Noether symmetric classical and quantum scalar field cosmology

TL;DR

This paper explores classical and quantum scalar field cosmology using Noether symmetries, identifying specific potentials that admit these symmetries and deriving exact solutions for both ordinary and phantom scalar fields.

Contribution

It introduces a method to determine scalar field potentials compatible with Noether symmetries in a two-dimensional minisuperspace cosmological model, providing exact solutions.

Findings

Identified potential functions with Noether symmetries.

Derived exact classical solutions for ordinary and phantom scalar fields.

Obtained quantum solutions corresponding to classical models.

Abstract

We study the evolution of a two dimensional minisuperspace cosmological model in classical and quantum levels by the Noether symmetry approach. The phase space variables turn out to correspond to the scale factor of a Friedmann-Robertson-Walker (FRW) model and a scalar field with which the action of the model is augmented. It is shown that the minisuperspace of such a model is a two dimensional manifold with vanishing Ricci scalar. We present a coordinate transformation which cast the corresponding minisuper metric to a Minkowskian or Euclidean one according to the choices of an ordinary or phantom model for the scalar field. Then, the Noether symmetry of such a cosmological model is investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. We explicitly calculate the form of the scalar field potential functions for which such symmetries exist. For these potential functions, the exact classical and quantum solutions in the cases where the scalar field is an ordinary or a phantom one, are presented and compared.