Role of matter in extended quasidilaton massive gravity

The extended quasidilaton theory is one of the simplest Lorentz-invariant massive gravity theories which can accommodate a stable self-accelerating vacuum solution. In this paper we revisit this theory and study the effect of matter fields. For a matter sector that couples minimally to the physical metric, we find hints of a Jeans type instability in the IR. In the analogue k-essence field setup, this instability manifests itself as an IR ghost for the scalar field perturbation, but this can be interpreted as a classical instability that becomes relevant below some momentum scale in terms of matter density perturbations. We also consider the effect of the background evolution influenced by matter on the stability of the gravity sector perturbations. In particular, we address the previous claims of ghost instability in the IR around the late time attractor. We show that, although the matter-induced modification of the evolution potentially brings tension to the stability conditions, one goes beyond the regime of validity of the effective theory well before the solutions become unstable. We also draw attention to the fact that the IR stability conditions are also enforced by the existence requirements of consistent background solutions.