Strong restriction on inflationary vacua from the local gauge invariance II: Infrared regularity and absence of the secular growth in Euclidean vacuum

Abstract

We investigate the initial state of the inflationary universe. In our recent publications, we showed that requesting the gauge invariance in the local observable universe to the initial state guarantees the infrared (IR) regularity of loop corrections in a general single clock inflation. Following this study, in this paper, we show that choosing the Euclidean vacuum ensures the gauge invariance in the local universe and hence the IR regularity of loop corrections. It has been suggested that loop corrections to inflationary perturbations may yield the secular growth, which can lead to the break down of the perturbative analysis in an extremely long term inflation. The absence of the secular growth has been claimed by picking up only the IR contributions, which we think is incomplete because the non-IR modes which are comparable to or smaller than the Hubble scale potentially can contribute to the secular growth. We prove the absence of the secular growth without neglecting these non-IR modes to a certain order in the perturbative expansion. We also discuss how the regularity of the n-point functions for the genuinely gauge invariant variable constrains the initial states of the inflationary universe. These results apply in a fully general single field model of inflation.