An analytical joint prior for effective spins for inference on the spin distribution of binary black holes

TL;DR

This paper derives an accurate analytical joint prior for effective spins of binary black holes, improving population inference reliability in gravitational wave data analysis.

Contribution

It introduces an analytical form of the joint prior for effective spins, replacing less accurate numerical methods used previously.

Findings

Analytical prior is accurate across the entire parameter space.

Reanalysis of GWTC-3 data shows minimal changes but highlights importance of prior accuracy.

Analytical prior reduces log-likelihood errors in population studies.

Abstract

We derive an analytical form of the joint prior of effective spin parameters, $\chi_\mathrm{eff}$ and $\chi_\mathrm{p}$, 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 $|\chi_\mathrm{eff}|$ and $\chi_\mathrm{p}$ are small. Our analytical 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 $\mathcal{O}(1)$. Since these errors accumulate with the increasing number of events, our analytical prior will be crucial in the future analyses.