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

TL;DR

This paper introduces a modified gravity theory based on a Khronon scalar field that aligns with MOND at galactic scales, recovers GR cosmology, and matches large-scale cosmological observations.

Contribution

It presents a relativistic extension of GR using a Khronon field that reproduces MOND, aligns with standard cosmology, and analyzes its stability and degrees of freedom.

Findings

Reproduces MOND at galactic scales

Matches standard cosmological observations

Contains a stable tensor mode and a scalar degree of freedom

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.