Template banks to search for compact binaries with spinning components in gravitational wave data
Abstract
Gravitational waves from coalescing compact binaries are one of the most promising sources for detectors such as LIGO, Virgo and GEO600. If the components of the binary posess significant angular momentum (spin), as is likely to be the case if one component is a black hole, spininduced precession of a binary’s orbital plane causes modulation of the gravitationalwave amplitude and phase. If the templates used in a matchedfilter search do not accurately model these effects then the sensitivity, and hence the detection rate, will be reduced. We investigate the ability of several search pipelines to detect gravitational waves from compact binaries with spin. We use the postNewtonian approximation to model the inspiral phase of the signal and construct two new template banks using the phenomenological waveforms of Buonanno, Chen and Vallisneri [A. Buonanno, Y. Chen and M. Vallisneri, Phys. Rev. D67, 104025 (2003)]. We compare the performance of these template banks to that of banks constructed using the stationary phase approximation to the nonspinning postNewtonian inspiral waveform currently used by LIGO and Virgo in the search for compact binary coalescence. We find that, at the same false alarm rate, a search pipeline using phenomenological templates is no more effective than a pipeline which uses nonspinning templates. We recommend the continued use of the nonspinning stationary phase template bank until the false alarm rate associated with templates which include spin effects can be substantially reduced.
pacs:
04.30.Db,04.25.Nx,04.80.Nn,95.55.Ymtoday
I Introduction
In 20052007, the Laser Interferometer GravitationalWave Observatory (LIGO) recorded two years of data at design sensitivity Abbott et al. (2007) and the LIGO, Virgo Acernese et al. (2005) and GEO600 Willke (2007) detectors now form a worldwide network of broadband gravitationalwave observatories. The LIGO and Virgo detectors are scheduled to resume operations in summer 2009 with a factor of – sensitivity increase over previous observations. The gravitational waves emitted during the inspiral and merger of binaries containing neutron stars (NS) and/or black holes (BH) are a primary target of this network. Binary neutron stars (BNS) can be observed up to 35 Mpc (70 Mpc) in the Initial (Enhanced) LIGO detectors and up to 450 Mpc in the Advanced LIGO detectors, which will begin observations in 2015 Abbott et al. (2009a). Binary black holes (BBH) with components should be visible at Mpc (350 Mpc) in the Initial (Enhanced) LIGO detectors, increasing to 2 Gpc in Advanced LIGO Abbott et al. (2009a). Population synthesis calculations constrained by radio observations of BNS systems containing pulsars predict BNS detection rates between – for Enhanced LIGO and – for Advanced LIGO, with the most likely values being and , respectively Kopparapu et al. (2008); Kalogera et al. (2004); Abbott et al. (2009a). Much less is known about the detection rates of BBH and NSBH coalescences, although it is plausible that Enhanced (Advanced) LIGO rates could be as high as for NSBH binaries and for BBH O’Shaughnessy et al. (2008); O’Shaughnessy et al. (2005a); Abbott et al. (2009a).
The sensitivities listed in the preceding paragraph are optimal: they assume accurate knowledge of the signal waveform in order to construct matched filters which can extract gravitationalwave signals buried in the noisy detector data Wainstein and Zubakov (1962); Thorne (1987). The gravitational waveform from the inspiral of two compact objects has been calculated using the postNewtonian (PN) approximation, which uses the characteristic velocity of the binary as an expansion parameter Blanchet et al. (1995a); Will and Wiseman (1996); Blanchet et al. (1996); Blanchet (1996); Blanchet et al. (1995b, 2002a, 2005a, 2004a); Arun et al. (2004); Blanchet et al. (2008); Kidder et al. (1993); Kidder (1995); Faye et al. (2006); Blanchet et al. (2006); Arun et al. (2008). Ongoing comparisons of PN waveforms with numerical simulations of binary black holes have thus far confirmed the accuracy of the PN solution in the late stages of inspiral Hannam et al. (2008); Buonanno et al. (2007a); Boyle et al. (2008), although optimal searches for sources with total mass in the firstgeneration detectors require waveforms that also model the merger and ringdown Buonanno et al. (2007b, 2009). If the components of the binary have negligible instrinsic angular momentum (spin) then it is straightforward to construct a bank of matched filters, parameterized by the two component masses of the binary, and use these filters to search for signals Owen (1996); Owen and Sathyaprakash (1999); Babak et al. (2006); Allen et al. (2005). However, if the components of a binary are spinning, then these spins can couple with the orbital angular momentum of the binary and with each other to cause amplitude and phase modulation of the gravitational waveform Apostolatos et al. (1994). Attempting to detect gravitational waves from spinning binaries with nonspinning templates will result in a suboptimal search and a corresponding reduction in the detection rate Apostolatos (1995, 1996). Since it is possible that a large fraction of astrophysical black holes have considerable spin Kalogera (2000); O’Shaughnessy et al. (2005b), it is important to consider the effect of spin in searches for gravitational waves from BBH and NSBH coalescences. However, optimal filters for spinning binaries are characterized by a much larger number of parameters than the ones for nonspinning binaries, complicating placement of filters in the bank and considerably increasing the computational cost of searches.
To mitigate the computational problem without compromising the sensitivity of the search, a phenomenological family of templates was proposed by Buonanno, Chen and Vallisneri Buonanno et al. (2003a) (we refer to these templates as BCVspin). Filters constructed from these templates are described by only four parameters and have good overlaps with the full PN waveforms Buonanno et al. (2003a). Moreover, constructing a bank of filters using BCVspin waveforms is straightforward, if cumbersome Buonanno et al. (2005); Abbott et al. (2008a). The first searches for binary black hole signals in LIGO data used nonspinning templates Abbott et al. (2006a, 2008b), however BCVspin templates were recently used to search for BBH and NSBH signals with spin in data from the third LIGO science run Abbott et al. (2008a). The sensitivity of the search described in Abbott et al. (2008a) was not as good as the main results of Ref. Buonanno et al. (2003a) might suggest. This was primarily due to the the response of the BCVspin template to the nongaussian noise transients present in real gravitationalwave detector data and the increase in the number of degrees of freedom associated with the detection statistic (due to the larger search parameter space) Abbott et al. (2008a). This was already anticipated in Buonanno et al. (2003a); here we provide a quantitative analysis.
In this paper, we present an improvement to the search pipeline described in Abbott et al. (2008a), by constructing banks that are much better suited to the BCVspin template family. We compare the sensitivity of this search to the search for gravitational waves from compact binary coalescence with nonspinning templates in LIGO data Babak et al. (2006); Allen et al. (2005). Our main conclusion is that while the BCVspin templates have rather good overlaps with the target waveforms, the current search pipeline needs further improvements before any gains from these increased overlaps can be realized. The false alarm rate of BCVspin filters in real detector data is larger than that of a nonspinning search. This makes a search using BCVspin templates less sensitive than a nonspinning search when looking for binaries with spin, since one has to use a higher detection threshold to obtain the same false alarm rate. The results of this work were used to guide the decision not to implement the BCVspin search on data from the fifth LIGO science run and instead to use nonspinning filters to search for binaries with spin Abbott et al. (2009b). The motivation for this decision was summarized in an appendix of Ref. Abbott et al. (2009b) and this paper can be seen as a companion to that work. Here we present a detailed account of how the BCVspin banks were constructed, and how the comparisons between the BCVspin and nonspinning searches were performed.
This paper is organized as follows. In Sec. II we give a description of our target signals, which are postNewtonian waveform models for signals from spinning black hole binaries, followed by a summary, in Sec. III, of the phenomenological BCVspin templates of Ref. Buonanno et al. (2003a). In Sec. IV we review the construction of the template bank used in Ref. Abbott et al. (2008a) and present two new methods to construct BCVspin template banks, relaxing the assumptions used in Ref. Abbott et al. (2008a): a “squarehexagonal” placement which generalizes the hexagonal placement developed in Cokelaer (2007) to a higherdimensional template manifold, and a stochastic placement technique proposed in Allen et al. (2009). In Sec. V we compute matches of these banks with target waveforms and compare the results with those obtained from twodimensional template banks based on the stationary phase approximation (SPA) Sathyaprakash and Dhurandhar (1991); Droz et al. (1999) and hexagonal template placement used in LIGO’s nonspinning searches Abbott et al. (2004, 2005a, 2005b, 2006a, 2006b, 2008b, 2009b). We compare the detection efficiency of the spinning banks with that of SPA banks in Sec. VI followed by concluding remarks in Sec. VII. Throughout this paper, we set unless otherwise stated.
Ii PostNewtonian waveforms from spinning binaries
Depending on their birth spins, BHs in binaries could accumulate significant spin through accretion Kalogera (2000); Belczynski et al. (2007). There is much uncertainty concerning the equation of state of a neutron star, but most models place an upper limit on the spin, above which the star would break up Cook et al. (1994). There is also an upper limit for the spin of a black hole due to torque caused by radiation from the accretion disk getting swallowed by the BH, leading to an expected bound of Thorne (1974). Most of the modeling of spin evolution in compact binaries has been confined to NSBH systems, in which case the spin tilt with respect to the orbital angular momentum can be considerable Kalogera (2000); Belczynski et al. (2007); this may also be the case for BBHs.
PN theory has achieved great success in modeling the adiabatic, quasicircular
phase of inspiral, during which the fractional change in the orbital frequency
over each orbital period will be negligible (see, e.g.,
Ref. Blanchet (2002) for a review). The orbital phasing has been
calculated to order (or 3.5PN in the usual
notation) Damour et al. (2001); Blanchet
et al. (1995b, a); Will and Wiseman (1996); Blanchet et al. (1996); Blanchet (1996); Blanchet
et al. (2002a, b); Damour et al. (2000); Blanchet
et al. (2004a, b); Blanchet and Iyer (2005); Blanchet
et al. (2005b, a)
while the gravitationalwave amplitude for nonspinning binaries has been calculated
to order Blanchet et al. (1996); Arun et al. (2004); Blanchet et al. (2008). The effect of spin on the
gravitationalwave phasing is known to order
Kidder et al. (1993); Kidder (1995); Faye et al. (2006); Blanchet et al. (2006) and
to order for the amplitude Arun et al. (2008). However,
since the matched filter is most sensitive to the phase evolution of the
binary, template waveforms amplitudes are typically computed only at leading
order in amplitude (the restricted waveform). Spinorbit
interaction enters the phasing at 1.5PN and 2PN order and spinspin
interaction at 2PN order. Spin effects influence the evolution of the orbital
frequency as a function of time. Including these effects, the adiabatic
evolution of the orbital frequency is given by
where is a unit vector in the direction of orbital angular momentum (and hence the unit normal to the orbital plane of the binary), are the spins, with the component masses, , is the total mass, and the symmetric mass ratio. is the EulerMascheroni constant. The spins and the direction of the orbital angular momentum evolve according to Kidder (1995); Apostolatos et al. (1994)
(2)  
(3)  
(4) 
where . The dynamics of the binary is governed by the nonlinear, coupled differential equations (LABEL:omegadot)–(4). It will not be possible to solve these exactly, but they can easily be treated numerically.
By numerically evolving one can obtain the orbital phase,
(5) 
which can be substituted into the usual expressions for the restricted PN waveform polarizations Wahlquist (1987). In the case of spinning binaries, we need to take into account the timedependence of the amplitudes through the inclination of the orbit with respect to the observer. The plus and cross polarizations of the gravitational wave are given by
(6) 
where the unit vector points from source to detector. The detector strain is
(7) 
where are detector antenna factors which depend on the the right ascension and declination of the source and a timedependent polarization angle (see, e.g., Ref. Apostolatos et al. (1994)).
As suggested by Eq. (4), the direction of orbital angular momentum and hence the plane of the inspiral will undergo precession, the effect being more pronounced for asymmetric systems. It will also be more prominent if the spins are large, and if they are significantly misaligned with the orbital angular momentum. The time evolution of spins and angular momentum will affect the phasing of the waveform through Eqns. (LABEL:omegadot)(5), and the precession of the orbital plane will modulate the amplitudes of the wave polarizations in Eq. (6). The waveforms given by Eqns. (LABEL:omegadot)–(7) will be the “targetsignal waveforms” for testing our template banks.
Iii A detection template family for spinning black hole binaries
The frequencydomain phenomenological detection template family proposed in Ref. Buonanno et al. (2003a) is designed to capture spininduced amplitude and frequency modulation in an approximate way. Specifically, for gravitationalwave frequencies , the BCVspin template is
(8) 
Here is the time of arrival, is the usual Heaviside step function and is an upper cutoff frequency beyond which the waveform is unlikely to be close to a true signal (due to breakdown of the adiabatic approximation to the inspiral regime). The detection statistic will be maximized analytically over the parameters in the linear combination (8), as well as over ; these parameters are referred to as extrinsic parameters because they do not need to be explicitly searched over.
The waveforms , are basis templates, which take the form
(9) 
where
(10) 
and captures the effect of spininduced amplitude modulation. The
(nonmodulated) phase takes the form
(11) 
It will not be possible to analytically maximize the detection statistic over the parameters , , and , and these must be explicitly searched over using a bank of templates; they are referred to as intrinsic parameters.
It will often be useful to approximately identify the intrinsic parameters with the physical masses and spins of a compact binary. By relating and to the 0PN and 1.5PN phase coefficients Arun et al. (2005), one has the correspondences
(12) 
Similarly, the parameter can be related to the rate of precession by Apostolatos et al. (1994)
(13) 
We stress that these mappings are only approximate, and for a given physical signal, the detection template that matches best may correspond to values of that differ significantly from the ones suggested by the identifications above.
The identifications (12) allow us to make a choice for . In the limit where one component mass goes to zero while total mass remains fixed, and assuming zero spins, the frequency of last stable orbit (LSO) of a test mass in the Schwarzschild spacetime is given by . For simplicity we set , where is computed from the correspondence (12).
Next one constructs an orthonormal basis from the basis templates (9) with respect to the usual inner product for waveforms , on the template manifold given by
(14) 
where tilde denotes a quantity computed directly in the frequency domain (as in the case of the BCVspin templates) or the Fourier transform of a timedomain quantity (such as the waveforms given in Eq. (7)). is the onesided power spectral density (PSD) of the detector data, and is some lower cutoff frequency associated with the detector; in the case of initial LIGO one sets . The orthonormalization of the basis templates can be effected using the GramSchmidt procedure as in Abbott et al. (2008a). In addition one demands that the templates themselves are normalized (denoted by ): . This leads to the requirement
(15) 
where the , , are the coefficients of when expressed into the orthonormal basis of templates resulting from the GramSchmidt procedure.
Finally, the signaltonoise ratio (SNR), which is used as the BCVspin detection statistic, is given by
(16) 
where represents the detector data stream, and the maximization over the is subject to the constraint (15).
Iv Template banks for spinning binaries
The template waveforms may not exactly model gravitationalwave signals . The loss in SNR due to differences between the template and signal waveforms is quantified by the fitting factor Apostolatos et al. (1994). If is a signal waveform and a template waveform, then
(17) 
where hat denotes normalization: . is the fractional loss in SNR resulting from the use of suboptimal template waveforms rather than the true signal waveforms. Since we do not a priori know the intrinsic parameters of any gravitationalwave signals we may detect, we decide on a target signal space and construct a discrete bank of templates to search for signals in this space. If is a normalized template waveform in the discrete bank and is a normalized waveform from the space used to construct the bank, then the minimum match of the bank is defined to be Owen (1996)
(18) 
A typical choice for the minimum match in gravitational wave searches is . When measuring the performance of a template bank, we are interested in the effective fitting factor given by Lindblom et al. (2008)
(19) 
If the signal waveforms are identical to those used to construct the bank, then the effective fitting factor will be bounded below by the minimum match. In practice, the true gravitationalwave signals will differ from the templates used to construct the bank, so the effective fitting factor may be smaller than the minimum match. The larger the effective fitting factor, the better the bank is at capturing the target signals.
If the parameters between two (normalized) templates differ by a small amount , the loss in SNR can be related to a distance defined by a metric given by
(20) 
where
(21) 
The standard method of constructing a template bank then consists of computing this metric in the intrinsic parameter space of a waveform family and using it to place templates such that the distance between any template waveform and the nearest template in the bank is greater than the desired minimum match . In searches for nonspinning binaries, the intrinsic parameters of the templates are the just component masses of the binary. In practice, we reparameterize the waveforms using the chirp times Owen and Sathyaprakash (1999). With respect to these variables the metric is almost Euclidean, and so template placement using the metric becomes a straightforward twodimensional hexagonal packing problem Babak et al. (2006).
As described in Sec. I, a search for gravitational waves using the BCVspin templates has been performed in S3 LIGO data Abbott et al. (2008a). The metric used in that analysis was computed using the “strong modulation approximation” where one assumes that the binary precesses many times while emitting in the most sensitive part of the detector’s band. This allows one to treat the basis templates of Eq. (9) as orthonormal, simplifying the calculation of the metric. However, the resulting template banks were only appropriate for fairly lowmass, asymmetric systems. We now present an improved algorithm for constructing a metric in which the assumptions of Abbott et al. (2008a) are dropped. In our case, the parameters of the waveform are
(22) 
The detection statistic can be maximized over the extrinsic parameters and , which, as shown in Pan et al. (2004), leads to a projected metric which only measures distances in the directions. However, the components of will still depend on the . This residual dependence on extrinsic parameters can be removed as follows:

Introduce some fiducial distance ;

Specify a large number of unit vectors (in the coordinate sense) in space;

For each , numerically maximize the metric length computed from , over values of the consistent with the constraint (15); i.e.,
(23) 
Rescale each vector by defining a new vector ;

Fit an ellipsoid in parameter space to the vectors ;

Define an “effective” metric by requiring that any point on the ellipsoid is at effective metric distance from the template we started with.
Note that this construction is independent of the fiducial length scale . In what follows, is the metric we will use to satisfy the criterion (18) through the relationship (20). An property of is that it is essentially independent of and and only has a weak dependence on .
It is important to note that given a short straight line segment in coordinate space with coordinate length , by construction associates almost the largest possible metric length to it consistent with the family of metrics parametrized by the . When generating template banks, in practice one specifies a minimum match which will then be used together with the metric to determine the spacing of templates. Since is too conservative in assigning lengths, neighboring templates will tend to have a larger match than needed, and the true minimum match defined by (18) will always be significantly larger than what was originally intended. As we shall see below, setting an a priori value of will be more than enough for a bank to obtain high overlaps () with target waveforms.
We would like to capture signals from binaries whose component masses are in the interval , with total masses . We do not need to worry about capturing BNS signals, since spin does not have a significant effect on waveforms from those sources. However, our template bank should have good overlap with NSBH and BBH signals. Taking neutron star masses to lie between and and black hole masses to be larger than , we impose . To capture these signals, we want an appropriately chosen bounding box in within which to place templates. Such a box can be specified using the correspondences (12)–(13). The suggested intervals for are then roughly
(24) 
where the upper bound for has been chosen generously. As to , the correspondence (13) suggests that should suffice, but to have good matches with a variety of physical signals, here too it turned out to be better to have a larger upper bound:
(25) 
We now present two methods for constructing template banks for BCVspin templates which cover this space.
iv.1 Squarehexagonal template bank
The metric depends only on , so it is natural to first define layers of constant , with a spacing determined by the minimum match. Within each of the twodimensional layers one can then lay out templates in a hexagonal pattern, which is the optimal placement in two dimensions. We will refer to this kind of placement as squarehexagonal. The construction of this bank is analogous to that described in Ref. Cokelaer (2007) which was used to construct template banks for search for binary black holes in data from the third and fourth LIGO science runs Abbott et al. (2008b) using nonspinning phenomenological templates Buonanno et al. (2003b). For the BCVspin templates, we have a 3metric, which in each layer is diagonalized by going to a new set of coordinates , where . After that a hexagonal placement in can be performed as in Cokelaer (2007). As explained above, the metric is overly conservative in specifying distances between templates, and setting an a priori minimum match of will suffice to obtain high matches with target waveforms.
iv.2 Stochastic template bank
We now consider a different bank placement for BCVspin, which we hope will reduce the overcoverage of the parameter space that is unavoidable with the squarehexagonal placement method defined above. This will lead to a smaller number of templates but will yield the same or better matches with target waveforms, and similar efficiencies. This template bank is created by the placement of a large number of randomly distributed templates, followed by a “pruning” stage in which unnecessary templates are discarded. This method is described in Allen et al. (2009) and summarized below. Other, similar methods for creating stochastic template banks were proposed in Babak (2008) and Messenger et al. (2008).
The stochastic placement algorithm we wish to use for BCVspin is very simple. We begin by generating a very large number of points in the parameter space, far more than would be needed to fill the space. We then iteratively cycle through these points, retaining a point only if it is not closer than some predefined metric distance to the points retained in previous iterations. The remaining points form our stochastically generated bank. Tests have shown Allen et al. (2009) that one should begin with at least points, where would be the number of templates remaining after filtering, to have a good coverage of the parameter space after “pruning”.
In testing this algorithm against lattice placement algorithms it was found Allen et al. (2009) that in a 2dimensional Cartesian space the stochastic algorithm produced a template bank with 1.5 times the number of templates that a square lattice algorithm would have generated. However, in the case of a 2dimensional nonspinning (nonCartesian) SPA bank (as described above) the stochastic algorithm was found to place % less templates than the square lattice algorithm and only % more templates than the hexagonal lattice placement, while achieving a similar degree of coverage. We emphasize here that this stochastic placement algorithm would be of most use in parameter spaces with more than 2 dimensions, where lattice placement becomes significantly subobtimal.
For the specific case of BCVspin the templates are sprinkled randomly over a rectangular box in space using the same bounding box as in the previous subsection. An estimate for the number of templates that will be needed is provided by the invariant volume of the box, divided by the volume taken up by an individual template:
(26) 
Once again it will suffice to set an a priori minimum match (i.e., setting the defined above to 0.2). Given the box in parameter space specified by (24,25), the number of sprinkled templates should then be about 500,000. When using a larger number of initial templates, we find that the final number of templates after pruning does not change significantly. With the Initial LIGO design PSD, the number of templates for stochastic BCVspin banks with is about 8000; SPA banks with have templates, and for BCVspin with squarehexagonal placement and more than templates are obtained (see Table 1).
V Comparison of BCVspin banks with spinning PN signals
We now study the performance of our banks against the target waveforms of Sec. II. In particular, for a variety of target waveforms corresponding to different masses and initial spins, we compute the effective fitting factor of the bank for the target waveforms, as given by Eq. (19).
Fig. 1 compares the effective fitting factor of templates in squarehexagonal and stochastic BCVspin banks with those of a nonspinning SPA bank. There is no discernable difference between squarehexagonal and stochastic placements, but both differ significantly from the nonspinning SPA bank. As one would expect, the difference is largest for binaries with large mass ratios, although there is improvement also for a variety of other target waveform masses. Depending on masses and spins, for the same target waveform , the difference in can be more than . The medians and means for the effective fitting factors are summarized in Table 1. We find that BCVspin with stochastic bank placement has marginally better effective fitting factors than the squarehexagonal bank, and both BCVspin banks have noticably higher effective fitting factors than the nonspinning bank. Given the small difference between the stochastic and squarehexagonal BCVspin banks we will subsequently only consider differences between the stochastic BCVspin bank and the nonspinning SPA bank.
Template  Bankplacement  SNR threshold  Minimum match  Number of templates  

SPA  SPA  5.5  0.95  11832  0.89  0.86 
BCVspin  Squarehexagonal  8  0.8  16431  0.96  0.92 
BCVspin  Stochastic  8  0.8  7913  0.96  0.93 
Vi Search performance of BCVspin template banks
The effective fitting factor of a target waveform over a template bank as defined in Eq. (19) indicates how similar the templates are to physical signals, but when searching for gravitationalwave signals in real detector data, other factors also come into play. The effective fitting factor of a template bank gives us a measure of how the signaltonoise ratio is reduced by not filtering with the true signal waveform, but to detect a signal we must be able to distingusih it from background noise in the detector. To determine the overall performance of a template bank, we have to consider both the effective fitting factor and the false alarm rate of the bank, i.e., the response of the filters to noise (both Gaussian and transient) in the detector. Once we establish the false alarm rate of a search, we measure the performance of a bank by its efficiency, i.e., the bank’s ability to find simulated target waveforms injected in the noise at a given false alarm rate. We will establish the false alarm rates and efficiencies of BCVspin and SPA banks by means of the dataanalysis pipeline used in searches by the LIGO Scientific Collaboration (LSC) for inspiral signals Abbott et al. (2008b, a, 2009b), which is available in the LSC Algorithm Library lal (). More details on this pipeline can be found in Ref. The LIGO Scientific Collaboration (2007).
vi.1 False alarm rates
The BCVspin detection statistic (16) involves maximization over six parameters (, and the with the constraint ), to be compared with only two for SPA. It should also be noted that the BCVspin detection template family consists of waveforms that are only approximate. As we shall see below, the larger number of degrees of freedom will make the BCVspin banks more prone to detecting instrumental noise transients with high SNR. Both for SPA and BCVspin one needs to set an SNR threshold below which no candidate events are accepted, and the higher false alarm rate with BCVspin will necessitate setting a higher threshold.
The pipeline used to search for gravitationalwave signals in the LIGO detectors demands that candidate events be coincident in two or more detectors The LIGO Scientific Collaboration (2007). If the noise sources in our detectors are uncorrelated (as in the case of the two km LIGO detectors), we can measure the false alarm (or background) rate of this pipeline by timeshifting the detector data by more than the gravitationalwave travel time between the detectors ( ms) and looking for coincident triggers; such triggers will be due to accidental coincidence of noise alone. We can repeat this with time steps of, say, s, and count the number of coincident triggers in each of the timeshifts to obtain a good estimate of the false alarm rate.
Before triggers are compared between detectors, they are clustered together, keeping only the trigger with the loudest SNR within a certain time window (in our case ms). Next, various methods can be used for declaring two clustered triggers to be coincident across detectors. Usually one demands not only coincidence in time, but also that the parameters of the template that gave the loudest SNR be similar in the different detectors. The simplest way of implementing this is the socalled boxcoincidence method, whereby two triggers are considered coincident if they occured within a certain time from each other (say, 100 ms), and the associated templates have parameters that differ only within certain tolerances The LIGO Scientific Collaboration (2007). In the case of BCVspin, these were chosen as and , with no restrictions on differences in Abbott et al. (2008a).
More recently, a more sophisticated technique was developed which has the potential to dramatically reduce the false alarm rate Robinson et al. (2008). In this method, the covariances between the signal parameters are used to define an error ellipsoid in parameter space around the triggers, and triggers in different detectors are considered coincident if their associated ellipsoids overlap. In the case of SPA banks, the size of the ellipsoids will depend strongly on the region of parameter space the triggers occur in. Generally they will be smaller for triggers associated with smaller masses, as waveforms will then spend more time in the detector band and errors will be smaller. This leads to a dramatic reduction in the number of spurious coincident triggers. By contrast, the boxcoincidence method described above uses the same parameter windows anywhere in parameter space.
The ellipsoid coincidence method has been successfully implemented for SPA banks. The technique is wellsuited for banks where the templates are simplified versions of target waveforms, so that one can assume template waveforms to be reasonably close to physical signals. It would be possible in principle to implement such a method also for the (phenomenological) BCVspin banks. However, in this case the metric is basically independent of and , the parameters that are most closely related to the masses. Hence, for a given value of , the associated ellipsoids would not differ in size across space, and only their orientations would differ with . This way, no great improvements can be expected in terms of reducing the false alarm rate.
Bank  (Mpc)  (Mpc)  (Mpc)  

SPA  97.3  8.7  40.1  33.9  15.9 
BCVspin  85.4  8.4  34.6  17.5  14.5 
Table. 2 shows the average number and variance of coincident triggers between the km LIGO Hanford and Livingston detectors for timeshifts within days of data from the fifth LIGO science run Abbott et al. (2007). The SNR threshold for SPA is 5.5, while for BCVspin it is 8. With these thresholds, SPA and BCVspin banks have approximately the same false alarm rates.
vi.2 Efficiencies
We are now in a position to compare the efficiencies of SPA and BCVspin banks. Given a large number of target waveforms injected in the data, the efficiency is the ratio of the number of found injections to the total number of injections made. For our purposes, an injection is considered found if it had an SNR above the chosen threshold with at least one template in the bank, within a certain time interval around the time when the injection was actually made. In the case of SPA, the width of this interval can be chosen to be 40 ms. BCVspin templates, being phenomenological, turn out to have a larger timing inaccuracy, and an interval of 100 ms was found to be more appropriate. This had already been noticed in Abbott et al. (2008a); presumably the larger timing uncertainties of BCVspin are related to its unphysical phasing (essentially, missing PN terms) as it is predominantly the phasing which affects timing errors.
It is important that efficiencies be compared for the same false alarm rate. And indeed, as we have just seen, SPA and BCVspin have essentially the same background rates if the SNR thresholds are set at 5.5 and 8, respectively.
We made 1124 injections distributed logarithmically in distance between 1 Mpc and 50 Mpc, with component masses randomly chosen between and , spin magnitudes , between 0.7 and 1, and arbitrary directions for the initial spin vectors. The efficiency of SPA then came out to be 0.93, versus 0.89 for BCVspin. These results have been summarized in Appendix I of Abbott et al. (2009b); here we have provided a detailed account of how they were obtained. We refer to the latter paper for plots of efficiency against distance; see Table 2 for the distances at which the efficiencies are 50%, 75%, and 90%, both for SPA and stochastic BCVspin.
We find that despite the fact that BCVspin banks have higher effective
fitting factors with the
target waveforms than SPA banks, in a more realistic dataanalysis comparison
the two waveform families have similar abilities to detect simulated signals.
The detection statistic for BCVspin involves more degrees of freedom and the
pipeline using BCVspin waveforms is more sensitive to nonstationary noise
transients in the data. Consequently, at the same false alarm rate the
detection threshold of the BCVspin bank is higher than the SPA bank,
negating the effect of the improved effective fitting factor of the BCVspin bank
Vii Conclusions
Past searches for low mass compact binary inspiral events in LIGO data (with the exception of Abbott et al. (2008a)) have used waveforms which do not attempt to model the spin effects, despite the fact that astrophysical black holes may be spinning rapidly. In this paper we have constructed template banks using the BCVspin waveform proposed in Buonanno et al. (2003a). Though phenomenological, these waveforms seek to capture the spininduced amplitude modulation one expects to see in a physical signal, and have high effective fitting factors with PN waveforms that include spin. We have improved on the search method of Abbott et al. (2008a), in two ways: (i) we have constructed a bank using the metric outlined in Ref. Buonanno et al. (2005), which is much better suited to the template family, and (ii) we have explored two new placement algorithms (squarehexagonal and stochastic). We used spinning PN signals to study the effective fitting factors of three different banks: an SPA bank, a BCVspin bank with squarehexagonal placement, and a BCVspin bank with stochastic placement. We found that the two BCVspin banks had a similar performance, but both did markedly better than SPA. However, search performance should be judged by detection efficiency at a given false alarm rate. The search pipeline for lowmass compact binaries () in data from the fifth LIGO science run used a nonspinning SPA bank with an SNR threshold of . We have demonstrated that to achieve a comparible false alarm rate with the currently available search pipelines using BCVspin templates requires an SNR threshold of and with this higher threshold, the detection efficiency of BCVspin for spinning PN signals becomes similar to that of the nonspinning SPA pipeline. Our findings, presented at length here and summarized in Ref. Abbott et al. (2009b), were used to guide the decision not to repeat the analysis of Ref. Abbott et al. (2008a) with data from the fifth LIGO science run.
In conclusion, the detection performance of the BCVspin pipeline is similar to that of the nonspinning SPA pipeline. We note, however, that our comparison is not entirely fair, because the SPA pipeline implements the metricbased coincidence algorithm of Ref. Robinson et al. (2008) which dramatically reduces the number of spurious coincident triggers. In principle such a technique could also be applied for BCVspin, but since the metric has essentially no dependence on and only a weak dependence on it is unlikely that implementation of the metriccoincidence algorithm would improve the sensitivity of the BCVspin pipeline. This justifies the use of nonspinning SPA pipelines rather than BCVspin pipelines in LIGO searches. Nevertheless, to search for spinning signals with nonspinning banks is still suboptimal, and work is ongoing to improve the performance of searches for spinning signals using templates determined by physical (rather than phenomenological) parameters proposed in Ref. Pan et al. (2004); Buonanno et al. (2004). In the mean time we recommend the continued use of nonspinning SPA banks in upcoming searches until more efficient template families designed to capture spinmodulated waveforms have been incorporated into a pipeline.
Acknowlegements
We would like to thank the members of the LIGO Scientific Collaboration Compact Binary Coalescence group for many helpful discussions, and in particular Michele Vallisneri for a critical reading of the manuscript. CVDB, TC, IH, GJ, and BSS are supported by PPARC grant PP/F001096/1. DB is supported by National Science Foundation grant NSF0847611. Hideyuki Tagoshi is supported by KAKENHI, Nos. 16540251 and 20540271. Hirotaka Takahashi is partially supported by the Uchida Energy Science Promotion Foundation, Sasagawa Scientific Research Grant and the JSPS GrantinAid for Scientific Research No. 20540260.
Footnotes
 At the time this work was started the spinorbit term at 2.5PN Faye et al. (2006); Blanchet et al. (2006) was not yet known and so is not included here.
 What is called here was denoted in Buonanno et al. (2003a).
 The problem had been anticipated in Buonanno et al. (2003a); here we have quantified it using real data.
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