A template bank to search for gravitational waves from inspiralling compact binaries: II. Phenomenological model
aa r X i v : . [ g r- q c ] S e p A template bank to search for gravitational waves from inspiralling compact binaries:II. Phenomenological model
T. Cokelaer School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, UK
Matched filtering is used to search for gravitational waves emitted by inspiralling compact binariesin data from ground-based interferometers. One of the key aspects of the detection process is thedeployment of a set of templates, also called a template bank, to cover the astrophysically interestingregion of the parameter space. In a companion paper, we described the template-bank algorithmused in the analysis of LIGO data to search for signals from non-spinning binaries made of neutronstar and/or stellar-mass black holes; this template bank is based upon physical template families.In this paper, we describe the phenomenological template bank that was used to search forgravitational waves from non-spinning black hole binaries (from stellar mass formation) in thesecond, third and fourth LIGO science runs. We briefly explain the design of the bank, whosetemplates are based on a phenomenological detection template family. We show that this templatebank gives matches greater than 95% with the physical template families that are expected to becaptured by the phenomenological templates.
PACS numbers: 02.70.-c, 07.05.Kf, 95.85.Sz, 97.80.-d
I. MOTIVATION
The Laser Interferometer Gravitational-Wave Obser-vatory (LIGO) detectors [1] have reached their designsensitivity curves. The fifth science run (S5) began inNovember 2005 and should be completed by the end of2007, with the goal of acquiring a year’s worth of datain coincidence between the three LIGO interferometers.Each successive LIGO science run has witnessed improve-ment from both experimental and data analyst’s point ofview. On the experimental side, better stationarity ofthe data and detector sensitivities closer to design sen-sitivity curve were achieved. On the data analysis side,the search pipeline was tuned, and new techniques weredeveloped to reduce the background rate while keepingdetection efficiencies high.Among the sources of gravitational waves that ground-based detectors are sensitive to, inspiralling compact bi-naries are among the most promising. Several searchesfor inspiralling compact binaries in the LIGO data havebeen pursued: primordial black holes (PBH) [2, 3], bi-nary neutron stars (BNS) [3, 4, 5], and intermediatemass binary black holes (BBH) [3, 6]. These searchesused matched filtering technique, which is the most effec-tive and commonly used method to search for inspirallingcompact binaries.Matched filtering computes cross correlation betweenthe detector output and a template waveform. If the tem-plate waveform is identical to the signal then the methodis optimal, in the sense of signal-to-noise ratio (SNR).However, in general, the template waveforms differ fromthe signals. Indeed, modelizations can only approximatethe exact solution of the two-body problem. In addi-tion, template waveforms are constructed with a subset ofthe signal’s parameters (e.g., the two component masseswhereas eccentricity and spins effects may be neglected).In this work, we consider only the case of non-spinning waveforms so that the signals are entirely defined by 4parameters: 2 mass components, the time of arrival andthe initial phase. Because the signal’s parameters are un-known, the detector output must be cross correlated witha set of template waveforms, which is called a templatebank . While the spacing between templates can be de-creased most certainly, and this is the insurance of a SNRclose to optimality, it also increases the size of the tem-plate bank (i.e., the computational cost). The distancebetween templates is governed by the trade-off betweencomputational cost and loss in detection rate; therefore,template bank placement is a key aspect of the detectionprocess.In a companion paper [7], we proposed a template bankwith a minimum match of 95%. We assume that bothtemplate and signal were based on the same physical tem-plate family (precisely, the stationary phase approxima-tion with the phase described at 2PN order [8, 9] ). Wehave shown that the template bank could be used, effec-tively, to search for BNS, PBH, black hole - neutron star(BHNS), and BBH systems. In this paper, we considerthe case of BBH systems only.There is a wide variety of techniques used to describethe gravitational flux and energy generated during thelate stage of the inspiralling phase (e.g., see [10]). How-ever, they lead to various physical template families, andoverlaps between them are not necessarily high. In thecase of heaviest systems, post-Newtonian (PN) expan-sion [10] begins to fail as the characteristic velocity v/c is not close to zero anymore (e.g., see [11]). In addi-tion, even though heavier BBH systems are accessibleeach time the detector sensitivity improves in the lowfrequency range, BBH waveforms remain short in theLIGO band. For instance, during the second science run(S2) [6], the lower cut-off frequency was set to 100 Hz,which restricted the total mass of the search to be be-low 40 M ⊙ , and the longest expected signal to last about0.60 s.There exist several template families, and there is noreason to select one in particular. A solution may be tofilter the detector output with a set of template banks,each of them associated with a different physical templatefamilies. We have shown in [17] that a unique templatebank placement could be used effectively with severaltemplate families. However, we investigate only 4 dif-ferent families at 2PN order. A different template bankmight be necessary for other template families. Moreimportantly, the number of template families could belarge, and computational cost not manageable. Insteadof searching for BBH signals using several physical tem-plate families, a single detection template family (DTF)was proposed by Buonanno, Chen, and Vallisneri [11](BCV) with the goal of embedding the different phys-ical approximations all into a single phenomenologicalmodel. This detection template family is known as BCVtemplate family and has been used to search for non-spinning BBH signals in LIGO data [3, 6].In this paper, we do not intend to compare a searchthat uses BCV templates and a search based upon phys-ical template family. Our main goal is to describe theBCV template bank that was developed and used tosearch for stellar-mass BBH signals in the second (S2),third (S3), and fourth (S4) LIGO science runs [3, 6]. Insection II, we briefly discuss the template parameters andthe filtering process related to BCV templates. In sec-tion III, we describe the BCV template bank, and thethe spacing between templates. In section IV, we testand validate the proposed template bank with exhaustivesimulated injections. Finally, in section V, we summarizethe results. II. THE BCV TEMPLATE FAMILY
The detection template family that was proposedin [11] is built directly from the Fourier transform [12]of gravitational-wave signals by writing the amplitudeand phase as polynomials in the gravitational-wave fre-quency law that appear in the stationary phase approx-imation [13]. In the frequency domain, the BCV tem-plates are defined to be h ( f ) = A ( f ) e iψ ( f ) , (2.1)where A ( f ) = f − / (cid:16) − αf / (cid:17) θ ( f cut − f ) , (2.2) ψ ( f ) = 2 πf t + φ + f − / N X k =0 ψ k f k/ . (2.3)The parameters t and φ are the standard time of arrivaland initial phase of the gravitational wave signal. Theparameter α is a shape parameter introduced to capturepost-Newtonian amplitude corrections. Because variousmodels predict different terminating frequencies, an end-ing cut-off frequency f cut is introduced. In the ampli- tude expression, the waveform is multiplied by a Heav-iside step function, θ ( f cut − f ). In the right hand sideof equation 2.3, we use only two parameters ψ and ψ ,which suffices to obtain a high match with most of thePN models [11]. The symbol ψ here is the same as thesymbol ψ / in [11].The ψ k parameters are the phase parameters of thephenomenological waveform, which cannot be directlylinked to the physical mass parameters; the BCV tem-plates are made for detection, not for parameter esti-mation. Nevertheless, a good approximation (for lowmasses) of the chirp mass M is given by ψ ≈ (cid:18) π M (cid:19) / . (2.4)In section IV D, we investigate the range of validity ofthis relation.The filtering of a data set using BCV templates is notas trivial as the one that uses physical template fami-lies. Indeed, the BCV filtering implies a search in sixdimensions ( ψ , ψ , α , φ , t , and f cut ). The SNR canbe analytically maximized over α , φ and t , which re-duces the search to three dimensions only. the maxi-mization over α and φ is not. In order to performthe filtering and the maximization over α and φ , weneed to construct orthonormal basis vectors n ˆ h k o k =1 ,.., for the 4-dimensional linear subspace of templates with φ ∈ [0 , π ) and α ∈ ( −∞ , ∞ ), and we want the basisvectors to satisfy D ˆ h i (cid:12)(cid:12)(cid:12) ˆ h j E = δ ij (see the Appendix fordetails). The SNR before maximization is given by ρ = x cos ω cos φ + x sin ω cos φ (2.5)+ x cos ω sin φ + x sin ω sin φ , where x i = D s (cid:12)(cid:12)(cid:12) ˆ h i E , and s is the data to be filtered . Theparameter ω is a function of α (see equation A8 and A9).The SNR ρ can be maximized over φ and ω ( α ). In [6],the maximization is done over the two new parameters A = ω + φ and B = ω − φ . The maximized SNR(independent of α and φ ), is denoted ρ U , and is givenby ρ U = 1 √ r V + q V + V ! , (2.6)where V k are function of x i (see Appendix and equationsA13, A14 and A14).The SNR provided in equation 2.6 is the unconstrained SNR that is independent of any constraint on the rangeof the parameter α UF = αf / (again, here the index U The expression of the SNR shows that the expected rate of falsealarm follows a chi-square distribution with 4 degrees of freedominstead of 2 in the case of physical template families represents the unconstrained case, and we shall use C forthe constrained case). Yet, in [11], the authors suggestedthat the parameter α UF should be restricted to the range[0 , α UF >
1, the amplitude in equation 2.2becomes negative, which corresponds to unphysical wave-forms. Moreover, when α UF <
0, the amplitude factor cansubstantially deviate from the predictions of PN theory.In S2 [6], many accidental triggers were found with α UF >
1, and the calculation of the SNR was uncon-strained (as in equation 2.6) leading to a high false alarmrate, which was decreased, a posteriori, by removing alltriggers for which α UF > α UF < α UF value into account, by using amaximization of equation 2.5 that leads to a constrained SNR denoted ρ C . The expression for the constrainedSNR depends now on the value of α UF . We have α CF = α UF if 0 ≤ α UF ≤
1, and no constraint is applied (i.e., ρ C = ρ U ). However, if α UF < α UF >
1, then a constrainedSNR is used so that the final α CF parameter is 0 and1, respectively, and ρ C ≤ ρ U . The expressions of theconstrained SNR are provided in the Appendix.Let us make the point clear. In the contrained-SNRcase, as explained above, α CF = α UF if 0 < α UF < α F takes only two values (0 or 1) but we know α UF (since it is the condition to apply the constraint ornot).It is worth noticing that for the study that follows,we always use a constrained SNR but using an uncon-strained SNR should not significantly change the resultsof our simulations and/or template bank placement. In-deed, simulated injections are generated with physicaltemplate families for which we do not expect α F to be un-physical (i.e., outside [0 − α F between ( −∞ , ∞ ) (and thereforein ( −∞ , ∪ ]1 , ∞ ) as well, where ρ C < ρ U ). Conse-quently, in general, for a given threshold λ , the SNR ofaccidental triggers have ρ C < ρ U , and the rate is there-fore lower with respect to a search with unconstrainedSNR. The number of triggers that needs to be stored islower by an order of magnitude. Nevertheless, the fi-nal rate of triggers between the two methods may beequivalent because of a posteriori cuts on α F when anunconstrained SNR is used as in [6]. III. BCV TEMPLATE BANK DESIGN
Template bank placement has been investigated in sev-eral papers [7, 14, 15, 16, 17, 18] in the context of physicaltemplate families. We refer the reader to the establishedliterature in this subject area.
A. Metric Computation in ψ – ψ Plane
In the case of BCV templates, the mismatch metric g ij [15] is known (see Appendix), and is constant overthe entire ψ – ψ parameter space. Nonetheless, the met-ric components are strongly related to the lower cut-offfrequency of the search, which affects the moments usedto calculate the metric (see equation B6). The momentcomputation also depends on the α parameter, as dis-cussed later. For now, let us suppose that the momentsare fixed.Because the metric is constant, the placement of tem-plates on the ψ – ψ parameter space is straightforward.In the first search for BBH signals [6], the template place-ment used a square lattice, and templates were placedparallel to the ψ axes. In S3 and S4 BBH searches, anoptimal placement was used (hexagonal lattice), whichreduced the requested computing resources (and triggerrate) by 30% with respect to S2. In this paper, we onlyconsider tests related to the hexagonal lattice case. In S3and S4 BBH searches, we placed the templates parallel tothe first eigenvector rather than parallel to the ψ axis.The target waveforms are BBH systems for which thelowest component mass is set to 3 M ⊙ and the high-est component mass is defined by the detector lowercut-off frequency (up to 80 M ⊙ in S4). Simulationsshow that to detect such target waveforms, the rangeof phenomenological parameters should be set to ψ ∈ [10000 , / and ψ ∈ [ − , −
10] Hz / .As explained in section III D, if we search for BBH sys-tems only, a significant fraction of those templates arenot needed and can be removed from the template bank. B. α B -dependence The moments used to estimate the metric componentsstrongly depend on the parameter α . We refer to thisparameter as α B to differentiate from the α parameter(or equivalently from α F ) that is used in the filteringprocess. As shown in figure 1, the number of templateschanges significantly when α B varies. There is a drop inthe number of templates around α B = 10 − . We wantto minimize the template bank size but we also need toconsider the efficiency of the bank as defined in [7, 17],and choose α B appropriately. Indeed, we expect the ef-ficiency of the template bank to be also affected by thisparameter. We performed simulated injections so as totest the efficiency of the template bank for various valuesof α B . Results are summarized in figure 2 for three typi-cal values of α B . Because efficiencies are very similar, wedecided to use an α B parameter such that the number oftemplates is close to a minimum, that is 10 − . In all thefollowing simulations and LIGO searches, α B = 10 − . α B T e m p l a t e ban k s i z e f s s FIG. 1: The size of the template bank function of the α B pa-rameter. The two curves show the template bank size versus α B parameter for two values of sampling frequency. The twocurves show the same pattern with a drop around α B = 10 − ,where template bank sizes are twice as low as compared to α B = 0 or α B = 2 . × − . This evolution of the templatebank size is directly linked to the moment computation (seeequation B6), where the parameter α B is used. Efficiency ofa template bank is not strongly related to this parameter (seefigure 2) so we choose a value that corresponds to the smallestbank size. In real analysis, we use 2048 Hz and for simplicitya value of α B = 10 − was chosen. In this example, we usedthe same simulation parameters as in section IV A. C. Template Bank using ending frequency layers
Starting from each template that is placed in the ψ – ψ plane, we need to lay templates along the third dimen-sion, which is the ending cut-off frequency f cut of the tem-plate. Because the mismatch is first order in ∆ f cut [11],it cannot be described by a metric.Using an exact formula, [11] proposed to lay tem-plates with different f cut values between f min and f end = f Nyquist that depend on the region searched for. We pop-ulate the f cut dimension as follows. First, we estimatethe frequency of the last stable orbit which we refer to as f min , and the frequency at the light ring which we refer toas f max . Between f min and f max , we place N cut layers oftemplates with the ending frequency chosen at equal dis-tance between f min and f max . The frequency at the laststable orbit and light ring are defined in terms of the totalmass ( f min = 1 / ( M π / ), f max = 1( M π / )). The to-tal mass is computed for each template using an empiricalexpression similar to equation 2.4: M ≈ − ψ / (32 π ψ ).This expression is an approximation. It underestimatesthe total mass for low mass range, however, it is suitablefor the wide range of mass that we are interested in: thefinal template bank gives high match with the variousphysical template families, as shown in section IV. In allour simulations and searches, we set the minimal match( M M ) [15] to 95%, so there is no guarantee that the rela-tion between M and ψ , is suitable for minimal matchesfar from 95%. −4 −3 −2 −1 Matches C u m u l a t i v e d i s t r i bu t i on p r obab ili t y α B =0 α B =0.01 α B =0.02 FIG. 2: Template bank efficiencies versus α B parameter. The α B parameter does not significantly affect the matches. Mostof them are above the minimal match of 95%, and more im-portantly the three distributions are close to each other. Inthis example, we used the same simulation parameters as insection IV A, and EOB injections. −4 −3 −2 −1 Matches C u m u l a t i v e d i s t r i bu t i on p r obab ili t y FIG. 3: Template bank efficiencies versus number of layers, N cut , in the f cut dimension. With the current template bankdesign, N cut does not affect the matches significantly. Thecumulative distribution of matches over 10,000 simulationsshows only small differences between 3 and 20 layers. Eventhe results obtained with 1 layer are not that far from N cut =3. In our real analysis and simulations, we used N cut = 3. Inthis example, we used the same simulation parameters as insection IV A, and EOB simulated injections. D. Polygon Fit
The boundaries of the template bank are defined bythe ranges of the parameters ψ i and the span of thecut-off frequency f cut in such a way that BBH systemswith component mass as low as 3 M ⊙ are detectable.The ψ i ranges provided in section III A cover a squaredarea that is actually too wide: a significant fraction ofthe templates are not targeting the BBH systems we aresearching for. Therefore, in order to reduce the templatebank size and optimize our searches, we introduce an ex-tra procedure that selects the pertinent templates only.This procedure is known as a polygon fit and works asfollows. First, we create a BCV template bank with therange of ψ and ψ parameters as large as possible, andfor our purpose, as quoted in section III A. This choiceof ranges allows us to not only detect BBH systems butalso BHNS systems. Since, we focus on the BBH systemsonly, we perform many BBH simulated injections and fil-ter them with the template bank that has been created.For each injection, we keep the ψ and ψ parametersof the template that gives the best match. We gatherall the final pairs of ψ , ψ parameters, and superposethem on top of the original template bank. It appearsthat only about a third of the templates are required todetect BBH systems with a high match. This sub-setof templates can be used to define a polygon area thatenclose all of them. The resulting polygon area definesthe boundaries of our new template bank and results in atemplate bank three times smaller than the original one.In figure 4, we show such a template bank that is withinthe boundary of a polygon constructed with our sim-ulated injections. The coordinates of this polygon arechosen empirically. For safety, the boundaries are chosenloosely, therefore the template bank has also the abilityto detect non-spinning BHNS. It is worth noticing thatwith this template bank designed to detect BBH manyBHNS systems are found with a match greater than therequested M M (See section IV C). −3500−3000−2500−2000−1500−1000−5000 ψ (Hz ) ψ ( H z / ) Template positionsPolygon fit
FIG. 4: Example of parameter space and template bank place-ment. This plot shows a projection of the templates onto the ψ /ψ plane. Simulations and equation 2.4 gives an estima-tion of the mapping between the phenomenological parame-ters and the chirp mass of the simulated injections. Low masssystems such as a (3 , M ⊙ are in the RHS, high mass sys-tems lie on the LHS and asymmetric systems in the bottomleft corner. In this example, we used the same parameters asin section IV B. IV. SIMULATIONS
In the following simulations, we fix the sampling fre-quency to 2048 Hz, α B = 10 − , N cut = 3, the ψ and ψ ranges are provided in section III A and a polygon fit asin figure 4 is used. The simulated injections are based onseveral physical template families that are labelled EOB,PadeT1, TaylorT1, and TaylorT3 [19, 20, 21, 22, 23] withthe phase expressed at 2PN order (see [17] for more de-tails). The population of simulated injections has a uni-form total mass. Although this choice is not based on anyastronomical observation, it is convenient to estimate theefficiency of our template banks. We use a noise modelthat mimics the design sensitivity curve of initial LIGO(see [12, 17]), and the minimal match is M M = 95%. Weperformed 2 simulations that are closely related to thethird and fourth LIGO science run’s BBH searches [3].
A. Example 1
The first set of simulations uses a lower cut-off fre-quency of 70 Hz, as in S3 BBH search [3]. The maximaltotal mass of the simulated injections is set to 40 M ⊙ and therefore the largest component mass to 37 M ⊙ .The template bank has 531 templates. The results aresummarized in figure 5 which shows the efficiency of thetemplate bank versus the total mass. There are a few in-jections found with a match as low as 93% for total mass M < . M ⊙ . Closer inspection shows that several issuesare linked to this feature. First, we used a sampling fre-quency of 2048 Hz, which reduces the template bank sizeby ≈
50% as compared to a sampling of 4096 Hz. Second,we set α B = 10 − , which reduces the template bank sizeby ≈
50% as compared to α B = 0. Finally, the numberof layers, N cut , is limited to 3. Therefore, this tuning sig-nificantly reduced the template bank size with the costof losing about only 1 to 2% SNR for a small fractionof the parameter space considered. From M = 6 . M ⊙ to about M = 20 M ⊙ , matches are above 95%. In thehigh mass range, a large fraction of the simulated in-jections are found below the minimal match (but largerthan 90%): 20% in the case of TaylorT1, TaylorT3, andPadeT1 models, and only 0.1% in the case of EOB injec-tions. This effect is expected because the lower cut-offfrequency is high, and therefore many of the high masssystems considered are very short (i.e., less than 100 ms).Because the final frequency of the EOB signals goes up tothe light ring, the matches are larger than in the case ofTaylorT1, TaylorT3, and PadeT1 approximants, whoselast frequencies stop at the last stable orbit. B. Example 2
The second set of simulations uses a lower cut-off fre-quency of 50 Hz, as in S4 BBH search [3]. The maximaltotal mass of the simulated injections is set to 80 M ⊙ and Total mas s ( M ⊙ ) M a t c he s
10 15 20 25 30 35 400.90.920.940.960.981 20406080100120
FIG. 5: Distribution of the efficiency versus the total mass.The simulation consisted of N s = 40 ,
000 injections. Thelower cut-off frequency for the injections and the BCV tem-plates was set to 70 Hz, as in S3 BBH search. therefore the largest component mass to 77 M ⊙ . Thetemplate bank has 1609 templates. The results are sum-marized in figure 6. Up to M ≈ M ⊙ , most of theinjected simulations are recovered with matches above95%. However, a small fraction is found with matchesbelow 95%, which represent 0.1% of the EOB, PadeT1,and TaylorT1 injections, and 3% of the TaylorT3 injec-tions. In the high mass region (up to 60 M ⊙ ), 20% of theinjections are below the required minimal match for theTaylorT1, TaylorT3, and PadeT1 injections, and only0.5% of the EOB injections. If we consider injectionswith total mass from 60 to 80 M ⊙ , almost 10% of EOBare below the minimal match (but above 92%). As forother models, matches drop quickly towards zero downto 40%, which is due to shorter and shorter duration ofthe injected waveforms. C. Example 3
As stated in section III D, although the template bankis designed to target BBH systems, it has the ability todetect some BHNS systems as well. The goal of this thirdsimulation is to demonstrate that indeed many BHNSsystems are detectable with a high match by using thetemplate designed to search for BBH systems in S3 andS4 data sets.The parameters used are exactly the same as in thesecond example. The maximal total mass of the simu-lated injections is set to 80 M ⊙ , the largest componentto 79 M ⊙ , and the lowest component mass is set to 1 M ⊙ .We impose the systems to be BHNS only (the mass of theneutron star is less than 3 M ⊙ , and the mass of the blackhole is larger than 3 M ⊙ ). The template bank is identicalto the second simulation (1609 templates). The resultsare summarized in figure 7, where we plot matches as afunction of the two component masses. We found that Total mas s ( M ⊙ ) M a t c he s
10 20 30 40 50 600.90.920.940.960.981 1020304050607080
FIG. 6: Distribution of the efficiency versus the total mass.The simulation consisted in N s = 40 ,
000 injections. Thelower cut-off frequency of the injections and the BCV tem-plates is 50 Hz, as in S4 BBH search. For convenience (samescale as in figure 5), we do not show simulated injections withtotal mass above 60 M ⊙ and matches below 90% (see text fordetails).
60% of the BHNS injections are recovered with the matchlarger than 95%, 77% with the match larger than 90%,and 98% with the match greater than 50%. Therefore,using the same bank as in S3 and S4 searches, whoseboundaries resulting from the polygon fit were deliber-ately chosen to be slightly wider than necessary, we candetect a significant fraction of the BHNS systems. It isalso clear from the figure that the lightest systems havea very low match. This was expected since the tem-plate bank aimed at detecting systems whose total massis greater than 6 M ⊙ , as defined by the maximum of the ψ range.We performed a second test where the polygon fit isnot applied anymore. The template bank is then muchlarger with 4635 templates but we found that 78% of theBHNS injections are recovered with a match larger than95%, 94% with a match larger than 90%, and 98% witha match greater than 50%. The size of such a templatebank is comparable to a template bank that uses phys-ical template families (e.g., with the same parametersas above, a hexagonal placement for physical templatefamilies [17] that covers a parameter space from 1 to 80solar mass has about 3000 templates if we exclude thetemplates for which both component mass are below 3 M ⊙ ).The events which are found with a low match (say, 60%or lower) correspond to low mass systems where the neu-tron star’s mass is less than 2 . M ⊙ and the BH’s massless than 7 M ⊙ which can be taken care of by increasingthe range of ψ . Black hole mass M ⊙ N e u t r o n s t a r m a ss M ⊙
10 20 30 40 50 60 701.21.41.61.822.22.42.62.8 0.10.20.30.40.50.60.70.80.9
FIG. 7: Matches between the BCV template bank used tosearch for BBH systems and BHNS injections. The simulationconsisted in N s = 100 ,
000 injections. We found that 77% ofthe BHNS injections have a match larger than 90%. See thetext for more details. −1 −0.5 0 0.5 1 1.5 2050010001500200025003000350040004500 α F N u m be r( ) FIG. 8: Distribution of the α UF parameter corresponding tosimulations made in section IV B. Most of the found sim-ulated injections have an α UF value between zero and unity.However, a significant fraction are distributed around α UF = 1.Those triggers correspond to M > M ⊙ , for which wave-forms cannot be differentiated from a transient noise (shortduration). D. Discussions
In this section, we use the results of section IV B tocheck (i) the range of validity of equation 2.4, which givesan estimation of the chirp mass, and (ii) the regime ofconstrained SNR (i.e., the value of α F ).Although we use a constrained SNR, we kept trackof the value of α UF before the maximization. We plotthe distribution of α UF in figure 8. About 83% of theinjections were found with a α UF value in the range ]0,1[.Therefore, as stated in section II, the results obtainedwith the constrained SNR are very similar to what wewould have obtained if we had used the unconstrainedSNR. The distribution has a first peak around 0.7 and −150−100−50050100150 M ( M ⊙ ) E rr o r ( % ) TaylorT1TaylorT3EOBPadeT1
FIG. 9: Errors in chirp mass estimation corresponding to sim-ulations made in section IV B. Errors increase significantly for M greater than 8 solar mass with errors larger than 50%. a second peak in the range [1, 1.1], which correspond toabout 15% of the injections; it corresponds to total massabove 60 M ⊙ .In figure 9, we plot the errors on the chirp mass (i.e.,( M i − M e ) / M i , where i stands for injected, and e forestimated). We used equation 2.4 to estimate the chirpmass. The errors are within 10% for BBH systems whenthe chirp mass is below ≈ M ⊙ . However, errors increasesignificantly when M & M ⊙ because (i) parameter es-timation of high mass BBH systems is intrinsically weak,even for physical template families and (ii) BCV tem-plates are known to be detection template families thatare not suitable for parameter estimation. V. CONCLUSIONS
The BCV template bank that we described in this pa-per was used to search for BBH systems in the S2, S3and S4 LIGO data sets. We described the significant im-provements that were made between the S2 search andthe S3/S4 searches: α B tuning, hexagonal lattice, andpolygon fit. These improvements reduce the templatebank size by an order of magnitude, while keeping the effi-ciency higher than 95% for most of the BBH systems con-sidered. Consequently, despite reducing the lower cut-offfrequency from 100Hz to 50Hz between the second andthe fourth science runs, the template bank size remainedsimilar.A principal motivation for the construction of a detec-tion template bank was to use a single template bankinstead of several physical template families. The tem-plate bank size is therefore an important aspect of a BCVsearch, and we have shown in this work how the num-ber of templates can be optimized to search for BBH.Remarkably, the same template bank has a high matchwith a wide range of BHNS.More importantly, the BCV template bank was de-signed to search for BBH systems in the context of theS2 LIGO search. That is, for a lower cut-off frequencyof 100 Hz for which most of the target waveforms areshort duration waveforms. However, LIGO detectors im-proved and are still improving at low frequency, makingthe waveforms longer. The advantage of using a BCVtemplate to search for systems as low as 3 M ⊙ is no longerevident, especially considering the absence of a well de-fined χ test for phenomenological templates. Thereforeif it were to be used, the author thinks that a BCV tem-plate bank should be used to search for a mass rangestarting at a higher value, such as 10 or 20 M ⊙ . Acknowledgments
We would like to acknowledge many useful discussionswith members of the LIGO Scientific Collaboration, inparticular of the LSC-Virgo Compact Binary Coalescenceworking group, which were critical in the formulation ofthe results described in this paper. This work has beensupported in part by Particle Physics and AstronomyResearch Council, UK, grant PP/B500731. This paperhas LIGO Document Number LIGO-P070089-01-Z.
APPENDIX A: FILTERING, α -MAXIMIZATION,AND CONSTRAINED SNR We define the inner product as follows D h (cid:12)(cid:12)(cid:12) h E = 4 Re (cid:18)Z ∞ h ( f ) h ∗ ( f ) S h ( f ) df (cid:19) , (A1)where S h ( f ) is the one-sided noise power spectral density.
1. Filtering
The BCV templates in the frequency domain are de-fined by equation 2.1. The amplitude part of a BCVtemplate A ( f ) can be decomposed into linear combina-tions of f − / and f − / . These expressions can be usedto construct an orthonormal basis { ˆ h k } k =1 .. . We wantthe basis vectors to satisfy D ˆ h i (cid:12)(cid:12)(cid:12) ˆ h j E = δ ij . (A2)First, we construct two real functions A ( f ) and A ( f )that satisfy (cid:10) A i (cid:12)(cid:12) A j (cid:11) = δ ij . Then, we define ˆ h , ( f ) = A , ( f ) e i ( ψ − φ ) , ˆ h , ( f ) = i A , ( f ) e i ( ψ − φ ) which willgive D ˆ h i (cid:12)(cid:12) ˆ h j E = δ ij , and the desired basis, { ˆ h k } . ψ isthe phase of the signal, as defined in equation 2.3, and φ the initial phase that we want to maximize. We canchoose the following basis functions (cid:20) A ( f ) A ( f ) (cid:21) = (cid:20) a a a (cid:21) (cid:20) f − / f − / (cid:21) , (A3)where the normalization factor are given by a = I − / / , a = − I / I / I − I / I / ! − / , (A4)and a = I − I / I / ! − / , (A5)and the integrals I k are defined by I k = 4 Z f cut f − k S h ( f ) df. (A6)The normalized template can be parametrized usingthe orbital phase φ and an angle ω ˆ h ( θ, ω ; f ) = (A7)ˆ h ( f ) cos ω cos φ + ˆ h ( f ) sin ω cos φ + ˆ h ( f ) cos ω sin φ + ˆ h ( f ) sin ω sin φ where w is related to α by (see [6])tan ω = − a αa + a α , (A8)which can be inverted to get αα = − a tan ωa tan ω + a . (A9)It follows that for any given signal s , the overlap is ρ = D s (cid:12)(cid:12)(cid:12) ˆ h E (A10)= x cos ω cos φ + x sin ω cos φ + x cos ω sin φ + x sin ω sin φ , where x i = D s (cid:12)(cid:12)(cid:12) ˆ h i E . We can then maximized over ω (i.e., α ), and φ without any constraint on the α parameter,which leads to the unconstrained SNR given by ρ U = 1 √ s(cid:18) V + q V + V (cid:19) (A11)where V = x + x + x + x , (A12) V = x + x − x − x , (A13) V = 2( x x + x x ) . (A14)The values of ω and φ that maximize ρ U are providedin [6] as function of the x i . We reformulate ω max usingthe V i and found this simple expression:tan 2 ω max = V V . (A15)It is then straightforward to obtain α max using equationA9, and α UF = α max f / .
2. constrained and unconstrained SNRs
Starting from equation A10, we can derive a con-strained SNR ρ C that depends upon the value of theparameter α F . Therefore, we need to maximize equa-tion A10 over the parameter φ only. This maximizationgives ρ ( ω ) = max ω √ p V + V cos 2 ω + V sin 2 ω . (A16)If α F <
0, we want to use a SNR calculation for which ω = 0, which means α CF = α = 0. Therefore, the con-strained SNR is ρ C = 1 √ p V + V . (A17)If α F >
1, we want to use a SNR calculation for which ω = ω max , which means that α CF = 1 (i.e., α = f − / ).Using equation A8, the angle ω = ω max is then a maxi-mum given by ω max = arctan ( − a f − / b + b f − / ) . (A18)The constrained SNR is then given by ρ C = 1 √ p V + V cos 2 ω max + V sin 2 ω max . (A19) Finally, using the relation V cos 2 ω + V sin 2 ω = p V + V cos (2 ω − θ ), where tan θ = V /V , we can re-write equation A16 in the general case where 0 < α F < ω = θ , which gives ρ C = 1 √ s(cid:18) V + q V + V (cid:19) , (A20)that is an identical expresion as in equation A11 (i.e., ρ C = ρ U ). So, equation A15 is also valid, and α CF = α UF .More details on this derivation can be found in [24]. APPENDIX B: METRIC
We can derive an expression for the match betweentwo BCV templates (described by equation 2.3, 2.1 and2.2). First, we consider templates with the same ampli-tude function (i.e., the same α and f cut parameter). Theoverlap h h ( ψ , ψ ) , h ( ψ + ∆ ψ , ψ + ∆ ψ ) i between tem-plates with close values of ψ and ψ can be described (tosecond order in ∆ ψ and ∆ ψ ) by the mismatch metric g ij [11]: h h ( ψ , ψ ) , h ( ψ + ∆ ψ , ψ + ∆ ψ ) i = 1 − X i,j =0 , g ij ∆ ψ i ∆ ψ j . (B1)The metric coefficients g ij can be evaluated analyti-cally [11], and are given by g ij = 12 (cid:2) M − M T M − M (cid:3) ij , (B2)where the M (1) ... (3) are the matrices defined by M (1) = (cid:20) J (2 n ) J ( n + n ) J ( n + n ) J (2 n ) (cid:21) , (B3) M (2) = (cid:20) J ( n ) J ( n ) J ( n − J ( n − (cid:21) , (B4) M (3) = (cid:20) J (0) J ( − J ( − J ( − (cid:21) , (B5) where n = 5 / n = 2 /
3, and J ( n ) ≡ (cid:20)Z |A ( f ) | S h ( f ) 1 f n df (cid:21) (cid:30) (cid:20)Z |A ( f ) | S h ( f ) df (cid:21) . (B6)Let us emphasize the fact that the mismatch h h ( ψ , ψ ) , h ( ψ + ∆ ψ , ψ + ∆ ψ ) i is translationally in-variant in the ψ – ψ plane, so the metric g ij is constanteverywhere since J ( n ) is independent of ψ , ψ parame-ters. [1] A. Abramovici et al. Science
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