Inflationary Fossils Beyond Perturbation Theory

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 (arXiv:1203.0302 and related works) that characterizes the power-spectrum of the inflaton field in presence of other long and non dynamical fossil fields, and a technique, appeared in arXiv:2103.09244, 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 five distinct cases 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.