Quantum phantom cosmology

We apply the formalism of quantum cosmology to models containing a phantom field. Three models are discussed explicitly: a toy model, a model with an exponential phantom potential, and a model with phantom field accompanied by a negative cosmological constant. In all these cases we calculate the classical trajectories in configuration space and give solutions to the Wheeler-DeWitt equation in quantum cosmology. In the cases of the toy model and the model with exponential potential we are able to solve the Wheeler-DeWitt equation exactly. For comparison, we also give the corresponding solutions for an ordinary scalar field. We discuss, in particular, the behavior of wave packets in minisuperspace. For the phantom field these packets disperse in the region that corresponds to the big-rip singularity. This thus constitutes a genuine quantum region at large scales, described by a regular solution of the Wheeler-DeWitt equation. For the ordinary scalar field, the big-bang singularity is avoided. Some remarks on the arrow of time in phantom models as well as on the relation of phantom models to loop quantum cosmology are given.

Figure 1

The classical trajectory in configuration space for the toy model with vanishing scalar field potential and vanishing cosmological constant. The diagram on the left-hand side shows the trajectory for the phantom field model. On the right-hand side the trajectory for the $\ell =1$ scalar field model is plotted.

Figure 2

The classical trajectories in configuration space for the models with cosh-potential and negative cosmological constant. On the left-hand side, the trajectory for the phantom field model is shown. The classical trajectory for the scalar field model $\ell =1$ is shown on the right-hand side. The similarity to the classical trajectories in the toy model in Sec. 2b is obvious.

Figure 3

The amplitude of the wave packet (50) for an exponential potential solution of the WDW equation for $\ell =1$ (left) and $\ell =-1$ (right). Here, $\hslash$ was set to unity and parameters $\sigma =0.1$ and $\lambda =\kappa /2$ have been chosen. The wave packet for the phantom field model is seen to spread near the classical singularity. For the scalar field model one has $\Psi \to 0$ at the origin. In each sector corresponding to one copy of the $(\alpha ,\varphi )$ plane, the same wave packet propagates.