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 ${\mathrm{post}}^{1}$-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 ${\mathit{M}}_{\mathrm{min}}$ of the least massive star in the binaries for which one searches by N\ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{5}$(0.1/L)(0.2${\mathit{M}}_{\mathrm{\ensuremath{\bigodot}}}$/${\mathit{M}}_{\mathrm{min}}$${)}^{2.7}$ for the first LIGO interferometers and by N\ensuremath{\sim}8\ifmmode\times\else\texttimes\fi{}${10}^{6}$(0.1/L)(0.2${\mathit{M}}_{\mathrm{\ensuremath{\bigodot}}}$/${\mathit{M}}_{\mathrm{min}}$${)}^{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 ${\mathit{M}}_{\mathrm{min}}$ 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 P\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}${10}^{10}$(0.1/L)(0.2${\mathit{M}}_{\mathrm{\ensuremath{\bigodot}}}$/${\mathit{M}}_{\mathrm{min}}$${)}^{2.7}$ for the first LIGO interferometers and by P\ensuremath{\sim}4\ifmmode\times\else\texttimes\fi{}${10}^{11}$(0.1/L)(0.2${\mathit{M}}_{\mathrm{\ensuremath{\bigodot}}}$/${\mathit{M}}_{\mathrm{min}}$${)}^{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. \textcopyright{} 1996 The American Physical Society.