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

¹ Max Planck Institute for Gravitationsphysik (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam, Germany
² Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
³ INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
⁴ School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
⁵ Institut d’Astrophysique de Paris, UMR 7095, CNRS and Sorbonne Université, 98 bis Boulevard Arago, 75014 Paris, France
⁶ SISSA – Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136 Trieste, Italy
⁷ INFN Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy
⁸ IFPU – Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy
⁹ Université Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
¹⁰ Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

^⋆ Corresponding author.

Received: 15 November 2024
Accepted: 23 June 2025

Abstract

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_* − M_BH 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.