On the flat galactic rotational curves in $f(\mathcal{R})$ gravity

A mysterious dark matter is supposed to exist in the galactic halos. In this contrast, we discuss the possibility of explaining the flat rotational velocity curves in f(R) gravity by solving field equations numerically in vacuum and for different matter distributions. For a spherically symmetric static space-time (as the galactic environment) we give metric for constant rotational velocity regions. For a constant rotational velocity region, we prove that all values of rotational velocities (most importantly observed rotational velocity ~200-300Km/s) do not lead to an analytic solution of the vacuum field equations. We then obtain numerical solutions of the field equations in vacuum and for three types of mass distributions named: (1) power law density profile, (2) simple model for galaxy with a core and, (3) Navarro, Frank and White (NFW) profile, for M31 and Milky way galaxy. The solutions suggest a slight modification from linear relations from R for vacuum whereas a significant deviation from R for the distributions can give flat rotational curves. Using Brans-Dicke theory, we also relate obtained modified gravity function with the equivalent scalar fields, the procedure gives us very interesting phenomena and behavior of dark matter in the galactic environment. We observe that the scalar dark matter, coming from different modified gravity functions of matter profiles, does not accumulate as the baryonic matter. These results then can be used to explain the spatial offset of the center of the total mass from the center of the baryonic mass peaks of the bullet cluster and Abell-520.