Highly asymmetric probability distribution from a finite-width upward step during inflation

We study a single-field inflation model in which the inflaton potential has an upward step between two slow-roll regimes by taking into account the finite width of the step. We calculate the probability distribution function (PDF) of the curvature perturbation P[R] using the delta N formalism. The PDF has an exponential-tail only for positive R whose slope depends on the step width. We find that the tail may have a significant impact on the estimation of the primordial black hole abundance. We also show that the PDF P[R] becomes highly asymmetric on a particular scale exiting the horizon before the step, at which the curvature power spectrum has a dip. This asymmetric PDF may leave an interesting signature in the large scale structure such as voids.