Search templates for gravitational waves from inspiraling binaries: Choice of template spacing

Gravitational waves from inspiraling, compact binaries will be searched for in the output of the LIGO-VIRGO interferometric network by the method of “matched filtering”-i.e., by correlating the noisy output of each interferometer with a set of theoretical wave form templates. These search templates will be a discrete subset of a continuous, multiparameter family, each of which approximates a possible signal. The search might be performed hierarchically, with a first pass through the data using a low threshold and a coarsely spaced, few-parameter template set, followed by a second pass on threshold-exceeding data segments, with a higher threshold and a more finely spaced template set that might have a larger number of parameters. Alternatively, the search might involve a single pass through the data using the larger threshold and finer template set. This paper extends and generalizes the Sathyaprakash-Dhurandhar (SD) formalism for choosing the discrete, finely spaced template set used in the final (or sole) pass through the data, based on the analysis of a single interferometer. The SD formalism is rephrased in geometric language by introducing a metric on the continuous template space from which the discrete template set is drawn. This template metric is used to compute the loss of signal-to-noise ratio and reduction of event rate which result from the coarseness of the template grid. Correspondingly, the template spacing and total number N of templates are expressed, via the metric, as functions of the reduction in event rate. The theory is developed for a template family of arbitrary dimensionality (whereas the original SD formalism was restricted to a single nontrivial dimension). The theory is then applied to a simple post¹-Newtonian template family with two nontrivial dimensions. For this family, the number of templates N in the finely spaced grid is related to the spacing-induced fractional loss L of event rate and to the minimum mass M_min of the least massive star in the binaries for which one searches by N2× 10⁵ (0.1/L)(0.2⊙/M_min)^2.7 for the first LIGO interferometers and by N8× 10⁶(0.1/L)(0.2M⊙/M_min)^2.7 for advanced LIGO interferometers. This is several orders of magnitude greater than one might have expected based on Sathyaprakash’s discovery of a near degeneracy in the parameter space, the discrepancy being due to that paper’s high choice of M_min and less stringent choice of L. The computational power P required to process the steady stream of incoming data from a single interferometer through the closely spaced set of templates is given in floating-point operations per second by P3×10¹⁰(0.1/L)(0.2M⊙/M_min)^2.7 for the first LIGO interferometers and by P4×10¹¹(0.1/L)(0.2M⊙/M_min)^2.7 for advanced LIGO interferometers. This will be within the capabilities of LIGO-era computers, but a hierarchical search may still be desirable to reduce the required computing power.