On Loops in Inflation III: Time Independence of zeta in Single Clock
  Inflation
  Studying loop corrections to inflationary perturbations, with particular
emphasis on infrared factors, is important to understand the consistency of the
inflationary theory, its predictivity and to establish the existence of the
slow-roll eternal inflation phenomena and its recently found volume bound. In
this paper we prove that the zeta correlation function is time-independent at
one-loop level in single clock inflation. While many of the one-loop diagrams
lead to a time-dependence when considered individually, the time-dependence
beautifully cancels out in the overall sum. We identify two subsets of diagrams
that cancel separately due to different physical reasons. The first
cancellation is related to the change of the background cosmology due to the
renormalization of the stress tensor. It results in a cancellation between the
non-1PI diagrams and some of the diagrams made with quartic vertices. The
second subset of diagrams that cancel is made up of cubic operators, plus the
remaining quartic ones. We are able to write the sum of these diagrams as the
integral over a specific three-point function between two very short
wavelengths and one very long one. We then apply the consistency condition for
this three-point function in the squeezed limit to show that the sum of these
diagrams cannot give rise to a time dependence. This second cancellation is
thus a consequence of the fact that in single clock inflation the attractor
nature of the solution implies that a long wavelength zeta perturbation is
indistinguishable from a trivial rescaling of the background, and so results in
no physical effect on short wavelength modes.
