Most general cubic-order Horndeski Lagrangian allowing for scaling solutions and the application to dark energy

In cubic-order Horndeski theories where a scalar field φ is coupled to nonrelativistic matter with a field-dependent coupling Q(φ), we derive the most general Lagrangian having scaling solutions on the isotropic and homogenous cosmological background. For constant Q including the case of vanishing coupling, the corresponding Lagrangian reduces to the form L = Xg2(Y) − g3(Y)□φ, where X = −∂µφ∂^µφ/2 and g2, g3 are arbitrary functions of Y = Xe^λφ with constant λ. We obtain the fixed points of the scaling Lagrangian for constant Q and show that the φ-matter-dominatedepoch (φMDE) is present for the cubic coupling g3(Y) containing inverse power-law functions of Y. The stability analysis around the fixed points indicates that the φMDE can be followed by a stable critical point responsible for the cosmic acceleration. We propose a concrete dark energy model allowing for such a cosmological sequence and show that the ghost and Laplacian instabilities can be avoided even in the presence of the cubic coupling.