Reconciling PTA and JWST and preparing for LISA with POMPOCO: a Parametrisation Of the Massive black hole POpulation for Comparison to Observations

We develop a parametrised model to describe the formation and evolution of massive black holes, designed for comparisons with both electromagnetic and gravitational wave observations. Using an extended Press-Schechter formalism, we generate dark matter halo merger trees. We then seed and evolve massive black holes through parameterised prescriptions. This approach, which avoids solving differential equations, is computationally efficient, enabling us to analyse observational data and infer the parameters of our model in a fully Bayesian framework. We find that observations of the black hole luminosity function are compatible with the nHz gravitational wave signal (likely) measured by PTAs, provided we allow for an increased luminosity function at high redshift ($4-7$), as recently suggested by JWST observations. Our model can simultaneously reproduce the bulk of the $M_*-M_{\rm BH}$ relation at $z-0$, as well as its outliers, something cosmological simulations struggle to do. The inferred model parameters are consistent with expectations from observations and more complex simulations: They favour heavier black hole seeds and short delays between halo and black hole mergers, while requiring supper-Edington accretion episodes lasting a few tens of million years, which in our model are linked to galaxy mergers. We find accretion to be suppressed in the most massive black holes below $z\simeq 2.5$, consistently with the anti-hierarchical growth hypothesis. Finally, our predictions for LISA, although fairly broad, are in agreement with previous models. Our model offers a new perspective on the apparent tensions between the black hole luminosity function and the latest JWST and PTA results. Its flexibility makes it ideal to fully exploit the potential of future gravitational wave observations of massive black hole binaries with LISA.