  The minimal theory of quasidilaton massive gravity with or without a
Horndeski-type kinetic term for the quasidilaton field propagates only three
physical modes: the two massive tensor polarizations and one scalar mode. This
reduced number of degrees of freedom is realized by a Lorentz symmetry
violation at cosmological scales and the presence of appropriate constraints
that remove unwanted modes. Vacuum cosmological solutions have been considered
in a previous work, and it has been shown that the late-time de Sitter
attractor is stable under inhomogeneous perturbations. In this work, we explore
the stability of cosmological solutions in the presence of matter fields.
Assuming for simplicity that the quasidilaton scalar is on an attractor at the
level of the background, we derive stability conditions in the subhorizon
limit, and find the scalar sound speeds, as well as the modification with
respect to general relativity to the gravitational potential in the quasistatic
approximation. We also find that the speed limit of gravitational waves
coincides with the speed of light for any homogeneous and isotropic
cosmological background, on or away from the attractor.
