Cosmology of bigravity with doubly coupled matter

We study cosmology in the bigravity formulation of the dRGT model where matter couples to both metrics. At linear order in perturbation theory two mass scales emerge: an hard one from the dRGT potential, and an environmental dependent one from the coupling of bigravity with matter. At early time, the dynamics is dictated by the second mass scale which is of order of the Hubble scale. The set of gauge invariant perturbations that couples to matter follow closely the same behaviour as in GR. The remaining perturbations show no issue in the scalar sector, while problems arise in the tensor and vector sectors. During radiation domination, a tensor mode grows power-like at super-horizon scales. More dangerously, the only propagating vector mode features an exponential instability on sub-horizon scales. We discuss the consequences of such instabilities and speculate on possible ways to deal with them.