Abstract
Abstract 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 ζ 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 ζ perturbation is indistinguishable from a trivial rescaling of the background, and so results in no physical effect on short wavelength modes.
Highlights
1.1 MotivationThe purpose of this paper is to prove that in single clock inflation, where there is only one relevant degree of freedom during inflation, the correlation function of the curvature perturbation ζ for separations outside the horizon is time independent at one loop level.We believe this to be a very important result to prove for several reasons
In this paper we prove that the ζ correlation function is time-independent at one-loop level in single clock inflation
To understand wether the diagrams considered so far can lead to a time dependence, it will turn out to be useful to first add the quartic diagrams that are associated to the rescaling of the spatial derivatives in the cubic vertices
Summary
The purpose of this paper is to prove that in single clock inflation, where there is only one relevant degree of freedom during inflation, the correlation function of the curvature perturbation ζ for separations outside the horizon is time independent at one loop level. The study of eternal inflation (usually of the false vacuum type) has been largely motivated by the fact that the universe is currently accelerating and by the apparent existence of a landscape of vacua in String Theory which put together suggest that the current acceleration can be understood as resulting from an anthropic selection of the vacuum energy made possible by an epoch of eternal inflation in our past Another piece of motivation to study eternal inflation relies on the perhaps mysterious connections between gravity and quantum mechanics in the presence of a horizon. Is a true fact of nature connected to the holographic interpretation of de Sitter space All these new results on Eternal Inflation assumed that the ζ two-point function at coincidence takes the form ζ(x)2 ∼ H3t ,. This motivates us to study the possible timedependence of ζ at loop level
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