TY - JOUR

T1 - Optimal template banks

AU - Allen, Bruce

PY - 2021/8/15

Y1 - 2021/8/15

N2 - 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, with a computational cost that is proportional to the number of these points. 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 Voronoï) cells have the smallest covering radius (smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost"because the template bank is discrete. We show that at fixed computational cost, 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 fraction of lost detections is proportional to a scale-invariant "quantizer constant"G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

AB - 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, with a computational cost that is proportional to the number of these points. 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 Voronoï) cells have the smallest covering radius (smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost"because the template bank is discrete. We show that at fixed computational cost, 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 fraction of lost detections is proportional to a scale-invariant "quantizer constant"G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

UR - http://www.scopus.com/inward/record.url?scp=85113766679&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2102.11254

DO - 10.48550/arXiv.2102.11254

M3 - Article

AN - SCOPUS:85113766679

VL - 104

JO - Physical Review D

JF - Physical Review D

SN - 2470-0010

IS - 4

M1 - 042005

ER -