Reconciling PTA and JWST, and preparing for LISA with

Aims. We develop a parametrised model to describe the formation and evolution of massive black holes. This model is designed for comparisons with observations of electromagnetic and gravitational waves. Methods. Using an extended Press-Schechter formalism, we generated dark matter halo merger trees. We then seeded and evolved massive black holes through parametrised prescriptions. This approach avoids solving differential equations and is computationally efficient. It enabled us to analyse observational data and infer the parameters of our model in a fully Bayesian framework. Results. Observations of the black hole luminosity function are compatible with the nanohertz gravitational-wave signal (that is likely) measured by pulsar-timing arrays when we allow for a higher luminosity function at high redshift (4 − 7), as was recently suggested based on observations with the James Webb Space Telescope. Our model can simultaneously reproduce the bulk of the M* − MBH relation at z − 0 and its outliers. Cosmological simulations struggle to do this. 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 super-Eddington accretion episodes that last a few dozen million years, which in our model are linked to galaxy mergers. Accretion is suppressed in the most massive black holes below z ≃ 2.5 in our model, which is consistent with the anti-hierarchical growth hypothesis. Finally, our predictions for LISA, although fairly broad, agree with previous models that assumed an efficient merging of massive black holes that formed from heavy seeds. Conclusions. Our model offers a new perspective on the apparent tensions between the black hole luminosity function and the latest results from the James Webb Space Telescope and pulsar-timing arrays. Its flexibility makes it ideal to fully exploit the potential of future gravitational-wave observations of massive black hole binaries with LISA.