Lie symmetries for the cosmological field equations in brane-world gravity with bulk scalar field

TL;DR

This paper uses Lie symmetry analysis to classify scalar field potentials in brane-world cosmology, finding that only exponential potentials admit symmetries and deriving exact solutions from these invariants.

Contribution

It applies Lie symmetry methods to classify scalar field potentials in five-dimensional brane-world models, identifying exponential potentials as unique in admitting symmetries and constructing solutions.

Findings

Exponential potential is the only one with Lie symmetries.

Lie invariants facilitate the derivation of exact solutions.

The analysis enhances understanding of scalar fields in brane-world cosmology.

Abstract

We address the group classification problem for gravitational field equations within the context of brane-world cosmology, considering the presence of a bulk scalar field. Our investigation revolves around a five-dimensional spacetime, with the four-dimensional Friedmann--Lema\^{\i}tre--Robertson--Walker geometry embedded within it. Additionally, we assume that the scalar field exists in this five-dimensional geometry (bulk) and possesses a nonzero mass. The resulting field equations constitute a system of nonlinear partial differential equations. We apply the Lie symmetry condition to identify all functional forms of the scalar field potential, ensuring that the field equations remain invariant under one-parameter point transformations. Consequently, we find that only the exponential potential exhibits Lie symmetries. Finally, the Lie invariants are used to construct similarity transformations, which enable us to derive exact solutions for the system.