Cosmological Solutions in Scalar-Tensor theory via the Eisenhart-Duval lift

TL;DR

The paper develops a geometric framework to obtain analytic cosmological solutions in scalar-tensor and modified gravity theories by applying the Eisenhart-Duval lift to an extended (three-dimensional) minisuperspace. By enforcing conformal-flatness of the extended space via the Cotton–York criterion, it derives differential constraints on the scalar potential $V(\phi)$ and the coupling $\omega(\phi)$, which render the field equations equivalent to linear geodesic equations. This yields explicit analytic solutions for Brans-Dicke-like models and maps to $f(R)$ gravity with $\phi=f_{,R}$ and to hybrid $f(\mathcal{R})$ gravity, including nonzero spatial curvature cases and their inter-frame relations through conformal transformations. The approach provides a unifying, integrability-driven route to closed-form cosmological solutions across scalar-tensor and modified gravity theories and points toward broader classes of integrable models when conformal-flatness is relaxed. These results offer new tools for exploring exact cosmological dynamics and the connections among different gravitational theories.

Abstract

We implement the Einsenhart-Duval lift in scalar-tensor gravity as a means to construct integrable cosmological models and analytic cosmological solutions. Specifically, we employ a geometric criterion to constrain the free functions of the scalar-tensor theory such that the field equations can be written in the equivalent form of linear equations. This geometric linearization is achieved by the introduction of an extended minisuperspace description. The results are applied to construct analytic solutions in modified theories of gravity such as the $f\left( R\right) $-theory and the hybrid metric-Palatini $f\left( \mathcal{R}\right) $-gravity.