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

We generalize the targeted -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 -statistic in the strong-signal limit ( ). By Taylor-expanding the weak-signal regime ( ), the new prior enables fully analytic amplitude marginalization, resulting in a simple, explicit statistic that is as computationally efficient as the maximum-likelihood -statistic, but significantly more robust. Numerical tests show that for day-long coherent searches, the weak-signal Bayes factor achieves sensitivities comparable to the -statistic, though marginally lower than the standard -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 -statistic and returns to the sensitivity of the (weighted) -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).