Revisiting the cuscuton as a Lorentz-violating gravity theory

The cuscuton is a scalar field with infinite speed of propagation. It was introduced in the context of cosmology but it has also been claimed to resemble Hořava gravity in a certain limit. Here we revisit the cuscuton theory as a Lorentz-violating gravity theory. We clarify its relation with Hořava gravity and Einstein-æther theory, analyze its causal structure, and consider its initial value formulation. Finally, we discuss to which extent the cuscuton theory can be used as a proxy for Hořava gravity in the context of gravitational collapse and formation of universal horizons.