Relativistic Khronon Theory in agreement with Modified Newtonian Dynamics and Large-Scale Cosmology

Abstract

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 $\tau$. 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 $\omega =0$. We find that the deconstrained Hamiltonian is bounded from below for wavenumbers larger than $\sim 10^{-31}\text{eV}$ and unbounded for smaller wavenumbers.

Small changes and clarifications in accordance with the referee comments. Added table of contents. Corrected typo in figure and caption. No change to overall results. Matches accepted (AAM) and published version. 35 pages, 1 figure

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