Instabilities of spherical solutions with multiple Galileons and SO(N) symmetry

The 4-dimensional effective theory arising from an induced gravity action for a codimension greater than one brane consists of multiple Galileon fields π1, I = 1,...,N, invariant under separate Galilean transformations for each scalar, and under an internal SO(N) symmetry. We study the viability of such models by examining spherically symmetric solutions.We find that for general, nonderivative couplings to matter invariant under the internal symmetry, such solutions exist and exhibit a Vainshtein screening effect. By studying perturbations about such solutions, we find both an inevitable gradient instability and fluctuations propagating at superluminal speeds. These findings suggest that more general, derivative couplings to matter are required for the viability of SOðNÞ Galileon theories.