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Relativistic MOND theory based on the Khronon scalar field

We investigate a model of modified gravity recovering the modified Newtonian dynamics (MOND) in the non-relativistic limit, based on the introduction of a preferred time foliation violating Lorentz invariance in the weak-field regime. Lorentz-invariance violation has been studied in the framework of Einstein-aether theory, the generalization of which, known as non-canonical Einstein-aether theory, having been proposed as a relativistic formulation of MOND. Our model can be seen as a minimal specialization to the hypersurface orthogonal case, which allows a different interpretation in terms of the preferred time : it can be either treated as a dynamical scalar field in a 4D formulation, or chosen as the time coordinate in a 3+1 formulation. We discuss the equivalence of the two points of view and the non-relativistic limit of the model.