Inflationary Fossils Beyond Perturbation Theory

TL;DR

This work connects the perturbative Fossils' approach for long-mode effects in inflation with a non-perturbative resummation method originally introduced for large background fluctuations. By applying the non-perturbative technique to toy models and a Maldacena-type inflationary model, the authors show that first-order expansions reproduce the Fossils' results, while the method inherently resums infinitely many in-in diagrams. They demonstrate this matching across scalar and tensor couplings, including setups that violate the consistency conditions, indicating the approach is robust beyond standard single-field expectations. The results yield concrete predictions such as quadrupolar distortions in the power spectrum and modified speeds of sound, with implications for non-Gaussianity and potentially Primordial Black Hole phenomenology. Overall, the paper argues that non-perturbative resummation extends the Fossils' framework to all orders and offers a versatile tool for exploring long-mode physics in inflation.

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 (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 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.

Figures (2)

• Figure 1: The thick horizontal line stands for the spatial hypersurface at the end of Inflation, and in the bulk full legs represent the $\sigma$-like field, dashed legs the $\chi$-like field. (a) Tree-level diagrams that are resummed in the effective theory thanks to the enhancement by the dashed legs (b) one-loop diagrams that are not resummed in the effective theory due to the suppression given by the mismatch between couplings and dashed legs, (c) tree-level diagrams that have more than two full legs, no matter the amount of dashed legs, are not resummed in the effective theory due to the excess of coupling with respect to the number of dashed legs. The effective theory is a free (i.e. with quadratic lagrangian) classical (i.e. without loops) theory for the $\sigma$-like field, and does not contemplate higher order correlators.
• Figure 2: A diagrammatical sketch of what we mean by resummation. The leftmost part is the free power spectrum of the $\sigma$-like field, in the middle we display the Fossils' correction, namely the bispectrum normalized by the power spectrum of the $\chi$-like field, and in the rightmost part the first of the infinitely many corrections that are not considered in the Fossils' approach, but that this non-perturbative technique does resum. We clarify that the thin horizontal lines are simply fraction lines.