Enhanced cosmological perturbations and the merger rate of PBH binaries

Abstract

The rate of merger events observed by LIGO/Virgo can be used in order to probe the fraction $f$ of dark mater in the form of Primordial Black Holes. Here, we consider the merger rate of PBH binaries, accounting for the effect of cosmological perturbations on their initial eccentricity $e$. The torque on the binaries receives significant contributions from a wide range of scales, that goes from the size of the horizon at the time when the binary forms, down to the co-moving size of the binary. In scenarios where PBH are formed from adiabatic perturbations, it is natural to expect an enhancement of the power spectrum $P_{\Phi }$ at small scales, where it is poorly constrained observationally. The effect can then be quite significant. For instance, a nearly flat spectrum with amplitude $P_{\Phi }\gtrsim 10^{-7}$ on scales smaller than $\sim 10Mp{c}^{-1}$ gives a contribution $⟨j^2⟩\sim 10^3{P}_{\Phi }$, where $j=(1-e^2{)}^{1/2}$ is the dimensionless angular momentum parameter of the binaries. This contribution can dominate over tidal torques from neighboring PBHs for any value of $f$. Current constraints allow for a power spectrum as large as $P_{\Phi }\sim 10^{-5}$ at the intermediate scales $10^3-10^{5}Mp{c}^{-1}$, comparable to the co-moving size of the binaries at the time of formation. In particular, this can relax current bounds on the PBH abundance based on the observed LIGO/Virgo merger rate, allowing for a fraction $f\sim 10\%$ of dark matter in PBH of mass $\sim 30M_{\odot }$. We investigate the differential merger rate $\Delta \Gamma (m_1,m_2)$, as a function of the masses of the binary components, and the corresponding ``universality" coefficient $\alpha =-(m_1+m_2{)}^2{\partial }^2\ln \Delta \Gamma /\partial m_1\partial m_2$, concluding that merger rates may provide valuable information on the spectrum of primordial cosmological perturbations at currently uncharted lengthscales.

arXiv comment38 pages, 12 figures

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