*8.2. Discussion*

Let us now try to summarize the physical insight that we can get from the analysis above, about the possible theoretical nature of the observed *H*0(*z*) behavior. We can keep as a reliably good starting point the idea that the origin of a modified scaling of the function *H*(*z*) with respect to the standard ΛCDM model can be identified in a slowly varying Einstein constant with the redshift. Furthermore, it is a comparably good assumption to search, in the framework of a scalar-tensor formulation of gravity, the natural explanation for such a varying Einstein constant. As shown in Section 6.2, a scalar-tensor formulation can reproduce the required scaling of the function *H*(*z*), which we observe as an *H*0(*z*) behavior in the standard ΛCDM model. Hence, we naturally explored one of the most interesting and well-motivated formulations of a scalar-tensor theory, namely the *f*(*R*) gravity in the JF. In this respect, in Section 6.2.2, we first evaluated the form of the scalar field potential inferred from the observed decreasing trend of *H*0(*z*), and our data analysis suggested a model described in Equations (26) and (29). Then, we investigated if, one of the most reliable models for reproducing the Dark Energy effect with modified gravity, i.e., the Hu–Sawicki proposal, was able to induce the requested luminosity distance to somehow remove the observed effect, thus accounting for its physical nature. The non-positive result of this investigation leads us to explore theoretically the question of reproducing simultaneously the Dark Energy contribution and the observed *H*0(*z*) effect, by a single *f*(*R*) model of gravity in the JF. In Section 8, it has been addressed this theoretical question, by establishing the conditions that a modified gravity model has to satisfy to reach the simultaneous aims mentioned above. Finally, we considered a specific model for the late universe, based on a slow-rolling picture for the scalar field near its today value *φ* ' 1. This model was successful in explaining the Dark Energy contribution and the necessary variation of the Einstein constant, but it seems hard to be reconciled with the earlier Universe behavior, when the role of the matter contribution becomes relevant. Thus, based on this systematic analysis, we can conclude that the explanation for *H*0(*z*) is probably to be attributed to modified gravity dynamics, but it appears more natural to separate its effect from the existence of a Dark Energy contribution. In other words, we are led to believe that what we discovered about the SNe Ia+BAOs binned analysis must be regarded as a modified gravity physics of the scalar-tensor type, but leaving on a standard Universe, well represented by a ΛCDM model a priori.
