Distorting General Relativity: Gravity's Rainbow and f(R) theories at work

TL;DR

This paper investigates how combining Gravity's Rainbow and f(R) theories affects the Zero Point Energy in Schwarzschild backgrounds, revealing finite shifts and potential implications for the cosmological constant.

Contribution

It introduces a novel approach by integrating Gravity's Rainbow with f(R) modifications to analyze Zero Point Energy in black hole backgrounds.

Findings

Zero Point Energy is finite and shifted due to combined theories.

The eigenvalue problem relates to the Wheeler-DeWitt equation.

Graviton fluctuations dominate the one-loop quantum corrections.

Abstract

We compute the Zero Point Energy in a spherically symmetric background combining the high energy distortion of Gravity's Rainbow with the modification induced by a f(R) theory. Here f(R) is a generic analytic function of the Ricci curvature scalar R in 4D and in 3D. The explicit calculation is performed for a Schwarzschild metric. Due to the spherically symmetric property of the Schwarzschild metric we can compare the effects of the modification induced by a f(R) theory in 4D and in 3D. We find that the final effect of the combined theory is to have finite quantities that shift the Zero Point Energy. In this context we setup a Sturm-Liouville problem with the cosmological constant considered as the associated eigenvalue. The eigenvalue equation is a reformulation of the Wheeler-DeWitt equation which is analyzed by means of a variational approach based on gaussian trial functionals. With the help of a canonical decomposition, we find that the relevant contribution to one loop is given by the graviton quantum fluctuations around the given background. A final discussion on the connection of our result with the observed cosmological constant is also reported.