Smoluchowski Coagulation Equation and the Evolution of Primordial Black Hole Clusters

TL;DR

This paper models the evolution of primordial black hole clusters using the Smoluchowski coagulation equation, providing insights into their merger history and mass distribution over cosmic time.

Contribution

It introduces a comprehensive simulation framework for PBH cluster evolution, incorporating mass segregation effects and advanced numerical methods.

Findings

Determines runaway timescales for PBH clusters.

Analyzes mass evolution of PBHs across redshifts.

Provides a detailed simulation approach for PBH mergers.

Abstract

In arXiv:2507.07171, we demonstrate that the high-redshift supermassive black holes in the so-called "little red dots" discovered by James Webb Space Telescope (JWST) can be explained by the primordial black hole (PBH) clustering on small scales. In this paper, we present a comprehensive simulation of the successive PBH mergers within a cluster by solving the Smoluchowski coagulation equation. We derive the coagulation kernel considering both cases with and without the effects of mass segregation. Then we employ the Monte Carlo method to solve the equation, implementing the full-conditioning scheme using the discrete inverse transformation method. Our simulations determine the runaway timescales of clusters and the mass population evolution of PBHs across a wide range of cosmic redshifts, depending on the number of PBHs within the cluster and the associated density.