Three fluid cosmological model using Lie and Noether symmetries

TL;DR

This paper constructs and analyzes a three-fluid cosmological model in flat spacetime, employing Lie and Noether symmetries to determine scalar field potentials and derive new analytic solutions for dark energy, dark matter, and perfect fluid interactions.

Contribution

It introduces a symmetry-based method to fix scalar field potentials and derive exact solutions in a three-fluid cosmological model, advancing analytical understanding of dark energy and matter interactions.

Findings

Identified exponential scalar field potentials via Lie symmetry.

Derived new analytic solutions for quintessence and phantom cosmologies.

Provided explicit expressions for cosmological functions like scale factor and Hubble rate.

Abstract

We employ a three fluid model in order to construct a cosmological model in the Friedmann Robertson Walker flat spacetime, which contains three types of matter dark energy, dark matter and a perfect fluid with a linear equation of state. Dark matter is described by dust and dark energy with a scalar field with potential V({\phi}). In order to fix the scalar field potential we demand Lie symmetry invariance of the field equations, which is a model-independent assumption. The requirement of an extra Lie symmetry selects the exponential scalar field potential. The further requirement that the analytic solution is invariant under the point transformation generated by the Lie symmetry eliminates dark matter and leads to a quintessence and a phantom cosmological model containing a perfect fluid and a scalar field. Next we assume that the Lagrangian of the system admits an extra Noether symmetry. This new assumption selects the scalar field potential to be exponential and forces the perfect fluid to be stiff. Furthermore the existence of the Noether integral allows for the integration of the dynamical equations. We find new analytic solutions to quintessence and phantom cosmologies which contain all three fluids. Using these solutions one is able to compute analytically all main cosmological functions, such as the scale factor, the scalar field, the Hubble expansion rate, the deceleration parameter etc.