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

TL;DR

This work proves that, in single clock inflation, the curvature perturbation zeta remains time-independent at one loop for modes outside the horizon. By organizing loop diagrams into CIS, CIM, and quartic contributions, and recasting the sum as integrals over a three-point function in the squeezed limit, the authors leverage the Maldacena consistency condition to show cancellations that prevent any secular time dependence. Key mechanisms include tadpole counterterm compensation of background renormalization and the use of a master three-point integral to encapsulate short-long mode interactions; derivative-containing quartic terms are shown to cancel against cubic-induced contributions. The result reinforces the robustness of inflationary predictions against infrared quantum corrections and supports the slow-roll eternal inflation volume bound as a property of the quantum-corrected theory. Overall, the paper provides a rigorous, diagrammatically complete demonstration that loop effects cannot spoil the time-invariance of superhorizon zeta in single-clock models, with implications for both theory and cosmological observables.

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 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.

Figures (7)

• Figure 1: Cut-in-the-middle ($CIM$) diagrams. Green continuos lines represent Green's functions, red dashed lines represent free fields, and red crosses circled by a blue dotted line represent correlations of free fields. Two crosses have to be contracted together in order for the diagram not to be zero. This diagram represents how vacuum correlation functions of quadratic operators $\zeta^{(1)}{}^2$, $\langle\zeta^{(1)}{}^2\zeta^{(1)}{}^2\rangle$ source perturbed correlation functions for $\zeta^{(2)}$: $\langle\zeta^{(2)}\zeta^{(2)}\rangle$.
• Figure 2: Cut-in-the-side quartic ($CIS_4$) diagrams. These diagrams represent how vacuum expectation values of quadratic operators $\langle\zeta^{(1)}{}^2\rangle$ affect the propagation of a mode $\zeta^{(3)}$, and therefore the $\zeta$ correlation function: $\langle\zeta^{(3)}\zeta^{(1)}\rangle$
• Figure 3: Non-1PI cut-in-the-side quartic ($CIS_{non-1PI}$) diagrams. These diagrams represent how vacuum expectation values of quadratic operators $\langle\zeta^{(1)}{}^2\rangle$ affect the propagation of the zero mode $\zeta^{(2)}_0$, and therefore the evolution of a mode by a non linear coupling $\zeta^{(3)}\sim \zeta^{(1)}\zeta_0^{(2)}$. This sources a correlation function of the form: $\langle\zeta^{(3)}\zeta^{(1)}\rangle$
• Figure 4: 1PI cut-in-the-side quartic ($CIS_{1PI}$) diagrams. These diagrams represent how the propagation of a mode is perturbed at two different times by two fluctuations that are correlated among themselves. This sources a correlation function of the form: $\langle\zeta^{(1)}\zeta^{(3)}\rangle$
• Figure 5: Cancellation between the tadpole diagram and the tadpole counterterm.