Stability of higher derivative modifications of Einstein - aether theory

A time-like unit vector field is used to generalize Einstein's gravity. The resulting theory, called the Einstein-aether theory, consists of a minimal coupling between an aether field and gravity. Inspired by the Bopp-Podolsky electrodynamics, which is well-known for removing the singularity at the point charge, we generalized the Einstein-aether theory by adding such a higher order self-interaction term. We show that the resulting theory can  explain the late time accelerated expansion of the universe. We then consider the cosmological perturbation theory on  the top of the de Sitter solution and try to answer the question whether an additional Lorentz breaking vector field can cure the small scales instability of the Bopp-Podolsky electrodynamics.