Geometric Approach to Analytic Marginalisation of the Likelihood Ratio for Continuous Gravitational Wave Searches

The likelihood ratio for a continuous gravitational wave signal is viewed geometrically as a function of the orientation of two vectors; one representing the optimal signal-to-noise ratio, the other representing the maximised likelihood ratio or $\mathcal{F}$-statistic. Analytic marginalisation over the angle between the vectors yields a marginalised likelihood ratio which is a function of the $\mathcal{F}$-statistic. Further analytic marginalisation over the optimal signal-to-noise ratio is explored using different choices of prior. Monte-Carlo simulations show that the marginalised likelihood ratios have identical detection power to the $\mathcal{F}$-statistic. This approach demonstrates a route to viewing the $\mathcal{F}$-statistic in a Bayesian context, while retaining the advantages of its efficient computation.