Scaling Relations for Binary Black Hole Merger Times from Cosmological Initial Conditions

Abstract

Recent evidence from Pulsar Timing Arrays (PTAs) for a nanohertz gravitational wave background is broadly consistent with theoretical expectations from a population of massive black hole binaries (MBHBs), although the inferred amplitude appears somewhat higher than predicted by standard models. Interpreting these observations requires a robust understanding of the merger timescales of MBHBs, and of their connection to host galaxy properties. In this work, we investigate the evolution of MBHBs selected from cosmological galaxy mergers in the IllustrisTNG simulation. We re-simulate these systems at high resolution using the N-body code Griffin to accurately resolve the dynamical friction and stellar hardening phases, and follow their evolution to coalescence with a semi-analytical model. We find that cosmological galaxy encounters and the resulting MBHBs are typically highly eccentric. We characterise the distribution of binary eccentricities at formation and at entry into the PTA band, and quantify the corresponding residence times. We identify the key parameters governing the duration of the different stages of MBHB evolution, and derive scaling relations linking galaxy and orbital properties to dynamical friction, hardening, and total coalescence times. These relations provide a framework for subgrid prescriptions in cosmological simulations. Applying these scaling relations to the full IllustrisTNG merger population, we infer the probability distributions of galaxy merger and black hole coalescence times. We find that galaxy mergers typically complete within $\sim 0.7$ Gyr, while the total black hole coalescence time is $\sim 1.0$ Gyr. These short timescales imply efficient binary evolution, consistent with current PTA constraints.