Regulating the infrared by mode matching: a massless scalar in expanding spaces with constant deceleration

Abstract

In this paper we consider a massless scalar field, with a possible coupling ξ to the Ricci scalar in a D dimensional Friedmann-Lemaître-Robertson-Walker space-time with a constant deceleration parameter q=ϵ-1, ϵ=-H˙/H2. Correlation functions for the Bunch-Davies vacuum of such a theory have long been known to be infrared divergent for a wide range of values of ϵ. We resolve these divergences by explicitly matching the space-time under consideration to a space-time without infrared divergencies. Such a procedure ensures that all correlation functions with respect to the vacuum in the space-time of interest are infrared finite. In this newly defined vacuum we construct the coincidence limit of the propagator and as an example calculate the expectation value of the stress-energy tensor. We find that this approach gives both in the ultraviolet and in the infrared satisfactory results. Moreover, we find that, unless the effective mass due to the coupling to the Ricci scalar ξR is negative, quantum contributions to the energy density always dilute away faster, or just as fast, as the background energy density. Therefore, quantum backreaction is insignificant at the one-loop order, unless ξR is negative. Finally we compare this approach with known results where the infrared is regulated by placing the Universe in a finite box. In an accelerating universe, the results are qualitatively the same, provided one identifies the size of the Universe with the physical Hubble radius at the time of the matching. In a decelerating universe, however, the two schemes give different late time behavior for the quantum stress-energy tensor. This happens because in this case the length scale at which one regulates the infrared becomes sub-Hubble at late times.