Primordial black hole formation with non-Gaussian random fields

In the context of transient constant-roll inflation, we compute the non-perturbative field redefinition that relates a Gaussian curvature mode with the real non-Gaussian perturbation. Our computation shows the emergence of a new critical amplitude $\zeta_*$, corresponding to perturbations that prevent the inflaton from overshooting the local maximum, thus becoming trapped in the false minimum of the potential. For potentials with a mild curvature at the local maximum (and thus small non-Gaussianity), we recover the known perturbative field redefinition. We apply these results to the formation of primordial black holes, and discuss the cases for which $\zeta_*$ is smaller or larger than the critical value for collapse of spherically symmetric overdensities.