Scalar Casimir Effect on a D-dimensional Einstein Static Universe

TL;DR

This paper calculates the renormalised energy-momentum tensor of a scalar field in a higher-dimensional Einstein Static Universe, revealing regulator-independent results, divergence structures, and potential for quantum stabilization of the universe.

Contribution

It provides a comprehensive analysis of the scalar Casimir effect in arbitrary dimensions, including regulator independence, divergence structure, and implications for universe stability.

Findings

Renormalised energy density depends on the dimension's parity.

Quantum effects can stabilize Einstein Static Universes.

Various regularization methods are shown to be equivalent within the framework.

Abstract

We compute the renormalised energy momentum tensor of a free scalar field coupled to gravity on an (n+1)-dimensional Einstein Static Universe (ESU), RxS^n, with arbitrary low energy effective operators (up to mass dimension n+1). A generic class of regulators is used, together with the Abel-Plana formula, leading to a manifestly regulator independent result. The general structure of the divergences is analysed to show that all the gravitational couplings (not just the cosmological constant) are renormalised for an arbitrary regulator. Various commonly used methods (damping function, point-splitting, momentum cut-off and zeta function) are shown to, effectively, belong to the given class. The final results depend strongly on the parity of n. A detailed analytical and numerical analysis is performed for the behaviours of the renormalised energy density and a quantity `sigma' which determines if the strong energy condition holds for the `quantum fluid'. We briefly discuss the quantum fluid back-reaction problem, via the higher dimensional Friedmann and Raychaudhuri equations, observe that equilibrium radii exist and unveil the possibility of a `Casimir stabilisation of Einstein Static Universes'.