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FINDCHIRP: an Algorithm for Detection of Gravitational Waves from Inspiraling Compact Binaries

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FINDCHIRP: an Algorithm for Detection of Gravitational Waves from Inspiraling Compact Binaries PHYSICAL REVIEW D 85, 122006 (2012) FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries Bruce Allen,1,2 Warren G. Anderson,1 Patrick R. Brady,1 Duncan A. Brown,1,3,4,5 and Jolien D. E. Creighton1 1Department of Physics, University of Wisconsin–Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201, USA 2Max-Planck-Institut fu¨r Gravitationsphysik (Albert-Einstein-Institut), Callinstraße 38, D-30167 Hannover, Germany 3Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 91125, USA 4LIGO Laboratory, California Institute of Technology, Pasadena, California 91125, USA 5Department of Physics, Syracuse University, Syracuse, New York 13244, USA (Received 4 August 2011; published 19 June 2012) Matched-filter searches for gravitational waves from coalescing compact binaries by the LIGO Scientific Collaboration use the FINDCHIRP algorithm: an implementation of the optimal filter with innovations to account for unknown signal parameters and to improve performance on detector data that has nonstationary and non-Gaussian artifacts. We provide details on the FINDCHIRP algorithm as used in the search for subsolar mass binaries, binary neutron stars, neutron starblack hole binaries, and binary black holes. DOI: 10.1103/PhysRevD.85.122006 PACS numbers: 06.20.Dk, 04.80.Nn The FINDCHIRP algorithm is the implementation of I. INTRODUCTION matched filtering used by the LIGO Scientific For the detection of a known signal it is well-known that, Collaboration (LSC) and Virgo’s offline searches for gravi- in the presence of stationary and Gaussian noise, the use of tational waves from low-mass (2M <M¼ m1 þ m2 < a matched filter is the optimal detection strategy [1]. Such 35M ), high-mass (25M <M<100M ), and primordial signals are the anticipated gravitational waveforms from black hole (0:2M <M<2M ) coalescing compact bi- the inspiral of binary systems containing compact objects, naries [14–25]. FINDCHIRP has also been used to search for i.e., binaries whose components are neutron stars and/or black supermassive black holes in data from the LISA Mock holes [2,3]. These gravitational-wave sources are among the Data Challenge [26–28] and for comparisons with numeri- most promising targets for ground-based gravitational-wave cal relativity waveforms for both high-mass and low-mass detectors, such as the Laser Interferometer Gravitational- binaries [29]. Several aspects of the algorithm have been wave Observatory (LIGO) [4]andVirgo[5]. described in passing in the above references, but here we Some practical complications arise for the gravitational- provide a detailed and comprehensive description of our wave detection problem, however, because: (i) the signal is algorithm as used in the LSC and Virgo’s search for not precisely known—it is parametrized by the binary coalescing compact binaries. companions’ masses, an initial phase, the time of arrival, The FINDCHIRP algorithm is the part of the search that and various parameters describing the distance and orien- (i) computes the matched-filter response to the interfer- tation of the system relative to the detector that can be ometer data for each template in a bank of templates, combined into a single parameter we call the ‘‘effective (ii) computes a chi-squared discriminant [13] (if needed) distance,’’ and (ii) the detector noise is not perfectly to reject instrumental artifacts that produce large spurious described as a stationary Gaussian process. Standard tech- responses of the matched filter but otherwise do not re- niques for extending the simple matched filter to search semble an expected signal, and (iii) selects candidate over the unknown parameters involve using a quadrature events or triggers based on the matched-filter and chi- sum of matched-filter outputs for orthogonal-phase wave- squared outputs. This is a fundamental part of the search forms (thereby eliminating the unknown phase), use for coalescing compact binaries, but the search also con- of Fourier transform to efficiently apply the matched sists of several other important steps such as data selection filters for different times of arrival, and use of a bank of and conditioning, template bank generation, rejection of templates to cover the parameter space of binary compan- candidate events by vetoes based on auxiliary instrumental ion masses [6–12]. Methods for making the matched filter channels and multidetector coincidence, and computation more robust against non-Gaussian noise artifacts, e.g., by of the false alarm rate of candidate triggers. examining the relative contributions of frequency-band- The entire search pipeline, which is a transformation of limited matched-filter outputs (vetoing those transients raw interferometer data into candidate events, contains all that produce large matched-filter outputs but have a time- these aspects. The specific details of the pipeline used to frequency decomposition that is inconsistent with the search for gravitational waves depends on the target popu- expected waveform), have also been explored [13] and lation (e.g., a triggered search for gamma-ray bursts or an found to be effective. all-sky search for low-mass compact binaries) and the 1550-7998=2012=85(12)=122006(19) 122006-1 Ó 2012 American Physical Society ALLEN et al. PHYSICAL REVIEW D 85, 122006 (2012) particular ‘‘science run’’ of the LIGO and/or Virgo detec- Fourier transform (FFT) algorithm. Thus, it is important tors. We refer to Refs. [14–16,18,19,21–25,30–32] for to write the most computationally demanding terms in the descriptions of the pipelines that have used the algorithms form of Eq. (2.3) so that this computation can be done most described here. This paper is not intended to provide efficiently. documentation for our implementation of the FINDCHIRP Throughout this paper we will reserve the indices j to be algorithm. (This can be found in Refs. [33,34].) Rather, this a time index (which labels a particular time sample), k to paper is intended to describe the algorithm itself. be a frequency index (which labels a particular frequency bin), m to be an index over a bank of templates, and n to be II. NOTATION an index over analysis segments. Thus, for example, the quantity z ½j will be the jth sample of analysis segment Our conventions for the Fourier transform are as follows. m;n n of the matched-filter output for the mth template, and For continuous quantities, the forward and inverse Fourier z~ ½k will be the kth frequency bin of the Fourier trans- transforms are given by m;n Z form of the matched-filter output for the same template and 1 analysis segment. x~ðfÞ¼ xðtÞeÀ2iftdt; (2.1a) 1 and III. WAVEFORM Z 1 For first-generation gravitational-wave detectors (such xðtÞ¼ x~ðfÞe2iftdf; (2.1b) 1 as Initial LIGO), the gravitational-wave signal from a compact binary inspiral waveform can be accurately respectively, so x~ðfÞ is the Fourier transform of xðtÞ. modeled by the restricted post-Newtonian waveform [35] If these continuous quantities are discretized so that below approximately M 12M [36] for nonspinning x½j¼xðjÁtÞ where 1=Át is the sampling rate and bodies. At higher masses, resummation techniques such j ¼ 0; ...;NÀ 1 are N sample points, then the discrete as the effective one body (EOB) waveforms better repro- approximation to the forward and inverse Fourier trans- duce the waveforms computed by numerically solving the forms are full nonlinear Einstein equations [37,38]. In either case, the NXÀ1 two polarizations of the gravitational-wave produced by À2ijk=N x~½k¼Át x½je ; (2.2a) such a system exhibit a monotonically-increasing fre- j¼0 quency and amplitude as the orbital motion radiates away NXÀ1 energy and decays. The waveform, often called a chirp x½j¼Áf x~½ke2ijk=N; (2.2b) waveform, is given for t<tc by k¼0 2 M À1=4 where Áf ¼ 1=ðNÁtÞ and x~½k is an approximation to 1 þ cos G tc À t hþðtÞ¼À the value of the continuous Fourier transform at fre- 2 c2D 5GM=c3 quency kÁf: x~½kx~ðkÁfÞ for 0 k bN=2c and x~½k  cos½2c þ 2ðt À tc; M; Þ; (3.1a) x~ððk À NÞÁfÞ for bN=2c <k<N (negative frequencies). GM t À t À1=4 Here bac means the greatest integer less than or equal to a. h ðtÞ¼Àcos c  2 M 3 The DC component is k ¼ 0 and, when N is even, k ¼ N=2 c D 5G =c corresponds to the Nyquist-frequency.  sin½2c þ 2ðt À tc; M; Þ; (3.1b) Notice that our convention is to have the Fourier com- ponents x~½k normalized so as to have the same units as the where D is the distance from the source, is the angle be- continuous Fourier transform x~ðfÞ, i.e., the discretized tween the direction to the observer and the orbital angular versions of the continuous forward and inverse Fourier momentum axis of the binary system, M ¼ 3=5M2=5 ¼ 3=5 transforms carry the normalization constants Át and Áf M (where M ¼ m1 þ m2 is the total mass of the respectively. Numerical packages instead compute the dis- two companions, the reduced mass is ¼ m1m2=M, and crete Fourier transform (DFT) ¼ =M) is the chirp mass, and ðt À tc; M; Þ is the orbital phase of the binary (whose evolution also depends NXÀ1 y½k¼ x½jeÇ2ijk=N (2.3) on the individual masses of the binary companions) j¼0 [39,40]. Here, tc and c are the time and phase of the binary coalescence when the waveform is terminated, where the minus sign in the exponential refers to the known as the coalescence time and coalescence phase 1 forward DFT and the positive sign refers to the reverse ð0; M; Þ¼0.
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