Template Bank Research Articles

Systematic errors in measuring parameters of non-spinning compact binary coalescences with post-Newtonian templates

We study the astrophysical impact of inaccurate and incomplete modeling of the gravitational waveforms from compact binary coalescences (CBCs). We do so by the matched filtering of phenomenological inspiral-merger-ringdown (IMR) signals with a bank of inspiral-phase templates modeled on the 3.5 post-Newtonian TaylorT1 approximant. The rationale for the choice of the templates is threefold. (1) The inspiral phase of the phenomenological IMR signals, which are an example of complete IMR signals, is modeled on the same TaylorT1 approximant. (2) In the low-mass limit, where the merger and ringdown phases are much shorter than the inspiral phase, the errors should tend to vanishingly small values and, thus, provide an important check on the numerical aspects of our simulations. (3) Since the binary black hole signals are not yet known for mass ratios above ten and since signals from CBCs involving neutron stars are affected by uncertainties in the knowledge of their equation of state, inspiral templates are still in use in searches for those signals. The results from our numerical simulations are compared with analytical calculations of the systematic errors using the Fisher matrix on the template parameter space. We find that the loss in signal-to-noise ratio (SNR) can be as large as 45% even for binary black holes with component masses m1 = 10 M⊙ and m2 = 40 M⊙. Also the estimated total mass for the same pair can be off by as much as 20%. Both of these are worse for some higher mass combinations. Even the estimation of the symmetric mass ratio η suffers a nearly 20% error for this example and can be worse than 50% for the mass ranges studied here. These errors significantly dominate their statistical counterparts (at a nominal SNR of 10). It may, however, be possible to mitigate the loss in SNR by allowing for templates with unphysical values of η.

Mock LISA data challenge for the Galactic white dwarf binaries

We present data analysis methods used in detection and the estimation of parameters of gravitational wave signals from the white dwarf binaries in the mock LISA data challenge. Our main focus is on the analysis of challenge 3.1, where the gravitational wave signals from more than 50 mln. Galactic binaries were added to the simulated Gaussian instrumental noise. Majority of the signals at low frequencies are not resolved individually. The confusion between the signals is strongly reduced at frequencies above 5 mHz. Our basic data analysis procedure is the maximum likelihood detection method. We filter the data through the template bank at the first step of the search, then we refine parameters using the Nelder-Mead algorithm, we remove the strongest signal found and we repeat the procedure. We detect reliably and estimate parameters accurately of more than ten thousand signals from white dwarf binaries.

Effect of eccentricity on searches for gravitational waves from coalescing compact binaries in ground-based detectors

Inspiralling compact binaries are expected to circularize before their gravitational-wave signals reach the sensitive frequency band of ground-based detectors. Current searches for gravitational waves from compact binaries using the LIGO and Virgo detectors therefore use circular templates to construct matched filters. Binary formation models have been proposed which suggest that some systems detectable by the LIGO--Virgo network may have non-negligible eccentricity. We investigate the ability of the restricted 3.5 post-Newtonian order TaylorF2 template bank, used by LIGO and Virgo to search for gravitational waves from compact binaries with masses $M \le 35 M_\odot$, to detect binaries with non-zero eccentricity. We model the gravitational waves from eccentric binaries using the $x$-model post-Newtonian formalism proposed by Hinder \emph{et. al.} [I. Hinder, F. Hermann, P. Laguna, and D. Shoemaker, arXiv:0806.1037v1]. We find that small residual eccentricities ($e_0 \lesssim 0.05$ at 40 Hz) do not significantly affect the ability of current LIGO searches to detect gravitational waves from coalescing compact binaries with total mass $2 M_\odot < M < 15 M_\odot$. For eccentricities $e_0 \gtrsim 0.1$, the loss in matched filter signal-to-noise ratio due to eccentricity can be significant and so templates which include eccentric effects will be required to perform optimal searches for such systems.

Cover art: Issues in the metric-guided and metric-less placement of random and stochastic template banks

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 close to others. In this article we explore several theoretical and practical issues in randomized placement: the size and performance of the resulting template banks; the effects of parameter-space boundaries; the use of quasi-random (self avoiding) number sequences; most important, the implementation of these algorithms in curved signal manifolds with and without the use of a Riemannian signal metric, which may be difficult to obtain. Specifically, we show how the metric can be replaced with a discrete triangulation-based representation of local geometry. We argue that the broad class of randomized placement algorithms offers a promising answer to many search problems, but that the specific choice of a scheme and its implementation details will still need to be fine-tuned separately for each problem.

Stochastic template placement algorithm for gravitational wave data analysis

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 wave data analysis.

Detecting white dwarf binaries in Mock LISA Data Challenge 3

We present a strategy for detecting gravitational wave signals from the Galactic white dwarf binaries in the Mock LISA Data Challenge 3 (MLDC3) and estimate their parameters. Our method is based on the matched filtering in the form of the -statistic. We perform the search on three-dimensional space (sky coordinate and frequency of gravitational wave) below 3 mHz and include the fourth parameter (frequency derivative) at high frequencies. A template bank is used to search for the strongest signal in the data, then we remove it and repeat the search until we do not have signals in the data above a preselected threshold. For the template bank, we construct an optimal grid that realizes the best lattice covering with a constraint such that the nodes of the grid coincide with the Fourier frequencies. This enables the use of the fast Fourier transform algorithm to calculate the -statistic.

Unmodeled search for black hole binary systems in the NINJA project

The gravitational-wave signature from binary black hole coalescences is an important target for ground-based interferometric detectors such as LIGO and Virgo. The Numerical INJection Analysis (NINJA) project brought together the numerical relativity and gravitational wave data analysis communities, with the goal to optimize the detectability of these events. In its first instantiation, the NINJA project produced a simulated data set with numerical waveforms from binary black hole coalescences of various morphologies (spin, mass ratio, initial conditions), superimposed to Gaussian colored noise at the design sensitivity for initial LIGO and Virgo. We analyzed the NINJA simulated data set with the Q-pipeline algorithm, designed for the all-sky detection of gravitational-wave bursts with minimal assumptions on the shape of the waveform. The algorithm filters the data with a bank of sine-Gaussians, sinusoids with Gaussian envelope, to identify significant excess power in the time-frequency domain. We compared the performance of this burst search algorithm with lalapps_ring, which match-filters data with a bank of ring-down templates to specifically target the final stage of a coalescence of black holes. A comparison of the output of the two algorithms on NINJA data in a single detector analysis yielded qualitatively consistent results; however, due to the low simulation statistics in the first NINJA project, it is premature to draw quantitative conclusions at this stage, and further studies with higher statistics and real detector noise will be needed.

A burst search for gravitational waves from binary black holes

Compact binary coalescence (CBC) is one of the most promising sources of gravitational waves. These sources are usually searched for with matched filters which require accurate calculation of the GW waveforms and generation of large template banks. We present a complementary search technique based on algorithms used in un-modeled searches. Initially designed for detection of un-modeled bursts, which can span a very large set of waveform morphologies, the search algorithm presented here is constrained for targeted detection of the smaller subset of CBC signals. The constraint is based on the assumption of elliptical polarization for signals received at the detector. We expect that the algorithm is sensitive to CBC signals in a wide range of masses, mass ratios and spin parameters. In preparation for the analysis of data from the fifth LIGO-Virgo science run (S5), we performed preliminary studies of the algorithm on test data. We present the sensitivity of the search to different types of simulated binary black hole waveforms. Also, we discuss how to extend the results of the test run into a search over all of the current LIGO-Virgo data set.

Template banks to search for compact binaries with spinning components in gravitational wave data

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 possess 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 pipelines to detect gravitational waves from compact binaries with spin. We use the post-Newtonian 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. D 67 104025 (2003)]. We compare the performance of these template banks to that of banks constructed using the stationary phase approximation to the nonspinning post-Newtonian 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.

Erratum: Template bank for gravitational waveforms from coalescing binary black holes: Nonspinning binaries [Phys. Rev. DPRVDAQ1550-799877, 104017 (2008)

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 inspiral, merger, and ring-down stages of the coalescence of non-spinning binary black holes that follow quasi-circular inspiral. This two-dimensional template family is explicitly parametrized by the physical parameters of the binary. We show that the template family is not only \emph{effectual} in detecting the signals from black hole coalescences, but also \emph{faithful} in estimating the parameters of the binary. We compare the sensitivity of a search (in the context of different ground-based interferometers) using all three stages of the black hole coalescence with other template-based searches which look for individual stages separately. We find that the proposed search is significantly more sensitive than other template-based searches for a substantial mass-range, potentially bringing about remarkable improvement in the event-rate of ground-based interferometers. As part of this work, we also prescribe a general procedure to construct interpolated template banks using non-spinning black hole waveforms produced by numerical relativity.

Random template banks and relaxed lattice coverings

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.

Improved analytical description of inspiralling and coalescing black-hole binaries

We present an analytical formalism, within the effective-one-body framework, which predicts gravitational-wave signals from inspiralling and coalescing black-hole binaries that agree, within numerical errors, with the results of the currently most accurate numerical-relativity simulations for several different mass ratios. In the equal-mass case, the gravitational-wave energy flux predicted by our formalism agrees, within numerical errors, with the most accurate numerical-relativity energy flux. We think that our formalism opens a realistic possibility of constructing a sufficiently accurate, large bank of gravitational-wave templates, as needed both for detection and data analysis of (nonspinning) coalescing binary black holes.

Searching for gravitational-wave signals emitted by eccentric compact binaries using a non-eccentric template bank: implications for ground-based detectors

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) in the context of initial LIGO detector. They ascertained that non-eccentric templates will be successful at detecting eccentric signals. We revisit their work by incorporating current and future ground-based detectors and precisely quantify the exact loss of SNR. In order to be more faithful to an actual search, we maximized the SNR over a template bank, whose minimal match is set to 95%. For initial LIGO detector, we claim that the initial eccentricity does not need to be taken into account in our searches for any system with total mass M ∊ [2–45]M⊙ if e0 ≲ 0.05 because the loss of SNR (about 5%) is consistent with the discreteness of the template bank. Similarly, this statement is also true for systems with M ∊ [6–35]M⊙ and e0 ≲ 0.10. However, by neglecting the eccentricity in our searches, significant loss of detection (larger than 10%) may arise as soon as e0 ⩾ 0.05 for neutron-star binaries. We also provide exhaustive results for VIRGO, Advanced LIGO and Einstein Telescope detectors. It is worth noting that for Einstein Telescope, neutron star binaries with e0 ⩾ 0.02 lead to a 10% loss of detection.

Adaptive and optimal difference operators in image processing

Differential operators are essential in many image processing applications. Previous work has shown how to compute derivatives more accurately by examining the image locally, and by applying a difference operator which is optimal for each pixel neighborhood. The proposed technique avoids the explicit computation of fitting functions, and replaces the function fitting process by a function classification process using a filter bank of feature detection templates. Both the feature detectors and the optimal difference operators have a specific shape and an associated cost, defined by a rigid mathematical structure, which can be described by Gröbner bases. This paper introduces a cost criterion to select the operator of the best approximating function class and the most appropriate template size so that the difference operator can be locally adapted to the digitized function. We describe how to obtain discrete approximates for commonly used differential operators, and illustrate how image processing applications can benefit from the adaptive selection procedure for the operators by means of two example applications: tangent computation for digitized object boundaries and the Laplacian of Gaussian edge detector.

Building a stochastic template bank for detecting massive black hole binaries

The coalescence of pairs of massive black holes are the strongest and most promising sources for LISA. In fact, the gravitational wave signal from the final inspiral and merger will be detectable throughout the universe. In this paper we describe the first step in a two-step hierarchical search for the gravitational wave signal from inspiraling massive BH binaries. It is based on a method routinely used in ground-based gravitational wave astronomy, namely filtering the data through a bank of templates. However we use a novel, Monte Carlo based (stochastic), method for laying a grid in the parameter space, and we use the likelihood maximized analytically over some parameters, known as the -statistic, as a detection statistic. We build a coarse template bank to detect gravitational wave signals and to make preliminary parameter estimation. The best candidates will be followed up using a Metropolis–Hasting stochastic search to refine the parameter estimates. We demonstrate the performance of the method by applying it to the Mock LISA data challenge 1B (training data set).

An implementation of matched filtering using the multi-dimensional Fourier transform

We present an implementation of matched filtering comprised three basic steps: (1) compute a fast Fourier transform (FFT) of the data, (2) map the FFT to a (data independent) one-dimensional track in a multi-dimensional space, and (3) take an inverse multi-dimensional Fourier transform. In contrast to the standard approach, this method does not require the explicit use of template waveforms. This implementation was found as a byproduct of an exploration of sampling theory in the context of the template placement problem. We show that the latter problem can be completely mapped, in all its details, onto that of sampling an analog multi-dimensional random field, including the use of anti-aliasing before the acquisition of samples. The last point is non-trivial since the analog signal in question, namely the output of a template bank, does not physically exist unless it is computed.

Improved search for galactic white-dwarf binaries in Mock LISA Data Challenge 1B using an -statistic template bank

We report on our -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 final candidates.

A hierarchical search for gravitational waves from supermassive black hole binary mergers

We present a method to search for gravitational waves from coalescing supermassive binary black holes in LISA data. The search utilizes the -statistic to maximize over, and determine the values of, the extrinsic parameters of the binary system. The intrinsic parameters are searched over hierarchically using stochastically generated multi-dimensional template banks to recover the masses, sky location and coalescence time of the binary. We present the results of this method applied to the Mock LISA Data Challenge 1B data set.

Analysis of spin precession in binary black hole systems including quadrupole-monopole interaction

We analyze in detail the spin precession equations in binary black hole systems, when the tidal torque on a Kerr black hole due to quadrupole-monopole coupling is taken into account. We show that completing the precession equations with this term reveals the existence of a conserved quantity at 2PN order when averaging over orbital motion. This quantity allows one to solve the (orbit-averaged) precession equations exactly in the case of equal masses and arbitrary spins, neglecting radiation reaction. For unequal masses, an exact solution does not exist in closed form, but we are still able to derive accurate approximate analytic solutions. We also show how to incorporate radiation-reaction effects into our analytic solutions adiabatically, and compare the results to solutions obtained numerically. For various configurations of the binary, the relative difference in the accumulated orbital phase computed using our analytic solutions versus a full numerical solution varies from $\ensuremath{\sim}0.3%$ to $\ensuremath{\sim}1.8%$ over $\ensuremath{\sim}80\char21{}140$ orbital cycles accumulated while sweeping over the orbital frequency range $\ensuremath{\sim}20\char21{}300\text{ }\text{ }\mathrm{Hz}$. This typically corresponds to a discrepancy of order $\ensuremath{\sim}5\char21{}6$ radians. While this may not be accurate enough for implementation in LIGO template banks, we still believe that our new solutions are potentially quite useful for comparing numerical relativity simulations of spinning binary black hole systems with post-Newtonian theory. They can also be used to gain more understanding of precession effects, with potential application to the gravitational recoil problem, and to provide semianalytical templates for spinning, precessing binaries.

Search of S3 LIGO data for gravitational wave signals from spinning black hole and neutron star binary inspirals

We report on the methods and results of the first dedicated search for gravitational waves emitted during the inspiral of compact binaries with spinning component bodies. We analyze 788 hours of data collected during the third science run (S3) of the LIGO detectors. We searched for binary systems using a detection template family specially designed to capture the effects of the spin-induced precession of the orbital plane. We present details of the techniques developed to enable this search for spin-modulated gravitational waves, highlighting the differences between this and other recent searches for binaries with nonspinning components. The template bank we employed was found to yield high matches with our spin-modulated target waveform for binaries with masses in the asymmetric range 1.0M⊙<m1<3.0M⊙ and 12.0M⊙<m2<20.0M⊙ which is where we would expect the spin of the binary’s components to have a significant effect. We find that our search of S3 LIGO data has good sensitivity to binaries in the Milky Way and to a small fraction of binaries in M31 and M33 with masses in the range 1.0M⊙<m1, m2<20.0M⊙. No gravitational wave signals were identified during this search. Assuming a binary population with spinning components and Gaussian distribution of masses representing a prototypical neutron star–black hole system with m1≃1.35M⊙ and m2≃5M⊙, we calculate the 90%-confidence upper limit on the rate of coalescence of these systems to be 15.9 yr−1L−110, where L10 is 1010 times the blue light luminosity of the Sun.