Eisenhart Lift for Scalar Fields in the FLRW Universe

TL;DR

This paper extends the Eisenhart lift to scalar fields in the FLRW universe, classifies symmetries, and provides methods for solving equations of motion using conformal Killing vectors.

Contribution

It generalizes the Eisenhart lift to scalar fields in cosmology and classifies symmetries for specific exponential potentials, enabling complete solutions.

Findings

Identified nontrivial conformal Killing vectors for certain potentials.

Classified symmetries of single and multiple scalar field systems.

Provided a method to solve equations of motion using symmetry analysis.

Abstract

The Eisenhart lift of Riemannian type describes the motion of a particle as a geodesic in a higher-dimensional Riemannian manifold with one additional coordinate. It has recently been generalized to a scalar field system by introducing one additional vector field. We apply this approach to a scalar field system in the Friedmann-Lemaitre-Robertson-Walker universe and classify the symmetries of the system. In particular, for a scalar field potential consisting of the square of a combination of exponential functions with specific index $e^{\pm\frac{\sqrt{6}}{4}\phi}$, we find nontrivial (conformal) Killing vector fields and Killing tensor fields. Moreover, for a potential written as an exponentiation of a combination of exponential potentials with general index, we find nontrivial conformal Killing vector fields. By introducing the coordinate along the conformal Killing vector field, we can solve the equations of motion completely. We also classify the symmetries of multiple scalar field system.