PBH abundance from random Gaussian curvature perturbations and a local
  density threshold
  The production rate of primordial black holes is often calculated by
considering a nearly Gaussian distribution of cosmological perturbations, and
assuming that black holes will form in regions where the amplitude of such
perturbations exceeds a certain threshold. A threshold $\zeta_{\rm th}$ for the
curvature perturbation is somewhat inappropriate for this purpose, because it
depends significantly on environmental effects, not essential to the local
dynamics. By contrast, a threshold $\delta_{\rm th}$ for the density
perturbation at horizon crossing seems to provide a more robust criterion. On
the other hand, the density perturbation is known to be bounded above by a
maximum limit $\delta_{\rm max}$, and given that $\delta_{\rm th}$ is
comparable to $\delta_{\rm max}$, the density perturbation will be far from
Gaussian near or above the threshold. In this paper, we provide a new plausible
estimate for the primordial black hole abundance based on peak theory. In our
approach, we assume that the curvature perturbation is given as a random
Gaussian field with the power spectrum characterized by a single scale, while
an optimized criterion for PBH formation is imposed, based on the locally
averaged density perturbation. Both variables are related by the full nonlinear
expression derived in the long-wavelength approximation of general relativity.
We do not introduce a window function, and the scale of the inhomogeneity is
introduced as a random variable in the peak theory. We find that the mass
spectrum is shifted to larger mass scales by one order of magnitude or so,
compared to a conventional calculation. The abundance of PBHs becomes
significantly larger than the conventional one, by many orders of magnitude,
mainly due to the optimized criterion for PBH formation and the removal of the
suppresion associated with a window function.
