Vector instabilities and self-acceleration in the decoupling limit of massive gravity

We investigate in detail the vector contributions to the Lagrangian of Λ 3 massive gravity in the decoupling limit, the less explored sector of this theory, with the main aim to study the stability of maximally symmetric self-accelerating solutions. Around self-accelerating configurations, vector degrees of freedom become strongly coupled since their kinetic terms vanish; their dynamics is controlled by contributions to the Lagrangian that arise at higher orders in perturbations. Even in the decoupling limit, the vector Lagrangian contains an infinite number of terms. We develop a systematic method to determine in a covariant way the vector Lagrangian at each order in perturbations, fully manifesting the symmetries of the system. We show that, around self-accelerating solutions, the structure of higher order p -form Galileons arise, avoiding the emergence of a sixth ghost Boulware-Deser mode. However, a careful analysis of the corresponding Hamiltonian shows that there are directions along which the Hamiltonian is unbounded from below, signaling an instability that can be interpreted as one of the available fifth modes behaving as a ghost. Hence, we conclude that self-accelerating configurations in the decoupling limit of Λ 3 massive gravity are generically unstable.