One Loop Back Reaction On Chaotic Inflation

TL;DR

This paper tests the claim that infrared quantum effects from long-wavelength modes slow chaotic inflation via a one-loop back-reaction. It extends inflaton-graviton Feynman rules to a general scalar potential and computes the leading late-time back-reaction for V(φ) = ½ m^2 φ^2, expressed as the time derivative of the scale factor in a gauge where ⟨gμν⟩ has FRW form. The authors show their covariant-quantization result agrees with the earlier canonical quantization result by Mukhanov et al., despite using different gauges, addressing Unruh's objections. They develop infrared-mode expansions to isolate the dominant contributions and provide explicit expressions for corrections to the Hubble rate and the scalar expectation value, and discuss implications for higher-loop effects and future work.

Abstract

We extend, for the case of a general scalar potential, the inflaton-graviton Feynman rules recently developed by Iliopoulos {\it et al.} As an application we compute the leading term, for late co-moving times, of the one loop back reaction on the expansion rate for $V(φ) = \frac12 m^2 φ^2$. This is expressed as the logarithmic time derivative of the scale factor in the coordinate system for which the expectation value of the metric has the form: $<0 | g_{μν}({\bar t},{\vec x}) | 0 > dx^μ dx^ν = - d{\bar t}^2 + a^2({\bar t}) d{\vec x} \cdot d{\vec x}$. This quantity should be a gauge independent observable. Our result for it agrees exactly with that inferred from the effect previously computed by Mukhanov {\it et al.} using canonical quantization. It is significant that the two calculations were made with completely different schemes for fixing the gauge, and that our computation was done using the standard formalism of covariant quantization. This should settle some of the issues recently raised by Unruh.