The debate on the physical relevance of conformal transformations can be faced by taking the Palatini approach into account in 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 the 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.

We show that a general, exact cosmological solution, where dynamics of scalar field is assigned by an exponential potential, fulfils all the issues of dark energy approach, both from a theoretical point of view and in comparison with available observational data. Moreover, tracking conditions are discussed, with a new treatment of the well known condition Γ > 1. We prove that the 1 currently used expression for Γ is wrong. PACS number(s): 98.80. Jk, 98.80.Cq, 98.80.Hw, 04.20.Jb

In an attempt to look for a viable mechanism leading to a present-day accelerated expansion, we investigate the possibility that the observed cosmic speed up may be recovered in the framework of the Rastall's theory, relying on the non-conservativity of the stress-energy tensor, i.e. T µ ν;µ = 0. We derive the modified Friedmann equations and show that they correspond to Cardassian-like equations. We also show that, under suitable assumptions on the equation of state of the matter term sourcing the gravitational field, it is indeed possible to get an accelerated expansion, in agreement with the Hubble diagram of both Type Ia Supernovae (SNeIa) and Gamma Ray Bursts (GRBs). Unfortunately, to achieve such a result one has to postulate a matter density parameter much larger than the typical ΩM ≃ 0.3 value inferred from cluster gas mass fraction data.

We review the dynamical equivalence between f (R) gravity in the metric formalism and scalartensor gravity, and use this equivalence to deduce the post-Newtonian parameters γ and β for a f (R) theory, obtaining a result that is different with respect to that known in the literature. Then, we obtain explicit expressions of these paremeters in terms of the mass of the scalar field (or, differently speaking, the mass of the additional scalar degree of freedom associated to a f (R) theory) which can be used to constrain f (R) gravity by means of current observations.