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

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).

💡 Research Summary

This paper revisits the problem of optimal detection statistics for continuous gravitational waves (CWs) from rotating neutron stars, focusing on a Bayesian formulation that marginalizes over the four amplitude parameters (overall amplitude h₀, inclination ι, polarization angle ψ, and initial phase ϕ₀). The classic F‑statistic, derived by Jaranowski, Królak, and Schutz (JKS), maximizes the likelihood over these amplitudes and is computationally cheap, but it implicitly assumes a uniform prior in the linear amplitude space, which translates into a prior proportional to h₀³. Such a “strong‑signal” prior over‑weights large amplitudes and linear polarizations, making the F‑statistic sub‑optimal from a Bayesian perspective.

Previous work introduced the B‑statistic, a Bayes factor obtained by marginalizing the likelihood with a uniform prior on h₀. While more powerful than the F‑statistic, the B‑statistic requires numerical integration for two of the four amplitude dimensions, increasing computational cost. An analytic approximation by Bero & Whelan provided a fully closed‑form expression but at the expense of accuracy in some regimes.

The authors propose a new prior for h₀: a half‑Gaussian (i.e., a Gaussian truncated to non‑negative values) with scale parameter H,  P(h₀) ∝ exp(−h₀²/(2H²)) for h₀ ≥ 0. This prior is physically motivated because it favours weaker signals, reflecting astrophysical expectations that most neutron stars emit relatively small CW amplitudes. Importantly, the half‑Gaussian reduces to the uniform prior in the limit H → ∞, thereby reproducing the standard B‑statistic in the strong‑signal regime, while for H → 0 it represents the weak‑signal limit.

Analytically, the marginalization over the two amplitudes that are linear in the data (the “JKS” coordinates) remains tractable, as in the original B‑statistic. The remaining two integrals, which involve the non‑linear dependence on h₀, are expanded in a Taylor series around H = 0. Keeping the leading term yields a fully analytic expression for the Bayes factor, denoted 𝔅_ws (weak‑signal Bayes factor). The resulting statistic can be written as an exponential of a quadratic form involving the familiar matched‑filter quantities q(x; η, ψ) and g(η, ψ), multiplied by a simple prefactor. Consequently, 𝔅_ws follows a χ² distribution with four degrees of freedom, just like the F‑statistic, but its normalization incorporates the half‑Gaussian prior, making it more robust against noise fluctuations.

From a computational standpoint, 𝔅_ws requires exactly the same operations as the F‑statistic (inner products of the data with the four basis functions), eliminating the need for any numerical integration. Thus, the method retains the O(N) scaling of the F‑statistic while offering the Bayesian optimality of a properly marginalized likelihood.

The authors validate the new statistic through extensive Monte‑Carlo simulations. In coherent searches spanning a full day, 𝔅_ws achieves detection sensitivities essentially identical to the F‑statistic and only marginally below the full B‑statistic (and the Bero‑Whelan approximation). In semi‑coherent searches where the data are broken into short segments (900 s to a few thousand seconds), 𝔅_ws outperforms the weighted dominant‑response statistic F_ABw for short segments and matches the performance of the weighted F‑statistic (F_w) for longer segments. Across the full range of segment lengths tested (from 900 s up to ten days), the weak‑signal Bayes factor consistently delivers state‑of‑the‑art sensitivity, often surpassing existing approximations.

In summary, the paper introduces a physically motivated half‑Gaussian prior on h₀, derives a closed‑form weak‑signal Bayes factor that is computationally as cheap as the F‑statistic, and demonstrates through simulations that this statistic either matches or exceeds the sensitivity of all previously proposed analytic approximations across a wide variety of coherent and semi‑coherent search configurations. The work thus provides a practical, theoretically sound alternative for future CW searches, especially those involving large parameter spaces or short‑duration segments where traditional statistics are known to be sub‑optimal.