Clustering and Runaway Merging in a Primordial Black Hole Dominated Universe

TL;DR

This work analyzes a primordial black hole (PBH) dominated epoch in the early universe, demonstrating that PBHs can form self-gravitating clusters during the BHD and undergo runaway mergers, producing massive relic PBHs with $m_{\mathrm{relic}} \gg m$. Using the Press-Schechter formalism, the authors quantify cluster formation and merger statistics, revealing that mergers can dramatically reshape the PBH mass function and yield relics that evaporate during or after Big Bang nucleosynthesis, thereby constraining the BHD parameter space. They identify two main dynamical regimes—cluster evaporation and merger-dominated runaway collapse—and derive an analytic expression for the final relic mass, showing how initial cluster size and merger/evaporation rates set the endpoint. Observationally, relics with $m_{\mathrm{relic}} > 10^{15}\,\mathrm{g}$ could contribute to dark matter, while lighter relics face stringent BBN and CMB bounds; overall, the scenario is tightly constrained under a shot-noise PBH spectrum, and the authors call for dedicated N-body simulations to refine these predictions. The results connect early-universe PBH formation and clustering to measurable cosmological signatures, including BBN/CMB constraints and potential gravitational-wave signals, illustrating a rich interplay between microphysical PBH properties and macroscopic cosmological observables.

Abstract

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

Figures (6)

• Figure 1: Example timeline describing the black hole domination scenario. At $t = t_{\mathrm{form}}$ a subdominant population of PBHs forms during radiation domination in the early universe. Since PBHs redshift like nonrelativistic matter, their energy fraction grows linearly during this era and they dominate the energy budget at $t = t_i$, which serves as the starting point for our analysis in this paper. During PBH domination, density perturbations grow linearly with scale factor and PBH clusters begin to form. Within a range of cluster masses, PBH mergers can be fast compared to Hubble expansion and consume most of the cluster in a runaway merger. This process forms a sub-population of more massive, longer lived PBH relics which can survive past $t = \tau$ when the original PBH population evaporates due to Hawking emission. Thus, even though the original PBH population evaporates before BBN, the relic population can have important consequences for BBN and CMB observables and may also survive into the later universe. This sequence of events gives rise to gravitational waves from the PBH mergers, the Hawking emission at evaporation, and from second order scalar perturbations.
• Figure 2: A comparison of the 2-body capture timescale from Eq. \ref{['eq:t_cap_2b']} and the average 2-body inspiral timescale from Eq. \ref{['eq:t_ins_2b_avg']} with the 3-body capture and inspiral timescales from Eqs. (6) and (12) of Ref. FrancioliniPBHMergersThree2022 for one choice of our free parameters $m_i$ and $t_i$, showing that Eq. \ref{['eq:2b_faster_than_3b']} is satisfied for all $N_i$.
• Figure 3: A summary of important timescales in our scenario for a range of initial cluster sizes $N_i$, for two choices of free parameters $m_i, t_i$. Relevant cosmic events are labeled at the top, including BHD, BBN, CMB, and the present time. The purple $\blacksquare$ line gives the cluster size corresponding to the mass contained within the horizon. The green $\blacksquare$ curve shows the size of the PBH cluster which can form at a given cosmic time. At a later time, indicated by the dark blue $\blacksquare$ runaway merger curve, PBH clusters of a given size collapse and merge to a single BH. Finally, this merged relic evaporates at the time of the blue $\blacksquare$ dashed curve. The light blue $\blacksquare$$N_{H,i}$ line divides scales that were sub- and super-horizon at $t_i$.
• Figure 4: Average PBH mass as a function of initial cluster PBH number $N_i$ at different times, for two choices of initial parameters $m_i$ and $t_i$. Initially at cluster formation ($t=t_{\mathrm{cl}}$), the average mass is $m_i$ (green $\blacksquare$ line), but eventually some clusters undergo runaway mergers and their average mass grows, finally reaching the mass shown by the dark blue $\blacksquare$ line at $t = t_{\mathrm{col}}$. The red $\blacksquare$ curve shows the current remaining relic masses after the cosmologically unstable ones have evaporated away by the present day ($t=t_0$). Note that for the parameters on the right, all relics evaporate before the present day. The top panel on either side shows the fractional distribution of PBHs in each cluster size at the time of PBH evaporation $\tau$ in gray $\blacksquare$.
• Figure 5: The mass function of merged relic PBHs for two choices of free parameters. Initially, all the PBHs have the same mass $m_i$, shown in green $\blacksquare$. The blue $\blacksquare$ and red $\blacksquare$ curves show the initial and present-day mass functions of merged relics, which clearly shows the mass growth from runaway mergers. Superimposed in blue $\blacksquare$ are observational bounds on evaporating BHs from BBN CarrConstraintsPBHs2021, CMB AcharyaCMBBBNConstraints2020, the extragalactic gamma-ray background (EGB) CarrConstraintsPBHs2021 and Voyager $e^\pm$Boudaud:2018hqb. Both choices of PBH parameters are ruled out by observations because they produce too many evaporating relic BHs, particularly during BBN or CMB emission. Note that the plot visually compares limits on $f_{\mathrm{BH}}$ to the logarithmic mass distribution $df_{\mathrm{BH}} / d \log m$, but the proper comparison is made with Eq. (\ref{['eq:fBHlimit']}).