Hierarchical Bayesian inference on an analytical model of the LISA massive black hole binary population

Massive black hole binary (MBHB) mergers will be detectable in large numbers by the Lisa Interferometer Space Antenna (LISA), which will thus provide new insights on how they form via repeated dark matter (DM) halo and galaxy mergers. Here we present a simple analytical model to generate a population of MBHB mergers based on a theoretical prescription that connects them to DM halo mergers. The high flexibility of our approach allows us to explore the broad and uncertain range of MBH seeding and growth mechanisms, as well as the different effects behind the interplay between MBH and galactic astrophysics. Such a flexibility is fundamental for the successful implementation and optimisation of the hierarchical Bayesian parameter estimation approach that here we apply to the MBHB population of LISA for the first time. Our inferred population hyper-parameters are chosen as proxies to characterise the MBH--DM halo mass scaling relation, the occupation fraction of MBHs in DM halos and the delay between halo and MBHB mergers. We find that LISA will provide tight constraints at the lower-end of the MBH-halo scaling relation, well complementing EM observations which are biased towards large masses. Furthermore, our results suggest that LISA will constrain some features of the MBH occupation fraction at high redshift, as well as merger time delays of the order of a few hundreds of Myr, opening the possibility to constrain dynamical evolution time scales such as the dynamical friction. The analysis presented here constitutes a first attempt at developing a hierarchical Bayesian inference approach to the LISA MBHB population, opening the way for several further improvements and investigations.