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

摘要

We investigate the possibility that the observed flatness of the rotationcurves of spiral galaxies is not an evidence for the existence of dark matterhaloes, but rather a signal of the breakdown of General Relativity. To thisaim, we consider power - law fourth order theories of gravity obtained byreplacing the scalar curvature $R$ with $f(R) = f_0 R^n$ in the gravityLagrangian. We show that, in the low energy limit, the gravitational potentialgenerated 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 thegravity Lagrangian and $r_c$ a scalelength depending on the gravitating systemproperties. In order to apply the model to realistic systems, we compute themodified potential and the rotation curve for spherically symmetric and forthin disk mass distributions. It turns out that the potential is stillasymptotically decreasing, but the corrected rotation curve, although not flat,is higher than the Newtonian one thus offering the possibility to fit rotationcurves without dark matter. To test the viability of the model, we consider asample of 15 low surface brightness (LSB) galaxies with combined HI andH$\alpha$ measurements of the rotation curve extending in the putative darkmatter dominated region. We find a very good agreement between the theoreticalrotation curve and the data using only stellar disk and interstellar gas.