Introducing the Slotheon: a slow Galileon scalar field in curved space-time

In this paper we define covariant Galilean transformations in curved spacetime and find all scalar field theories invariant under this symmetry. The Slotheon is a Galilean invariant scalar field with a modified propagator such that, whenever gravity is turned on and energy conditions are not violated, it moves "slower" than in the canonical set-up. This property is achieved by a non-minimal derivative coupling of the Slotheon to the Einstein tensor. We prove that spherically symmetric black holes cannot have Slotheonic hairs. We then notice that in small derivative regimes the theory has an asymptotic local shift symmetry whenever the non-canonical coupling dominates over the canonical one.