Stability Conditions for the Horndeski Scalar Field Gravity Model

We constrain the viable models of Horndeski gravity, written in its equivalent Generalised Galileon version, by resorting to the Witten positive energy theorem. We find that the free function $G_3(\varphi ,X)$ in the Lagrangian is constrained to be a function solely of the scalar field, $G_3(\varphi )$, and relations among the free functions are found. Other criterion for stability are also analysed, such as the attractiveness of gravity, the Dolgov-Kawasacki instability and the energy conditions. Some applications for Cosmology are discussed