Optimal Template Banks
Published 2021-02-22Version 1
When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space. The computational cost is proportional to the number of these points, and if that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Voronoi) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, we show that at fixed computational cost, for realistic populations of signal sources, the minimum thickness template bank does NOT maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the most detections are obtained for a template bank which minimizes the average squared distance from the nearest template, i.e., the average expected mismatch. Mathematically, such a template bank is said to be the "optimal quantizer". We review the optimal quantizers for template banks that are built as $n$-dimensional lattices, and show that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.