Low surface brightness galaxies rotation curves in the low energy limit of $R^n$ gravity : no need for dark matter?

Abstract

We investigate the possibility that the observed flatness of the rotation curves of spiral galaxies is not an evidence for the existence of dark matter haloes, but rather a signal of the breakdown of General Relativity. To this aim, we consider power - law fourth order theories of gravity obtained by replacing the scalar curvature $R$ with $f(R) = f_0 R^n$ in the gravity Lagrangian. We show that, in the low energy limit, the gravitational potential generated by a pointlike source may be written as $\Phi(r) \propto r^{-1} \left [ 1 + (r/r_c)^{\beta} \right ]$ with $\beta$ a function of the slope $n$ of the gravity Lagrangian and $r_c$ a scalelength depending on the gravitating system properties. In order to apply the model to realistic systems, we compute the modified potential and the rotation curve for spherically symmetric and for thin disk mass distributions. It turns out that the potential is still asymptotically decreasing, but the corrected rotation curve, although not flat, is higher than the Newtonian one thus offering the possibility to fit rotation curves without dark matter. To test the viability of the model, we consider a sample of 15 low surface brightness (LSB) galaxies with combined HI and H$\alpha$ measurements of the rotation curve extending in the putative dark matter dominated region. We find a very good agreement between the theoretical rotation curve and the data using only stellar disk and interstellar gas.