Non-perturbative Wavefunction of the Universe in Inflation with (Resonant) Features

We study the statistics of scalar perturbations in models of inflation with small and rapid oscillations in the inflaton potential (resonant non-Gaussianity). We do so by deriving the wavefunction $\Psi[\zeta(\boldsymbol{x})]$ non-perturbatively in $\zeta$, but at first order in the amplitude of the oscillations. The expression of the wavefunction of the universe (WFU) is explicit and does not require solving partial differential equations. One finds qualitative deviations from perturbation theory for $ |\zeta| \gtrsim \alpha^{-2}$, where $\alpha \gg 1$ is the number of oscillations per Hubble time. Notably, the WFU exhibits distinct behaviours for negative and positive values of $\zeta$ (troughs and peaks respectively). While corrections for $\zeta <0$ remain relatively small, of the order of the oscillation amplitude, positive $\zeta$ yields substantial effects, growing exponentially as $e^{\pi\alpha/2}$ in the limit of large $\zeta$. This indicates that even minute oscillations give large effects on the tail of the distribution.