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

We generalize the targeted B-statistic for continuous gravitational

waves by modeling the h0 -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 B-statistic in the strong-signal limit (H→∞). By

Taylor-expanding the weak-signal regime (H→0), the new prior enables

fully analytic amplitude marginalization, resulting in a simple,

explicit statistic that is as computationally efficient as the

maximum- likelihood 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

F-statistic, though marginally lower than the standard 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 F b ABw -statistic and

returns to the sensitivity of the (weighted) F b 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 900 s to 10 days).