Gravitational wave signal templates, pattern recognition, and reciprocal Eulerian gamma functions

The direct detection of Gravitational Waves (GWs) is one of the most challenging problems in experimental gravitation today. It necessitates the use of highly advanced large laser interferometers such as LIGO, VIRGO, LISA, TAMA 300, GEO 600 and AIGO. The analysis of the data from such instruments requires and combines the expertise from a multitude of scientific disciplines. The verification of a detected signal demands an effective way to distinguish the source signal from the background noise. Such a study is required for an all-sky search to determine the ϕ and θ angles on the sky of gravitational wave sources and their frequencies. In this paper, we present analytical solutions and associated numerical approximations for the inner products employed in matched filtering a GW signal using templates. An exact closed-form expression for the inner products is rigourously derived using the special functions of mathematical physics. The inner products involve reciprocal Eulerian gamma functions, which occur in the study of many diverse phenomena. The spectral noise density of the VIRGO GW detector is shown to be amenable to our analysis. Spectral noise densities like those for LIGO and GEO 600, although different and in a slightly more restricted frequency band, are likewise amenable. We study numerical computation of the inner products, estimate the computational time of the solution on serial and parallel computers, and show the efficiency of the resulting algorithms. The fitting factor that indicates the goodness of fit between a signal and a template is given in closed-form and computed numerically. The numerical plots display an approximate symmetry in the template ϕ and θ domain.