Template-based searches for gravitational waves are often limited by the computational cost associated with searching large parameter spaces. The study of efficient template banks, in the sense of using the smallest number of templates, is therefore of great practical interest. The "traditional" approach to template-bank construction requires every point in parameter space to be covered by at least one template, which rapidly becomes inefficient at higher dimensions. Here we study an alternative approach, where any point in parameter space is covered only with a given probability < 1. We find that by giving up complete coverage in this way, large reductions in the number of templates are possible, especially at higher dimensions. The prime examples studied here are "random template banks", in which templates are placed randomly with uniform probability over the parameter space. In addition to its obvious simplicity, this method turns out to be surprisingly efficient. We analyze the statistical properties of such random template banks, and compare their efficiency to traditional lattice coverings. We further study "relaxed" lattice coverings (using Zn and An* lattices), which similarly cover any signal location only with probability < 1. The relaxed An* lattice is found to yield the most efficient template banks at low dimensions (n < 10), while random template banks increasingly outperform any other method at higher dimensions.
Matched filtering is used to search for gravitational waves emitted by inspiralling compact binaries in data from ground-based interferometers. One of the key aspects of the detection process is the deployment of a set of templates, also called a template bank, to cover the astrophysically interesting region of the parameter space. In a companion paper, we described the template-bank algorithm used in the analysis of LIGO data to search for signals from non-spinning binaries made of neutron star and/or stellar-mass black holes; this template bank is based upon physical template families. In th…
This paper presents an algorithm for constructing matched-filter template banks in an arbitrary parameter space. The method places templates at random, then removes those which are "too close" together. The properties and optimality of stochastic template banks generated in this manner are investigated for some simple models. The effectiveness of these template banks for gravitational wave searches for binary inspiral waveforms is also examined. The properties of a stochastic template bank are then compared to the deterministically placed template banks that are currently used in gravitational…
Recent discoveries of gravitational wave (GW) signals from astrophysical compact binary systems of neutron stars and black holes have firmly established them as prime sources for advanced GW detectors. Theoretical templates of expected signals from such systems have been used to filter the detector data using the \emph{matched filtering} technique. An efficient grid over the parameter space at a fixed minimal match has a direct impact on improving the computational efficiency of these searches. We present the construction of three-dimensional template banks (in component masses and an effectiv…
We construct efficient banks of templates suitable for all-sky narrow-band searches of almost monochromatic gravitational waves originating from spinning neutron stars in our Galaxy in data collected by interferometric detectors. We consider waves with one spindown parameter included and we assume that both the position of the gravitational-wave source in the sky and the wave's frequency together with spindown parameter are unknown. In the construction we employ simplified model of the signal with constant amplitude and phase which is a linear function of unknown parameters. Our template banks…
Gravitational waves from coalescing compact binaries are searched using the matched filtering technique. As the model waveform depends on a number of parameters, it is necessary to filter the data through a template bank covering the astrophysically interesting region of the parameter space. The choice of templates is defined by the maximum allowed drop in signal-to-noise ratio due to the discreteness of the template bank. In this paper we describe the template-bank algorithm that was used in the analysis of data from the Laser Interferometer Gravitational Wave Observatory (LIGO) and GEO 600 d…
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, spin-induced precession of a binary's orbital plane causes modulation of the gravitational-wave amplitude and phase. If the templates used in a matched-filter 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 pipelin…
The construction of optimal template banks for matched-filtering searches is an example of the sphere covering problem. For parameter spaces with constant-coefficient metrics a (near-) optimal template bank is achieved by the A_n* lattice, which is the best lattice-covering in dimensions n <= 5, and is close to the best covering known for dimensions n <= 16. Generally this provides a substantially more efficient covering than the simpler hyper-cubic lattice. We present an algorithm for generating lattice template banks for constant-coefficient metrics and we illustrate its implementation by ge…
Matched filtering is used to search for gravitational waves emitted by inspiralling compact binaries in data from the ground-based interferometers. One of the key aspects of the detection process is the design of a template bank that covers the astrophysically pertinent parameter space. In an earlier paper, we described a template bank that is based on a square lattice. Although robust, we showed that the square placement is over-efficient, with the implication that it is computationally more demanding than required. In this paper, we present a template bank based on an hexagonal lattice, whic…
We describe the methodology and novel techniques used to construct a set of waveforms, or template bank, applicable to searches for compact binary coalescences in Advanced LIGO's second observing run. This template bank is suitable for observing systems composed of two neutron stars, two black holes, or a neutron star and a black hole. The Post-Newtonian formulation is used to model waveforms with total mass less than 4 $M_{\odot}$ and the most recent effective-one-body model, calibrated to numerical relativity to include the merger and ringdown, is used for total masses greater than 4 $M_{\od…
The general problem of computing the false-alarm rate vs. detection-threshold relationship for a bank of correlators is addressed, in the context of maximum-likelihood detection of gravitational waves, with specific reference to chirps from coalescing binary systems. Accurate (lower-bound) approximants for the cumulative distribution of the whole-bank supremum are deduced from a class of Bonferroni-type inequalities. The asymptotic properties of the cumulative distribution are obtained, in the limit where the number of correlators goes to infinity. The validity of numerical simulations made on…
When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, called a template bank. These may be constructed systematically as a lattice, or alternatively, by placing templates at randomly selected points in parameter space. Here, we calculate the fraction of signals lost by an $n$-dimensional random template bank (compared to a very finely spaced bank). This fraction is compared to the corresponding loss fraction for the be…
Recent gravitational-wave observations have begun to constrain the internal physics of neutron stars. However, current detection searches for neutron star systems assume that potential neutron stars are low-mass black holes, ignoring any affect on the gravitational-wave signal due to the internal neutron-star physics. We wish to create a template bank of binary neutron star waveforms including the effect of tidal deformability. However, we find that the Fisher matrix, which is commonly used to approximate match calculations when placing template banks, is unsuitable to predict the match betwee…
Coalescence of two massive black holes is the strongest and most promising source for LISA. In fact, gravitational signal from the end of inspiral and merger will be detectable throughout the Universe. In this article we describe the first step in the two-step hierarchical search for gravitational wave signal from the inspiraling massive BH binaries. It is based on the routinely used in the ground base gravitational wave astronomy method of filtering the data through the bank of templates. However we use a novel Monte-Carlo based (stochastic) method to lay a grid in the parameter space, and we…
We report the construction of a three-dimensional template bank for the search for gravitational waves from inspiralling binaries consisting of spinning compact objects. The parameter space consists of two dimensions describing the mass parameters and one "reduced-spin" parameter, which describes the secular (non-precessing) spin effects in the waveform. The template placement is based on an efficient stochastic algorithm and makes use of the semi-analytical computation of a metric in the parameter space. We demonstrate that for "low-mass" ($m_1 + m_2 \lesssim 12\,M_\odot$) binaries, this temp…
Astrophysical compact binary systems consisting of neutron stars and blackholes are an important class of gravitational wave (GW) sources for advanced LIGO detectors. Accurate theoretical waveform models from the inspiral, merger and ringdown phases of such systems, are used to filter detector data under the template based matched filtering paradigm. An efficient grid over the parameter space at a fixed minimal match has a direct impact on the overall time taken by these searches. We present a new hybrid geometric-random template placement algorithm for signals described by two masses and one …
We report on our F-statistic search for white-dwarf binary signals in the Mock LISA Data Challenge 1B (MLDC1B). We focus in particular on the improvements in our search pipeline since MLDC1, namely refinements in the search pipeline and the use of a more accurate detector response (rigid adiabatic approximation). The search method employs a hierarchical template-grid based exploration of the parameter space, using a coincidence step to distinguish between primary (``true'') and secondary maxima, followed by a final (multi-TDI) ``zoom'' stage to provide an accurate parameter estimation of the f…
Gravitational-wave searches for signals from inspiralling compact binaries have relied on matched filtering banks of waveforms (called template banks) to try to extract the signal waveforms from the detector data. These template banks have been constructed using four main considerations, the region of parameter space of interest, the sensitivity of the detector, the matched filtering bandwidth, and the sensitivity one is willing to lose due to the granularity of template placement, the latter of which is governed by the minimal match. In this work we describe how the choice of the lower freque…
We construct efficient banks of templates suitable for directed searches of almost monochromatic gravitational waves originating from spinning nuetron stars in our Galaxy in data being collected by currently operating interferometric detectors. We thus assume that the position of the gravitational-wave source in the sky is known, but we do not assume that the wave's frequency and its derivatives are a priori known. In the construction we employ simplified model of the signal with constant amplitude and phase which is a polynomial function of time. All our template banks enable usage of the fas…
One strategy for reducing the online computational cost of matched-filter searches for gravitational waves is to introduce a compressed basis for the waveform template bank in a grid-based search. In this paper, we propose and investigate several tunable compression schemes for a general template bank. Through offline compression, such schemes are shown to yield faster detection and localisation of signals, along with moderately improved sensitivity and accuracy over coarsened banks at the same level of computational cost. This is potentially useful for any search involving template banks, and…
Placing signal templates (grid points) as efficiently as possible to cover a multi-dimensional parameter space is crucial in computing-intensive matched-filtering searches for gravitational waves, but also in similar searches in other fields of astronomy. To generate efficient coverings of arbitrary parameter spaces, stochastic template banks have been advocated, where templates are placed at random while rejecting those too close to others. However, in this simple scheme, for each new random point its distance to every template in the existing bank is computed. This rapidly increasing number …
Most of the inspiralling compact binaries are expected to be circularized by the time their gravitational-wave signals enter the frequency band of ground-based detectors such as LIGO or VIRGO. However, it is not excluded that some of these binaries might still possess a significant eccentricity at a few tens of hertz. Despite this possibility, current search pipelines based on matched filtering techniques consider only non-eccentric templates. The effect of such an approximation on the loss of signal-to-noise ratio (SNR) has been investigated by Martel and Poisson (1999 Phys. Rev. D 60 124008)…
The final inspiral phase in the evolution of a compact binary consisting of black holes and/or neutron stars is among the most probable events that a network of ground-based interferometric gravitational wave detectors is likely to observe. Gravitational radiation emitted during this phase will have to be dug out of noise by matched-filtering (correlating) the detector output with a bank of several $10^5$ templates, making the computational resources required quite demanding, though not formidable. We propose an interpolation method for evaluating the correlation between template waveforms and…
The efficient placement of signal templates in source-parameter space is a crucial requisite for exhaustive matched-filtering searches of modeled gravitational-wave sources. Unfortunately, the current placement algorithms based on regular parameter-space meshes are difficult to generalize beyond simple signal models with few parameters. Various authors have suggested that a general, flexible, yet efficient alternative can be found in randomized placement strategies such as random placement and stochastic placement, which enhances random placement by selectively rejecting templates that are too…
Matched filtering is a traditional method used to search a data stream for signals. If the source (and hence its $n$ parameters) are unknown, many filters must be employed. These form a grid in the $n$-dimensional parameter space, known as a template bank. It is often convenient to construct these grids as a lattice. Here, we examine some of the properties of these template banks for $\mathbb{Z}^n$ and $A_n^*$ lattices. In particular, we focus on the distribution of the mismatch function, both in the traditional quadratic approximation and in the recently-proposed spherical approximation. The …
When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, with a computational cost that is proportional to the number of these points. If that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Voronoï) cells have the smallest covering radius ($\equiv$ smallest maximal m…
Gravitational wave searches to date have largely focused on non-precessing systems. Including precession effects greatly increases the number of templates to be searched over. This leads to a corresponding increase in the computational cost and can increase the false alarm rate of a realistic search. On the other hand, there might be astrophysical systems that are entirely missed by non-precessing searches. In this paper we consider the problem of constructing a template bank using stochastic methods for neutron-star--black-hole binaries allowing for precession, but with the restrictions that …
All-sky surveys for isolated continuous gravitational waves present a significant data-analysis challenge. Semicoherent search methods are commonly used to efficiently perform the computationally-intensive task of searching for these weak signals in the noisy data of gravitational-wave detectors such as LIGO and Virgo. We present a new implementation of a semicoherent search method, Weave, that for the first time makes full use of a parameter-space metric to generate banks of search templates at the correct resolution, combined with optimal lattices to minimize the required number of templates…
The most sensitive search pipelines for gravitational waves from compact binary mergers use matched filters to extract signals from the noisy data stream coming from gravitational wave detectors. Matched-filter searches require banks of template waveforms covering the physical parameter space of the binary system. Unfortunately, template bank construction can be a time-consuming task. Here we present a new method for efficiently generating template banks that utilizes automatic differentiation to calculate the parameter space metric. Principally, we demonstrate that automatic differentiation e…
Gravitational waves (GW) from coalescing stellar-mass black hole binaries (BBH) are expected to be detected by the Advanced Laser Interferometer Gravitational-wave Observatory and Advanced Virgo. Detection searches operate by matched-filtering the detector data using a bank of waveform templates. Traditionally, template banks for BBH are constructed from intermediary analytical waveform models which are calibrated against numerical relativity simulations and which can be aluated for any choice of BBH parameters. This paper explores an alternative to the traditional approach, namely the constru…
Searching for gravitational waves (GWs) from binary black holes (BBHs) with LIGO and Virgo involves matched-filtering data against a set of representative signal waveforms --- a template bank --- chosen to cover the full signal space of interest with as few template waveforms as possible. Although the component black holes may have significant angular momenta (spin), previous searches for BBHs have filtered LIGO and Virgo data using only waveforms where both component spins are zero. This leads to a loss of signal-to-noise ratio for signals where this is not the case. Combining the best availa…
Gravitational waveforms from the inspiral and ring-down stages of the binary black hole coalescences can be modelled accurately by approximation/perturbation techniques in general relativity. Recent progress in numerical relativity has enabled us to model also the non-perturbative merger phase of the binary black-hole coalescence problem. This enables us to \emph{coherently} search for all three stages of the coalescence of non-spinning binary black holes using a single template bank. Taking our motivation from these results, we propose a family of template waveforms which can model the inspir…
Low-latency gravitational wave search pipelines such as GstLAL take advantage of low-rank factorization of the template matrix via singular value decomposition (SVD). With unprecedented improvements in detector bandwidth and sensitivity in advanced-LIGO and Virgo detectors, one expects several orders of magnitude increase in the size of template banks. This poses a formidable computational challenge in factorizing huge template matrices. Previously, [in Kulkarni et al. [6]], we introduced the idea of random projection (RP)-based matrix factorization as a computationally viable alternative to S…
We introduce an algorithm for placing template waveforms for the search of compact binary mergers in gravitational wave interferometer data. We exploit the smooth dependence of the amplitude and unwrapped phase of the frequency-domain waveform on the parameters of the binary. We group waveforms with similar amplitude profiles and perform a singular value decomposition of the phase profiles to obtain an orthonormal basis for the phase functions. The leading basis functions span a lower-dimensional linear space in which the unwrapped phase of any physical waveform is well approximated. The optim…
We describe an F-statistic search for continuous gravitational waves from galactic white-dwarf binaries in simulated LISA Data. Our search method employs a hierarchical template-grid based exploration of the parameter space. In the first stage, candidate sources are identified in searches using different simulated laser signal combinations (known as TDI variables). Since each source generates a primary maximum near its true "Doppler parameters" (intrinsic frequency and sky position) as well as numerous secondary maxima of the F-statistic in Doppler parameter space, a search for multiple source…
Coalescing binary black holes (BBHs) are among the most likely sources for the Laser Interferometer Gravitational-wave Observatory (LIGO) and its international partners Virgo and KAGRA. Optimal searches for BBHs require accurate waveforms for the signal model and effectual template banks that cover the mass space of interest. We investigate the ability of the second-order post-Newtonian TaylorF2 hexagonal template placement metric to construct an effectual template bank, if the template waveforms used are effective one body waveforms tuned to numerical relativity (EOBNRv2). We find that by com…
Background estimation is important for determining the statistical significance of a gravitational-wave event. Currently, the background model is constructed numerically from the strain data using estimation techniques that insulate the strain data from any potential signals. However, as the observation of gravitational-wave signals become frequent, the effectiveness of such insulation will decrease. Contamination occurs when signals leak into the background model. In this work, we demonstrate an improved background estimation technique for the searches of gravitational waves (GWs) from binary…
We demonstrate a new geometric method for fast template placement for searches for gravitational waves from the inspiral, merger and ringdown of compact binaries. The method is based on a binary tree decomposition of the template bank parameter space into non-overlapping hypercubes. We use a numerical approximation of the signal overlap metric at the center of each hypercube to estimate the number of templates required to cover the hypercube and determine whether to further split the hypercube. As long as the expected number of templates in a given cube is greater than a given threshold, we sp…
New approach to template banks of gravitational waves with higher harmonics: Reducing matched-filtering cost by over an order of magnitude
Institute for Advanced Study（高等研究院）；University of California at Santa Barbara（加州大学圣巴巴拉分校）；Tata Institute of Fundamental Research（塔塔基础研究院）；California Institute of Technology（加州理工学院）；Princeton University（普林斯顿大学）；Weizmann Institute of Science（魏茨曼科学研究所）
Matched-filtering for the identification of compact object mergers in gravitational-wave antenna data involves the comparison of the data stream to a bank of template gravitational waveforms. Typically the template bank is constructed from phenomenological waveform models since these can be evaluated for an arbitrary choice of physical parameters. Recently it has been proposed that singular value decomposition (SVD) can be used to reduce the number of templates required for detection. As we show here, another benefit of SVD is its removal of biases from the phenomenological templates along wit…
The space-borne gravitational wave detectors will observe a large population of double white dwarf binaries in the Milky Way. However, the search for double white dwarfs in the gravitational wave data will be time-consuming due to the large number of templates involved and antenna response calculation. In this paper, we implement an iterative combinatorial algorithm to search for double white dwarfs in MLDC-3.1 data. To quickly determine the rough parameters of the target sources, the following algorithms are adopted in a coarse search process: (1) using the downsampling method to reduce the n…
We study the validity of inspiral templates in gravitational wave data analysis with Advanced LIGO sensitivity for low mass binary black holes with total masses of $M \leq 30 Msun$. We mainly focus on the nonspinning system. As our complete inspiral-merger-ringdown waveform model ($IMR$), we assume the phenomenological model, "PhenomA", and define our inspiral template model ($Imerg$) by taking the inspiral part into account from $IMR$ up to the merger frequency (fmerg). We first calculate the {\it true} statistical uncertainties using $IMR$ signals and $IMR$ templates. Next, using $IMR$ signa…
Matched filtering is a popular data analysis framework used to search for gravitational wave signals emitted by compact object binaries. The templates used in matched filtering searches are constructed predominantly from the quadrupolar mode because this mode is the energetically most dominant channel. However, for highly precessing binaries or binaries with moderately large mass ratios, significant power is also carried by higher-order modes. We investigate how the inclusion of higher modes in the templates increases the prospects for detecting gravitational waves. Specifically, we use numeri…
All-sky searches for gravitational-wave pulsars are generally limited in sensitivity by the finite availability of computing resources. Semicoherent searches are a common method of maximizing search sensitivity given a fixed computing budget. The work of Wette and Prix [Phys. Rev. D 88, 123005 (2013)] and Wette [Phys. Rev. D 92, 082003 (2015)] developed a semicoherent search method which uses metrics to construct the banks of pulsar signal templates needed to search the parameter space of interest. In this work we extend the range of validity of the parameter-space metrics using an empirically…
We extend the description of gravitational waves emitted by binary black holes during the final stages of inspiral and merger by introducing in the third post-Newtonian (3PN) effective-one-body (EOB) templates seven new ``flexibility'' parameters that affect the two-body dynamics and gravitational radiation emission. The plausible ranges of these flexibility parameters, notably the parameter characterising the fourth post-Newtonian effects in the dynamics, are estimated. Using these estimates, we show that the currently available standard 3PN bank of EOB templates does ``span'' the space of si…
Current searches for gravitational waves from compact-object binaries with the LIGO and Virgo observatories employ waveform models with spins aligned (or anti-aligned) with the orbital angular momentum. Here, we derive a new statistic to search for compact objects carrying generic (precessing) spins. Applying this statistic, we construct banks of both aligned- and generic-spin templates for binary black holes and neutron-star--black-hole binaries, and compare the effectualness of these banks towards simulated populations of generic-spin systems. We then use these banks in a pipeline analysis o…
Searches for gravitational-wave signals are often based on maximizing a detection statistic over a bank of waveform templates, covering a given parameter space with a variable level of correlation. Results are often evaluated using a noise-hypothesis test, where the background is characterized by the sampling distribution of the loudest template. In the context of continuous gravitational-wave searches, properly describing said distribution is an open problem: current approaches focus on a particular detection statistic and neglect template-bank correlations. We introduce a new approach using …
Gravitational-wave searches for the merger of compact binaries use matched-filtering as the method of detecting signals and estimating parameters. Such searches construct a fine mesh of filters covering a signal parameter space at high density. Previously it has been shown that singular value decomposition can reduce the effective number of filters required to search the data. Here we study how the basis provided by the singular value decomposition changes dimension as a function of template bank density. We will demonstrate that it is sufficient to use the basis provided by the singular value…
