Merger History of Clustered Primordial Black Holes

TL;DR

This work addresses how initially clustered PBHs influence binary formation and mergers, introducing a formalism to evolve the orbital-parameter distribution under continuous perturbations. It shows that perturbations destroy or elongate many binaries, causing a suppression of the merger rate starting at a characteristic time t_sup and producing a merger-history that scales as R ∝ t^{-45/14} during active perturbations, in contrast to the unclustered R ∝ t^{-34/37}. The authors connect this to the stochastic gravitational-wave background, predicting a broken power-law spectrum with Ω_gw ∝ ν^{-65/28} above a turnover ν_sup, and demonstrate that space-based detectors like LISA could detect signatures of PBH clustering. Overall, the results provide a concrete observational handle on PBH clustering scenarios and their viability as dark-matter candidates via future GW observations.

Abstract

Primordial black hole (PBH) binaries experience strong gravitational perturbations in the case of their initial clustering, which significantly affects the dynamics of their mergers. In this work, we develop a new formalism to account for these perturbations and track the evolution of the binary orbital parameters distribution. Based on this approach, we calculate the merger rate of PBH binaries and demonstrate that its temporal evolution differs greatly from that of isolated binary systems. Moreover, PBH clustering produces distinctive features in the stochastic gravitational-wave background: the canonical $2/3$ spectral slope transforms to $Ω_{\rm gw} \propto ν^{-65/28}$ in a certain frequency band. These predictions can be probed in future gravitational wave observations, opening up new opportunities to test the clustering of PBHs and their contribution to dark matter.

Figures (6)

• Figure 1: The example of the orbital parameter space of PBH binaries with $\delta_{\rm PBH} =0.5$, $\sigma = 10$ km s$^{-1}$ and $m = 10\, M_{\odot}$. Left: Binaries merging by time $\leq t_0/10$ are shaded in gray. The hatched area (which emphasizes that binaries with these parameters "drop out" of the merger process) schematically shows binaries that have been perturbed at the same time, for which the semimajor axis exceeds the value given by $\tau(a) = t_0/10$. For reasons given in the text, perturbations of binaries with a nearly circular orbit ($j \geq 1/2$) are assumed not to affect their merger time. The dashed horizontal line indicates the critical angular momentum of binaries $j_p$\ref{['eq:jp']}, separating tight systems that avoid perturbation to $t_0/10$ and wide perturbed binaries. Right: Similar to the left parameter space, but on the $(t,j)$ plane, where the transformation from $a$ to $t$ is done using Eq. \ref{['eq:tgw']}. At fixed time $t$, the merging binaries have angular momenta from $j_{\rm min}$ to $1$. However, systems with small angular momenta (corresponding to wide binaries) are perturbed, leading to an effective constraint from below on the value of $j_p$.
• Figure 2: Time evolution of the merger rate for different values of the density contrast $\delta_{\rm PBH}$ and velocity dispersion in the cluster $\sigma = 10$ km s$^{-1}$ is shown by the solid lines. The purple dot-dashed line shows the case of a cluster with $\sigma = 10^3$ km s$^{-1}$ and $\delta_{\rm PBH} = 0.1$. The red dotted line shows the analytical estimate of the merger rate given by Eq. \ref{['eq:mr_est']}. The dashed purple line shows the PBH merger rate when the effects of binary perturbations are negligible (corresponding to the unclustered case). The vertical dashed lines show the moment of time $t_{\rm sup}$, determined by Eq. \ref{['eq:tsup']}, for $\delta_{\rm PBH} = 2$ and $\delta_{\rm PBH} = 1$, respectively.
• Figure 3: Contour plot of the current merger rate of PBH binaries $\mathcal{R}_0$ (in units of [Gpc$^{ -3}$ yr$^{-1}$]) calculated for the PBH fraction $f = 0.01$ for different clustering parameters $\delta_{\rm PBH}$ and the velocity dispersion $\sigma$ in the cluster, the PBH mass $m = 10\,M_{\odot}$. Regions in parameter space incompatible with the LIGO-Virgo-KAGRA merger rate $\mathcal{R}_0 = 18 - 44$ Gpc$^{ -3}$ yr$^{-1}$KAGRA:2021duu are ruled out by GW observations. For other PBH fractions $f \neq 0.01$, the merger rate should be rescaled by a factor of $f/0.01$ (also the condition $f \leq \delta_{\rm PBH}$ should be satisfied).
• Figure 4: Spectral energy density of the stochastic GW background from mergers of PBHs with $m = 10\,M_{\odot}$ mass for different values of the clustering parameter $\delta_{\rm PBH}$ and $\sigma = 10$ km s$^{-1}$. For illustration purposes we adopt $f = 0.1$ here, though microlensing constraints typically favor $f \sim 0.01$. The dashed line shows the analytic approximation is given by Eq. \ref{['eq:Ogw3']}, with the minimum plotting frequency determined by $\nu_{\rm sup}$ from Eq. \ref{['eq:nu_sup']}. The dash-dotted line line shows the case of the Poisson (with no initial clustering) distribution of PBHs normalized such that the current merger rate $\mathcal{R}_{\rm PBH}(t_0) = 10$ Gpc$^{-3}$ yr$^{-1}$ and changes with redshift as $\mathcal{R} \propto t^{-34/37}$. The red dotted line shows the case of astrophysical black holes (see Appendix B for details) and $\mathcal{R}_{\rm ABH}(t_0) = 40$ Gpc$^{-3}$ yr$^{-1}$. The shaded areas show the projected sensitivity of some future GW detectors, as well as the upper limit from the O3 run results KAGRA:2021kbb.
• Figure 5: The current merger rate as a function of the angular momentum distribution parameter $n$ in Eq. \ref{['dP_per']}. Different curves correspond to different velocity dispersions $\sigma$.