Class of solutions of the Wheeler-DeWitt equation in the Friedmann-Robertson-Walker universe

TL;DR

This paper derives solutions to the Wheeler-DeWitt equation in various Friedmann-Robertson-Walker universe models, revealing specific functional forms and energy spectra for different energy conditions.

Contribution

It provides explicit solutions using triconfluent Heun functions for all three geometries, including polynomial solutions and energy spectra in matter-dominated universes.

Findings

Polynomial solutions exist in matter-dominated universes.

No polynomial solutions in radiation- and vacuum-dominated universes.

Explicit energy density spectra derived for matter-dominated cases.

Abstract

We show that the solutions of the Wheeler-DeWitt equation in a homogeneous and isotropic universe are given by triconfluent Heun functions for the spatially closed, flat, and open geometries of the Friedmann-Robertson-Walker universe filled with different forms of energy. In a matter-dominated universe, we find the polynomial solution and the energy density spectrum. In the cases of radiation-dominated and vacuum universes, there are no polynomial solutions as shown.