Zeta function regularization in de Sitter space: the Minkowski limit

TL;DR

This paper investigates the zeta function regularization of a scalar field in de Sitter space, demonstrating how the effective potential's properties approach those in Minkowski space as the de Sitter radius becomes large.

Contribution

It introduces a contour deformation method for the zeta function integral representation, enabling the analysis of the Minkowski limit in de Sitter space.

Findings

Explicitly shows $\, ext{zeta}(0) ext{ and } ext{zeta}'(0)$ approach Minkowski values in D=2.

Provides a suitable integral form for the zeta function for large de Sitter radius.

Validates the regularization method for the effective potential in curved spacetime.

Abstract

We study an integral representation for the zeta function of the one-loop effective potential for a minimally coupled massive scalar field in D-dimensional de Sitter spacetime. By deforming the contour of integration we present it in a form suitable for letting the de Sitter radius tend to infinity, and we demonstrate explicitly for the case D=2 that the quantities $\zeta( 0 )$ and $\zeta'( 0 )$ have the appropriate Minkowski limits.