Covariant Galileon

We consider the recently introduced Galileon field in a dynamical spacetime. When the Galileon is assumed to be minimally coupled to the metric, we underline that both field equations of the Galileon and the metric involve up to third-order derivatives. We show that a unique nonminimal coupling of the Galileon to curvature eliminates all higher derivatives in all field equations, hence yielding second-order equations, without any extra propagating degree of freedom. The resulting theory breaks the generalized Galilean invariance of the original model.