Abundance of Primordial Black Holes Depends on the Shape of the Inflationary Power Spectrum

Abstract

In this letter, combining peak theory and the numerical analysis of gravitational collapse in the radiation dominated era, we show that the abundance of primordial blacks holes, generated by an enhancement in the inflationary power spectrum, is extremely dependent on the shape of the peak. Given the amplitude of the power spectrum, we show that the density of primordial black holes generated from a narrow peak, is exponentially smaller than in the case of a broad peak. Specifically, for a top-hat profile of the power spectrum in Fourier space, we find that for having primordial black holes comprising all of the dark matter, one would only need a power spectrum amplitude an order of magnitude smaller than suggested previously whereas in the case of a narrow peak, one would instead need a much larger power spectrum amplitude, which in many cases would invalidate the perturbative analysis of cosmological perturbations. Finally, we show that, although critical collapse gives a broad mass spectrum, the density of primordial black holes formed is dominated by masses roughly equal to the cosmological horizon mass measured at horizon crossing.

Nonlinear statistics of primordial black holes from Gaussian curvature perturbations

“…The linear relation between curvature perturbations and over-densities has been used in [4] to calculate the abundance of PBHs via the statistics of high (gaussian) peaks [5] of over-densities. This was done under the assumption that the non-linearities, relating the over-density to curvature perturbations, should be statistically irrelevant in the distributions of the peak amplitudes, given the fact that they are proportional to higher correlators of the curvature perturbations.…”

“…The linear theory approach to PBHs abundance used in [4] has been questioned in [8]. There, the authors have considered the role of the skewness obtained from the non-linear relation between the over-density and the (gaussian) curvature perturbations.…”

“…There, the authors have considered the role of the skewness obtained from the non-linear relation between the over-density and the (gaussian) curvature perturbations. Those non-gaussianities are typically very small with respect to the two-point functions, thus one would be tempted to discard them [4]. However, as already mentioned, the statistics of PBHs is related to the (exponentially suppressed) tail area of the over-density probability distribution.…”

“…In [10], the authors used the probability distribution of rare peaks in δ (i.e. using the methods of [4]) to estimate PBH abundances. Although this approach ameliorates the linear approach of [4], it still suffers from the same main problems.…”

“…using the methods of [4]) to estimate PBH abundances. Although this approach ameliorates the linear approach of [4], it still suffers from the same main problems. Knowing the probability distribution of rare peaks in δ is not enough: in order to perform the averaging of the over-density, the (statistical) condition that R is the compaction function maximum must be imposed.…”