Abstract: In this work we provide the missing link between two approaches aimed at characterizing the effect of long perturbation modes in Inflation. We consider the "Inflationary Fossils'" approach, that characterizes the power-spectrum of the inflaton field in presence of other long and non dynamical "fossil" fields, and a technique that computes, beyond perturbation theory, the power-spectrum of a scalar field in presence of a "large" fluctuation of a second field.

We clarify a few points on the applicability of the non-perturbative technique. We prove in six distinct cases, one involving a violation of the consistency conditions, that the non-perturbative approach, once expanded to first order in the coupling, matches the perturbative result following the Fossils' approach. 

We believe that this non-perturbative technique extends to all orders the Fossils' approach, resumming infinitely many diagrams of standard in-in perturbation theory.