Eisenhart-Duval lift, Nonlocal Conservation laws and Painlev\'{e} Analysis in Scalar field Cosmology

TL;DR

This paper explores nonlocal conservation laws and integrability of scalar field cosmology equations using Eisenhart-Duval lift and Painlevé analysis, revealing conditions for solutions and their qualitative behavior.

Contribution

It introduces a novel approach to identify nonlocal conservation laws and analyzes the integrability of scalar field cosmology models with specific potentials.

Findings

Scalar field potential admits nontrivial conservation laws.

Field equations are integrable for certain parameter ranges.

Analytic solutions are obtained and their evolution discussed.

Abstract

We investigate the existence of nonlocal conservation laws for the gravitational field equations of scalar field cosmology in an FLRW background with a dust fluid source. We perform such analysis by using a novel approach for the Eisenhart-Duval lift. It follows that the scalar field potential $V\left( \phi \right) =\alpha \left( e^{\lambda \phi }+\beta \right) $ admits nontrivial conservation laws. Furthermore, we employ the Painlev\'{e} analysis to examine the integrability of the field equations. For the quintessence model, we establish that the cosmological field equations possess the Painlev\'{e} property and are integrable for $\lambda ^{2}>6$. In contrast, for the phantom scalar field, the cosmological field equations exhibit the Painlev\'{e} property for any value of the parameter $% \lambda $. We present analytic solutions expressed in terms of Right Laurent expansions for various values of the parameter $\lambda $. Finally, we discuss the qualitative evolution of the effective equation of state parameter for these analytic solutions.