Square-torsion gravity, dark matter halos and the baryonic Tully-Fisher relation

Abstract: Square-torsion gravity is applied to the long standing dark matter problem. In this context the theory reduces to General Relativity complemented by a dark stress-energy tensor due to the torsion of spacetime and is studied under the simplifying assumption of spherical symmetry. The dark stress-energy tensor is found to satisfy an anisotropic structure equation. In vacuum this is shown to be equivalent to a wave equation with sources. A natural class of exact solutions is found which explicitly perturbs any seed spacetime metric by a conformal factor satisfying a (1+1)-dimensional wave equation. This leads to the concept of dark coating. The static solutions are then used to construct structures that model dark matter halos surrounding baryonic bodies. In the Newtonian r\'egime the baryonic mass $m_b$ and the flat rotation curve velocity $v_f$ are found to be related by the baryonic Tully-Fisher relation $m_b\propto v_f^4$. The present work proposes thus a possible theoretical motivation of this hitherto purely empirical result. The example of a dark halo on the Schwarzschild geometry is made as a toy model for a galaxy. All qualitative an quantitative features of galactic rotation curves are recovered. A dark halo surrounding a Schwarzschild black hole is found to possess a boundary of staticity called torsion sphere placed between the photon sphere and the event horizon. The phenomenon of dark radiation is briefly exposed. The way for cosmological applications is then opened by showing how Hubble expansion is a natural feature of the theory.