Smoluchowski Coagulation Equation and the Evolution of Primordial Black Hole Clusters

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.

Figures (10)

• Figure 1: Schematic plot of the domain of the function $P(\tau,i,j)$
• Figure 2: Dimensionless runaway timescales as a function of $N_{\rm cl}$. Orange circles show results without mass segregation, brown squares correspond to the Gaussian model, and cyan diamonds correspond to the Plummer model. The corresponding dashed lines are linear fits for each model. The dimensionless runaway timescale is defined as the time at which the central SMBH attains approximately half of the total cluster mass.
• Figure 3: PBH mass population evolution in a cluster of $N_{\rm cl}=10^5$ from redshift $z\simeq11.9\textup{--}5.8$ with $n_{\rm cl}=2.0\times10^8{\rm\,pc^{-3}}$ and $v_{\rm vir}=443{\rm\,km\,s^{-1}}$.The evolution of the mass distribution in a PBH cluster can be divided into three stages. During the first stage, the cluster comprises predominantly small-mass PBHs, and the merger is slow. In the second stage, intermediate-mass black holes emerge, and the merger rate increases. In the final stage, a SMBH forms, and mass increases dramatically within a short period corresponding to the runaway-merger phase. During this phase, intermediate-mass black holes merge into the central SMBH, leaving only small-mass PBHs, and an EMRI system naturally arises.
• Figure 4: PBH mass population evolution in a cluster of $N_{\rm cl}=10^5$ from redshift $z\simeq22.0\textup{--}13.3$ with $n_{\rm cl}=2.0\times10^8{\rm\,pc^{-3}}$ and $v_{\rm vir}=443{\rm\,km\,s^{-1}}$ with mass segregation of Gaussian model.
• Figure 5: PBH mass population evolution in a cluster of $N_{\rm cl}=10^5$ from redshift $z\simeq30.0\textup{--}20.2$ with $n_{\rm cl}=2.0\times10^8{\rm\,pc^{-3}}$ and $v_{\rm vir}=443{\rm\,km\,s^{-1}}$ with mass segregation of Plummer model.