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We reconsider the recently proposed mimetic gravity, focusing in particularon whether the theory is able to reproduce the inferred flat rotation curves ofgalaxies. We extend the theory by adding a non-minimal coupling between matterand mimetic field. Such coupling leads to the appearance of an extra forcewhich renders the motion of test particles non-geodesic. By studying the weakfield limit of the resulting equations of motion, we demonstrate that in theNewtonian limit the acceleration law induced by the non-minimal couplingreduces to a Modified Newtonian Dynamics (MOND)-like one. In this way, it ispossible to reproduce the successes of MOND, namely the explanation for theflat galactic rotation curves and the Tully-Fisher relation, within theframework of mimetic gravity, without the need for particle dark matter. Thescale-dependence of the recovered acceleration scale opens up the possibilityof addressing the missing mass problem not only on galactic but also on clusterscales: we defer a full study of this issue, together with a complete analysisof fits to spiral galaxy rotation curves, to an upcoming companion paper.

Recovering a MOND-like acceleration law in mimetic gravity