The Merger Rate of Primordial Black Holes

TL;DR

This work develops a numerical framework to compute the total PBH merger rate by combining unperturbed binaries evolving solely via gravitational waves with dynamical evolution inside dark matter halos, including binary-single interactions and GW capture channels. The authors derive the unperturbed rate from GW-driven evolution, model halo-driven hardening/softening across halo shells, and compute per-halo merger rates that are integrated over the halo mass function to obtain the total comoving rate. Their results show that environmental effects inside halos significantly enhance mergers at late times, with the total rate at z ≤ 2 being about 50% higher than estimates assuming fully isolated binaries; the framework also reveals how halo mass and formation history shape the contributions from different channels. The study provides publicly available PBH merger rates and offers a versatile tool for interpreting gravitational-wave observations and constraining PBH dark matter scenarios.

Abstract

The merger rate of primordial black hole (PBH) binaries can be used to understand the source population of the merging black hole binaries observable through gravitational-waves (GWs) and also to constrain the possible contribution of PBHs to dark matter. In the literature, the PBH merger rate is calculated analytically, assuming that PBH binaries stay in isolation (i.e. are unperturbed) and evolve solely via GW emission during their entire lifetime. However, if some or all of dark matter consists of PBHs, then as cosmic structures grow, PBH binaries and single PBHs fall inside dark matter halos. In those halos, the PBH binaries' interactions with their environment significantly affect the subsequent evolution of their orbital properties. In this paper, we present a numerical framework that accurately calculates the total PBH merger rate by combining the evolution of isolated binaries outside halos with the dynamics of binaries inside halos. In our work we have found that the isolated binary channel is suppressed at low redshifts and dynamical interactions in halos reshape the merger rate evolution with time, accelerating some mergers. At redshifts of $\lesssim 2$ the total merger rate is a factor of $\simeq 50 \%$ higher than the results assuming that all PBH binaries effectively stay unperturbed until their merger. Our simulations provide a definitive calculation on the total PBH merger rates, that are currently being probed and constrained from gravitational-wave observations. We make our merger rates publicly available at Zenodo

Figures (11)

• Figure 1: The comoving merger rate of unperturbed early PBH binaries outside halos assuming $f_{\textrm{PBH binaries}}(z=3400)=0.5$ and $m_{\textrm{PBH}} = 30,M_\odot$. The red dashed-dotted curve shows the comoving merger rate of unperturbed binaries rescaled by the actual fraction of dark matter present outside halos $f_{\rm{DM,outside}}(z)$, while the gray dashed curve takes all PBH binaries to remain in isolation (unperturbed) setting $f_{\rm{DM,outside}}(z)=1$. The blue solid curve (right $y$-axis) shows the fraction of dark matter outside halos as a function of redshift.
• Figure 2: Examples (colored dashed, dotted, and dot-dashed lines) of how binary-single interactions inside dark matter halos can significantly alter the merger timescale for PBH binaries compared to the assumption that those binaries evolve in isolation (black solid lines). The top panel gives examples of binary-single interactions softening a binary and delaying its merger. The middle and bottom panels show instead examples of binary-single interactions causing hardening and accelerating a binary's merger. We give examples of different PBH binary's redshift of entry into the dark matter halo ($z_{\rm entry}$) and different local environmental assumptions (shell of the dark matter halo). See text for more details.
• Figure 3: The distribution of the semi-major axes $a_{\rm init}$ and eccentricities $1-e_{\rm init}$ at $z=12$, for PBH binaries that merge by $z=0$. Top: all binaries are evolved in isolation via GW emission. Bottom: binaries evolved for simplicity all in a halo of mass $M=10^{6} M_\odot$ in one of its mid-radial distance shells ($i=3$ out of 5 shells). There are binaries that would merge over very long timescales ($t_{\rm merge} \sim 10^4~\rm Gyr$) in isolation but merge by $z=0$ due to dynamical interactions inside the halo. The binaries begin to fall inside the dark matter halo around $z= 12$ but continue to grow the halo up to $z=0$ (see text for more details).
• Figure 4: Histogram of the merger times for different PBH binary populations. We show results for $t_{\textrm{merge}}$ up to the age of the universe. Green: unperturbed binaries evolved in isolation, assuming $f_{\rm DM \, outside}(z)=1$. Solid red: binaries evolved inside halos, showing the acceleration of mergers due to binary–single interactions. Dashed red: unperturbed binaries evolved in isolation properly weighted by $f_{\rm DM \, outside}(z)$. Purple: total population once the dark matter halo evolution is considered.
• Figure 5: The merger rate per halo as a function of redshift for a halo with $\rm M(z=0)=1.15 \times 10^{12} M_{\odot}$. We have assumed $f_{\textrm{PBH\, binaries}}=0.5$ and $f_{\mathrm{PBH}}=1$.