point of view, we can avoid the constraints coming from the standard model, by considering a brane world scenario instead of a compactified dimension, that is, we are living in a hypersurface (the brane) in a higher dimensional spacetime (the bulk). From the theoretical point of view, string theory predicts a boundary layer, a brane, on which edges of open strings stand. The possibility that we may be living in a brane generates many questions and possible solutions to numerous debated problems such as the hierarchy problem [@Gogberashvili:1998vx].

Among brane models, DGP [@Dvali:2000hr] attracted a lot of attention even though an induced curvature term can’t be motivated by higher energy theories and the self-accelerating branch is plagued by an instability [@Luty:2003vm; @Nicolis:2004qq; @Koyama:2005tx; @Koyama:2005br; @Charmousis:2006pn]. However, the analysis of the model continued, and an effective theory in 4D has been derived by integrating out the extra dimension, giving rise to the so-called galileon, [@Luty:2003vm] where the scalar field stands for the brane-bending mode (i.e., a longitudinal graviton). It has been extended with additional terms with the same symmetry in Minkowski background [@Nicolis:2008in]. The model has been first promoted to a covariant form [@Deffayet:2009wt] by abandoning the galilean symmetry and finally 