An Analytic Approximation to the Bayesian Detection Statistic for Continuous Gravitational Waves

TL;DR

This paper introduces an analytical approximation to the Bayesian detection statistic for continuous gravitational waves, simplifying calculations while maintaining effectiveness over traditional methods in long-term observations.

Contribution

The authors derive a closed-form approximation to the Bayesian $ ext{B}$-statistic using a first-order Taylor expansion, enabling efficient detection in continuous gravitational wave searches.

Findings

The approximation matches the full $ ext{B}$-statistic in detection power.

It outperforms the $ ext{F}$-statistic in several scenarios.

Performance declines for short, near-instantaneous observations.

Abstract

We consider the Bayesian detection statistic for a targeted search for continuous gravitational waves, known as the $\mathcal{B}$-statistic. This is a Bayes factor between signal and noise hypotheses, produced by marginalizing over the four amplitude parameters of the signal. We show that by Taylor-expanding to first order in certain averaged combinations of antenna patterns (elements of the parameter space metric), the marginalization integral can be performed analytically, producing a closed-form approximation in terms of confluent hypergeometric functions. We demonstrate using Monte Carlo simulations that this approximation is as powerful as the full $\mathcal{B}$-statistic, and outperforms the traditional maximum-likelihood $\mathcal{F}$-statistic, for several observing scenarios which involve an average over sidereal times. We also show that the approximation does not perform well for a near-instantaneous observation, so the approximation is suited to long-time continuous wave observations rather than transient modelled signals such as compact binary inspiral.