Mixed Galileons and Spherically Symmetric Solutions

It was previously found that in a certain parameter subspace of scalar-tensor theories emerging from massive gravity, the only stable field configuration created by static spherically symmetric sources was one with cosmological asymptotics. Moreover, these backgrounds were shown to be sub-luminal everywhere in the space; in contrast to the common believe that these theories are necessarily superluminal in the vicinity of a static source. In this work we complete that analysis by extending it to cover the whole parameter space of these scalar-tensor theories. We find that the stability argument renders the asymptotically flat backgrounds unrealizable, forcing once again for cosmological asymptotics. In the case of pressureless sources these backgrounds are stable. However, they get destabilized in the presence of positive pressure, larger than a critical density. Even on the self-accelerated background, on which the scalar mode decouples from sources, in the region occupied by the source it acquires an elliptic equation of motion. Therefore, we conclude that the only parameter space which is not ruled out, by solar system measurements, is the one considered in Berezhiani {\it et al.} (arXiv:1302.0549), namely the one for which the scalar and tensor modes can be diagonalized via local transformations. We also reinvestigate the scale at which perturbation theory breaks down in a general Galileon theory. We show that the Vainshtein mechanism successfully redresses the strong coupling scale to a small one, just like in the cubic Galileon, despite the cancellations occurring in the special spherically symmetric case. We emphasize that even if these tests were performed at scales at which perturbation theory broke down, these could not be interpreted as a lower bound for the graviton mass.