Wightman function and vacuum densities in de Sitter spacetime with toroidally compactified dimensions

TL;DR

This paper analyzes quantum field vacuum effects in higher-dimensional de Sitter spacetime with compactified dimensions, deriving recurrence relations and exploring their behavior during different cosmological epochs.

Contribution

It introduces recurrence formulae for vacuum expectation values in de Sitter spacetime with arbitrary compactification, considering both periodic and antiperiodic boundary conditions.

Findings

Topological parts dominate early universe vacuum expectation values.

Behavior of topological contributions is independent of curvature coupling for massless fields.

At late times, uncompactified de Sitter contributions dominate the vacuum expectation values.

Abstract

We investigate the Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor for a scalar field with general curvature coupling parameter in $(D+1)$-dimensional de Sitter spacetime with an arbitrary number of compactified spatial dimensions. Both cases of periodicity and antiperiodicity conditions along the compactified dimensions are considered. Recurrence formulae are derived which express the vacuum expectation values for the dS spacetime of topology $\mathrm{R}^{p}\times (\mathrm{S}^{1})^{q}$ in the form of the sum of the vacuum expectation values in the topology $\mathrm{R}^{p+1}\times (\mathrm{S}^{1})^{q-1}$ and the part induced by the compactness of the $(p+1)$th spatial dimension. The behavior of the topological parts is investigated in various asymptotic regions of the parameters. In the early stages of the cosmological evolution the topological parts dominate the contribution in the expectation values due to the uncompactified dS part. In this limit the behavior of the topological parts does not depend on the curvature coupling parameter and coincides with that for a conformally coupled massless field. At late stages of the cosmological expansion the expectation values are dominated by the part corresponding to uncompactified dS spacetime. The vanishing of the topological parts is monotonic or oscillatory in dependence of the mass and the curvature coupling parameter of the field.