Emulating the complex galactic-scale orbital dynamics of LISA massive black hole pairs with normalizing flows

Abstract

Massive black hole (MBH) pairs, formed in galaxy mergers, may coalesce in a burst of gravitational waves. Estimating the coalescence time-scales and rates is a long-standing astrophysical problem, essential to inform predictions for future gravitational-wave detectors such as the Laser Interferometer Space Antenna, but remains challenging: MBH orbital decay in realistic galactic environments is complex and stochastic, and non-axisymmetric structures such as stellar bars can perturb MBH pair dynamics, delaying or accelerating binary formation and undermining the assumption that dynamical friction alone sets the inspiral duration. Capturing this evolution requires simulations too expensive to run at population scale. Here we present an artificial-intelligence framework that emulates the galactic-scale orbital decay of an inspiralling MBH using conditional normalizing flows trained on a large suite of semi-analytical orbital integrations. Our model captures the evolution of secondary MBHs orbiting within multi-component galactic merger remnants featuring rotating stellar discs and bars, across a broad range of MBH masses, orbital configurations, and bar properties. The trained emulator reproduces the simulations' decay-time distributions while reducing computational cost by orders of magnitude. For the first time, we apply this model to galaxy populations drawn from a cosmological simulation, exploiting morphological information on barred and non-barred galaxies to infer MBH binary formation time-scales across cosmic time. Our results show that stellar bars can alter the distribution of MBH binary formation times. More broadly, this demonstrates how simulation-based, surrogate machine-learning emulators can unlock a class of astrophysical problems where the physics is well understood system-by-system but intractable at scale.