Mean field squared and energy-momentum tensor for the hyperbolic vacuum in dS spacetime

TL;DR

This paper computes the vacuum expectation values of a scalar field in de Sitter spacetime with hyperbolic slicing, revealing thermal properties of the hyperbolic vacuum and its relation to other vacua like Bunch-Davies.

Contribution

It provides integral representations for the Hadamard function differences and analyzes the energy-momentum tensor in hyperbolic vacuum, connecting it to flat spacetime vacua via conformal relations.

Findings

Bunch-Davies state is thermal relative to hyperbolic vacuum.

Derived integral formulas for vacuum expectation values.

Compared hyperbolic vacuum properties with flat spacetime vacua.

Abstract

We evaluate the Hadamard function and the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor for a massless conformally coupled scalar field in $(D+1)$-dimensional de Sitter (dS) spacetime foliated by spatial sections of negative constant curvature. It is assumed that the field is prepared in the hyperbolic vacuum state. An integral representation for the difference of the Hadamard functions corresponding to the hyperbolic and Bunch-Davies vacua is provided that is well adapted for the evaluation of the expectation values in the coincidence limit. It is shown that the Bunch-Davies state is interpreted as thermal with respect to the hyperbolic vacuum. An expression for the corresponding density of states is provided. The relations obtained for the difference in the VEVs for the Bunch-Davies and hyperbolic vacua are compared with the corresponding relations for the Fulling-Rindler and Minkowski vacua in flat spacetime. The similarity between those relations is explained by the conformal connection of dS spacetime with hyperbolic foliation and Rindler spacetime. As a limiting case, the VEVs for the conformal vacuum in the Milne universe are discussed.