Stochastic inflation and non-perturbative power spectrum beyond slow roll

Stochastic inflation, together with the $\Delta N$ formalism, provides a powerful tool for estimating the large-scale behaviour of primordial fluctuations. In this work, we develop a numerical code to capture the non-perturbative statistics of these fluctuations and validate it to obtain the exponential non-Gaussian tail of the curvature perturbations. We present a numerical algorithm to compute the non-perturbative curvature power spectrum and apply it to both slow-roll (SR) and ultra-slow-roll (USR) single-field models of inflation. We accurately generate a non-perturbative scale-invariant power spectrum in the SR scenario. In the USR case, we obtain a peak in the power spectrum that, in the time-independent regime, aligns with the structure of its perturbative counterpart. Additionally, We underscore how the evolving nature of the super-Hubble perturbations in the USR model complicates the numerical computation of the non-perturbative spectrum.