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 ( PBH ) .
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 may receive 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 .
Extrapolating the observed plateau in the power spectrum P _ { \Phi } \approx 10 ^ { -9 } from cosmological scales down to the co-moving size of binaries , the torque from perturbations is small .
In this case , for f \gtrsim 10 ^ { -2 } , the distribution of eccentricities is dominated by tidal torques from neighboring PBHs .
On the other hand , in scenarios where PBH are formed from adiabatic perturbations , it is natural to expect an enhancement of 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 [ [ 1 ] ] \alpha = - ( m _ { 1 } + m _ { 2 } ) ^ { 2 } \partial ^ { 2 } \ln \Delta \Gamma / \partial m _ { 1 } \partial m _ % { 2 } .
For an enhanced power spectrum with spectral index p we find that \alpha \approx 30 / ( 32 - 7 p ) for 0 < p \lesssim 2 , and \alpha \approx 5 / 3 for p \gtrsim 2 .
Such values may lie well outside the narrow range \alpha \approx 1 \pm 0.05 characteristic of tidal forces from neighboring PBHs .
We conclude that , given a large enough sample of events , merger rates may provide valuable information on the spectrum of primordial cosmological perturbations at currently uncharted lengthscales .