PPN rotation curves in static distributions with spherical symmetry
  From a Parametrized Post-Newtonian (PPN) [46-51] perspective, we address the
question of whether or not the new degrees of freedom, represented by the PPN
potentials, can lead to significant modifications in the dynamics of galaxies
in the direction of rendering dark matter obsolete. Here, we focus on the study
of rotation curves associated with spherically symmetric configurations. The
values for the post-Newtonian parameters, which help us to classify the
different metric theories of gravity, are tightly constrained mainly by solar
system experiments [47,48]. However, in recent years, metric theories
characterized by screening mechanisms [4,10,22,25] have become popular, due to
the fact that they lead to the possibility of modifications in larger scales
than the solar system while retaining the success of GR on it, allowing for
violations of the constraints of the Post-Newtonian parameters. In such a
context, we consider here two kinds of solutions for field equations: (i)
Vacuum solutions and (ii) in the presence of a politropic distribution of
matter. For the case (i) we find that the post-Newtonian corrections do not
lead to modifications significant enough to be considered an alternative to the
dark matter hypothesis. In the case (ii) we find that for a wide range of
values for the PPN parameters {\gamma}, {\beta}, {\xi}, {\alpha}_3 , {\zeta}_1
and {\zeta}_2 , the need for dark matter is unavoidable, in order to find flat
rotation curves. It is only for theories in which {\zeta}_3>0 that some
resemblance of flat rotation curves is found. The latter suggests, at least for
the models considered, that these are the only theories capable of replacing
dark matter as a possible explanation for the dynamics of galaxies.
