Title: Exact solutions of the Wheeler–DeWitt equation with ordering term in a dark energy scenario
Abstract: We investigate the quantum evolution of the universe in the presence of two types of dark energies. First, we consider the phantom class (ω<−1) which would be responsible for a super-accelerated cosmic expansion, and then we apply the procedure to an ordinary Λ>0 vacuum (ω=−1). This is done by analytically solving the Wheeler–DeWitt equation with ordering term (WdW) in the cosmology of Friedmann–Robertson–Walker. In this paper, we find exact solutions in the scale factor a and the ordering parameter q. For q=1 it is shown that the universe has a high probability of evolving from a big bang singularity. On the other hand, for q=0 the solution indicates that an initial singularity is unlikely. Instead, the universe has maximal probability of starting with a finite well-defined size which we compute explicitly at primordial times. We also study the time evolution of the scale factor by means of the Hamilton–Jacobi equation and show that an ultimate big rip singularity emerges explicitly from our solutions. The phantom scenario thus predicts a dramatic end in which the universe would reach an infinite scale factor in a finite cosmological time as pointed by Caldwell et al. in a classical setup (Caldwell et al., 2003). Finally, we solve the WdW equation with ordinary constant dark energy and show that in this case the universe does not rip apart in a finite era.