Proving the absence of large one-loop corrections to the power spectrum of curvature perturbations in transient ultra-slow-roll inflation within the path-integral approach

TL;DR

This paper demonstrates that one-loop quantum corrections to the curvature perturbation power spectrum during transient ultra-slow-roll inflation are negligible, ensuring the validity of perturbation theory and supporting primordial black hole formation models.

Contribution

The authors derive a simplified path-integral formalism for one-loop corrections and prove the absence of large corrections in USR inflation, clarifying previous conflicting claims.

Findings

One-loop corrections are small compared to tree-level.

Consistency relations ensure the stability of super-Hubble modes.

Perturbation theory remains valid during USR inflation.

Abstract

We revisit one-loop corrections to the power spectrum of curvature perturbations $\zeta$ in an inflationary scenario containing a transient ultra-slow-roll (USR) period. In Ref.[1], it was argued that one-loop corrections to the power spectrum of $\zeta$ can be larger than the tree-level one within the parameter region generating the seeds of primordial black holes during the USR epoch, which implies the breakdown of perturbation theory. We prove that this is not the case by using a master formula for one-loop corrections to the power spectrum obtained in Ref.[2]. We derive the same formula within the path-integral formalism, which is simpler than the original derivation in [2]. To show the smallness of one-loop corrections, the consistency relations and the effective constancy of tree-level mode functions of $\zeta$ for super-Hubble modes play essential roles, with which the master formula gives a simple expression for one-loop corrections. For concreteness, we provide a reduced set of interactions including the leading-order one, while establishing the consistency relations in a self-consistent manner. We also show how the consistency relations of various operators hold explicitly, which plays a key role in proving the absence of large one-loop corrections.