Covariant Galileon

TL;DR

The paper shows that naive covariantization of galileon Lagrangians ${\cal L}_4$ and ${\cal L}_5$ introduces third derivatives in both the scalar and metric sectors. It identifies unique nonminimal curvature couplings that cancel these higher derivatives, producing second-order equations of motion and derivative-free energy-momentum tensors, thereby defining a ghost-free scalar-tensor extension with a single scalar degree of freedom. These constructions break the original Galilean symmetry in curved spacetime and yield compact action forms, offering a path toward stable covariant galileon theories with potential cosmological applications. Stability and phenomenology in a cosmological context remain to be explored in future work.

Abstract

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.