Clustering and Runaway Merging in a Primordial Black Hole Dominated Universe

If primordial black holes (PBH) are present in the early universe, their contribution to the energy budget grows relative to that of radiation and generically becomes dominant unless the initial abundance is exponentially small. This black hole domination scenario is largely unconstrained for PBHs with masses $\lesssim 10^9\,\mathrm{g}$, which evaporate prior to Big Bang nucleosynthesis. However, if the era of PBH domination is sufficiently long, the PBHs form clusters and can merge appreciably within these objects. We calculate the population statistics of these clusters within the Press-Schechter formalism and find that, for a wide range of PBH masses and Hubble rates at the onset of PBH domination, the mergers within PBH clusters can exhibit runaway behavior, where the majority of the cluster will eventually form a single black hole with a mass much greater than the original PBH mass. These mergers can dramatically alter the PBH mass distribution and leave behind merged relic black holes that evaporate after Big Bang nucleosynthesis and yield various observational signatures, excluding parameter choices previously thought to be viable