  We study cosmological perturbations for a ghost free massive gravity theory
formulated with a dynamical extra metric that is needed to massive deform GR.
In this formulation FRW background solutions fall in two branches. In the
dynamics of perturbations around the first branch solutions, no extra degree of
freedom with respect to GR ispresent at linearized level, likewise what is
found in the Stuckelberg formulation of massive gravity where the extra metric
isflat and non dynamical. In the first branch, perturbations are probably
strongly coupled. On the contrary, for perturbations around the second branch
solutions all expected degrees of freedom propagate. While tensor and vector
perturbations of the physical metric that couples with matter follow closely
the ones of GR, scalars develop an exponential Jeans-like instability on
sub-horizon scales. On the other hand, around a de Sitter background there is
no instability. We argue that one could get rid of the instabilities by
introducing a mirror dark matter sector minimally coupled to only the second
metric.
