Eisenhart-Duval lift, Nonlocal Conservation laws and Painlevé Analysis in Scalar field Cosmology
Andronikos Paliathanasis
TL;DR
This work addresses integrability questions in scalar-field cosmology within a spatially flat FLRW background with dust by employing the Eisenhart-Duval lift to generate nonlocal conservation laws for a potential $V(φ)=α(e^{λφ}+β)$. The authors classify conformal symmetries in the extended minisuperspace, obtain nonlocal conserved quantities in the original system, and apply the ARS Painlevé algorithm to quintessence and phantom cases. They find that phantom models are integrable for all $λ$, while quintessence is integrable only for $λ^2>6$, with analytic Right Laurent expansions and explicit EOS behavior in both cases. These results extend the catalog of analytically tractable cosmological models and illustrate how lifted symmetries and Painlevé analysis illuminate integrability and solution structure in scalar-field cosmology.
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( φ\right) =α\left( e^{λφ}+β\right) $ admits nontrivial conservation laws. Furthermore, we employ the Painlevé analysis to examine the integrability of the field equations. For the quintessence model, we establish that the cosmological field equations possess the Painlevé property and are integrable for $λ^{2}>6$. In contrast, for the phantom scalar field, the cosmological field equations exhibit the Painlevé property for any value of the parameter $% λ$. We present analytic solutions expressed in terms of Right Laurent expansions for various values of the parameter $λ$. Finally, we discuss the qualitative evolution of the effective equation of state parameter for these analytic solutions.