The Status of Horava Gravity

Horava gravity is a proposal for a UV completion of gravitation obtained by endowing the space-time manifold with a preferred foliation in space-like hypersurfaces. This allows for a power-counting renormalizable theory free of ghosts, at the cost of breaking local Lorentz invariance and diffeomorphism invariance down to foliation preserving transformations. In this updated review, we report the main successes and challenges of the proposal, discussing the main features of the projectable and non-projectable versions of Ho\v rava gravity. We focus in three main aspects: (i) the UV regime, discussing the renormalizability and renormalization group flow of the projectable theory, as well as the obstacles towards similar results in the non-projectable case; (ii) the low energy phenomenology of both models, including the PN regime, the most updated constraints in the parameter space of the theory, the structure of black holes at low energies, and the possibility of dark matter emerging from gravitational dynamics in the projectable model; and (iii) the specific phenomena induced by higher derivatives, such as the possibility of regularizing singularities, the dynamical behavior of solutions to dispersive equations, and the emission of Hawking radiation by universal horizons.