Post-Newtonian constraints on Lorentz-violating gravity theories with a MOND phenomenology

We study the post-Newtonian expansion of a class of Lorentz-violating gravity theories that reduce to khronometric theory (i.e. the infrared limit of Hořava gravity) in high-acceleration regimes and reproduce the phenomenology of modified Newtonian dynamics (MOND) in the low-acceleration, nonrelativistic limit. Like in khronometric theory, Lorentz symmetry is violated in these theories by introducing a dynamical scalar field (the "khronon") whose gradient is enforced to be timelike. As a result, hypersurfaces of constant khronon define a preferred foliation of the spacetime, and the khronon can be thought of as a physical absolute time. The MOND phenomenology arises as a result of the presence, in the action, of terms depending on the acceleration of the congruence orthogonal to the preferred foliation. We find that if the theory is forced to reduce exactly to general relativity (rather than to khronometric theory) in the high-acceleration regime, the post-Newtonian expansion breaks down at low accelerations, and the theory becomes strongly coupled. Nevertheless, we identify a sizeable region of the parameter space where the post-Newtonian expansion remains perturbative for all accelerations, and the theory passes both Solar System and pulsar gravity tests, besides producing a MOND phenomenology for the rotation curves of galaxies. We illustrate this explicitly with a toy model of a system containing only baryonic matter but no dark matter.