Relativistic Khronon theory in agreement with modified Newtonian dynamics and large-scale cosmology

We propose an extension of General Relativity (GR) based on a space-time foliation by three-dimensional space-like hypersurfaces labeled by the Khronon scalar field τ. We show that this theory (i) leads to modified Newtonian dynamics (MOND) at galactic scales for stationary systems; (ii) recovers GR plus a cosmological constant in the strong field regime; (iii) is in agreement with the standard cosmological model and the observed cosmic microwave background anisotropies at linear cosmological scales, where the theory reduces to a subset of the generalized dark matter (GDM) model. We compute the second order action on a Minkowski background and show that it contains the usual tensor modes of GR and a scalar degree of freedom with dispersion relation ω=0. We find that the deconstrained Hamiltonian is bounded from below for wavenumbers larger than ∼ 10-31 eV and unbounded for smaller wavenumbers.