Non-linear metric perturbations and production of primordial black holes

TL;DR

The paper addresses primordial black hole formation from inflationary perturbations featuring a plateau in the inflaton potential. It introduces a coarse-grained metric formalism to derive a non-Gaussian distribution for the metric perturbation h and shows that large perturbations are more probable than in Gaussian theory, potentially boosting PBH abundances. By applying Carr and Nadezhin–Novikov–Polnarev PBH criteria to the non-linear distribution, it demonstrates that non-Gaussian effects raise the required plateau amplitude delta_pl by about 50% relative to Gaussian estimates, though the overall qualitative conclusions remain similar. The work provides a semi-analytic framework connecting plateau-like inflationary features to PBH statistics and highlights the importance of non-linear, non-Gaussian effects, with plans to extend to more realistic two-field models.

Abstract

We consider the simple inflationary model with peculiarity in the form of "plateau" in the inflaton potential. We use the formalism of coarse-grained field to describe the production of metric perturbations $h$ of an arbitrary amplitude and obtain non-Gaussian probability function for such metric perturbations. We associate the spatial regions having large perturbations $h\sim 1$ with the regions going to primordial black holes after inflation. We show that in our model the non-linear effects can lead to overproduction of the primordial black holes.

Figures (2)

• Figure 1: We plot the dependence of plateau parameter $\delta_{pl}$ on PBH's mass $M_{pbh}$ assuming that the PBH's abundance is given by the eqn. $(44)$. The solid line represents the solution of eqn. $(45)$ (i.e we calculate $\delta_{pl}$ taking into account the non-Gaussian effects in this case). The dashed line represents $\delta_{pl}$ calculated in the standard Gaussian theory. The PBH's masses lie in the range: $10^{-18}M_{\odot} < M_{pbh} < 10^{6}M_{pbh}$. The PBH's of the mass $10^{-18}M_{\odot}\sim 10^{15}g$ should be evaporated at the present time. Actually, the abundance of these PBH's is limited much stronger than is assumed in our calculations.
• Figure 2: The dependence of probability density ${\cal P}(h)$ on the metric amplitude $h$. The non-Gaussian curve (solid line) is calculated with help of eqn. $(39)$ assuming PBH's abundance $\beta(M_{\odot})\approx 10^{-8}$. That gives $\delta_{1pl}(M_{\odot})\approx 0.089$. The dashed line is the reference Gaussian probability density calculated for the same abundance. For that curve we have $\delta_{2pl}(M_{\odot})\approx 0.134$. The dotted curve represents the Gaussian distribution taken with $\delta_{1pl}(M_{\odot})\approx 0.089$. This distribution strongly under-produces PBH's, and in this case we have $\beta \sim 10^{-17}$.