The balancing act of template bank construction: inspiral waveform template banks for gravitational-wave detectors and optimizations at fixed computational cost
TThe Balancing Act Of Template Bank Construction:
Inspiral Waveform Template Banks For Gravitational-Wave DetectorsAnd Optimizations At Fixed Computational Cost
Drew Keppel
1, 2, ∗ Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universit¨at Hannover, D-30167 Hannover, Germany
Gravitational-wave searches for signals from inspiralling compact binaries have relied on matchedfiltering banks of waveforms (called template banks ) to try to extract the signal waveforms fromthe detector data. These template banks have been constructed using four main considerations, theregion of parameter space of interest, the sensitivity of the detector, the matched filtering bandwidth,and the sensitivity one is willing to lose due to the granularity of template placement, the latterof which is governed by the minimal match. In this work we describe how the choice of the lowerfrequency cutoff, the lower end of the matched filter frequency band, can be optimized for detection.We also show how the minimal match can be optimally chosen in the case of limited computationalresources. These techniques are applied to searches for binary neutron star signals that have beenpreviously performed when analyzing Initial LIGO and Virgo data and will be performed analyzingAdvanced LIGO and Advanced Virgo data using the expected detector sensitivity. By followingthe algorithms put forward here, the volume sensitivity of these searches is predicted to improvewithout increasing the computational cost of performing the search.
I. INTRODUCTION
For the past decade, large scale interferometricgravitational-wave (GW) detectors have operated, allow-ing searches for signals from inspiralling compact bina-ries to be performed [1–18]. These searches have thusfar detected no GW signals, however once the detectorsare upgraded to their advanced configurations, multipleevents are expected to be detected each year [19].Searches for inspiral signals in detector data dependon matched filtering the data with template waveformsto produce signal-to-noise ratio (SNR) time-series, themaxima of which are used to produce GW “triggers”. Im-portant criteria in constructing banks of template wave-forms (i.e., template banks) for these searches are theregion of parameter space to be searched, the sensitivityof the detector, the lower and upper frequency cutoffsassociated with matched filtering the data, and the max-imum fractional loss of SNR (the complement of whichis more commonly know as the minimal match) that oneis willing to tolerate due to granularity of the templateplacement. Of these criteria, one is free to tune the lowerfrequency cutoff and the minimal match due to sensitiv-ity and computational cost considerations.In [20], the authors discuss the issue of balancing com-putational cost versus SNR gain while decreasing thelower frequency cutoff. However, they do not venture sofar as to derive the optimal choices. Instead, they chooseto set the lower frequency cutoff at a level such that onewould lose less than 1% of the SNR by the cutoff beingdifferent from 0. In addition, they choose the minimalmatch of the template bank to be
M M = 95%; largeenough that the metric estimate of the fractional SNR ∗ [email protected] loss is still valid but small enough for computational costconsiderations. Recent searches for GW from inspirallingcompact binaries have chosen a larger value for the min-imal match, M M = 97%, so that less than 10% of thesignals at the worst mismatch locations of the templatebank would be lost.In this paper, we further investigate the effects of dif-ferent lower frequency cutoff and minimal match choices.In Sec. III we look at how decreasing the lower frequencycutoff both increases the amount of raw SNR one is ableto extract from a signal and increases the trials factorby increasing the number of templates required to searchfor the waveforms. Sec. IV goes on to describe how tochoose the optimal combination of lower frequency cutoffand minimal match for a fixed computational cost. Ex-amples of both these choices are given in Sec. V wherethe methods are applied to previous and future searchesof GW detector data.
II. PRELIMINARIES
In searching for signals from inspiralling compact ob-jects in GW data, a commonly used event identificationalgorithm relies on matched filtering, where the data is“whitened” and filtered with the template waveform be-ing searched for. Specifically, the square SNR is givenby ρ = ( s | h c ) + ( s | h s ) σ , (1)where s is the data from a detector that may containa GW signal of unknown strength, h c and h s are thetarget waveforms associated with the same source anddiffer in phase by π/ σ := ( h c | h c ) is the sensitivityof our detector to a waveform at a reference distance, a r X i v : . [ phy s i c s . d a t a - a n ] M a r typically chosen to be 1 Mpc, and the inner product ( x | y )is defined as ( x | y ) := 4 (cid:60) (cid:90) f high f low ˜ x ˜ y ∗ S n ( f ) df. (2)Here ˜ x is the Fourier transform of x , () ∗ denotes thecomplex conjugate operator, and S n ( f ) is the one-sidedpower spectral density (PSD) of the detector’s noise.As can be seen from (2), the SNR recovered when thereis a signal present in the data will depend on the limits ofthe integration. The upper frequency cutoff f high is setby the lower of either the Nyquist frequency of the dataor the maximum frequency of the template waveform. Incontrast, the lower frequency cutoff f low is a parameterthat can be tuned in optimizing the search algorithm.To search a region of parameter space, many templatewaveforms from points spread throughout the region needto be matched filtered. The locations of these points arechosen by constructing a metric on the parameter space g ij [21–24]. This metric describes the distance betweenpoints based on the fractional loss of SNR associated withmatched filtering a signal waveform from one point inparameter space with a template waveform from anotherpoint. To second order in the parameter differences ∆ λ i ,the fractional loss of SNR, or mismatch m , is given by m = 12 g ij ∆ λ i ∆ λ j , (3)where the metric is given by projecting out dimensionsof the parameter space from normalized Fisher matrix, g µν := ( ∂ µ h | ∂ ν h )( h | h ) , (4)that are associated with extrinsic parameters, which canbe maximized either analytically or efficiently. Here ∂ µ isthe partial derivative with respect to parameter λ µ . Thedensity of templates is then governed by the maximumamount of mismatch one is willing to tolerate, or thecomplement of this, referred to as the minimal match M M = 1 − m . III. SIGNAL POWER VERSUS TRIALSFACTOR: OPTIMIZING THE LOWERFREQUENCY CUTOFF FOR MAXIMUMSENSITIVITY
The goal of designing a search is to maximize the vol-ume at which we are sensitive to signals for a fixed falsealarm probability (FAP). The first parameter we tunewith this in mind is the lower frequency cutoff. We startwith the distance out to which we can see an inspiralsignal with a fixed SNR ρ , D = σρ . (5)Changing the lower frequency cutoff changes the powerof the signal that we could possibly recover. If one were to recover a signal with the same SNR, the distance towhich one could see a signal would vary when the lowerfrequency cutoff was changed from f ref to f low , D ( f low ) D ( f ref ) = σ ( f low ) σ ( f ref ) . (6)Let us now look at how the observable distance of asignal is affected when the signal is recovered with a mis-matched template. The observed SNR ρ will be reducedfrom the SNR obtained by a template that matches thesignal ρ ref by ρ = ρ ref (1 − m ) , (7)where m is the mismatch between the template that re-covers the signal and the actual signal. Eq. 5 implies thatthe distance to which such a signal will be observable isreduced by the same factor D ( f ref , m ) D ( f ref ,
0) = (1 − m ) . (8)So far we have focused on the obserable distance of asignal at fixed SNR. However it is actually the obserabledistance of a signal at fixed FAP that we are interestedin. The FAP associated with a single observation of SNR ρ is given by FAP ∝ exp[ − ρ ] . (9)The recovered FAP is subject to a trials factor related tothe number of independent trials N we use in looking fora signal, FAP (cid:48) = 1 − (1 − FAP) N ≈ N FAP . (10)We can translate a single observation of ρ observed among N independent trials to a reference SNR ρ ref among adifferent number of trials N ref at the same FAP by com-bining (9) and (10). ρ = ρ + ln NN ref . (11)When searching a non-zero measure region of param-eter space, additional trials are accrued proportional tothe volume of the parameter space. The volume of pa-rameter space is in turn given by the number of templatesneeded to cover the parameter space M templates [25], N trials ∝ (cid:90) (cid:112) | g | dλ d = M templates m d/ θ (12)where (cid:112) | g | is the square root of the determinant of themetric on the space, θ is a geometrical quantity asso-ciated with how the template bank tiles the parameterspace, m = 1 − M M is maximum mismatch allowed inthe template bank covering the parameter space, and d is the dimensionality of the parameter space being tiled(i.e., two for templates associated with waveforms fromnon-spinning objects that are laid out in the two dimen-sional mass space).Since the metric (4) is defined in terms of the innerproducts from (2), the full metric itself is a function ofthe lower frequency cutoff, which in turn implies that themetric density of the mass subspace is also a function of f low , (cid:112) | g | = (cid:112) | g ( f low ) | . (13)The total volume we can observe is proportional tothe cube of the distance, thus the ratio of the volumewe can observe for a mismatched signal at a given valueof f low to the volume we could observe a matched signalwith a reference lower frequency cutoff f ref is found bycombining (6), (8), (11), (12), and (13), V ( f low , m ) V ( f ref ,
0) = σ ( f low ) σ ( f ref ) × (1 − m ) (cid:18) ρ ( f ref ) ln (cid:20) (cid:82) √ | g ( f low ) | dλ d (cid:82) √ | g ( f ref ) | dλ d (cid:21)(cid:19) / . (14)We call this the relative volume. For two-dimensionaltemplate banks, a hexagonal covering of templates fol-lowing the A ∗ lattice will result in a distribution of mis-matches that is essentially flat between 0 and the maxi-mum mismatch [25]. Using this fact, the average relativevolume is found to be (cid:104) V ( f low , m ) (cid:105)(cid:104) V ( f ref , (cid:105) = σ ( f low ) σ ( f ref ) × (cid:68) (1 − m ) (cid:69)(cid:18) ρ ( f ref ) ln (cid:20) (cid:82) √ | g ( f low ) | dλ d (cid:82) √ | g ( f ref ) | dλ d (cid:21)(cid:19) / , (15)where the average of the mismatch term in the numeratoris given by (cid:68) (1 − m ) (cid:69) = 1 − m + m − m . (16)The average relative volume can be maximized with theproper choice of f low for a fixed value of the templatebank maximum mismatch. IV. WIDER OR DENSER?: MAXIMIZINGSENSITIVITY AT FIXED COMPUTATIONALCOST
In the face of limited computational resources, we mustconsider not only how to maximize the sensitivity of asearch through the choice of the lower frequency cutoff,but we must also ensure that our choices of the lowerfrequency cutoff and the minimal match satisfy the con-straint on the total computational cost C total . This con-straint can be viewed as a combination of two effects: the computational cost of filtering the data with a single tem-plate waveform C filter multiplied by the computationalcost associated with N templates such filters C total ( f low ) = N templates ( f low ) C filter ( f low )= C filter ( f low ) θm − d/ (cid:90) (cid:112) | g ( f low ) | dλ d . (17)Using this constraint, we seek to maximize the con-strained average relative volume (cid:104) V ( f low ) (cid:105)(cid:104) V ( f ref ) (cid:105) = σ ( f low ) σ ( f ref ) (cid:68) (1 − m ( f low )) (cid:69)(cid:68) (1 − m ( f ref )) (cid:69) × (cid:18) ρ ( f ref ) ln (cid:20) (cid:82) √ | g ( f low ) | dλ d (cid:82) √ | g ( f ref ) | dλ d (cid:21)(cid:19) / , (18)with the proper choice of f low and m ( f low ).Assuming one is able to computationally preform thesearch for a given combination of lower frequency cutoff f ref and maximum mismatch m ( f ref ), the maximum mis-match at any other choice of lower frequency cutoff f low satisfying the constraint on the computational cost canbe solved for easily, m ( f low ) = m ( f ref ) (cid:32) C filter ( f low ) (cid:82) (cid:112) | g ( f low ) | dλ d C filter ( f ref ) (cid:82) (cid:112) | g ( f ref ) | dλ d (cid:33) /d (19)The computational cost of filtering data with a singletemplate will depend intrinsically on the implementationof a search. As a first example, it could be independentof the choice of f low , as is the case in the FINDCHIRP algorithm [26] where data is processed with fast Fouriertransforms using fixed length chunks.In a different algorithm where data is analyzed in thetime domain using finite impulse response (FIR) filters,the computational cost would be set by the number oftaps in the FIR filter. This is proportional to the lengthof the waveform T , given to Newtonian order by T ( f low ) = 5256 M / (cid:104) (4 πf low ) − / − (4 πf high ) − / (cid:105) , (20)where M = ( m m ) / ( m + m ) / is the chirp mass ofthe binary system.Alternatively, if one were able to change the samplingrate associated with the template filter continuously, onecould reduce the computational cost by filtering the datawith a changing local sampling rate such that the fre-quency of the signal at any time was always equal to thelocal Nyquist frequency of the filter. In this approach, thecomputational cost would be proportional to the numberof cycles in the signal waveform, N cycles ( f low ) = 164 π / M / (cid:16) f − / − f − / (cid:17) . (21)Finally, as an application of this method to a pipelineproposed to search for binary neutron star (BNS) signalswith low latency in the Advanced LIGO (aLIGO) sen-sitive band, we consider the computational cost of the LLOID algorithm [27]. The
LLOID algorithm partitionsthe waveforms into S time-slices and filters the wave-form portions of slice s at a power of two sampling rate f s such that the Nyquist frequency of the slice is justgreater than the largest frequency of any of the portionsof the waveform in that slice. In addition, for each slice,the LLOID algorithm decomposes the N templates templatewaveform portions into L s bases basis vectors using singularvalue decomposition (SVD) [28]. These basis vectors areused as FIR filters of N s taps taps for slice s . The compu-tational cost of filtering a bank of waveforms with thisalgorithm is dominated by the filtering costs of the basisvectors and the reconstruction costs of turning the basisfilter outputs into outputs of template filters, N FLOPS = 2 S − (cid:88) s =0 f s L s bases ( N s taps + N templates ) . (22)Let us look at how the different pieces of LLOID ’s com-putational cost will change with varying f low . For a par-ticular slice, as f low is reduced, the template waveformsthat go into the SVD matrix will more densely coverthe region of parameter space, resulting in a larger num-ber of waveforms that will need to be reconstructed (i.e., N templates will increase). However, the number of bases N s bases needed reconstruct the template waveforms to aspecific accuracy is invariant for the minimal matcheswe are interested in [29]. Finally, the number of sliceskept will depend on f low as each slice covers a differentfrequency range of the waveforms. Thus, the total com-putational cost of the LLOID algorithm can be writtenas N FLOPS = A ( f low ) N templates + B ( f low ) , (23)where A ( f low ) and B s ( f low ) are defined appropriatelywith respect to (22). For this algorithm, (19) takes adifferent form, m ( f low ) = m ( f ref ) A ( f low ) N ref (cid:82) √ | g ( f low ) | dλ d (cid:82) √ | g ( f ref ) | dλ d A ( f ref ) N ref + B ( f ref ) − B ( f low ) /d , (24)where N ref := N templates ( f low ). V. EXAMPLES
In this section we apply the methods from Secs. IIIand IV to the (expected) sensitivities of several pastand future detectors. In particular, we investigate theLIGO and Virgo PSDs from S5/VSR1 [30], S6/VSR2-3 [31, 32], and the expected advanced detector PSDs for f (Hz)10 − − − − − − − p S n ( f )( / √ H z ) S5S6aLIGO (a) H1 PSDs10 f (Hz)10 − − − − − − − p S n ( f )( / √ H z ) VSR1VSR2-3aVirgo (b) V1 PSDs10 f (Hz)10 − − − − − − − p S n ( f )( / √ H z ) S5/VSR1S6/VSR2-3aLIGO/aVirgo (c) Harmonic Sum PSDs
FIG. 1: (a) shows different PSDs associated withdifferent eras of the H1 LIGO detector. (b) showsdifferent PSDs associated with different eras of theVirgo detector. (c) shows different PSDs associatedwith detector networks from different eras. TheH1H2L1V1 network PSD associated with the S5/VSR1era is given by the harmonic sum of individualdetectors’ PSDs. For the S6/VSR2-3 and aLIGO/AdVeras, an H1L1V1 network is used.aLIGO [33] and AdV [34]. We also consider joint de-tector analyses where the individual detectors PSDs arecombined by taking the harmonic sum, which yields thesame combined SNR as either the coherent network SNRor the sum-of-squares SNR associated with a coincidentsearch [35, 36]. These PSDs can be seen in Fig. 1. Theparameter space we focus on for these comparisons isthat associated with searches for BNS signals from non-spinning objects. Using the stationary phase approxima-tion, we expand the template waveforms to Newtonianorder in the amplitude and 3.5 post-Newtonian (PN) or-der in the phase. The metric for these waveforms is givenin [24]. With this focus, we approximate the ratio of theintegrated metric density by a point estimate such thatthe mass of each object is 1.4 M (cid:12) , (cid:82) (cid:112) | g ( f low ) | dλ d (cid:82) (cid:112) | g ( f ref ) | dλ d ≈ (cid:115) | g BNS ( f low ) || g BNS ( f ref ) | . (25)It should also be noted that, in this approximation, weassume that the effects from the bulk of parameter spacedominate over effects from the boundaries. For param-eter spaces where the effects of the boundaries are non-negligible, more care will be needed in computing theratio of the integrated metric densities and how they re-late to the trials factor and computational cost. A. Choice of f low First we optimize the choice of the lower frequencycutoff of an inspiral search for different detectors with-out regard to the computational cost. Table I summa-rizes the results for all of the detector combinations men-tioned, compared to the “standard” choice of the lowerfrequency cutoff. For the most part, this is a very smalleffect, as can be anticipated through the logarithmic de-pendence of the effect of the trials factor in (15). Thelargest differences between the standard choice and theoptimal choice occur for the Virgo detector during VSR1,which increases the sensitivity of the search by 15%. Thisseems to be attributed to a rapid decrease in the recov-erable SNR that is seen between about 55Hz and 60Hz.Table II makes a similar comparison, although here thestandard lower frequency cutoff choice is replaced by theminimum reported frequency associated with a particu-lar PSD. It is particularly interesting to see the trialsfactor effect associated with the Virgo detector duringVSR1. In that case, the difference between the minimumchoice of 10Hz and the optimal choice of 38.1Hz is a fewparts in 10 . What is interesting about this comparison isthe large difference in the lower frequency cutoff choices.As Virgo detector’s PSD from VSR1 had a very shallowslope at the low frequency end, it provides a good exam-ple of how the effect of the trials factor can grow morequickly than the SNR gain as the lower frequency cutoffis lowered. More detailed sensitivity comparisons can befound in Figs. 2-4, which separately show the effect of
30 35 40 45 50 55 60 f low . . . . . V V total V σ V trials (a) H1 S5 PSD10 20 30 40 50 60 f low . . . . . V V total V σ V trials (b) V1 VSR1 PSD10 20 30 40 50 60 f low . . . . . . . . . V V total V σ V trials (c) H1H2L1V1 S5/VSR1 Harmonic Sum PSD FIG. 2: (a) shows the average relative volume V total as afunction of lower frequency cutoff for the H1 LIGOdetector during the S5 era. (b) and (c) show the samefor the Virgo detector and H1H2L1V1 detector networkfor the VSR1 and S5/VSR1 eras, respectively. Eachpanel also contains traces for the contributions to theaverage relative volume from the recoverable SNR V σ and the trials factor V trials .
40 45 50 55 60 65 70 f low . . . . . . . V V total V σ V trials (a) H1 S6 PSD10 20 30 40 50 60 70 80 f low . . . . . . . V V total V σ V trials (b) V1 VSR2-3 PSD10 20 30 40 50 60 70 80 f low . . . . . . . . V V total V σ V trials (c) H1L1V1 S6/VSR2-3 Harmonic Sum PSD FIG. 3: (a) shows the average relative volume as afunction of lower frequency cutoff for the H1 LIGOdetector during the S6 era. (b) and (c) show the samefor the Virgo detector and H1L1V1 detector network forthe VSR2-3 and S6/VSR2-3 eras, respectively. Eachpanel also contains traces for the contributions to theaverage relative volume from the recoverable SNR V σ and the trials factor V trials . f low . . . . . . . . . . V V total V σ V trials (a) H1 aLIGO PSD10 15 20 25 30 35 40 f low . . . V V total V σ V trials (b) V1 AdV PSD5 10 15 20 25 30 35 40 f low . . . . . . . . . V V total V σ V trials (c) H1L1V1 aLIGO/AdV Harmonic Sum PSD FIG. 4: (a) shows the average relative volume as afunction of lower frequency cutoff for the proposed H1LIGO detector during the aLIGO era. (b) and (c) showthe same for the proposed Virgo detector and H1L1V1detector network for the AdV and aLIGO/AdV eras,respectively. Each panel also contains traces for thecontributions to the average relative volume from therecoverable SNR V σ and the trials factor V trials . Era Detector f standardlow f optimallow Volume GainS5 H1 40Hz 37.3Hz 6 . × − VSR1 V1 60Hz 38.1Hz 1 . × − S5/VSR1 H1H2L1V1 40Hz 37.8Hz 4 . × − S6 H1 40Hz 43.7Hz 2 . × − VSR2-3 V1 50Hz 16.8Hz 1 . × − S6/VSR2-3 H1L1V1 40Hz 34.0Hz 7 . × − aLIGO H1 10Hz 9.6Hz 1 . × − AdV V1 10Hz 17.6Hz 1 . × − aLIGO/AdV H1L1V1 10Hz 10.1Hz 8 . × − TABLE I: We show the increase in the average relativevolume (15) that can be achieved by switching from thestandard lower frequency cutoff to the optimal lowerfrequency cutoff. The minimal match in either case isset to be 3%. The volume increase compared to thestandard choice is very small, except for the V1 VSR1PSD, where a higher than normal lower frequency cutoffwas employed.
Era Detector f minimumlow f optimallow Volume GainS5 H1 30Hz 37.3Hz 2 . × − VSR1 V1 10Hz 38.1Hz 3 . × − S5/VSR1 H1H2L1V1 10Hz 37.8Hz 1 . × − S6 H1 40Hz 43.7Hz 2 . × − VSR2-3 V1 10Hz 16.8Hz 1 . × − S6/VSR2-3 H1L1V1 10Hz 34.2Hz 5 . × − aLIGO H1 9Hz 9.6Hz 2 . × − AdV V1 10Hz 17.6Hz 1 . × − aLIGO/AdV H1L1V1 9Hz 10.1Hz 2 . × − TABLE II: Similar to Table I, we show the increase inthe average relative volume (15) that can be achievedby switching to the optimal lower frequency cutoff.However, here the reference lower frequency cutoff is setto the minimum frequency at which a detector’s PSD isreported.varying lower frequency cutoff on the recovered SNR andon the trials factor effect as a function of the lower fre-quency cutoff. The example described above associatedwith the Virgo VSR1 PSD can be seen in Fig. 2b.
B. Fixed Computational Cost
We now consider the task of choosing optimal val-ues for both the lower frequency cutoff and the minimalmatch of the template bank subject to the constraint offixed computational cost. Table III shows a compari-son between the standard values and the optimal valueschosen using the algorithm proposed in this paper. Inaddition to the detector/era associated with a particular PSD and the standard and optimal choices for the mini-mal match and lower frequency cutoff, this table also liststhe computational cost algorithm that is appropriate fora given search.We see that including the constraint on the computa-tional cost produces a larger effect than optimizing thelower frequency cutoff alone without the constraint. It isinteresting to note that for the majority of the cases in-vestigated, the optimal choice involves reducing the com-putational cost through raising the lower frequency cutoffand then reinvesting the computational savings into in-creasing the density of the template bank.As before, we also show a more detailed comparison ofthe constrained optimization of the lower frequency cutoffand minimal match as a function of the lower frequencycutoff. This can be found in Figs. 5-8. In this situation,the largest increase in sensitivity is a few percent, com-ing from the proposed AdV detector’s PSD. Figure 7bshows that the majority of the effect here is coming fromdecreasing the maximum mismatch (i.e., increasing theminimal match) of the template bank from 3% maxi-mum mismatch to 0.56% maximum mismatch. In thissituation, the drive toward larger lower frequency cutoffsseems to come from the reduction in the computationalcost per template associated with the total number of cy-cles contained in the waveform, as opposed to reducingthe trials factor effect.Finally, we also compare the previous choice of lowerfrequency cutoff and minimal match suggested in [20](i.e., m max = 5% and lower frequency cutoff such thatfractional SNR loss is 1%) to the optimal choice at thesame computational cost. This comparison can be foundin Table IV. This choice is closer to the optimal choice, al-though the optimal choice still provides sensitivity gainsas large as one percent for the aLIGO/AdV detector net-work. Detector Era Cost f standardlow , m standardmax f optimallow , m optimalmax Volume GainS5 H1 Fixed 40Hz, 3% 49.1Hz, 2.4% 4 . × − VSR1 V1 Fixed 60Hz, 3% 50.1Hz, 2.8% 6 . × − S5/VSR1 H1H2L1V1 Fixed 40Hz, 3% 50.2Hz, 2.3% 5 . × − S6 H1 Fixed 40Hz, 3% 55.7Hz, 2.6% 3 . × − VSR2-3 V1 Fixed 50Hz, 3% 37.8, 6.2% 1 . × − S6/VSR2-3 H1L1V1 Fixed 40Hz, 3% 51.2Hz, 2.2% 7 . × − aLIGO H1 Cycles 10Hz, 3% 14.4Hz, 1.1% 2 . × − AdV V1 Cycles 10Hz, 3% 22.0Hz, 0.56% 3 . × − aLIGO/AdV H1L1V1 Cycles 10Hz, 3% 15.0Hz, 1.0% 2 . × − aLIGO/AdV H1L1V1 LLOID 9.7Hz, 3% 14.2Hz, 0.84% 2 . × − TABLE III: We show the gain in the constrained average relative volume (18) that can be obtained by changingfrom the standard choice of lower frequency cutoff and maximum mismatch to the optimal choice. Thecomputational cost for each of these calculations is set using the algorithm listed under “Cost”. Most of thesesearches are optimized by increasing the lower frequency cutoff and decreasing the maximum mismatch (i.e.,increasing the density) of the template bank.
Era Detector Cost f previouslow , m previousmax f optimallow , m optimalmax Volume GainS5 H1 Fixed 57.8Hz, 5% 58.4Hz, 4.9% 5 . × − VSR1 V1 Fixed 60.0Hz, 5% 50.1Hz, 5.4% 6 . × − S5/VSR1 H1H2L1V1 Fixed 60.0Hz, 5% 59.7Hz, 5.0% 3 . × − S6 H1 Fixed 67.1Hz, 5% 63.2Hz, 5.7% 2 . × − VSR2-3 V1 Fixed 44.7Hz, 5% 43.4Hz, 5.2% 3 . × − S6/VSR2-3 H1L1V1 Fixed 62.8Hz, 5% 60.3Hz, 5.3% 6 . × − aLIGO H1 Cycles 17.0Hz, 5% 19.5Hz, 3.1% 9 . × − AdV V1 Cycles 28.8Hz, 5% 31.3Hz, 3.7% 5 . × − aLIGO/AdV H1L1V1 Cycles 18.0Hz, 5% 20.8Hz, 3.1% 1 . × − TABLE IV: We show the gain in the constrained average relative volume (18) that can be obtained by changingfrom the choice of lower frequency cutoff and maximum mismatch proposed in [20] to the optimal choice. Again, thecomputational cost for each of these calculations is set using the algorithm listed under “Cost”. The choices of [20]are close to optimal, although the advanced detector network search can be improved by of order one percent whenswitching to the optimal choices.
30 35 40 45 50 55 60 f low . . . . . . . . V V total V σ V trials h V m i (a) H1 S5 PSD10 20 30 40 50 60 f low . . . . . . . . . . V V total V σ V trials h V m i (b) V1 VSR1 PSD10 20 30 40 50 60 f low . . . . . . V V total V σ V trials h V m i (c) H1H2L1V1 S5/VSR1 Harmonic Sum PSD FIG. 5: (a) shows the constrained average relativevolume as a function of lower frequency cutoff for theH1 LIGO detector during the S5 era. (b) and (c) showthe same for the Virgo detector and H1H2L1V1detector network for the VSR1 and S5/VSR1 eras,respectively. The computational cost of the searchesassociated with these eras is given by the fixed costalgorithm. Each panel also contains traces for thecontributions to the constrained average relative volumefrom the recoverable SNR V σ , the trials factor V trials ,and the average template bank mismatch (cid:104) V m (cid:105) .
40 45 50 55 60 65 70 f low . . . . . . . . V V total V σ V trials h V m i (a) H1 S6 PSD10 20 30 40 50 60 70 80 f low . . . . . . . V V total V σ V trials h V m i (b) V1 VSR2-3 PSD10 20 30 40 50 60 70 80 f low . . . . . . . . . V V total V σ V trials h V m i (c) H1L1V1 S6/VSR2-3 Harmonic Sum PSD FIG. 6: (a) shows the constrained average relativevolume as a function of lower frequency cutoff for theH1 LIGO detector during the S6 era. (b) and (c) showthe same for the Virgo detector and H1L1V1 detectornetwork for the VSR2-3 and S6/VSR2-3 eras,respectively. The computational cost of the searchesassociated with these eras is given by the fixed costalgorithm. Each panel also contains traces for thecontributions to the constrained average relative volumefrom the recoverable SNR V σ , the trials factor V trials ,and the average template bank mismatch (cid:104) V m (cid:105) .0 f low . . . . . . . . . . V V total V σ V trials h V m i (a) H1 aLIGO PSD10 15 20 25 30 35 40 f low . . . . . V V total V σ V trials h V m i (b) V1 AdV PSD5 10 15 20 25 30 35 40 f low . . . . . . . . . V V total V σ V trials h V m i (c) H1L1V1 aLIGO/AdV Harmonic Sum PSD FIG. 7: (a) shows the constrained average relativevolume as a function of lower frequency cutoff for theproposed H1 LIGO detector during the aLIGO era. (b)and (c) show the same for the proposed Virgo detectorand H1L1V1 detector network for the AdV andaLIGO/AdV eras, respectively. The computational costof the searches associated with these eras is given bythe cycles cost algorithm, (21). Each panel also containstraces for the contributions to the constrained relativeaverage volume from the recoverable SNR V σ , the trialsfactor V trials , and the average template bankmismatch (cid:104) V m (cid:105) . f low . . . . V V total V σ V trials h V m i FIG. 8: We show the constrained average relativevolume as a function of low frequency cutoff for theproposed H1L1V1 detector network during theaLIGO/AdV era. The computational cost of this searchis given by the
LLOID algorithm. The contributions tothe constrained average relative volume from therecoverable SNR V σ , the trials factor V trials , and theaverage template bank mismatch (cid:104) V m (cid:105) are also shown.It is interesting to see that the optimal choices for thissearch are similar to that of a search where thecomputational cost is given by the number of cycles inthe template waveform.1 VI. CONCLUSION
We have presented an analysis of the two tunable vari-ables that affect searches for inspiral signals in GW data.We find that with the minimal match of the templatebank held fixed, there is an optimal choice for the lowerfrequency cutoff below which reducing this parameter re-duces the sensitivity of a search that employs a maximumlikelihood ratio estimate of the SNR. This could be seenas the following inverse result. Even though decreas-ing the lower frequency cutoff does not gain significantamounts of SNR, it still provides discriminating powerin determining the parameters, thus increasing the trialsfactor associated with a fixed region of parameter space.In addition, through careful balancing of the computa-tional cost associated with the lower frequency cutoff andthe minimal match of the template bank, we show thatimproved performance can be achieved at fixed computa-tional cost. This is the first work that has laid out a pro-cedure for determining the optimal choice of these param- eters for searches for BNS GW signals from non-spinningobjects. As searches for inspiral GW signals from othersystems can involve additional waveform parameters, andthus larger computational cost, it will be important toapply this method to other parameter spaces (e.g., theparameter space of waveforms from binary systems thatincluding effects from the objects’ spins) in order to max-imize the sensitivity of those searches.
ACKNOWLEDGMENTS
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