Massless scalar fields and infrared divergences in the inflationary brane world

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 ( with the Ricci scalar) is infrared divergent except for the minimally coupled case. The usual procedure for taming this divergence is replacing 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 divergent everywhere when the lowest lying mode becomes massless except for the massless minimal coupling case. A simple extension of the AF vacuum to the present case avoids this global divergence, but remains 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 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.