Massless scalar fields and infrared divergences in the inflationary brane world

TL;DR

This paper investigates quantum effects of bulk scalar fields in a de Sitter brane world, revealing infrared divergences in the stress tensor and analyzing their regularization, with implications for bulk inflaton models and perturbation growth.

Contribution

It extends the analysis of infrared divergences and vacuum regularization techniques from de Sitter space to brane world scenarios with extra dimensions.

Findings

Infrared divergences occur for massless scalar fields in the brane world.

A generalized Allen Follaci vacuum can mitigate global divergences.

Perturbations grow linearly on the brane and are bounded in the bulk.

Abstract

We study the quantum effects induced by bulk scalar fields in a model with a de Sitter (dS) brane in a flat bulk (the Vilenkin-Ipser-Sikivie model) in more than four dimensions. In ordinary dS space, it is well known that the stress tensor in the dS invariant vacuum for an effectively massless scalar ($m_\eff^2=m^2+\xi {\cal R}=0$ with ${\cal R}$ the Ricci scalar) is infrared divergent except for the minimally coupled case. The usual procedure to tame this divergence is to replace the dS invariant vacuum by the Allen Follaci (AF) vacuum. The resulting stress tensor breaks dS symmetry but is regular. Similarly, in the brane world context, we find that the dS invariant vacuum generates $\tmn$ divergent everywhere when the lowest lying mode becomes massless except for massless minimal coupling case. A simple extension of the AF vacuum to the present case avoids this global divergence, but $\tmn$ remains to be divergent along a timelike axis in the bulk. In this case, singularities also appear along the light cone emanating from the origin in the bulk, although they are so mild that $\tmn$ stays finite except for non-minimal coupling cases in four or six dimensions. We discuss implications of these results for bulk inflaton models. We also study the evolution of the field perturbations in dS brane world. We find that perturbations grow linearly with time on the brane, as in the case of ordinary dS space. In the bulk, they are asymptotically bounded.