Non-Perturbative Hamiltonian and Higher Loop Corrections in USR Inflation

Calculating the action and the interaction Hamiltonian at higher orders in cosmological perturbation theory is a cumbersome task. We employ the formalism of EFT of inflation in the decoupling limit for single-field ultra slow-roll (USR) inflation and obtain a non-perturbative Hamiltonian in terms of the Goldstone field $\pi$. To complete the dictionary, a non-linear relation between the curvature perturbations and $\pi$ is presented. Using these results, we compute higher-order loop corrections in USR models with a sharp transition to the attractor phase, relevant for PBHs formation. It is shown that in the idealized picture in which the transition from the USR phase to SR phase is instantaneous and sharp, the loop corrections on long CMB scales increase rapidly with the number of loops $L$ and the setup may go out of perturbative control.