Template Banks for Binary black hole searches with Numerical Relativity waveforms

Template Banks for Binary black hole searches with Numerical Relativity waveforms

Prayush Kumar,1, ∗ Ilana MacDonald,2, 3 Duncan A. Brown,1, 4 Harald P. Pfeiffer,2, 5 Kipp Cannon,2 MichaelBoyle,6 Lawrence E. Kidder,6 Abdul H. Mroue,2 Mark A. Scheel,7 Bela Szilagyi,7 and Anıl Zenginoglu7

1Department of Physics, Syracuse University, Syracuse, NY 13244, USA2Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON M5S 3H8, Canada

3Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada4LIGO Laboratory, California Institute of Technology, Pasadena CA 91125, USA

5Canadian Institute for Advanced Research, 180 Dundas St. West, Toronto, ON M5G 1Z8, Canada6Center for Radiophysics and Space Research, Cornell University, Ithaca, New York 14853, USA

7Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA(Dated: October 31, 2013)

Gravitational waves (GW) from coalescing stellar-mass black hole binaries (BBH) are expected to be detectedby the Advanced Laser Interferometer Gravitational-wave Observatory and Advanced Virgo. Detection searchesoperate by matched-filtering the detector data using a bank of waveform templates. Traditionally, template banksfor BBH are constructed from intermediary analytical waveform models which are calibrated against numericalrelativity simulations and which can be evaluated for any choice of BBH parameters. This paper explores analternative to the traditional approach, namely the construction of template banks directly from numerical BBHsimulations. Using non-spinning BBH systems as an example, we demonstrate which regions of the mass-parameter plane can be covered with existing numerical BBH waveforms. We estimate the required number andrequired length of BBH simulations to cover the entire non-spinning BBH parameter plane up to mass-ratio 10,thus illustrating that our approach can be used to guide parameter placement of future numerical simulations.We derive error bounds which are independent of analytical waveform models; therefore, our formalism can beused to independently test the accuracy of such waveform models. The resulting template banks are suitable foradvanced LIGO searches.

PACS numbers: 04.80.Nn, 95.55.Ym, 04.25.Nx, 04.25.dg, 04.30.Db, 04.30.-w,

I. INTRODUCTION

Upgrades to the LIGO and Virgo observatories are under-way [1, 2], with first observation runs planned for 2015 [3].The construction of the Japanese detector KAGRA has alsobegun [4]. The advanced detectors will be sensitive to grav-itational waves at frequencies down to ∼ 10Hz, with an or-der of magnitude increase in sensitivity across the band. Thisis a significant improvement over the lower cutoff of 40Hzfor initial LIGO. Estimates for the expected rate of detec-tion have been placed between 0.4 − 1000 stellar-mass bi-nary black hole (BBH) mergers a year [5]. The uncertaintyin these estimates comes from the uncertainties in the variousfactors that govern the physical processes in the BBH forma-tion channels [6, 7]. In sub-solar metallicity environments,stars (in binaries) are expected to lose relatively less mass tostellar winds and form more massive remnants [8–10]. Popu-lation synthesis studies estimate that sub-solar metallicity en-vironments within the horizon of advanced detectors could in-crease the detection rates to be as high as a few thousand peryear [11, 12]. On the other hand, high recoil momenta duringcore-collapse and merger during the common-envelope phaseof the binary star evolution could also decrease the detectionrates drastically [10, 11].

Past GW searches have focused on GW bursts [13–15]; coa-lescing compact binaries [16–22], and ringdowns of perturbedblack holes [23], amongst others [24–28]. For coalescing

∗ [email protected]

BBHs, detection searches involve matched-filtering [29, 30]of the instrument data using large banks of theoretically mod-eled waveform templates as filters [31–36]. The matched-filter is the optimal linear filter to maximize the signal-to-noise ratio (SNR), in the presence of stochastic noise [37]. Itrequires an accurate modeling of the gravitational waveformemitted by the source binary. Early LIGO-Virgo searchesemployed template banks of Post-Newtonian (PN) inspiralwaveforms [16–20], while more recent searches targeting highmass BBHs used complete inspiral-merger-ringdown (IMR)waveform templates [21, 22].

Recent developments in Numerical Relativity (NR) haveprovided complete simulations of BBH dynamics in thestrong-field regime, i.e. during the late-inspiral and mergerphases [38–42]. These simulations have contributed unprece-dented physical insights to the understanding of BBH merg-ers (see, e.g., [43–46] for recent overviews of the field). Dueto their high computational cost, fully numerical simulationscurrently span a few tens of inspiral orbits before merger. Formass-ratios q = m1/m2 = 1, 2, 3, 4, 6, 8, the multi-domainSpectral Einstein code (SpEC) [47] has been used to sim-ulate 15–33 inspiral merger orbits [48–50]. These simula-tions have been used to calibrate waveform models, for ex-ample, within the effective-one-body (EOB) formalism [51–54]. Alternately, inspiral waveforms from PN theory canbe joined to numerical BBH inspiral and merger waveforms,to construct longer hybrid waveforms [55–59]. NR-PN hy-brids have been used to calibrate phenomenological waveformmodels [60, 61], and within the NINJA project [62, 63] tostudy the efficacy of various GW search algorithms towardsrealistic (NR) signals [64, 65].

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Constructing template banks for gravitational wavesearches has been a long sought goal for NR. Traditionally,intermediary waveform models are calibrated against numer-ical simulations and then used in template banks for LIGOsearches [21, 22]. In this paper we explore an alternative tothis traditional approach, proposing the use of NR waveformsthemselves and hybrids constructed out of them as search tem-plates. For a proof of principle, we focus on the non-spinningBBH space, with the aim of extending to spinning binaries infuture work. We investigate exactly where in the mass spacecan the existing NR waveforms/hybrids be used as templates,finding that only six simulations are sufficient to cover bina-ries with m1,2 & 12M� upto mass-ratio 10. This methodcan also be used as a guide for the placement of parametersfor future NR simulations. Recent work has shown that ex-isting PN waveforms are sufficient for aLIGO searches forM = m1 + m2 . 12M� [66, 67]. To extend the NR/hybridbank coverage down to M ' 12M�, we demonstrate thata total of 26 simulations would be sufficient. The templatebanks are constructed with the requirement that the net SNRrecovered for any BBH signal should remain above 96.5% ofits optimal value. Enforcing this tells that that these 26 sim-ulations would be required to be ∼ 50 orbits long. This goalis achievable, given the recent progress in simulation tech-nology [57, 68, 69]. Our template banks are viable for GWsearches with aLIGO, and the framework for using hybridswithin the LIGO-Virgo software framework has been demon-strated in the NINJA-2 collaboration [70]. In this paper, wealso derive waveform modeling error bounds which are inde-pendent of analytical models. These can be extended straight-forwardly to assess the accuracy of such models.

First, we construct a bank for purely-NR templates, re-stricting to currently available simulations [48, 49, 57, 68].We use a stochastic algorithm similar to Ref. [71–73], andplace a template bank grid constrained to q = m1/m2 ={1, 2, 3, 4, 6, 8}. The bank placement algorithm uses the EOBmodel from Ref. [53] (EOBNRv2), which was calibratedagainst NR for 5 out of these 6 mass-ratios. To demon-strate the efficacy of the bank, we measure its fitting-factors(FFs) [74] over the BBH mass space. We simulate a popula-tion of 100, 000 BBH waveforms with masses sampled uni-formly over 3M� ≤ m1,2 ≤ 200M� and M = m1 +m2 ≤ 200M�, and filter them through the template bankto characterize its SNR recovery. For a bank of NR tem-plates, any SNR loss accrued will be due to the coarsenessof the bank grid. We measure this requiring both signalsand templates to be in the same manifold, using the EOB-NRv2 model for both. We find that for systems with chirpmassMc ≡ (m1 +m2)−1/5(m1m2)3/5 above ∼ 27M� and1 ≤ q ≤ 10, this bank has FFs ≥ 97% and is sufficientlyaccurate to be used in GW searches. We also show that thecoverage of the purely-NR bank can be extended to include10 ≤ q ≤ 11, if we instead constrain it to templates withmass-ratios q = {1, 2, 3, 4, 6, 9.2}.

Second, we demonstrate that currently available PN-NR hy-brid waveforms can be used as templates to search for BBHswith much lower masses. The hybrids used correspond tomass-ratios q = {1, 2, 3, 4, 6, 8}. We use two distinct meth-

ods of bank placement to construct a bank with these mass-ratios, and compare the two. The first method is the stochasticalgorithm we use for purely-NR templates. The second is adeterministic algorithm, that constructs the two-dimensionalbank (in M and q) through a union of six one-dimensionalbanks, placed separately for each allowed value of mass-ratio.Templates are placed over the total mass dimension by re-quiring that all pairs of neighboring templates have the samenoise weighted overlap. As before, we measure the SNR lossfrom both banks, due to the discrete placement of the tem-plates, by simulating a population of 100, 000 BBH signals,to find the SNR recovered. We measure the intrinsic hybriderrors using the method of Ref. [56, 57], and subsequentlyaccount for them in the SNR recovery fraction. We find thatthe NR-PN hybrid bank is effectual for detecting BBHs withm1,2 ≥ 12M�, with FFs ≥ 96.5%. The number of templatesrequired was found to be close to that of a bank constructedusing the second-order TaylorF2 hexagonal bank placementalgorithm [31–36]. We note that by pre-generating the tem-plate for the least massive binary for each of the mass-ratiosthat contribute to the bank, we can re-scale it on-the-fly todifferent total masses in the frequency domain [75]. Used indetection searches, such a bank would be computationally in-expensive to generate relative to a bank of time-domain mod-eled waveforms.

Finally, we determine the minimal set of NR simulationsthat we would need to extend the bank down to M ' 12M�.We find that a bank that samples from the set of 26 mass-ratios listed in Table II would be sufficiently dense, even atthe lowest masses, for binaries with mass-ratios 1 ≤ q ≤ 10.We show that this bank recovers more than 98% of the optimalSNR, not accounting for hybrid errors. To restrict the loss inevent detection rate below 10%, we restrict the total SNR lossbelow 3.5%. This implies the hybrid error mismatches staybelow 1.5%, which constrains the length of the NR part foreach hybrid. We find that NR simulations spanning about 50orbits of late-inspiral, merger and ringdown would suffice toreduce the PN truncation error to the desired level. With sucha bank of NR-PN hybrids and purely-PN templates for lowermasses, we can construct GW searches for stellar-mass BBHswith mass-ratios q ≤ 10.

The paper is organized as follows, in Sec. II A, we dis-cuss the NR waveforms used in this study, in Sec. II B wedescribe the PN models used to construct the NR-PN hybrids,and in Sec. II C we describe the construction of hybrid wave-forms. In Sec. II D we describe the EOB model that we use toplace and test the template banks. In Sec. III we describe theaccuracy measures used in quantifying the loss in signal-to-noise ratio in a matched-filtering search when using a discretebank of templates and in the construction of hybrid wave-forms. In Sec. IV we describe the construction and efficacyof the NR-only banks, while in Sec. V we discuss the samefor the NR-PN hybrid template banks constructed with cur-rently available NR waveforms. In Sec. VI, we investigate theparameter and length requirements for future NR simulationsin order to cover the entire non-spinning parameter space with12M� ≤ M ≤ 200M�, m1,2 ≥ 3M�, and 1 ≤ q ≤ 10. Fi-nally, in Sec. VII we summarize the results.

3

η q Length (in orbits)0.25 1 33

0.2222 2 150.1875 3 180.1600 4 150.1224 6 200.0988 8 25

TABLE I. SpEC BBH simulations used in this study. Given are sym-metric mass-ratio η, mass-ratio q = m1/m2, and the length in orbitsof the simulation.

II. WAVEFORMS

In the sections that follow, we will describe the constructionof template banks for NR or NR-PN hybrid waveform tem-plates. The NR waveforms that we use correspond to mass ra-tios q = {1, 2, 3, 4, 6, 8}, and were simulated using the SpECcode [47]. The construction of hybrid waveforms involvesjoining a long inspiral portion, modeled using PN theory, tothe merger-ringdown waveform from NR. In this section wedescribe both, the NR waveforms and the PN models usedin our study. Measuring the effectualness of these banks in-volves simulating a population of BBH signals. We use the re-cently published EOBNRv2 model [53] to obtain waveformsfor BBHs with arbitrary masses. This model was calibratedagainst five out of the six NR simulations we use to constructour banks, and is expected to be faithful at comparable massratios [53]. In this section, we briefly summarize the construc-tion of EOBNRv2 waveforms as well.

A. Numerical Relativity simulations

The numerical relativity waveforms used in this paper wereproduced with the SpEC code [47], a multi-domain pseu-dospectral code to solve Einsteins equations. SpEC uses Gen-eralized Harmonic coordinates, spectral methods, and a flexi-ble domain decomposition, all of which contribute to it beingone of the most accurate and efficient codes for computing thegravitational waves from binary black hole systems. High ac-curacy numerical simulations of the late-inspiral, merger andringdown of coalescing binary black-holes have been recentlyperformed for mass-ratios q ≡ m1/m2 ∈ {1, 2, 3, 4, 6, 8}[48, 49, 76, 77].

The equal-mass, non-spinning waveform covers 33 inspiralorbits and was first discussed in [49, 57]. This waveform wasobtained with numerical techniques similar to those of [48].The unequal-mass waveforms of mass ratios 2, 3, 4, and 6were presented in detail in Ref. [48]. The simulation withmass ratio 6 covers about 20 orbits and the simulations withmass ratios 2, 3, and 4 are somewhat shorter and cover about15 orbits. The unequal mass waveform with mass ratio 8 waspresented as part of the large waveform catalog in [49, 68]. Itis approximately 25 orbits in length. We summarize the NRsimulations used in this study in Table I.

B. Post-Newtonian waveforms

Post-Newtonian (PN) theory is a perturbative approach todescribing the motion of a compact object binary, during theslow-motion and weak-field regime, i.e. the inspiral phase.The conserved energy of a binary in orbit, E, has been cal-culated to 3PN order in literature [78–84]. Using the adia-batic approximation, we treat the course of inspiral as a se-ries of radially shrinking circular orbits. This is valid duringthe inspiral when the angular velocity of the binary evolvesmore slowly than the orbital time-scale. The radial separationshrinks as the binary loses energy to gravitational radiationthat propagates outwards from the system. The energy fluxfrom a binary F is known in PN theory to 3.5PN order [85–88]. Combining the energy balance equation, dE/dt = −F ,with Kepler’s law gives a description of the radial and orbitalphase evolution of the binary. We start the waveform wherethe GW frequency enters the sensitive frequency band of ad-vanced LIGO, i.e. at 15Hz. Depending on the way the ex-pressions for orbital energy and flux are combined to obtainthe coordinate evolution for the binary, we get different Tay-lor{T1,T2,T3,T4} time-domain approximants. Using the sta-tionary phase approximation [89], frequency-domain equiva-lents of these approximants, i.e. TaylorFn, can be constructed.Past GW searches have extensively used the TaylorF2 approx-imant, as it has a closed form and mitigates the computationalcost of generating and numerically fourier-transforming time-domain template [16–20]. We refer the reader to Ref. [90, 91]for an overview. From the coordinate evolution, we obtainthe emitted gravitational waveform; approximating it by thequadrupolar multipole h2,±2 which is the dominant mode ofthe waveform.

C. PN-NR hybrid waveforms

The hybridization procedure used for this investigation isdescribed in Sec. 3.3 of Ref. [56]: The PN waveform, hPN(t),is time and phase shifted to match the NR waveform, hNR(t),and they are smoothly joined together in a GW frequency in-terval centered at ωm with width δω:

ωm −δω

2≤ ω ≤ ωm +

δω

2. (1)

This translates into a matching interval tmin < t < tmax be-cause the GW frequency continuously increases during theinspiral of the binary. As argued in Ref. [56], we chooseδω = 0.1ωm because it offers a good compromise of sup-pressing residual oscillations in the matching time, while stillallowing hPN(t) to be matched as closely as possible to thebeginning of hNR(t).

The PN waveform depends on a (formal) coalescence time,tc, and phase, Φc. These two parameters are determined byminimizing the GW phase difference in the matching interval[tmin, tmax] as follows:

t′c,Φ′c = argmin

tc,Φc

∫ tmax

tmin

(φPN(t; tc,Φc)− φNR(t)

)2dt, (2)

4

where t′c and Φ′c are the time and phase parameters for the bestmatching between hPN(t) and hNR(t), and φ(t) is the phase ofthe (2,2) mode of the gravitational radiation. Since we con-sider only the (2,2) mode, this procedure is identical to timeand phase shifting the PN waveform until it has best agree-ment with NR as measured by the integral in Eq. (2). Thehybrid waveform is then constructed in the form

hH(t) ≡ F(t)hPN(t; t′c,Φ′c) +

[1−F(t)

]hNR(t), (3)

where F(t) is a blending function defined as

F(t) ≡

1, t < tmin

cos2 π(t−tmin)2(tmax−tmin) , tmin ≤ t < tmax

0. t ≥ tmax.

(4)

In this work, we construct all hybrids using the same proce-dure, Eqs. (1)–(4), and vary only the PN approximant and thematching frequency ωm.

D. Effective-One-Body model

Full numerical simulations are available for a limited num-ber of binary mass combinations. We use a recently proposedEOB model [53], which we refer to as EOBNRv2, as a sub-stitute to model the signal from binaries with arbitrary com-ponent masses in this paper. This model was calibrated tomost of the numerical simulations that we use to constructtemplates banks, which span the range of masses we considerhere well. So we expect this approximation to hold. We de-scribe the model briefly here.

The EOB formalism maps the dynamics of a two-body system onto an effective-mass moving in a deformedSchwarzschild-like background [51]. The formalism hasevolved to use Pade-resummations of perturbative expansionscalculated from PN theory, and allows for the introduction ofhigher (unknown) order PN terms that are subsequently cal-ibrated against NR simulations of BBHs [52, 92–95]. TheEOB model proposed recently in Ref. [53] has been cali-brated to SpEC NR waveforms for binaries of mass-ratiosq = {1, 2, 3, 4, 6}, where q ≡ m1/m2, and is the one that weuse in this paper (we will refer to this model as EOBNRv2).

The dynamics of the binary enters in the metric coefficientof the deformed Schwarzschild-like background, the EOBHamiltonian [51], and the radiation-reaction force. Theseare known to 3PN order [51, 96] from PN theory. The4PN & 5PN terms were introduced in the potential A(r),which was Pade resummed and calibrated to NR simulations[52, 53, 94, 97, 98]. We use the resummed potential calibratedin Ref. [53] (see Eq. (5-9)). The geodesic dynamics of the re-duced mass µ = m1m2/M in the deformed background isdescribed by the Hamiltonian Heff given by Eq. (3) in [53].The Hamiltonian describing the conservative dynamics of thebinary (labeled the real Hamiltonian Hreal) is related to Heff

as in Eq. (4) of [53]. The inspiral-merger dynamics can beobtained by numerically solving the Hamiltonian equations ofmotion for Hreal, see e.g. Eq.(10) of [53].

The angular momentum carried away from the binary bythe outwards propagating GWs results in a radiation-reactionforce that causes the orbits to shrink. This is due to the flux ofenergy from the binary, which is obtained by summing overthe contribution from each term in the multipolar decompo-sition of the inspiral-merger EOB waveform. Complete re-summed expressions for these multipoles [95] can be read offfrom Eq.(13)-(20) of Ref. [53]. In this paper, as for PN wave-forms, we model the inspiral-merger part by summing overthe dominant h2,±2 multipoles.

The EOB merger-ringdown part is modeled as a sum ofN quasi-normal-modes (QNMs), where N = 8 for EOB-NRv2 [52, 97–99]. The ringdown frequencies depend on themass and spin of the BH that is formed from the coalescenceof the binary. The inspiral-merger and ringdown parts areattached by matching them at the time at which the ampli-tude of the inspiral-merger waveform peaks. [52, 53]. Thematching procedure followed is explained in detail Sec. II Cof Ref. [53]. By combining them, we obtain the completewaveform for a BBH system.

III. QUANTIFYING WAVEFORM ACCURACY & BANKEFFECTUALNESS

To assess the recovery of SNR from template banks withNR waveforms or NR-PN hybrids as templates, we use themeasures proposed in Ref. [31, 74, 100]. The gravitationalwaveform emitted during and driving a BBH coalescence isdenoted as h(t), or simply h. The inner product between twowaveforms h1 and h2 is

(h1|h2) ≡ 4

∫ fNy

fmin

h1(f)h∗2(f)

Sn(f)df, (5)

where Sn(f) is the one-sided power spectral density (PSD)of the detector noise, which is assumed to be stationary andGaussian with zero mean; fmin is the lower frequency cut-off for filtering; fNy is the Nyqyuist frequency correspondingto the waveform sampling rate; and h(f) denotes the Fouriertransform of h(t). In this paper, we take Sn(|f |) to be thezero-detuning high power noise curve for aLIGO, for bothbank placement and overlap calculations [101]; and set thelower frequency cutoff fmin = 15 Hz. The peak GW fre-quency for the lowest binary masses that we consider, i.e. form1 +m2 ' 12M�, is ∼ 2.1 kHz during the ringdown phase.We sample the waveforms at 8192 Hz, preserving the infor-mation content up to the Nyquist frequency fNy = 4096 Hz.A waveform, h, is normalized (made to be a unit vector) byh = h/

√h|h. In addition to being senstive to their intrinsic

mass parameters, the inner product of two normalized wave-forms is sensitive to phase and time shift differences betweenthe two, φc and tc. These two parameters (φc and tc) can beanalytically maximized over to obtain the maximized overlapO,

O(h1, h2) = maxφc,tc

(h1|h2(φc, tc)

), (6)

5

which gives a measure of how “close” the two waveforms arein the waveform manifold, disregarding differences in overallamplitude. The mismatchM between the two waveforms isthen

M(h1, h2) = 1−O(h1, h2). (7)

Matched-filtering detection searches employ a discretebank of modeled waveforms as filters. The optimal signal-to-noise ratio (SNR) is obtained when the detector strains(t) ≡ htr(t) + n(t) is filtered with the true waveform htr

itself, i.e.

ρopt = maxφc,tc

(htr|htr(φc, tc)

)=∣∣∣∣htr

∣∣∣∣ , (8)

where ||htr|| ≡√

(htr, htr) is the noise weighted norm of thewaveform. With a discrete bank of filter templates, the SNRwe recover

ρ ' O(htr, hb)∣∣∣∣htr

∣∣∣∣ = O(htr, hb) ρopt, (9)

where hb is the filter template in the bank (subscript b) thathas the highest overlap with the signal htr. The furthest dis-tance to which GW signals can be detected is proportional tothe matched-filter SNR that the search algorithm finds the sig-nal with. Note that 0 ≤ O(htr, hb) ≤ 1, so the recoveredSNR ρ ≤ ρopt (c.f. Eq. (9)). For a BBH population uni-formly distributed in spacial volume, the detection rate woulddecrease as O(htr, hb)

3. Searches that aim at restricting theloss in the detection rate strictly below 10% (or 15%), wouldrequire a bank of template waveforms that have O above0.965 (or 0.947) with any incoming signal [102, 103].

Any template bank has two sources for loss in SNR: (i) thediscreteness of the bank grid in the physical parameter spaceof the BBHs, and, (ii) the disagreement between the actualGW signal htr and the modeled template waveforms used asfilters. We de-coupled these to estimate the SNR loss. Sig-nal waveforms are denoted as htr

x in what follows, where thesuperscript tr indicates a true signal, and the subscript x indi-cates the mass parameters of the corresponding binary. Tem-plate waveforms are denoted as hMb , where M denotes thewaveform model, and b indicates that it is a member of thediscrete bank. Fig. 1 shows the signal htr

x in its manifold, andthe bank of templates hMb residing in the model waveformmanifold, both being embedded in the same space of all pos-sible waveforms. The point hM⊥ is the waveform which hasthe smallest mismatch in the entire (continuous) model man-ifold with htr

x , i.e. hM⊥ : M(htrx , h

M⊥ ) = min

yM(htr

x , hMy ).

The fraction of the optimal SNR recovered at different pointsx in the binary mass space can be quantified by measuring thefitting factor FF of the bank [74],

FF(x) = 1−minbM(htr

x , hMb ). (10)

For two waveforms h1 and h2 close to each other in the wave-form manifold: ||h1|| ' ||h2||, and mutually aligned in phaseand time such that the overlap between them is maximized,

||h1 − h2||2 ' 2 (h1|h1)

(1− (h1|h2)√

(h1|h1)√

(h1|h1)

).

(11)

FIG. 1. We show the true (upper) and the hybrid (lower) waveformmanifolds here, with the signal residing in the former, and a discretebank of templates placed along lines of constant mass-ratio in thelatter. Both manifolds are embedded in the same space of all possiblewaveforms. The true signal waveform is denoted as htr

x , while thetemplates in the bank are labelled hM

b . The hybrid waveform thatmatches the signal Htr

x best is shown as hM⊥ . Also shown is the

“distance” between the signal and the hybrid template that has thehighest overlap with it. This figure is qualitatively similar to Fig. 3of Ref. [102].

The mismatch can, hence, be written as (c.f. Eq. (7))

M (h1, h2) =1

2 ||h1||2||h1 − h2||2 . (12)

We note that this equation is an upper bound for Eq. (25) ofRef. [104]. From this relation, and treating the space embed-ding the true and model waveform manifolds as Euclidean atthe scale of template separation, we can separate out the ef-fects of bank coarseness and template inaccuracies as

FF(x) = 1−minb

1

2 ||htrx ||

2

∣∣∣∣htrx − hMb

∣∣∣∣2 , (13a)

= 1− ΓHyb(x)− Γbank(x); (13b)

where

ΓHyb(x) ≡ 1

2 ||htrx ||

2

∣∣∣∣htrx − hM⊥

∣∣∣∣2 =M(htrx , h

M⊥ ) (14)

is the SNR loss from model waveform errors out of the mani-fold of true signals; and

Γbank(x) ≡ minb

1

2 ||htrx ||

2

∣∣∣∣hM⊥ − hMb ∣∣∣∣2 = minbM(hM⊥ , h

Mb )

(15)is the loss in SNR from the distant spacing of templates inthe bank. The decomposition in Eq. (13b) allows for themeasurement of the two effects separately. NR-PN hybridshave the inspiral portion of the waveform, from PN theory,

6

joined to the available late-inspiral and merger portion fromNR (as described in Sec. II C). Towards the late inspiral, thePN waveforms accumulate phase errors, contaminating thehybrids [56, 57]. For each hybrid, we constrain this effectusing mismatches between hybrids constructed from the sameNR simulation and different PN models, i.e.

ΓHyb(x) ≤M(htrx , h

Hybx ) . max

(i,j)M(hMi

x , hMjx ), (16)

where Mi = TaylorT[1,2,3,4]+NR. However, this is only pos-sible for a few values of mass-ratio for which NR simulationsare available. We assume ΓHyb to be a slowly and smoothlyvarying quantity over the component-mass space at the scaleof template grid separation. At any arbitrary point x in themass space we approximate ΓHyb with its value for the “clos-est” template, i.e.

ΓHyb(x) ≤ max(i,j)

M(hMix , hMj

x ) ' max(i,j)

M(hMi

b , hMj

b ),

(17)where hMb is the hybrid template in the bank with the highestoverlap with the signal at x.

The other contribution to SNR loss comes from the discreteplacement of templates in the mass space. In Fig. 1, this isshown in the manifold of the template model. As NR wave-forms (or hybrids) are available for a few values of mass-ratio,we measure this in the manifold of EOBNRv2 waveforms.The EOBNRv2 model reproduces most of the NR simulationsthat were consider here well [53], allowing for this approxi-mation to hold. For the same reason, we expect hEOBNRv2

x tobe close to hEOBNRv2

⊥ , with an injective mapping between thetwo. This allows us to approximate (c.f. Eq. (15))

Γbank(x) ' minbM(hEOBNRv2

x , hEOBNRv2b ). (18)

In Sec. IV, we construct template banks that use purely-NRtemplates, which have negligible waveform errors. The SNRrecovery from such banks is characterized with

FF(x) = 1− Γbank(x), (19)

where the SNR loss from bank coarseness is obtained usingEq. (18). In Sec. V, VI, we construct template banks aimed atusing NR-PN hybrid templates. Their SNR recovery is char-acterized using Eq. (13b), where the additional contributionfrom the hybrid waveform errors are obtained using Eq. (17).

IV. CONSTRUCTING A TEMPLATE BANK FOR NRWAVEFORMS

In this section we demonstrate the effectualness of a tem-plate bank viable for using NR waveforms as templates. Thegravitational-wave phase of the dominant waveform multi-pole extracted from runs at different resolutions was foundto converge within ∼ 0.3 rad for q = 3, 4, 6, and within∼ 0.06 rad for q = 1, 2 at merger (see Fig. (6) of Ref. [48],and Fig. (6,7) of Ref. [53] for a compilation). Most of thisphase disagreement accumulates over a relatively short dura-tion of ∼ 50M − 100M before merger, and is significantly

FIG. 2. The color at each point gives the number of waveform cy-cles Ncyc, for that particular binary, which contain 99% of the signalpower in the aLIGO sensitivity band. The figure is trucated to ex-clude the region where Ncyc > 40. The solid curve shows the lowerbounding edge of the region withMc = 27M�.

lower over the preceding inspiral and plunge. As the matched-filter SNR accumulates secularly over the entire waveform,these numerical phase errors are negligible in terms of mis-matches. We set ΓHyb = 0 while computing the fitting fac-tors, so one is left with considering Γbank to determine thefidelity of the bank (c.f. Eq. (13b)).

With NR simulations as templates, the region that the bankcan cover is restricted to binaries that have approximately thesame number of waveform cycles within the sensitive fre-quency band of the detectors as the simulations themselves.We take their fiducial length to be ∼ 40 GW cycles [105].For BBHs with 3M� ≤ m1,m2 ≤ 200M� and m1 + m2 ≤200M� we map out the region with 99% of the signal powerwithin 40 cycles as the target region of the purely-NR bank.For samples taken over the mass space, we determine the fre-quency interval [f1, f2] for which∫ f2

f1

df|h(f)|2

Sn(|f |)= 0.99×

∫ fNy

fmin

df|h(f)|2

Sn(|f |). (20)

This is done by finding the peak of the integrand in Eq. (20)and integrating symmetrically outwards from there, in time,till the interval [f1, f2] is found. The number of waveformcycles in this interval is

Ncyc =Φ(t(f2))− Φ(t(f1))

2π, (21)

where Φ(t) is the instantaneous phase of the waveform,h+(t) − ih×(t) = A(t) e−iΦ(t), un-wrapped to be a mono-tonic function of time. We find that for a significant portionof the mass-region, the signal power is contained within 40waveform cycles. This is shown in Fig. 2, where the colorat each point gives Ncyc for that system, and the region withNcyc > 40 is excluded. Conservatively, this region is boundedbyMc = 27M�, as shown by the solid curve in the figure.

We place a bank over this region, using a stochastic methodsimilar to Ref. [71–73]. The algorithm begins by taking an

7

FIG. 3. The color at each point in the figure gives the value of FF '1 − Γbank of the bank for that binary, for the NR bank restricted toSq = {1, 2, 3, 4, 6, 8}. This is the same as the fraction of the optimalSNR, for the binary, that the template bank recovers. The black dotsshow the location of the templates in the bank. We note that they alllie along straight lines of constant q passing through the origin. Theregion shaded light-grey (towards the bottom of the figure) is wherethe FF drops sharply below 97%.

empty bank, corresponding to step 0. At step i, a proposalpoint (q,M) is picked by first choosing a value for q from therestricted set Sq = {1, 2, 3, 4, 6, 8}. The total mass M is sub-sequently sampled from the restricted interval correspondingto the pre-drawn q. The proposal is accepted if the waveformat this point has overlaps O < 0.97 with all the templates inthe bank from step i − 1. This gives the bank at step i. Theprocess is repeated till the fraction of proposals being acceptedfalls below ∼ 10−4, and the coverage fraction of the bank is& 99%. To complete the coverage, 100, 000 points are sam-pled over the region of mass space depicted in Fig. 2, and FFof the bank is computed at each point. With the islands of un-dercoverage isolated, the points sampled in these regions areadded to the bank, pushing their mass-ratios to the two neigh-boring mass-ratio from Sq . This helps accelerate the conver-gence of the bank, albeit at the cost of over-populating it, asthe algorithm for computing the FF for the sampled points isparallelizable.

We asses the effectualness of the bank, as discussed inSec. III, using Eq. (19). We draw a population of 100, 000BBH signals, uniformly from the binary mass space, and filterthem through the bank. Fig. 3 shows the FF, or the fractionof the optimal SNR recovered by the bank. The region shownis restricted to binaries with Ncyc ≤ 40. The black dots in thefigure show the position of templates in the bank. The bankrecovers ≥ 97% of the optimal SNR over the entire region ofinterest for q ≤ 10. We propose an additional simulation forq = 9.2, to increase the coverage to higher mass-ratios. Sub-stituting this for q = 8 in the set of allowed mass-ratios Sq , weplace another bank as before, with Sq = {1, 2, 3, 4, 6, 9.2}.The SNR loss from this bank is shown in Fig. 4. This bankrecovers ≥ 97% of the SNR for systems with q ≤ 11 andNcyc ≤ 40. The choice of the additional simulation at q = 9.2was made by choosing a value close-to the highest possiblevalue of q that does not lead to under-coverage in the regionbetween q = 6 and that value. The exact highest allowed

FIG. 4. This figure is similar to Fig. 3. The color at each point givesthe value of FF ' 1 − Γbank of the bank for that binary, for theNR bank restricted to Sq = {1, 2, 3, 4, 6, 9.2}. The black dots showthe location of the templates in the bank. The region shaded light-grey (towards the bottom of the figure) is where the FF drops below97%. We note that with an additional NR waveform for mass ratioq = 9.2, the coverage of the bank is extended to include binarieswith 10 ≤ q ≤ 11.

value was not chosen to reduce the sensitivity of the coverageof the bank to fluctuations in detector sensitivity.

We conclude that with only six NR waveforms for non-spinning BBHs, that are ∼ 20 orbits (or 40 GW cycles) inlength, a template bank can be constructed that is effectual fordetecting binaries with chirp mass above 27M� and 1 ≤ q ≤10. With an additional simulation for q = 9.2, this bank canbe extended to higher mass-ratios, i.e. to 1 ≤ q ≤ 11.

V. CONSTRUCTING A TEMPLATE BANK FOR NR-PNHYBRIDS

The template bank contructed in Sec. IV is effectual forGW detection searches focussed at relatively massive bina-ries withMc & 27M�. As the NR waveforms are restrictedto a small number of orbits, it is useful to consider NR-PNhybrids to bring the lower mass limit down on the templatebank. PN waveforms can be generated for an arbitrarily largenumber of inspiral orbits, reasonably accurately and relativelycheaply. Thus, a hybrid waveform comprised of a long PNearly-inspiral and an NR late-inspiral, merger, and ringdowncould also be arbitrarily long. There are, however, uncertain-ties in the PN waveforms, due to the unknown higher-orderterms. During the late-inspiral and merger phase, these termsbecome more important and the PN description becomes lessaccurate. In addition, when more of the late-inspiral is inthe detector’s sensitivity frequency range, hybrid waveformmismatches due to the PN errors become increasingly large,and reduce the recovered SNR. Thus, when hybridizing PNand NR waveforms, there must be enough NR orbits that thePN error is sufficiently low for the considered detector noise-curve. In this section, we construct an NR + PN hybrid tem-plate bank, for currently available NR waveforms, and deter-mine the lowest value of binary masses to which it covers.

The hybrids we use are constructed by joining the PN and

8

20 40 60 80 100Mtotal(Msun)

0

0.04

0.08

0.12

0.16

0.2

0.24

0.28m

ism

atch

1 2 3 4 5 6 7 8q

20

40

60

80

100

Mm

in

NR only

hybrids

mismatch = 1.5%

q = 8

q=1

FIG. 5. Bounds on mismatches of PN-NR hybrid waveforms, forthe currently existing NR simulations. The PN error is for hybridsmatched at Mωm = 0.025 for q = 1, Mωm = 0.038 for q = 2,and Mωm = 0.042 for q = 3, 4, 6, 8. The black circles indicate thelower bound of the template bank in Sec. IV. The black square showthe lower bound with a hybrid error of 1.5%. The inset shows theselower bounds as a function of mass ratio.

NR portions, as described in Sec. II C. The number of or-bits before merger at which they are joined depends on thelength of the available NR waveforms. We estimate the PNwaveform errors using hybridization mismatches ΓHyb, asdiscussed in Sec. III. Fig. 5 shows the same for all the hybrids,as a function of total mass. In terms of orbital frequency, theseare matched at Mωm = 0.025 for q = 1, Mωm = 0.038 forq = 2, and Mωm = 0.042 for q = 3, 4, 6, 8. In terms ofnumber of orbits before merger, this is 31.9 orbits for q = 1,17.8 orbits for q = 2, 16.9 orbits for q = 3, 18.4 orbits forq = 4, 21.6 orbits for q = 6, and 25.1 orbits for q = 8.The dotted line indicates a mismatch of 1.5%, a comparativelytight bound that leaves flexibility to accommodate errors dueto template bank discreteness. The black circles show the hy-brid mismatches at the lower mass bound of the NR-only tem-plate bank in Sec. IV, which are negligible. The inset showsthis minimum mass as a function of mass ratio, as well as theminimum attainable mass if we accept a hybrid error of 1.5%.At lower masses, the mismatches increase sharply with moreof the PN part moving into the Advanced LIGO sensitivityband. This is due to the nature of the frequency dependenceof the detector sensitivity. The detectors will be relativelyvery sensitive to a relatively short frequency band. As thehybridization frequency sweeps through that band, the hybriderrors rise sharply. They fall again at the lowest masses, forwhich mostly the PN portion stays within the sensitive band.

We now consider template banks viable for hybrids con-structed from currently available NR waveforms at mass ra-tios q = 1, 2, 3, 4, 6, 8. The lower mass limit, in this case,

is extended down to masses where the hybridization error ex-ceeds 3%. We demonstrate two independent methods of lay-ing down the bank grid. First, we use the stochastic place-ment method that proceeds as described in Sec. IV. The tem-plates are sampled over the total mass - mass-rato (M, q)coordinates, sampling q from the restricted set. The totalmass M is sampled from the continuous interval betweenthe lower mass limit, which is different for each q, and theupper limit of 200M�. To assess the SNR loss from thesparse placement of the templates, we simulate a population of100, 000 BBH signal waveforms, with masses sampled with3M� ≤ m1,2 ≤ 200M� and M ≤ 200M�, and filter themthrough the bank. This portion of the SNR loss needs to bemeasured with both signals and templates in the same wave-form manifold. We use the EOBNRv2 approximant [53] tomodel both, as it has been calibrated to most of the NR wave-forms we consider here, and it allows us to model waveformsfor arbitrary systems. The left panel of Fig. 6 shows the frac-tion of the optimal SNR that the bank recovers, accountingfor its discreteness alone. We observe that, with just six mass-ratios, the bank can be extended to much lower masses beforeit is limited by the restricted sampling of mass-ratios for thetemplates. For binaries with both black-holes more massivethan ∼ 12M�, the spacing between mass-ratios was found tobe sufficiently dense. The total SNR loss, after subtractingout the hybrid mismatches from Fig. 5, are shown in the rightpanel of Fig. 6. At the lowest masses, the coverage shrinksbetween the lines of constant q over which the templates areplaced, due to the hybrid errors increasing sharply. We con-clude that this bank is viable for hybrid templates for GWsearches for BBHs with m1,2 ≥ 12M�, 1 ≤ q ≤ 10, andM ≤ 200M�. Over this region the bank will recover morethan 96.5% of the optimal SNR. This is a significant increaseover the coverage allowed for with the purely-NR bank, the re-gion of coverage of which is shown in the right panel of Fig. 7,bounded at lowest masses by the magenta (solid) curve.

Second, we demonstrate a non-stochastic algorithm of bankplacement, with comparable results. We first construct sixindependent bank grids, each restricted to one of the mass-ratios q = 1, 2, 3, 4, 6, 8, and spanning the full range of totalmasses. The spacing between neighboring templates is givenby requiring that the overlap between them be 97%. We takethe union of these banks as the final two-dimensional bank.As before, we measure the SNR loss due to discreteness ofthe bank and the waveform errors in the templates separately.To estimate the former, we simulate a population of 100, 000BBH systems, and filter them through the bank. The sig-nals and the templates are both modeled with the EOBNRv2model. The left panel of Fig. 7 reveals the fraction of SNRrecovered over the mass space, accounting for the sparsity ofthe bank alone, i.e. 1−Γbank. At lower masses, we again startto see gaps between the lines of constant mass ratio which be-come significant at m1,2 ≤ 12M�. The right panel of Fig. 7shows the final fraction of the optimal SNR recovered, i.e. theFF as defined in Eq. (13b). As before, these are computed bysubtracting out the hybrid mismatches ΓHyb in addition to thediscrete mismatches, as described in Sec. III.

The efficacy of both methods of template bank construc-

9

FIG. 6. These figures show fitting factors FF obtained when using a discrete mass-ratio template bank for q = 1, 2, 3, 4, 6, 8. For eachmass-ratio, the templates are extended down to a total mass where the NR-PN hybridization mismatch becomes 3%. The bank is placed usingthe stochastic algorithm, similar to Ref. [71–73]. The black dots show the location of the templates. The fitting factor on the left plot doesnot take into account the hybridization error, and therefore shows the effect of the sparse placement of the templates alone. The right plotaccounts for the hybridization error and gives the actual fraction of the optimal SNR that would be recovered with this bank of NR-PN hybridtemplates. The region bounded by the magenta (solid) line in both plots indicates the lower end of the coverage of the bank of un-hybridizedNR waveforms. Lastly, the shaded grey dots show the points where the fitting factor was below 96.5%.

FIG. 7. These figures are similar to Fig. 6. The figures show fitting factors FF obtained when using a discrete mass-ratio template bank forq = 1, 2, 3, 4, 6, 8. For each mass-ratio, the templates are extended down to a total mass where the NR-PN hybridization mismatch becomes3%. Templates are placed independently for each mass-ratio, and span the full range of total masses. For each mass-ratio, neighboringtemplates are required to have an overlap of 97%. The union of the six single-q one-dimensional banks is taken as the final bank. The blackdots show the location of the templates. The fitting factor on the left plot does not take into account the hybridization error, and thereforeshows the effect of the sparse placement of the templates alone. The right plot accounts for the hybridization error and gives the actual fractionof the optimal SNR that would be recovered with this bank of NR-PN hybrid templates. The region bounded by the magenta (solid) line inboth plots indicates the lower end of the coverage of the bank of un-hybridized NR waveforms. Lastly, the shaded grey dots show the pointswhere the fitting factor was below 96.5%.

tion can be compared from Fig. 6 and Fig. 7. We observe thatthe final banks from either of the algorithms have very sim-ilar SNR recovery, and are both effectual over the range ofmasses we consider here. Both were also found to give a verysimilar number of templates. The uniform-in-overlap methodyields a grid with 2, 325 templates. The stochastic bank, onthe other hand, was placed with a requirement of 98% minimalmismatch, and had 2, 457 templates. This however includestemplates with m1,2 < 12M�. Restricted to provide cover-age over the region with m1,2 ≥ 12M�, 1 ≤ q ≤ 10, andM ≤ 200M�, the two methods yield banks with 627 and 667

templates respectively. The size of these banks is compara-ble to one constructed using the second-order post-NewtonianTaylorF2 hexagonal template placement method [32, 34–36],which yields a grid of 522 and 736 templates, for a minimalmatch of 97% and 98%, respectively.

Finally, we test the robustness of these results using Tay-lorT4+NR hybrids as templates. As before, we simulate apopulation of 100, 000 BBH signal waveforms. As we do nothave hybrids for arbitary binary masses, we model the signalsas EOBNRv2 waveforms. This population is filtered against abank of hybrid templates. The SNR recovered is shown in the

10

FIG. 8. This figure is similar to Fig. 7. The figures show fitting factors FF obtained when using a discrete mass-ratio template bank forq = 1, 2, 3, 4, 6, 8. Templates are placed independently for each mass-ratio, and span the range of total masses, down to the region where thehybrid errors become 3%. For each mass-ratio, neighboring templates are required to have an overlap of 97%. The union of the six single-qone-dimensional banks is taken as the final bank. The black dots show the location of the templates. The GW signals are modeled using theEOBNRv2 approximant [53], while TaylorT4+NR hybrids are used as templates. The fitting factor on the left plot shows the combined effectof the sparse placement of the templates, and the (relatively small) disagreement between the hybrid and EOBNRv2 waveforms. The right plotexplicitly accounts for the hybridization error and gives the (conservative) actual fraction of the optimal SNR that would be recovered withthis bank of NR-PN hybrid templates. The region bounded by the magenta (solid) line in both plots indicates the lower end of the coverage ofthe bank of un-hybridized NR waveforms. Lastly, the shaded grey dots show the points where the fitting factor was below 96.5%.

left panel of Fig. 8. Comparing with the left panels of Fig. 6, 7,we find that the EOBNRv2 manifold is a reasonable approx-imation for the hybrid manifold; and that, at lower masses,there is a small systematic bias in the hybrids towards EOB-NRv2 signals with slightly higher mass-ratios. The right panelof Fig. 8 shows the fraction of optimal SNR recovered af-ter subtracting out the hybrid mismatches from the left panel.The similarity of the FF distribution between the right panelsof Fig. 8 and Fig. 6, 7 is remarkable. This gives us confidencethat the EOBNRv2 model is a good approximation for testingNR/hybrid template banks, as we do in this paper; and that atemplate bank of NR+PN hybrids is indeed effectual for bina-ries with m1,2 ≥ 12M�, M ≤ 200M� and q ≤ 10.

VI. COMPLETE NR-PN HYBRID BANK FORNON-SPINNING BBH

The last sections outlined properties of template banks ofNR waveforms (and their hybrids) which are available today.We also investigate the parameter and length requirements forfuture NR simulations, that would let us contruct detectiontemplate banks all the way to M = m1 +m2 = 12M�. Thislower limit was chosen following Ref. [66, 67] which showedthat the region with M . 12M� can be covered with banksof post-Newtonian inspiral-only waveforms.

Constructing such a bank is a two-step process. First, wepick mass-ratios that allow construction of such a bank givenlong enough waveforms for these mass-ratios. Second, oneneeds to determine the necessary length of the NR portion ofthe waveforms, such that the PN-hybridization error is suffi-ciently low for all masses of interest.

To motivate the first step, Fig. 9 shows the coverage of

q (≡ m1/m2)1, 1.5, 1.75,

2, 2.25, 2.5, 2.75,3, 3.25, 3.5, 3.8,

4.05, 4.35, 4.65, 4.95,5.25, 5.55, 5.85,

6.2, 6.55,7, 7.5,8, 8.5,9, 9.6

TABLE II. List of mass-ratios, a template bank restricted to whichwill be effectual over the region of the non-spinning BBH mass spacewhere m1 +m2 & 12M� and 1 ≤ q ≤ 10. The fraction of optimalSNR recovered by such a bank, accounting for discreteness losses,remains above 98%. This is shown in Fig. 10.

banks that sample from a single mass-ratio each (from left toright: q = 1, 4, 8). We see that the resolution in q required atlower values ofM increases sharply belowM ∼ 60M�. Thisfollows from the increase in the number of waveform cycles inaLIGO frequency band as the total mass decreases, which, inturn, increases the discriminatory resolution of the matched-filter along the q axis. To determine the least set of mass-ratios which would sample the q axis sufficiently densely atlower masses, we iteratively add mass-ratios to the allowedset and test banks restricted to sample from it. We find that,constrained to the set Sq given in Table II, a template bank canbe constructed that has a minimal match of 98% at the lowestmasses. The left panel of Fig. 10 shows the loss in SNR dueto bank grid coarseness, i.e. 1 − Γbank. This loss remainsbelow 2% for mass-ratios 1 ≤ q ≤ 10, even at M = 12M�.This leaves a margin of 1.5% for the hybrid mismatches thatwould incur due to the hybridization of the NR merger wave-

11

FIG. 9. These figures show the coverage of template banks restricted to single mass-ratios, i.e. (from left to right) q = 1, 4, 8. We note thatat higher total masses, the templates are correlated to simulated signals for considerably different mass-ratios, than at lower total masses. Thisagrees with what we expect as with decreasing total mass, the number of cycles in the sensitive frequency band of Advanced LIGO increases.

FIG. 10. This figure shows fitting factors for a hybrid template bank which samples from the 26 mass ratios q = 1, 1.5, 1.75, 2, .., 9.6, andallows coverage to masses down to m1 + m2 = 12M� and 1 ≤ q ≤ 10, with a minimal-match of 98% at the lowest masses. The left andright panel show the same on M − q and m1−m2 axes, respectively. The magenta lines, in both panels, shows the upper bound in total mass,below which frequency-domain PN waveforms can be used to construct template banks for aLIGO searches [66, 67]. The dash-dotted line inthe right panel shows the lower mass limit on the smaller component object, to which a bank of currently available NR-PN hybrids can cover,i.e. min(m1,m2) = 12M� (see Sec. V). The blue (solid) curve in the right panel gives the lower mass limit to which a bank of currentlyavailable NR waveforms can cover (see Sec. IV). Thus, between the simulations listed in Table II, and frequency domain PN waveforms, wecan search for the entire range of BBH masses.

forms with long PN inspirals. The right panel of Fig. 10 showsthe same data in the m1-m2 plane. In this figure, the re-gion covered by the NR-only bank is above the blue (solid)curve, while that covered by a bank of the currently avail-able NR-PN hybrids is above the line of m2 = 12M� (withm2 ≤ m1). The region from Ref. [66, 67] that can be coveredby PN templates is in the bottom left corner, bounded by themagenta (solid) line. Our bank restricted to the set of 26 mass-ratios, as above, provides additional coverage for binaries withM ≥ 12M�, m2 ≤ 12M� and 1 ≤ q ≤ 10. Thus betweenpurely-PN and NR/NR-PN hybrid templates, we can constructeffectual searches for non-spinning BBHs with q ≤ 10.

Having the set of required mass-ratios Sq determined, weneed to decide on the length requirements for the NR simu-lations, in order to control the PN hybridization error. For aseries of matching frequencies, we construct NR-PN hybridswith Taylor{T1,T2,T3,T4} inspirals, and compute their pair-wise mismatch as a function of total mass. The maximumof these mismatches serves a conservative bound on the PN-

hybridization error for that hybrid (c.f. Eq. (17)). Fig. 11shows the results of this calculation. Each panel of Fig. 11focuses on one mass-ratio. Within each panel, each line rep-resents one matching-frequency, with lines moving down to-ward earlier hybridization with smaller mismatches. Becausethe hybridization frequency is not particularly intuitive, thelines are labeled by the number of orbits of the NR portion ofthe hybrid-waveform. For a short number of orbits this calcu-lation is indeed done with NR waveforms, whereas for largenumber of orbits, we substitute EOBNRv2 waveforms. Thedashed lines represent the earliest one can match a NR+PNhybrid given the currently available NR waveforms, and arethe same as the q = 1, 4, and 8 lines in Fig 5. The solid curvesshow the results using EOB hybrids, while the dotted curves(just barely visible) show the results with NR hybrids. Theyare virtually identical, which is a confirmation that EOB hy-brids can act as a good proxy for NR hybrids in this case. Thehorizontal dotted line indicates a mismatch of 1.5%, while thevertical dotted line shows a lower mass limit for each mass

12

10 20 30 40 50Mtotal (Msun)

0

0.05

0.1

mis

mat

ch

mismatch = 1.5%

12 Msun q = 1

31.924.3

19.4

17.4 15.213.6 12.6

10 20 30 40 50Mtotal (Msun)

0

0.1

0.2

mis

mat

ch

mismatch = 1.5%

M2 = 3Msun

q = 4

43.8

32.1

24.8 21.6

18.4

15.914.5

10 20 30 40 50Mtotal (Msun)

0

0.1

0.2

mis

mat

ch

mismatch = 1.5%

M2 = 3Msun q = 8

98.3

65.546.9

35.2

30.2

25.1

21.218.9

FIG. 11. The maximum mismatch between different PN approximants for hybrid waveforms plotted against the total mass for at differentmatching frequencies (Mωm). The dotted lines indicate a mismatch of 1.5% and a lower total mass limit, 12M� for q = 1, and M2 = 3M�for q = 4, 8. The thick dashed lines indicate the currently possible matching frequency for hybrids based on the length of NR waveforms. Thenumbers next to each line indicate the number of orbits before merger where the PN and NR (or EOB) waveforms were stitched together.

FIG. 12. This plot shows the lower mass limit of a template bank constructed with hybrid waveforms in terms of the number of NR orbits(left panel) and initial gravitational wave frequency (right panel) needed to have a PN error below 1.5% (solid curves) or 3% (dashed curves).The dotted line indicates the lower total mass limit when one component mass is 3M�.

ratio: 12M� for q = 1, which is the point at which one canconstruct a template bank with only PN inspirals, 15M� forq = 4, and 27M� for q = 8, which are the lower mass limitsif both component masses are ≥ 3M�.

Fig. 12 presents the information obtained in the previousparagraph in a different way. Given NR-PN hybrids with Norbits of NR, the shaded areas in the left panel of Fig. 12 in-dicate the region of parameter space for which such hybridshave hybridization errors smaller than 1.5%. As before, wesee that for high masses, comparatively few NR orbits are suf-ficient (e.g. the purple N = 15 region), whereas lower totalmasses require increasingly more NR orbits. The dashed lines

indicate the region of parameter space with hybrid error be-low 3%. The black dotted line designates the point where onecomponent mass is greater than 3M�, which is a reasonablelower mass limit for a physical black hole. The right panelshows this same analysis instead with initial GW frequencyindicated by the solid and dashed lines. Thus, for the regionof parameter space we’re interested in, no more than ∼ 50NR orbits, or an initial GW frequency of Mω = 0.025 wouldbe necessary to construct a detection bank with hybrid mis-matches below 1.5%.

13

VII. CONCLUSIONS

The upgrades currently being installed to increase the sensi-tivity of the ground based interferometric gravitational-wavedetectors LIGO and Virgo [1, 2] are scheduled to completewithin the next two years. The second generation detectorswill have a factor of 10 better sensitivity across the sensitivefrequency band, with the lower frequency limit being pushedfrom 40Hz down to∼ 10Hz. They will be able to detect GWsfrom stellar-mass BBHs up to distances of a few Gpc, with theexpected frequency of detection between 0.4−1000 yr−1 [5].

Gravitational-wave detection searches for BBHs operate bymatched-filtering the detector data against a bank of mod-eled waveform templates [31–36, 106]. Early LIGO-Virgosearches employed PN waveform template banks that spannedonly the inspiral phase of the coalescence [16–20]. Recentwork has shown that a similar bank of PN templates would beeffectual for the advanced detectors, to detect non-spinningBBHs with m1 + m2 . 12M� [66, 67]. Searches from theobservation period between 2005 − 07 and 2009 − 10 em-ployed templates that also included the late-inspiral, mergerand ringdown phases of binary coalescence [21, 22].

Recent advancements in Numerical Relativity have led tohigh-accuracy simulations of the late-inspiral and mergers ofBBHs. The multi-domain SpEC code [47] has been usedto perform simulations for non-spinning binaries with mass-ratios q = 1, 2, 3, 4, 6, 8 [48–50]. Owing to their high com-putational complexity, the length of these simulations variesbetween 15−33 orbits. Accurate modeling of the late-inspiraland merger phases is important for stellar mass BBHs, asthey merge at frequencies that the advanced detectors wouldbe sensitive to [67]. Analytic models, like those withinthe Effective-One-Body formalism, have been calibrated tothe NR simulations to increase their accuracy during thesephases [51–54]. Other independent models have also beendeveloped using information from NR simulations and theirhybrids [60, 61, 107, 108]. An alternate derived prescriptionis that of NR+PN hybrid waveforms, that are constructed byjoining long PN early-inspirals and late-inspiral-merger sim-ulations from NR [55–59].

NR has long sought to contribute template banks forgravitational-wave searches. Due to the restrictions on thelength and number of NR waveforms, this has been conven-tionally pursued by calibrating intermediary waveform mod-els, and using those for search templates. In this paper, we ex-plore the alternative of using NR waveforms and their hybridsdirectly in template banks. We demonstrate the feasibility ofthis idea for non-spinning binaries, and extending it to spin-ning binaries would be the subject of a future work. We findthat with only six non-spinning NR simulations, we can coverdown to m1,2 & 12M�. We show that with 26 additionalNR simulations, we can complete the non-spinning templatebanks down to M ' 12M�, below which existing PN wave-forms have been shown to suffice for aLIGO. From templatebank accuracy requirements, we are able to put a bound on therequired length and initial GW frequencies for the new sim-ulations. This method can therefore be used to lay down theparameters for future simulations.

First, we construct a bank for using pure-NR waveforms astemplates, using a stochastic algorithm similar to Ref. [71–73]. The filter templates are constrained to mass-ratiosfor which we have NR simulations available, i.e. q =1, 2, 3, 4, 6, 8. We assume that the simulations available tous are ≥ 20 orbits in length. To test the bank, we simulatea population of 100, 000 BBH signals and filter them throughthe bank. The signals and templates are both modeled withthe EOBNRv2 model [53]. We demonstrate that this bank isindeed effectual and recovers ≥ 97% of the optimal SNR forGWs from BBHs with mass-ratios 1 ≤ q ≤ 10 and chirp-massMc ≡ (m1 +m2)−1/5(m1m2)3/5 above 27M�. Fig. 3shows this fraction at different simulated points over the bi-nary mass space. With an additional simulation for q = 9.2,we are able to extend the coverage to higher mass-ratios.We show that a bank viable for NR waveform templates forq = 1, 2, 3, 4, 6, 9.2, would recover ≥ 97% of the optimalSNR for BBHs with 10 ≤ q ≤ 11. The SNR recovery frac-tion from such a bank is shown in Fig. 4.

Second, we construct effectual banks for currently avail-able NR-PN hybrid waveform templates. We derive a boundon waveform model errors, which is independent of analyti-cal models and can be used to independently assess the errorsof such models (see Sec. III for details). This allows us toestimate the hybrid waveform mismatches due to PN error,which are negligible at high masses, and become significantat lower binary masses. We take their contribution to the SNRloss into account while characterizing template banks. Forhybrid banks, we demonstrate and compare two independentalgorithms of template bank construction. First, we stochas-tically place a bank grid, as for the purely-NR template bank.Second, we lay down independent sub-banks for each mass-ratio, with a fixed overlap between neighboring templates,and take their union as the final bank. To test these banks,we simulate a population of 100, 000 BBH signals and filterthem through each. We simulate the GW signals and the tem-plates using the recently developed EOBNRv2 model [53].The fraction of the optimal SNR recovered by the two banks,before and after accounting for the hybrid errors, are shownin the left and right panels of Fig. 6 and Fig. 7 (respectively).We observe that for BBHs with m1,2 ≥ 12M� hybrid tem-plate banks will recover ≥ 96.5% of the optimal SNR. Fortesting the robustness of our conclusions, we also test thebanks using TaylorT4+NR hybrid templates. The SNR re-covery from a bank of these is shown in Fig. 8. We concludethat, the currently available NR+PN hybrid waveforms canbe used as templates in a matched-filtering search for GWsfrom BBHs with m1,2 ≥ 12M� and 1 ≤ q ≤ 10. Thenumber of templates required to provide coverage over thisregion was found to be comparable to a bank constructedusing the second-order post-Newtonian TaylorF2 hexagonaltemplate placement method [32, 34–36]. The two algorithmswe demonstrate yield grids of 667 and 627 templates, respec-tively; while the metric based placement method yields a gridof 522 and 736 templates, for 97% and 98% minimal match,respectively.

At lower mass, the length of the waveform in the sensitivefrequency band of the detectors increases, increasing the reso-

14

lution of the matched-filter. We therefore see regions of under-coverage between mass ratios for which we have NR/hybridtemplates (see, e.g. Fig. 7 at the left edge). For M . 12M�,existing PN waveforms were shown to be sufficient for aLIGOsearches. We find the additional simulations that would beneeded to extend the hybrid tempalte bank down to 12M�.We show that a bank of hybrids restricted to the 26 mass-ratios listed in Table II would be sufficiently dense at 12M�.This demonstrates that the method proposed here can be usedto decide which NR simulations should be prioritized for thepurpose of the GW detection problem. By filtering a popu-lation of 100, 000 BBH signals through this bank, we showthat the SNR loss due to its discreteness stays below 2% overthe entire relevant range of masses. The fraction of optimalSNR recovered is shown in Fig. 10. Constraining the detectionrate loss below 10% requires that detection template banks re-cover more than 96.5% of the optimal SNR. Therefore ourbank would need hybrids with hybridization mismatches be-low 1.5%. From this accuracy requirement, we obtain thelength requirement for all the 26 simulations. This is depictedin the left panel of Fig. 12, where we show the region of themass space that can be covered with hybrids, as the length oftheir NR portion varies. We find that for 1 ≤ q ≤ 10 thenew simulations should be about 50 orbits in length. In theright panel of Fig. 12 we show the corresponding initial GWfrequencies. The requirement of ∼ 50 orbit long NR simula-tions is ambitious, but certainly feasible with the current BBHsimulation technology [69].

In summary, we refer to the right panel of Fig. 10. The re-gion above the dashed (red) line and above the solid (blue)line can be covered with a bank of purely-NR waveforms cur-rently available. The region above the dashed (red) and thedash-dotted (black) line can be covered with the same sim-ulations hybridized to long PN inspirals. With an additionalset of NR simulations summarized in Table II, the coverage ofthe bank can be extended down to the magenta (solid) line inthe lower left corner of the figure. Thus between hybrids andPN waveforms, we can cover the entire non-spinning BBHspace. The ability to use hybrid waveforms within the soft-ware infrastructure of the LIGO-Virgo collaboration has beendemonstrated in the NINJA-2 collaboration [70]. The tem-plate banks we present here can be directly used in aLIGOsearches. This work will be most useful when extended toaligned spin and precessing binaries [109, 110], which is thesubject of a future work.

The detector noise power is modeled using the zero-detuning high-power noise curve for Advanced LIGO [101].The construction of our template banks is sensitive to thebreadth of the frequency range that the detector would be sen-sitive to. The noise curve we use is the broad-band final designsensitivity estimate. For lower sensitivities at the low/high

frequencies, our results would become more conservative, i.e.the template banks would over-cover (and not under-cover).

We finally note that in this paper we have only consideredthe dominant (2, 2) mode of the spherical decomposition ofthe gravitational waveform. For high mass-ratios and highbinary masses, other modes would also become important,both for spinning as well as non-spinning black hole bina-ries [67, 111, 112]. Thus, in future work, it would be rele-vant to examine the sub-dominant modes of the gravitationalwaves. Lastly, though we have looked at the feasibility ofusing this template bank for Advanced LIGO as a single de-tector, this instrument will be part of a network of detectors,which comes with increased sensitivity and sky localization.For this reason, in subsequent studies it would be useful toconsider a network of detectors.

ACKNOWLEDGMENTS

We thank Steve Privitera for useful code contributions andIan Harry, Alex Nitz, Stefan Ballmer and the Gravitational-Wave group at Syracuse University for productive discus-sions. We also thank Thomas Dent for carefully readingthrough the manuscript and providing feedback. DAB, PKand HPP are grateful for hospitality of the TAPIR groupat the California Institute of Technology, where part of thiswork was completed. DAB and PK also thank the LIGOLaboratory Visitors Program, supported by NSF coopera-tive agreement PHY-0757058, for hospitality during the com-pletion of this work. KC, IM, AHM and HPP acknowl-edge support by NSERC of Canada, the Canada Chairs Pro-gram, and the Canadian Institute for Advanced Research. Wefurther acknowledge support from National Science Foun-dation awards PHY-0847611 (DAB and PK); PHY-0969111and PHY-1005426 (MB, LEK); and PHY-1068881, PHY-1005655, and DMS-1065438 (MAS, BS, AZ). We are gratefulfor additional support through a Cottrell Scholar award fromthe Research Corporation for Science Advancement (DAB)and from the Sherman Fairchild Foundation (MB, LEK, MAS,BS, AZ). Simulations used in this work were performed withthe SpEC code [47]. Calculations were performed on theZwicky cluster at Caltech, which is supported by the ShermanFairchild Foundation and by NSF award PHY-0960291; on theNSF XSEDE network under grant TG-PHY990007N; on theSyracuse University Gravitation and Relativity cluster, whichis supported by NSF awards PHY-1040231 and PHY-1104371and Syracuse University ITS; and on the GPC supercomputerat the SciNet HPC Consortium [113]. SciNet is funded by: theCanada Foundation for Innovation under the auspices of Com-pute Canada; the Government of Ontario; Ontario ResearchFund–Research Excellence; and the University of Toronto.

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