Exact solution to the Wheeler-DeWitt equation: Early and current Universe

Abstract

In this paper, we present exact solutions to the Wheeler-DeWitt equation in two different scenarios: the early Universe, where the ordering parameter of kinetic energy is important, and the current Universe, where the ordering parameter effect is negligible. To make the exact solutions as general as possible, we incorporate as many different types of energy density as possible into the Hamiltonian, including baryonic and non-baryonic matter (dark matter), radiation, vacuum, and quintessence (dark energy). In the early Universe scenario, we obtain exact solutions in terms of the Biconfluent Heun functions, whereas in the current Universe, the exact solutions are given in terms of the Triconfluent Heun functions. Furthermore, by applying the polynomial conditions to each case, we obtain a constraint equation that supports the notion that the Wheeler-DeWitt equation can be viewed as an eigenvalue problem for the cosmological constant.