Geometric origin of the galaxies' dark side

We hereby show that Einstein’s conformal gravity is able to explain simply on the geometric ground the galactic rotation curves without need to introduce any modification in both the gravitational as well as in the matter sector of the theory. Our result applies to any Weyl conformal invariant theory that admits the Schwarzschild’s metric as an exact solution. However, we here mostly consider the Einstein’s scalar-tensor theory, which is surely ghost-free. The geometry of each galaxy is described by a metric obtained making a singular rescaling of the Schwarzschild’s spacetime. The new exact solution, which is asymptotically Anti-de Sitter, manifests an unattainable singularity at infinity that can not be reached in finite proper time, namely, the spacetime is geodetically complete. It deserves to be notice that we here think different from the usual. Indeed, instead of making the metric singularity-free, we make it apparently but harmlessly even more singular then the Schwarzschild’s one. Finally, it is crucial to point that the Weyl’s conformal symmetry is spontaneously broken to the new singular vacuum rather then the asymptotically flat Schwarzschild’s one. The metric is unique according to: the null energy condition, the zero acceleration for photons in the Newtonian regime, and the homogeneity of the Universe at large scales. Once the matter is conformally coupled to gravity, the orbital velocity turns out to be asymptotically constant consistently with the observations and the Tully-Fisher relation. Hence, in the properly identified effective Newtonian theory, we consider the effect of all stars in a galaxy on a probe star to finally get the physical velocity profile that we compare with the observations in order to fit the only free parameter in the metric and the mass to luminosity ratio for each galaxy. Our fit is based on a sample of 175 galaxies and shows that our velocity profile very well interpolates the galactic rotation-curves data for the most of spiral galaxies. The fitting results for the the mass to luminosity ratio turn out to be close to 1 consistently with the absence of dark matter.