Bayesian parameter estimation for targeted anisotropic gravitational-wave background

Extended sources of the stochastic gravitational backgrounds have been conventionally searched on the spherical harmonics bases. The analysis during the previous observing runs by the ground-based gravitational-wave detectors, such as LIGO and Virgo, have yielded the constraints on the angular power spectrum ${C}_{\ensuremath{\ell}}$, yet it lacks the capability of estimating other parameters such as a spectral index. In this paper, we introduce an alternative Bayesian formalism to search for such stochastic signals with a particular distribution of anisotropies on the sky. This approach provides a Bayesian posterior of model parameters and also enables selection tests among different signal models. While the conventional analysis fixes the highest angular scale a priori, here we show a more systematic and quantitative way to determine the cutoff scale based on a Bayes factor, which depends on the amplitude and the angular scale of observed signals. Also, we analyze the third observing runs of LIGO and Virgo for the population of millisecond pulsars and obtain the 95% constraints of the signal amplitude, $\ensuremath{\epsilon}<2.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$.