A study of phantom scalar field cosmology using Lie and Noether symmetries

TL;DR

This paper explores phantom scalar field cosmology within Einstein gravity by applying Lie and Noether symmetries to determine coupling functions, potentials, and exact solutions, enhancing understanding of scalar field dynamics in cosmology.

Contribution

It introduces a novel application of Lie and Noether symmetries to derive potential functions and exact solutions in phantom scalar field cosmology.

Findings

Determined coupling and potential functions using Lie symmetry.

Derived general potential form from Noether symmetry.

Obtained exact solutions via conserved charges and cyclic variables.

Abstract

The paper deals with phantom scalar field cosmology in Einstein gravity. At first using Lie symmetry, the coupling function to the kinetic term and the potential function of the scalar field and the equation of state parameter of the matter field are determined and a simple solution is obtained. Subsequently, Noether symmetry is imposed on the Lagrangian of the system. The symmetry vector is obtained and the potential takes a very general form from which potential using Lie Symmetry can be obtained as a particular case. Then we choose a point transformation $(a,\phi)\rightarrow(u,v)$ such that one of the transformed variables (say u) is a cyclic for the Lagrangian. Using conserved charge (corresponding to the cyclic coordinate) and the constant of motion, solutions are obtained.