Quintom cosmological model and some possible solutions using Lie and Noether symmetries

TL;DR

This paper explores a quintom dark energy model within a flat FLRW universe, using Lie and Noether symmetries to determine potentials, simplify the system, and derive cosmological solutions.

Contribution

It introduces a method to determine potentials via Lie and Noether symmetries and simplifies the system to obtain explicit cosmological solutions.

Findings

Lie algebra of Noether symmetry is a sub-algebra of Lie symmetry algebra

A point transformation simplifies the system and yields conserved quantities

Explicit cosmological solutions are derived and analyzed

Abstract

The present work deals with a quintom model of dark energy in the framework of a spatially flat isotropic and homogeneous Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. At first, Lie point symmetry is imposed to the system and the unknown coupled potential of the model is determined. Then Noether symmetry, which is also a point like symmetry of the Lagrangian, is imposed on the physical system and the potential takes a general form. It is shown that the Lie algebra of Noether symmetry is a sub-algebra of the corresponding Lie algebra of the Lie symmetry. Finally, a point transformation in the three dimensional augmented space is performed suitably so that one of the variables become cyclic and as a result there is considerable simplification to the physical system. Hence conserved quantities (i.e, constants of motion) are expressed in a compact form and cosmological solutions are evaluated and analyzed in the present context.