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Free keywords:
Astrophysics, Instrumentation and Methods for Astrophysics, astro-ph.IM,General Relativity and Quantum Cosmology, gr-qc,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP
Abstract:
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.
