Eisenhart-Duval lift, Nonlocal Conservation laws and Painlevé Analysis 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( \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.