Models of non-relativistic quantum gravity: the good, the bad and the healthy

Horava's proposal for non-relativistic quantum gravity introduces a preferred time foliation of space-time which violates the local Lorentz invariance. The foliation is encoded in a dynamical scalar field which we call 'khronon'. The dynamics of the khronon field is sensitive to the symmetries and other details of the particular implementations of the proposal. In this paper we examine several consistency issues present in three non-relativistic gravity theories: Horava's projectable theory, the healthy non-projectable extension, and a new extension related to ghost condensation. We find that the only model which is free from instabilities and strong coupling is the non-projectable one. We elaborate on the phenomenology of the latter model including a discussion of the couplings of the khronon to matter. In particular, we obtain the parameters of the post-Newtonian expansion in this model and show that they are compatible with current observations.