The exponential-tail behaviors of the probability density function (PDF) of the primordial curvature perturbation are confirmed in the mild-waterfall variants of hybrid inflation with the use of the stochastic formalism of inflation. On top of these tails, effective upper bounds on the curvature perturbation are also observed, corresponding to the exact hilltop trajectory during the waterfall phase. We find that in the model where the leading and higher-order terms in the expansion of the inflaton potential around the critical point are fine-tuned to balance, this upper bound can be significantly reduced, even smaller than the primordial black hole (PBH) threshold, as a more novel perturbation-reduction mechanism than the one proposed by Tada and Yamada. It makes PBH formation much more difficult compared to the Gaussian or exponential-tail approximation. We also introduce Johnson’s $S_U$ distribution as a useful fitting function for the PDF, which reveals a nonlinear mapping between the Gaussian field and the curvature perturbation.

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• Inflation
• Primordial black holes
• Stochastic processes & statistics