Modified gravity approach based on a preferred time foliation

We propose, in a heuristic way, a relativistic modified gravity model as an alternative to particle dark matter at galactic scales. The model is based on a postulated preferred time foliation described by a dynamical scalar field called the Khronon. In coordinates adapted to the foliation, it appears as a modification of general relativity violating local Lorentz invariance in a regime of weak gravitational fields. The model is a particular case of noncanonical Einstein-{}ther theory, but in which the {}ther vector field is hypersurface-orthogonal. We show that this model recovers the phenomenology of the modified Newtonian dynamics (MOND) in the nonrelativistic limit, and predicts the same gravitational lensing as general relativity but with a modified Poisson-type potential.