Running Cosmological Constant and Running Newton Constant in Modified Gravity Theories

TL;DR

This paper investigates how the Wheeler-DeWitt equation can be used to derive the cosmological constant and Newton's constant in modified gravity theories, employing a one-loop approximation, variational methods, and renormalization techniques.

Contribution

It introduces a method to extract the cosmological constant and Newton's constant from the Wheeler-DeWitt equation using a variational approach and renormalization in de Sitter and Anti-de Sitter backgrounds.

Findings

No ghosts appear in the evaluation of the cosmological constant.

A renormalization group equation is derived to handle divergences.

The approach can be extended to $f(R)$ gravity theories.

Abstract

We discuss how to extract information about the cosmological constant from the Wheeler-DeWitt equation, considered as an eigenvalue of a Sturm-Liouville problem in a de Sitter and Anti-de Sitter background. The equation is approximated to one loop with the help of a variational approach with Gaussian trial wave functionals. A canonical decomposition of modes is used to separate transverse-traceless tensors (graviton) from ghosts and scalar. We show that no ghosts appear in the final evaluation of the cosmological constant. A zeta function regularization is used to handle with divergences. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. We apply this procedure on the induced cosmological constant $\Lambda$ and, as an alternative, on the Newton constant $G$. A brief discussion on the extension to a $f(R)$ theory is considered.