Spherical ansatz for parameter-space metrics

Abstract

A fundamental quantity in signal analysis is the metric $g_{ab}$ on parameter space, which quantifies the fractional "mismatch" $m$ between two (time- or frequency-domain) waveforms. When searching for weak gravitational-wave or electromagnetic signals from sources with unknown parameters $\lambda$ (masses, sky locations, frequencies, etc.) the metric can be used to create and/or characterize "template banks". These are grids of points in parameter space; the metric is used to ensure that the points are correctly separated from one another. For small coordinate separations $d{\lambda }^a$ between two points in parameter space, the traditional ansatz for the mismatch is a quadratic form $m=g_{ab}d{\lambda }^{a}d{\lambda }^b$. This is a good approximation for small separations but at large separations it diverges, whereas the actual mismatch is bounded. Here we introduce and discuss a simple "spherical" ansatz for the mismatch $m={\sin }^2(\sqrt{g_{ab}d{\lambda }^{a}d{\lambda }^b})$. This agrees with the metric ansatz for small separations, but we show that in simple cases it provides a better (and bounded) approximation for large separations, and argue that this is also true in the generic case. This ansatz should provide a more accurate approximation of the mismatch for semi-coherent searches, and may also be of use when creating grids for hierarchical searches that (in some stages) operate at relatively large mismatch.

8 pages, 2 figures, will be submitted to PRD

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