Analytic joint priors of effective spin parameters and inference of the spin distribution of binary black holes

We derive an analytic form of the joint prior of effective spin parameters, χeff and χp, assuming an isotropic and uniform-in-magnitude spin distribution. This is a vital factor in performing hierarchical Bayesian inference for studying the population properties of merging compact binaries observed with gravitational waves. In previous analyses, this was evaluated numerically using kernel density estimation (KDE). However, we find that this numerical approach is inaccurate in certain parameter regions, where both |χeff| and χp are small. Our analytic approach provides accurate computations of the joint prior across the entire parameter space and enables more reliable population inference. Employing our analytic prior, we reanalyze binary black holes in the gravitational-wave transient catalog 3 (GWTC-3) by the LIGO-Virgo-KAGRA collaboration. While the results are largely unchanged, log-likelihood errors due to the use of the inaccurate prior evaluations are O(1), implying the bias is already at the concerning level if one adopts a log-likelihood variance cut. Since the systematic bias from the numerical method accumulates with the increasing number of events, our analytic prior will be more crucial in future analyses. Published by the American Physical Society 2025