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

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.