Modified gravity with R-matter couplings and (non-)geodesic motion

We consider alternative theories of gravity with a direct coupling between matter and the Ricci scalar. We study the relation between these theories and ordinary scalar-tensor gravity, or scalar-tensor theories which include non-standard couplings between the scalar and matter. We then analyze the motion of matter in such theories, its implications for the equivalence principle, and the recent claim that they can alleviate the dark matter problem in galaxies