Analytic weak-signal approximation of the Bayes factor for continuous gravitational waves

TL;DR

This paper introduces a new analytic approximation for the Bayes factor in continuous gravitational wave searches, improving computational efficiency and robustness across various segment lengths.

Contribution

It generalizes the targeted $ ext{B}$-statistic with a half-Gaussian prior, enabling fully analytic marginalization and improved sensitivity in weak-signal regimes.

Findings

Achieves sensitivity comparable to the $ ext{F}$-statistic in day-long searches.

Outperforms existing semi-coherent search statistics for short segments.

Demonstrates state-of-the-art sensitivity across diverse segment lengths.

Abstract

We generalize the targeted $\mathcal{B}$-statistic for continuous gravitational waves by modeling the $h_0$-prior as a half-Gaussian distribution with scale parameter $H$. This approach retains analytic tractability for two of the four amplitude marginalization integrals and recovers the standard $\mathcal{B}$-statistic in the strong-signal limit ($H\rightarrow\infty$). By Taylor-expanding the weak-signal regime ($H\rightarrow0$), the new prior enables fully analytic amplitude marginalization, resulting in a simple, explicit statistic that is as computationally efficient as the maximum-likelihood $\mathcal{F}$-statistic, but significantly more robust. Numerical tests show that for day-long coherent searches, the weak-signal Bayes factor achieves sensitivities comparable to the $\mathcal{F}$-statistic, though marginally lower than the standard $\mathcal{B}$-statistic (and the Bero-Whelan approximation). In semi-coherent searches over short (compared to a day) segments, this approximation matches or outperforms the weighted dominant-response $\mathcal{F}_{\mathrm{ABw}}$-statistic and returns to the sensitivity of the (weighted) $\mathcal{F}_{\mathrm{w}}$-statistic for longer segments. Overall the new Bayes-factor approximation demonstrates state-of-the-art or improved sensitivity across a wide range of segment lengths we tested (from 900s to 10days).