It has been suggested that Einsteins theory of General Relativity can be modified to accomodate mismatches between the gravitational field and luminous matter on a wide range of scales. Covariant theories of modified gravity generically predict the existence of extra degrees of freedom which may be interpreted as dark matter. We study a subclass of these theories where the overall energy density in these extra degrees of freedom is subdominant relative to the baryon density and show that they favour the presence of massive neutrinos. For some specific cases (such as a flat Universes with a cosmological constant) one finds a conservative lower bound on the neutrinos mass of $m_ u>0.31$ eV.
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