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