# Quantum relativistic cosmology: dynamical interpretation and tunneling universe

## TL;DR

This paper derives wave functions for the quantum relativistic universe using the Wheeler-DeWitt equation, analyzing its expansion and tunneling behavior across different energy densities with solutions involving Heun functions.

## Contribution

It provides explicit solutions for the wave functions of the quantum universe with various energy components, and explores their implications for cosmic expansion and tunneling phenomena.

## Key findings

- Asymptotic exponential expansion of the universe for large times
- Early universe behavior follows a t^{1/2} law
- Transmission coefficients for tunneling through potential barriers are calculated

## Abstract

In this work the wave functions associated to the quantum relativistic universe, which is described by the Wheeler-DeWitt equation, are obtained. Taking into account different kinds of energy density, namely, matter, radiation, vacuum, dark energy, and quintessence, we discuss some aspects of the quantum dynamics. In all these cases, the wave functions of the quantum relativistic universe are given in terms of the triconfluent Heun functions. We investigate the expansion of the universe using these solutions and found that the asymptotic behavior for the scale factor is $a(t) \sim \mbox{e}^{t}$ for whatever the form of energy density is. On the other hand, we analyze the behavior at early stages of the universe and found that $a(t) \sim t^{1/2}$. We also calculate and analyze the transmission coefficient through the effective potential barrier.