Untargeted Bayesian search of anisotropic gravitational-wave backgrounds through the analytical marginalization of the posterior

We develop a method to perform an untargeted Bayesian search for anisotropic gravitational-wave backgrounds that can efficiently and accurately reconstruct the background intensity map. Our method employs an analytic marginalization of the posterior of the spherical-harmonic components of the intensity map, without assuming the background possesses any specific angular structure. The key idea is that the likelihood function of the spherical-harmonic components is a multivariate Gaussian when the intensity map is expressed as a linear combination of the spherical-harmonic components and the noise is stationary and Gaussian. If a uniform and wide prior of these spherical-harmonic components is prescribed, the marginalized posterior and the Bayes factor can be well approximated by a high-dimensional Gaussian integral. The analytical marginalization allows us to regard the spherical-harmonic components of the intensity map of the background as free parameters and to construct their individual marginalized posterior distribution in a reasonable time, even though many spherical-harmonic components are required. The marginalized posteriors can, in turn, be used to accurately construct the intensity map of the background. By applying our method to mock data, we show that we can recover precisely the angular structures of various simulated anisotropic backgrounds, without assuming prior knowledge of the relation between the spherical-harmonic components predicted by a given model. Our method allows us to bypass the time-consuming numerical sampling of a high-dimensional posterior, leading to a more model-independent and untargeted Bayesian measurement of the angular structures of the gravitational-wave background.