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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.