Hadamard renormalization, conformal anomaly and cosmological event horizons

TL;DR

This paper derives a two-dimensional analog of the renormalized stress tensor in Schwarzschild-de Sitter space using Hadamard renormalization, linking it to the conformal anomaly, horizon temperature, and proposing a cut-off hypothesis near the black hole horizon.

Contribution

It introduces a novel two-dimensional analog of the stress tensor incorporating the conformal anomaly and proposes a cut-off hypothesis to relate the stress tensor to the black hole's outwards signals.

Findings

The stress tensor is determined by the anomalous trace and additional parameters.

A cut-off scale related to the cosmological horizon temperature is proposed.

The correction to Hawking temperature due to the cosmological horizon is established.

Abstract

The Hadamard renormalization prescription is used to derive a two dimensional analog of the renormalized stress tensor for a minimally coupled scalar field in Schwarzschild-de Sitter space time. In the two dimensional analog the minimal coupling reduces to the conformal coupling and the stress tensor is found to be determined by the (nonlocal) contribution of the anomalous trace and some additional parameters in close relation to the work [1]. To properly relate the stress tensor to the state of outwards signals coming from the direction of the black hole at late times we propose a cut-off hypothesis which excludes the contribution of the anomalous trace close to the black hole horizon. The corresponding cut-off scale is found to be related to the equilibrium-temperature of the cosmological horizon in a leading order estimate. Finally, we establish a relation between the radiation-temperature of the black hole horizon at large distance from the hole and the the anomalous trace and determine the correction term to the Hawking temperature due to the presence of the cosmological horizon.