time. This approach can be generalized considering scalar-tensor theories of gravity where also a self-interaction potential comes into the game or more than one scalar field are taken into account. In general, any gravitational theory not simply linear in the Ricci scalar can be reduced to a scalar-tensor one. In the case of $f(R)$-gravity, it is straightforward to show that an [*O’Hanlon representation*]{} by scalar fields is possible [@ohanlon]. In this case, a scalar field is non-minimally coupled to the Ricci scalar and a self-interacting potential is present while there is no kinetic term. Scalar field dynamics is guaranteed by the non-minimal coupling and the potential (see [@report] and references therein). In other words, the further gravitational degrees of freedom, coming from the fact that $f(R)\neq R$, can be figured out as a scalar field. As soon as we are considering non-minimal couplings or higher-order terms in the Lagrangian, we are in the so-called [*Jordan frame*]{}, after Jordan who first introduced this notion [@Jordan38; @Jordan52].

In this article we would like to address the following basic question: *In the framework of the scalar field cosmology, is it possible to relate the available Lagrangians in a conformal way?*

The structure of 