Smoluchowski Coagulation Equation and the Evolution of Primordial Black Hole Clusters

Borui Zhang, Wei-Xiang Feng, Haipeng An

First listed 2026-04-02 | Last updated 2026-04-12

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.

Short digest

The authors model hierarchical growth inside small-scale primordial black hole clusters by casting mergers as a Smoluchowski coagulation problem, deriving a gravitational-wave–capture kernel with and without mass segregation and solving it via a full-conditioning Monte Carlo using the discrete inverse transform. The simulations chart runaway-merger timescales and the evolving PBH mass function across cosmic time as functions of cluster richness and density, including the track of the maximum mass and the merger-rate history. They identify the cluster conditions that trigger rapid runaway capable of assembling a central massive black hole early, supporting PBH-cluster seeding of the little-red-dot SMBHs proposed previously. The framework is a computationally efficient alternative to Fokker–Planck treatments while retaining the essential GW-driven binary physics in virialized clusters.

Key figures to inspect

• Merger kernel and GW-capture cross section: plot of K(m1,m2) averaged over the virial velocity distribution, highlighting the scaling from their Eq. (2.6)–(2.7) and how velocity dispersion suppresses/enhances capture relative to gravitational (Coulomb) scattering.
• Runaway timescale versus cluster parameters: curves of t_run as a function of PBH number per cluster and cluster density, shown both with and without mass segregation to quantify how segregation accelerates coalescence.
• Mass-function evolution snapshots: n(m) at successive times/redshifts starting from an initial monochromatic distribution, illustrating the emergence of a top-heavy tail and the fraction of mass locked in the most massive bin(s).
• Maximum-mass growth track: M_max(t or z) for representative cluster setups, used to read off when a central massive BH forms and how quickly it approaches SMBH scales.
• Merger-rate history: cluster-integrated merger rate versus time/redshift from the Smoluchowski solution, with notes on implications for stochastic GW backgrounds and EMRI progenitors.