Runaway Merging of Black Holes: Analytical Constraint on the Timescale

TL;DR

The paper investigates whether an intermediate-mass black hole (≥10^3 M☉) can form within a star cluster via runaway mergers of stellar-mass BHs in the context of a starburst (as suggested by the M82 BH). It develops an analytical framework using the Smoluchowski equation to track the evolving mass distribution under two-body mergers driven by gravitational radiation, incorporating mass segregation effects that boost merger probabilities for massive BHs. A finite-time divergence of the second moment in the mass distribution yields an upper bound on the runaway timescale t_rm, which, for favorable cluster conditions (m0 ≈ 30 M☉, n0 ≈ 10^6 pc^-3, v0 ≈ 1 km s^-1), lies around 2–4 × 10^7 years, compatible with observed starburst ages. The results show that intermediate-mass BH formation in starburst environments is plausible under rare, favorable conditions and provide a useful analytic constraint for understanding the link between starburst activity and the growth of massive BHs.

Abstract

Following the discovery of a black hole (BH) with a mass of 10^3-10^6 M(sun) in a starburst galaxy M82, we study formation of such a BH via successive merging of stellar-mass BHs within a star cluster. The merging has a runaway characteristic. This is because massive BHs sink into the cluster core and have a high number density, and because the merging probability is higher for more massive BHs. We use the Smoluchowski equation to study analytically the evolution of the BH mass distribution. Under favorable conditions, which are expected for some star clusters in starburst galaxies, the timescale of the runaway merging is at most of order 10^7 yr. This is short enough to account for the presence of a BH heavier than 10^3 M(sun) in an ongoing starburst region.