The Cosmological Constant as an Eigenvalue of the Hamiltonian constraint in a Varying Speed of Light theory

TL;DR

This paper explores how the cosmological constant can be viewed as an eigenvalue in a Varying Speed of Light theory, using the Wheeler-DeWitt equation and Sturm-Liouville problem techniques to analyze vacuum states.

Contribution

It demonstrates the equivalence of the Wheeler-DeWitt equation to a Sturm-Liouville problem in this context and computes eigenvalues using a variational approach with Bessel function trial wave functionals.

Findings

Existence of eigenvalues related to negative powers of the scale factor.

No such eigenvalues are found at the inflationary scale.

The cosmological constant is identified as an eigenvalue in this framework.

Abstract

In the framework of a Varying Speed of Light theory, we study the eigenvalues associated with the Wheeler-DeWitt equation representing the vacuum expectation values associated with the cosmological constant. We find that the Wheeler-DeWitt equation for the Friedmann-Lema\^{\i}tre-Robertson-Walker metric is completely equivalent to a Sturm-Liouville problem provided that the related eigenvalue and the cosmological constant be identified. The explicit calculation is performed with the help of a variational procedure with trial wave functionals related to the Bessel function of the second kind $K_{\nu }\left( x\right) $. We find the existence of a family of eigenvalues associated to a negative power of the scale. Furthermore, we show that at the inflationary scale such a family of eigenvalues does not appear.