Two-loop corrections in power spectrum in models of inflation with PBHs formation

TL;DR

This paper computes two-loop quantum corrections to the primordial power spectrum in inflation models with PBH formation, highlighting the importance of transition smoothness for perturbative validity.

Contribution

It provides the first detailed calculation of two-loop corrections in these inflation models and shows their dependence on the transition sharpness.

Findings

Two-loop corrections scale with the square of one-loop corrections.

Sharp transitions lead to large loop corrections, challenging perturbative control.

Mild transitions are necessary for reliable perturbative analysis.

Abstract

We calculate the two-loop corrections in primordial power spectrum in models of single field inflation incorporating an intermediate USR phase employed for PBHs formation. Among the total eleven one-particle irreducible Feynman diagrams, we calculate the corrections from the "double scoop" two-loop diagram involving two vertices of quartic Hamiltonians. We demonstrate that the fractional two-loop correction in power spectrum scales like the square of the fractional one-loop correction. We confirm our previous findings that the loop corrections become arbitrarily large in the setup where the transition from the intermediate USR to the final slow-roll phase is very sharp. This suggests that in order for the analysis to be under perturbative control against loop corrections, one requires a mild transition with a long enough relaxation period towards the final attractor phase.