Phantom energy versus cosmological constant: Comparative results via analytical solutions of the Wheeler-DeWitt equation

Abstract

In the present paper we investigate how the phantom class of dark energy, presumably responsible for a super-accelerated cosmic expansion and here described by the state parameter \(\omega=-5/3\), influences the wave function of the Universe. This is done by analytically solving the Wheeler-DeWitt (WdW) equation in the cosmology of Friedmann-Robertson-Walker with an ambiguity term arising from the ordering of the conjugate operators associated with the scale factor \(a\). Its solutions depend on an additional parameter \(q\) related to that ordering and show that the Universe presents maximal probability to come into existence with a well-defined size for \(q = 0\). The amplitude of the wavefunction is higher the higher is the phantom energy content so an initial singularity of the type \(a = 0\) is very unlikely. In this semi-classical approach we also study how the scale factor evolves with time via the Hamilton-Jacobi equation assuming a flat Universe. We show that the ultimate big rip singularity emerges explicitly from our solutions predicting a dramatic end where the Universe reaches an infinite scale factor in a finite cosmological time. Next, we solve the WdW equation with ordinary dark energy related to a positive cosmological constant. In this case, we show that the Universe does not rip apart in a finite era.