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Solving peak theory in the presence of local non-gaussianities

We compute the probability density distribution of maxima for a scalar random field in the presence of local non-gaussianities. The physics outcome of this analysis is the following. If we focus on maxima whose curvature is larger than a certain threshold for gravitational collapse, our calculations illustrate how the fraction of the Universe's mass in the form of primordial black holes (PBHs) changes in the presence of local non-gaussianities. We find that previous literature on the subject overestimates, by many orders of magnitude, the impact of local non-gaussianities on the PBH abundance. We explain the origin of this discrepancy, and conclude that, in realistic single-field inflationary models with ultra slow-roll, one can obtain the same abundance found with the gaussian approximation simply changing the peak amplitude of the curvature power spectrum by no more than a factor of two. We comment about the relevance of non-gaussianities for second-order gravitational waves.