PPN rotation curves in static distributions with spherical symmetry

Abstract

From a parametrized post-Newtonian (PPN) 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. Such restrictions render the modifications of gravitational effects, with respect to general relativity (GR), to be insignificant, making attempts to find alternative metrical theories rather fruitless. However, in recent years, metric theories characterized by screening mechanisms 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 (i.e., when no matter fields are present) and (ii) fields in the presence of a polytropic distribution of matter. For 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 case (ii), we find that for a wide range of values for the PPN parameters $\ensuremath{\gamma}$, $\ensuremath{\beta}\ensuremath{\le}1$, $\ensuremath{\xi}\ensuremath{\ge}0$, ${\ensuremath{\alpha}}_{3}$, ${\ensuremath{\zeta}}_{1}$, and ${\ensuremath{\zeta}}_{2}$, the need for dark matter is unavoidable in order to find flat rotation curves. It is only for theories in which ${\ensuremath{\zeta}}_{3}g0$, or $\ensuremath{\beta}g1$, or $\ensuremath{\xi}l0$ that some resemblance of flat rotation curves is found. But the latter two require some direct fine-tuning of the screening radius ${r}_{c}$, while ${\ensuremath{\zeta}}_{3}g0$ implies the most sound modifications. The latter suggests, at least for the models considered, that these are the only theories (consistent with the usual PPN approach) capable of replacing dark matter as a possible explanation for the dynamics of galaxies.