Zeta Functions in Brane World Cosmology

TL;DR

This paper calculates the zeta function and functional determinant for a scalar field in a higher-dimensional de Sitter brane background, directly working with the wave operator and generalizing previous conformal methods.

Contribution

It introduces a direct method for evaluating the zeta function in higher dimensions without conformal transformations, including a generic mass and curvature coupling.

Findings

Recovered known results in the conformally coupled limit

Validated the method by matching the large AdS radius limit with known determinants

Provided an alternative approach to compute the ball determinant

Abstract

We present a calculation of the zeta function and of the functional determinant for a Laplace-type differential operator, corresponding to a scalar field in a higher dimensional de Sitter brane background, which consists of a higher dimensional anti-de Sitter bulk spacetime bounded by a de Sitter section, representing a brane. Contrary to the existing examples, which all make use of conformal transformations, we evaluate the zeta function working directly with the higher dimensional wave operator. We also consider a generic mass term and coupling to curvature, generalizing previous results. The massless, conformally coupled case is obtained as a limit of the general result and compared with known calculations. In the limit of large anti-de Sitter radius, the zeta determinant for the ball is recovered in perfect agreement with known expressions, providing an interesting check of our result and an alternative way of obtaining the ball determinant.