The cosmological constant as an eigenvalue of f(R)-gravity Hamiltonian constraint

TL;DR

This paper investigates the cosmological constant as an eigenvalue of the Hamiltonian constraint in f(R) gravity theories, deriving explicit calculations for Schwarzschild metrics using zeta function regularization.

Contribution

It introduces a method to determine the cosmological constant as an eigenvalue in f(R) gravity within the ADM formalism, including explicit calculations for Schwarzschild metrics.

Findings

Derived a renormalized running cosmological constant Lambda.

Calculated one-loop energy for Schwarzschild metric.

Established a link between eigenvalues and vacuum states in f(R) gravity.

Abstract

In the framework of ADM formalism, it is possible to find out eigenvalues of the WDW equation with the meaning of vacuum states, i.e. cosmological constants, for f(R) theories of gravity, where f(R) is a generic analytic function of the Ricci curvature scalar R. The explicit calculation is performed for a Schwarzschild metric where one-loop energy is derived by the zeta function regularization method and a renormalized running Lambda constant is obtained.