Conformal aspects of the Palatini approach in Extended Theories of Gravity

The debate on the physical relevance of conformal transformations can be faced by taking the Palatini approach into account gravitational theories. We show that conformal transformations are not only a mathematical tool to disentangle gravitational and matter degrees of freedom (passing from the Jordan frame to the Einstein frame) but they acquire a physical meaning considering the bi-metric structure of Palatini approach which allows to distinguish between spacetime structure and geodesic structure. These facts are relevant at least at cosmological scales, while at small scale (i.e. in the spacetime regions relevant for observations) the conformal factor is slowly varying and its effects are not relevant. Examples of higher-order and non-minimally coupled theories are worked out and relevant cosmological solutions in Einstein frame and Jordan frame are discussed showing that also the interpretation of cosmological observations can drastically change depending on the adopted frame.