. The figures 11 and 12 summarize thedistribution of errors in parameters of CBC for all theinjections that pass the corresponding thresholds basedon search and the detector network. The error estimate8 m m m m m m M Chirp M Chirp M Chirp η η η R e l a t i v e e rr o r ( % ) Coincident SNR threshold: 10 (HL), 12.25 (HLV), 14.25 (HLVK) X eff Error N u m b e r o f E v e n t s HLVK: . HLV: . HL: . FIG. 12. Extending the number of detectors in the network and performing PSO based coincident search over the 20k BBHinjections. The thresholds for the network are scaled by the square root of the number of detectors in the network. All possiblecombinations of detectors are considered in the search. That is, for HLVK detector network we consider all possible combinationof two detector and three detector events that cross the corresponding thresholds. The plot above shows the error in differentparameters of the CBC in HL (blue), HLV(Yellow1) and HLVK (Pink). The fraction of events recovered is indicated in thelegend of the plot and in table II. The results obtained are as per the expectations with the increase in the number of detectorsthe fraction of recovered events increases. The distribution of error is comparable as the majority of events in higher detectornetworks arise from two detectors in that network. is almost similar for a higher number of detectors as amajority of events recovered in HLV and HLVK arisefrom two detectors in that network.
VI. RESULTS
We use PSO to set up an aligned-spin coincidentsearch, which uses 48k MFOs per detector and isan effective algorithm to search for CBCs. For anetwork of two detectors HL, we compare an aligned-spinPSO search with a template-bank search using the O2-template bank of Canton et al. [21]. TheO2-template bank uses approximately 400k MFOs perdetector compared to 48k MFOs used by aligned-spinPSO-based coincident search. As evident from the figure8 we see that at a lower computational cost, PSO recoversapproximately 28 % more events as compared to templatebank above the SNR of 10 and at the same time, theestimation of parameters is also better with lower error ininjected parameters in the detection step as compared tothe template bank. The improvement in the SNR (higherSNR achieved with PSO algorithm) in the detectionstage helps BAYESTAR, a Bayesian algorithm for rapid9
Network Coincident Search Coincident Search Spinning Coherent Spinning CoherentTemplate Bank PSO-based Search Low, HighHL (FAP < − ) 21.0 (10) % 26.9 % (10) 30.4 % (9.5) 32.0 %, 28.7 %HLV (FAP < − ) - 38.9 % (12.25) 40.8 % (11.7) 43.4 %, 38.1 %HLVK (FAP < − ) - 50.6 % (14.25) 52.4 % (13.4) 55.1 %, 49.7 %TABLE II. The above table summarizes the fraction of events (of the 20k BBH injections) that passed the thresholds a whichcorrespond to the same false alarm probability from background triggers in gaussian noise. The thresholds for a two detectornetwork are obtained from Fig. 7. For higher detectors in the network, the threshold is rescaled by the square root of thenumber of detectors in the network. We find that coherent search outperforms coincident search in terms of the number ofevents recovered and this trend is consistent with a higher number of detectors in the network. The last column shows thediscrepancy in incoherent search. We see that coherent search under-performs for higher-spin injections compared to low-spininjections. However, overall coherent search seems to outperform coincident search. a indicated within brackets localization, to localize the source in a smaller region inthe sky. The localization capabilities of BAYESTAR isdependent on the recovered SNR in the detection stage.Chi-squared discriminator [13, 24] for glitches can beeasily incorporated in the PSO based searches.Next, we extend the parameter space of the searchto incorporate precession. The dimensionality of thesearch space now includes the component masses, theorbital inclination and the component spin vectors ofthe two objects. To help PSO cope with the higherdimensionality of this extended parameter space, weincrease the number of particles and the number ofswarms used by the PSO algorithm. Using 10 swarmswith 800 particles each we set up a precessing coincidentsearch, which uses a total number of 320k MFOsper detector. The precessing PSO search recoversapproximately 18 % more injections as compared to analigned-spin PSO search with the same PSO parameters.Another striking feature of the precessing search is theaccuracy in the estimation of the χ eff , among otherparameters. Our work provides a scheme for backgroundestimation in PSO-based searches, whether they becoincident or coherent, which can be readily applied toreal data.In our work, we also use PSO to implement anall-sky blind coherent search. We study the effect ofintrinsic spins of component masses on the performanceof coherent search and find that high-spin systemsaffect the performance of coherent search negatively, ascompared to the low-spin system. On comparing the twosubgroups of injections – aligned high-spin and alignedlow-spin – we find aligned spin coherent search recovers,for the same FAP, a higher number of events from thelow-spin injection set. However, we find that in eachof the two-subgroups – high-spin and low-spin CBCs– the coherent search outperforms coincident searchby recovering a higher number of injections above thethreshold corresponding to the same FAP in each search(coincident/coherent). This trend is consistent with ahigher number of detectors as summarized in the table II.In conclusion, the effectiveness of PSO in a blind all-sky coherent search was demonstrated here using 576k MFOsper detector, and with parallelization techniques for thePSO algorithm [41, 46], a low-latency coherent searchpipeline using PSO can be developed as well in the future. VII. ACKNOWLEDGEMENTS
Special thanks are due to Bhooshan Gadre for hisinputs on the manuscript and helpful discussions. Wealso thank Anuradha Samajdar and Steven Reyes forhelpful discussions, and Jayanti Prasad for technicalsupport. The work is partially funded by the NSFAward-1352511 (P.I. Stefan Ballmer) and the NavajbaiRatan Tata Trust. We thank Derek Davis forcarefully reading the manuscript. We acknowledgethe use of IUCAA LDG cluster Sarathi for thecomputational/numerical work.0 [1] Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016,Classical and Quantum Gravity, 33, 134001[2] —. 2016, Phys. Rev. Lett., 116, 241103[3] —. 2016, Physical Review Letters, 116, 1[4] —. 2016, Phys. Rev. Lett., 116, 221101[5] —. 2017, Phys. Rev. Lett., 118, 221101[6] —. 2017, The Astrophysical Journal Letters, 851, L35[7] —. 2017, Phys. Rev. Lett., 119, 141101[8] —. 2017, Phys. Rev. Lett., 119, 161101[9] —. 2017, The Astrophysical Journal Letters, 848, L12[10] —. 2018, Classical and Quantum Gravity, 35, 65010[11] —. 2018, arXiv preprint arXiv:1805.11581[12] Ajith, P., & Bose, S. 2009, Physical Review D, 79, 84032[13] Allen, B. 2005, Phys. Rev. D, 71, 62001[14] Apostolatos, T. A., Cutler, C., Sussman, G. J., &Thorne, K. S. 1994, Phys. Rev. D, 49, 6274[15] Arun, K. G., Buonanno, A., Faye, G., & Ochsner, E.2009, Phys. Rev. D, 79, 104023[16] Berti, E., Barausse, E., Cardoso, V., et al. 2015, Classicaland Quantum Gravity, 32, 243001[17] Boh´e, A., Shao, L., Taracchini, A., et al. 2017, Phys. Rev.D, 95, 044028[18] Bose, S., Chakravarti, K., Rezzolla, L., Sathyaprakash,B. S., & Takami, K. 2018, Phys. Rev. Lett., 120, 031102[19] Bose, S., Dayanga, T., Ghosh, S., & Talukder, D. 2011,arXiv, 134009, 12[20] Buonanno, A., Faye, G., & Hinderer, T. 2013, Phys. Rev.D, 87, 44009[21] Canton, T. D., & Harry, I. W. 2017, arXiv preprintarXiv:1705.01845[22] Damour, T., Nagar, A., & Villain, L. 2012, Phys. Rev.D, 85, 123007[23] De, S., Finstad, D., Lattimer, J. M., et al. 2018, arXivpreprint arXiv:1804.08583[24] Dhurandhar, S., Gupta, A., Gadre, B., & Bose, S. 2017,Phys. Rev., D96, 103018[25] Gossan, S., Veitch, J., & Sathyaprakash, B. S. 2012,Physical Review D, 85, 124056[26] Hannam, M., Schmidt, P., Boh´e, A., et al. 2014, Phys.Rev. Lett., 113, 151101[27] Harry, I. W., & Fairhurst, S. 2011, Phys. Rev. D, 83,84002[28] Harry, I. W., Nitz, A. H., Brown, D. A., et al. 2014, Phys.Rev. D, 89, 24010 [29] Kennedy, J., & Eberhart, R. 1995, Proceedings ofICNN’95 - International Conference on Neural Networks,1942[30] Khan, S., Husa, S., Hannam, M., et al. 2016, Phys. Rev.D, 93, 044007[31] Macleod, D., Harry, I. W., & Fairhurst, S. 2016, Phys.Rev. D, 93, 064004[32] Margalit, B., & Metzger, B. D. 2017, The AstrophysicalJournal Letters, 850, L19[33] Messick, C., Blackburn, K., Brady, P., et al. 2017, Phys.Rev. D, 95, 042001[34] Metzger, B. D. 2016, 1[35] Nicholl, M., Berger, E., Kasen, D., et al. 2017, TheAstrophysical Journal Letters, 848, L18[36] Nitz, A., Harry, I., Biwer, C. M., et al. 2017,ligo-cbc/pycbc: Test Pre-Release to Validate E@H TravisBuild, , , doi:10.5281/zenodo.376277[37] Nitz, A. H., Dal Canton, T., Davis, D., & Reyes, S. 2018,Phys. Rev. D, 98, 024050[38] Owen, B. J. 1996, Physical Review D, 53, 6749[39] Owen, B. J., & Sathyaprakash, B. S. 1999, PhysicalReview D, 60, 22002[40] Schmidt, P., Ohme, F., & Hannam, M. 2015, Phys. Rev.D, 91, 024043[41] Schutte, J. F., Reinbolt, J. A., Fregly, B. J., Haftka,R. T., & George, A. D. 2004, International journal fornumerical methods in engineering, 61, 2296[42] Shoemaker, D. 2009, Advanced LIGO anticipatedsensitivity curves, Technical notes LIGO-T0900288-v2,LIGO, note: the high-power detuned model used in thispaper is given in the data file