Integrability of Conformal Killing Vectors in the Eisenhart Lift of Scalar-Field FLRW Cosmology

TL;DR

This paper investigates the conditions under which conformal Killing vectors exist in the Eisenhart lift of scalar-field FLRW cosmology, showing that previously found potentials are the most general for this symmetry.

Contribution

It proves that the potential identified earlier is the most general local potential admitting a non-trivial conformal Killing vector in this context.

Findings

The potential found earlier is the most general local potential with this symmetry.

The integrability condition reduces to a nonlinear differential equation for h=V'/V.

Two solution branches are found: a regular one matching previous results, and a singular one incompatible with the full equations.

Abstract

We study the integrability conditions of the conformal Killing equations for the Eisenhart lift of a scalar field in a flat Friedmann-Lema\^\i tre-Robertson-Walker universe. We show that the potential found in our earlier work is already the most general local potential that admits a non-trivial conformal Killing vector in the sector independent of the cyclic Eisenhart coordinate. The determinant condition of the prolonged conformal Killing equations reduces to a nonlinear second-order differential equation for $h=V'/V$. We solve this equation locally and find two branches. The regular branch reproduces exactly the family of potentials obtained before, while the singular branch lies on the locus where the determinant equation cannot be written locally in normal form with respect to $h''$ and is incompatible with the full conformal Killing equations. Hence the ansatz used in our earlier work is exhaustive.