Template banks based on ${\mathbb{Z}}^n$ and $A_n^{\ast }$ lattices

Abstract

Matched filtering is a traditional method used to search a data stream for signals. If the source (and hence its $n$ parameters) are unknown, many filters must be employed. These form a grid in the $n$-dimensional parameter space, known as a template bank. It is often convenient to construct these grids as a lattice. Here, we examine some of the properties of these template banks for ${\mathbb{Z}}^n$ and $A_n^{\ast }$ lattices. In particular, we focus on the distribution of the mismatch function, both in the traditional quadratic approximation and in the recently-proposed spherical approximation. The fraction of signals which are lost is determined by the even moments of this distribution, which we calculate. Many of these quantities we examine have a simple and well-defined $n\to \infty$ limit, which often gives an accurate estimate even for small $n$. Our main conclusions are the following: (i) a fairly effective template-based search can be constructed at mismatch values that are shockingly high in the quadratic approximation; (ii) the minor advantage offered by an $A_n^{\ast }$ template bank (compared to ${\mathbb{Z}}^n$) at small template separation becomes even less significant at large mismatch. So there is little motivation for using template banks based on the $A_n^{\ast }$ lattice.

17 pages, 8 figures

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• Banks of templates for directed and all-sky narrow-band searches of continuous gravitational waves from spinning neutron stars with several spindowns
• Performance of Random Template Banks