A relativistic description of MOND using the Palatini formalism in an extended metric theory of gravity

We construct a relativistic metric description of MOND using the Palatini formalism following the \( f(\chi)=\chi^b \) description of \citet{mendozatula}. We show that in order to recover the non-relativistic MOND regime where, for circular orbits the Tully-Fisher law replaces Kepler's third law, the value of the parameter $ b = 3/2 $, which is coincident with the value found using a pure metric formalism Capozziello et al. (2011). Unlike the pure metric formalism, which yields 4th order field equations, the Palatini approach yields second order field equations, which is a desirable requirement from a theoretical perspective. Thus, the phenomenology associated to astrophysical phenomena with Tully-Fisher scalings can be accounted for using this proposal, without the need to introduce any non-baryonic dark matter particles.