Hierarchical Bayesian Inference on an Analytical Toy Model of the LISA MBHB Population

ABSTRACT Massive black hole binary (MBHB) mergers detected by the Laser Interferometer Space Antenna (LISA) will provide insights on their formation via dark matter (DM) halo and galaxy mergers. We present a novel Bayesian inference pipeline to infer the properties of an analytical model describing the MBHB population. The flexibility of our approach allows for exploring the uncertain range of MBH seeding and growth, as well as the interplay between MBH and galactic astrophysics. This flexibility is fundamental for the successful implementation and optimization of hierarchical Bayesian parameter estimation that we apply to the LISA MBHB population for the first time. Our inferred population hyper-parameters are chosen as proxies to characterize the MBH–DM halo mass scaling relation, the occupation fraction of MBHs in DM haloes and the delay between halo and MBHB mergers. We find that LISA will provide tight constraints at the lower-end of the mass scaling relation, complementing EM observations which are biased towards large masses. Furthermore, our results suggest that LISA will constrain features of the MBH occupation fraction at high redshift, as well as time delays around a few hundreds of Myr. Although our analysis clearly shows that results are affected by a degeneracy between the efficiency of time delays and the overall abundance of MBH that can potentially merge, they open the possibility to constrain dynamical evolution times such as the dynamical friction. Our analysis is a first attempt at developing hierarchical Bayesian inference to the LISA MBHB population, opening the way for further investigations.