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Studying loop corrections to inflationary perturbations, with particularemphasis on infrared factors, is important to understand the consistency of theinflationary theory, its predictivity and to establish the existence of theslow-roll eternal inflation phenomena and its recently found volume bound. Inthis paper we prove that the zeta correlation function is time-independent atone-loop level in single clock inflation. While many of the one-loop diagramslead to a time-dependence when considered individually, the time-dependencebeautifully cancels out in the overall sum. We identify two subsets of diagramsthat cancel separately due to different physical reasons. The firstcancellation is related to the change of the background cosmology due to therenormalization of the stress tensor. It results in a cancellation between thenon-1PI diagrams and some of the diagrams made with quartic vertices. Thesecond subset of diagrams that cancel is made up of cubic operators, plus theremaining quartic ones. We are able to write the sum of these diagrams as theintegral over a specific three-point function between two very shortwavelengths and one very long one. We then apply the consistency condition forthis three-point function in the squeezed limit to show that the sum of thesediagrams cannot give rise to a time dependence. This second cancellation isthus a consequence of the fact that in single clock inflation the attractornature of the solution implies that a long wavelength zeta perturbation isindistinguishable from a trivial rescaling of the background, and so results inno physical effect on short wavelength modes.

On Loops in Inflation III: Time Independence of zeta in Single Clock Inflation