Abstract: 
The curvature perturbation R originating from the quantum fluctuation of the inflaton field δφ seeds the cosmic microwave background anisotropies and cosmic structures on large scales, which is nearly Gaussian, as the quantum fluctuation δφ is Gaussian, and R(δφ) is linear. On small scales, enhanced curvature perturbation may generate primordial black holes (PBHs) and scalarinduced gravitational waves (GWs), which depends crucially on the high tails of the probability distribution function (PDF) of R. By studying a singlefield inflation model with a typical piecewise quadratic potential, we found a new fullynonlinear relation R(δφ) for the curvature perturbation, which is a sum of dual logarithmic functions of δφ, δφ', and their values at the junction of the potentials. We clarify the condition under which R(δφ, δφ') reduces to a single logarithm, which yields either the renowned “exponential tail” in the PDF of R or a Gumbeldistributionlike tail, of which the latter is helpful in suppressing the seemingly overproduced PBHs when we try to use the scalarinduced GWs to explain the recent discovery of nHz GW signal by the pulsar timing array observations.
