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Title: Improving the efficiency of the detection of gravitational wave signals from inspiraling compact binaries: Chebyshev interpolation

Abstract

Inspiraling compact-object binary systems are promising gravitational wave sources for ground and space-based detectors. The time-dependent signature of these sources is a well-characterized function of a relatively small number of parameters; thus, the favored analysis technique makes use of matched filtering and maximum likelihood methods. As the parameters that characterize the source model vary, so do the templates against which the detector data are compared in the matched filter. For small variations in the parameters, the filter responses are closely correlated. Current analysis methodology samples a bank of filters whose parameter values are chosen so that the correlation between successive samples from successive filters in the bank is 97%. Correspondingly, the additional information available with each successive template evaluation is, in a real sense, only 3% of that already provided by the nearby templates. The reason for such a dense coverage of parameter space is to minimize the chance that a real signal, near the detection threshold, will be missed by the parameter space sampling. Here we investigate the use of Chebyshev interpolation for reducing the number of templates that must be evaluated to obtain the same analysis sensitivity. Additionally, rather than focus on the 'loss' of signal-to-noise associated withmore » the finite number of filters in the template bank, we evaluate the receiver operating characteristic (ROC) as a measure of the effectiveness of an analysis technique. The ROC relates the false alarm probability to the false dismissal probability of an analysis, which are the quantities that bear most directly on the effectiveness of an analysis scheme. As a demonstration, we compare the present 'dense sampling' analysis methodology with the 'interpolation' methodology using Chebyshev polynomials, restricted to one dimension of the multidimensional analysis problem by plotting the ROC curves. We find that the interpolated search can be arranged to have the same false alarm and false dismissal probabilities as the dense sampling strategy using 25% fewer templates. Generalized to the two-dimensional space used in the computationally limited current analyses, this suggests a factor of 2 increase in computational efficiency; generalized to the full seven-dimensional parameter space that characterizes the signal associated with an eccentric binary system of spinning neutron stars or black holes, it suggests an order of magnitude increase in computational efficiency. Since the computational cost of the analysis is driven almost exclusively by the matched filter evaluations, a reduction in the number of template evaluations translates directly into an increase in computational efficiency; additionally, since the computational cost of the analysis is large, the increased efficiency translates also into an increase in the size of the parameter space that can be analyzed and, thus, the science that can be accomplished with the data.« less

Authors:
; ;  [1];  [2]
  1. Inter-University Centre for Astronomy and Astrophysics, Ganeshkhind, Pune-411 007 (India)
  2. (United States)
Publication Date:
OSTI Identifier:
20711515
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 72; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.72.102001; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND TECHNOLOGY; BINARY STARS; BLACK HOLES; COMPARATIVE EVALUATIONS; CORRELATIONS; DETECTION; EFFICIENCY; GRAVITATIONAL WAVE DETECTORS; GRAVITATIONAL WAVES; INTERPOLATION; MAXIMUM-LIKELIHOOD FIT; NEUTRON STARS; POLYNOMIALS; PROBABILITY; SAMPLING; SENSITIVITY; SIGNAL-TO-NOISE RATIO; SIGNALS; SPACE; TIME DEPENDENCE; VARIATIONS

Citation Formats

Mitra, S., Dhurandhar, S.V., Finn, L.S., and Center for Gravitational Wave Physics, Pennsylvania State University, University Park, Pennsylvania 16802. Improving the efficiency of the detection of gravitational wave signals from inspiraling compact binaries: Chebyshev interpolation. United States: N. p., 2005. Web. doi:10.1103/PhysRevD.72.102001.
Mitra, S., Dhurandhar, S.V., Finn, L.S., & Center for Gravitational Wave Physics, Pennsylvania State University, University Park, Pennsylvania 16802. Improving the efficiency of the detection of gravitational wave signals from inspiraling compact binaries: Chebyshev interpolation. United States. doi:10.1103/PhysRevD.72.102001.
Mitra, S., Dhurandhar, S.V., Finn, L.S., and Center for Gravitational Wave Physics, Pennsylvania State University, University Park, Pennsylvania 16802. Tue . "Improving the efficiency of the detection of gravitational wave signals from inspiraling compact binaries: Chebyshev interpolation". United States. doi:10.1103/PhysRevD.72.102001.
@article{osti_20711515,
title = {Improving the efficiency of the detection of gravitational wave signals from inspiraling compact binaries: Chebyshev interpolation},
author = {Mitra, S. and Dhurandhar, S.V. and Finn, L.S. and Center for Gravitational Wave Physics, Pennsylvania State University, University Park, Pennsylvania 16802},
abstractNote = {Inspiraling compact-object binary systems are promising gravitational wave sources for ground and space-based detectors. The time-dependent signature of these sources is a well-characterized function of a relatively small number of parameters; thus, the favored analysis technique makes use of matched filtering and maximum likelihood methods. As the parameters that characterize the source model vary, so do the templates against which the detector data are compared in the matched filter. For small variations in the parameters, the filter responses are closely correlated. Current analysis methodology samples a bank of filters whose parameter values are chosen so that the correlation between successive samples from successive filters in the bank is 97%. Correspondingly, the additional information available with each successive template evaluation is, in a real sense, only 3% of that already provided by the nearby templates. The reason for such a dense coverage of parameter space is to minimize the chance that a real signal, near the detection threshold, will be missed by the parameter space sampling. Here we investigate the use of Chebyshev interpolation for reducing the number of templates that must be evaluated to obtain the same analysis sensitivity. Additionally, rather than focus on the 'loss' of signal-to-noise associated with the finite number of filters in the template bank, we evaluate the receiver operating characteristic (ROC) as a measure of the effectiveness of an analysis technique. The ROC relates the false alarm probability to the false dismissal probability of an analysis, which are the quantities that bear most directly on the effectiveness of an analysis scheme. As a demonstration, we compare the present 'dense sampling' analysis methodology with the 'interpolation' methodology using Chebyshev polynomials, restricted to one dimension of the multidimensional analysis problem by plotting the ROC curves. We find that the interpolated search can be arranged to have the same false alarm and false dismissal probabilities as the dense sampling strategy using 25% fewer templates. Generalized to the two-dimensional space used in the computationally limited current analyses, this suggests a factor of 2 increase in computational efficiency; generalized to the full seven-dimensional parameter space that characterizes the signal associated with an eccentric binary system of spinning neutron stars or black holes, it suggests an order of magnitude increase in computational efficiency. Since the computational cost of the analysis is driven almost exclusively by the matched filter evaluations, a reduction in the number of template evaluations translates directly into an increase in computational efficiency; additionally, since the computational cost of the analysis is large, the increased efficiency translates also into an increase in the size of the parameter space that can be analyzed and, thus, the science that can be accomplished with the data.},
doi = {10.1103/PhysRevD.72.102001},
journal = {Physical Review. D, Particles Fields},
number = 10,
volume = 72,
place = {United States},
year = {Tue Nov 15 00:00:00 EST 2005},
month = {Tue Nov 15 00:00:00 EST 2005}
}
  • A data-analysis strategy based on the maximum-likelihood method (MLM) is presented for the detection of gravitational waves from inspiraling compact binaries with a network of laser-interferometric detectors having arbitrary orientations and arbitrary locations around the globe. For simplicity, we restrict ourselves to the Newtonian inspiral wave form. However, the formalism we develop here is also applicable to a wave form with post-Newtonian (PN) corrections. The Newtonian wave form depends on eight parameters: the distance r to the binary, the phase {delta}{sub c} of the wave form at the time of final coalescence, the polarization-ellipse angle {psi}, the angle of inclinationmore » {epsilon} of the binary orbit to the line of sight, the source-direction angles {l_brace}{theta},{phi}{r_brace}, the time of final coalescence t{sub c} at the fiducial detector, and the chirp time {xi}. All these parameters are relevant for a chirp search with multiple detectors, unlike the case of a single detector. The primary construct on which the MLM is based is the network likelihood ratio (LR). We obtain this ratio here. For the Newtonian inspiral wave form, the LR is a function of the eight signal parameters. In the MLM-based detection strategy, the LR must be maximized over all of these parameters. Here, we show that it is possible to maximize it analytically with respect to four of the eight parameters, namely, {l_brace}r,{delta}{sub c},{psi},{epsilon}{r_brace}. Maximization over the time of arrival is handled most efficiently by using the fast-Fourier-transform algorithm, as in the case of a single detector. This not only allows us to scan the parameter space continuously over these five parameters but also cuts down substantially on the computational costs. The analytical maximization over the four parameters yields the optimal statistic on which the decision must be based. The value of the statistic also depends on the nature of the noises in the detectors. Here, we model these noises to be mainly Gaussian, stationary, and uncorrelated for every pair of detectors. Instances of non-Gaussianity, as are present in detector outputs, can be accommodated in our formalism by implementing vetoing techniques similar to those applied for single detectors. Our formalism not only allows us to express the likelihood ratio for the network in a very simple and compact form, but also is at the basis of giving an elegant geometric interpretation to the detection problem. Maximization of the LR over the remaining three parameters is handled as follows. Owing to the arbitrary locations of the detectors in a network, the time of arrival of a signal at any detector will, in general, be different from those at the others and, consequently, will result in signal time delays. For a given network, these time delays are determined by the source-direction angles {l_brace}{theta},{phi}{r_brace}. Therefore, to maximize the LR over the parameters {l_brace}{theta},{phi}{r_brace} one needs to scan over the possible time delays allowed by a network. We opt for obtaining a bank of templates for the chirp time and the time delays. This means that we construct a bank of templates over {xi}, {theta}, and {phi}. We first discuss 'idealized' networks with all the detectors having a common noise curve for simplicity. Such an exercise nevertheless yields useful estimates about computational costs, and also tests the formalism developed here. We then consider realistic cases of networks comprising the LIGO and VIRGO detectors: These include two-detector networks, which pair up the two LIGOs or VIRGO with one of the LIGOs, and the three-detector network that includes VIRGO and both the LIGOs. For these networks we present the computational speed requirements, network sensitivities, and source-direction resolutions.« less
  • The network of interferometric detectors that is under construction at various locations on Earth is expected to start searching for gravitational waves in a few years. The number of search templates that is needed to be cross correlated with the noisy output of the detectors is a major issue since computing power capabilities are restricted. By choosing higher and higher post-Newtonian order expansions for the family of search templates we make sure that our filters are more accurate copies of the real waves that hit our detectors. However, this is not the only criterion for choosing a family of searchmore » templates. To make the process of detection as efficient as possible, one needs a family of templates with a relatively small number of members that manages to pick up any detectable signal with only a tiny reduction in signal-to-noise ratio. Evidently, one family is better than another if it accomplishes its goal with a smaller number of templates. Following the geometric language of Owen, we have studied the performance of the post{sup 1.5}-Newtonian family of templates on detecting post{sup 2}-Newtonian signals for binaries. Several technical issues arise from the fact that the two types of waveforms cannot be made to coincide by a suitable choice of parameters. In general, the parameter space of the signals is not identical with the parameter space of the templates, although in our case they are of the same dimension, and one has to take into account all such peculiarities before drawing any conclusion. An interesting result we have obtained is that the post{sup 1.5}-Newtonian family of templates happens to be more economical for detecting post{sup 2}-Newtonian signals than the perfectly accurate post{sup 2}-Newtonian family of templates itself. The number of templates is reduced by 20-30%, depending on the acceptable level of reduction in signal-to-noise ratio due to discretization of the family of templates. This makes the post{sup 1.5}-Newtonian family of templates more favorable for detecting gravitational waves from inspiraling, compact, nonspinning, binaries. Apart from this useful quantitative result, this study constitutes an application of the template-numbering technique, introduced by Owen, for families of templates that are not described by the same mathematical expression as the assumed signals. For example, this analysis will be very useful when constructing sufficiently simple templates for detecting precessing spinning binaries.« less
  • No abstract prepared.
  • 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 overefficient, with the implication that it is computationally more demanding than required. In this paper, we present a template bank based on an hexagonal lattice, which size is reduced by 40% withmore » respect to the proposed square placement. We describe the practical aspects of the hexagonal template bank implementation, its size, and computational cost. We have also performed exhaustive simulations to characterize its efficiency and safeness. We show that the bank is adequate to search for a wide variety of binary systems (primordial black holes, neutron stars, and stellar-mass black holes) and in data from both current detectors (initial LIGO, Virgo and GEO600) as well as future detectors (advanced LIGO and EGO). Remarkably, although our template bank placement uses a metric arising from a particular template family, namely, stationary phase approximation, we show that it can be used successfully with other template families (e.g., Pade resummation and effective one-body approximation). This quality of being effective for different template families makes the proposed bank suitable for a search that would use several of them in parallel (e.g., in a binary black hole search). The hexagonal template bank described in this paper is currently used to search for nonspinning inspiralling compact binaries in data from the Laser Interferometer Gravitational-Wave Observatory (LIGO)« less
  • We provide ready-to-use time-domain gravitational waveforms for spinning compact binaries with precession effects through 1.5 post-Newtonian (PN) order in amplitude, and compute their mode decomposition using spin-weighted -2 spherical harmonics. In the presence of precession, the gravitational-wave modes (l,m) contain harmonics originating from combinations of the orbital frequency and precession frequencies. We find that the gravitational radiation from binary systems with large mass asymmetry and large inclination angle can be distributed among several modes. For example, during the last stages of inspiral, for some maximally spinning configurations, the amplitude of the (2, 0) and (2, 1) modes can be comparablemore » to the amplitude of the (2, 2) mode. If the mass ratio is not too extreme, the l=3 and l=4 modes are generally 1 or 2 orders of magnitude smaller than the l=2 modes. Restricting ourselves to spinning, nonprecessing compact binaries, we apply the stationary-phase approximation and derive the frequency-domain gravitational waveforms including spin-orbit and spin(1)-spin(2) effects through 1.5PN and 2PN order, respectively, in amplitude, and 2.5PN order in phase. Since spin effects in the amplitude through 2PN order affect only the first and second harmonics of the orbital phase, they do not extend the mass reach of gravitational-wave detectors. However, they can interfere with other harmonics and lower or raise the signal-to-noise ratio depending on the spin orientation. These ready-to-use waveforms could be employed in the data analysis of the spinning, inspiraling binaries as well as in comparison studies at the interface between analytical and numerical relativity.« less