Enhanced cosmological perturbations and the merger rate of PBH binaries

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 $\sim 10 Mpc^{-1}$ gives a contribution $\langle j^2 \rangle \sim 10^3 P_\Phi$, 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 $10^3-10^5 Mpc^{-1}$, 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 $\sim 30 M_\odot$. 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.