Exact self-accelerating cosmologies in the ghost-free massive gravity -- the detailed derivation

We present the detailed derivation of the recently announced most general cosmological solution with homogeneous and isotropic metric in the ghost-free massive gravity theory. We use the standard parametrization of the theory in terms of the matrix square root, and then show how the same results are recovered within the tetrad formulation. The solution obtained includes the matter source, it exists for generic values of the theory parameters, and it describes a universe that can be spatially open, closed, or flat, and that shows the late time acceleration due to the effective cosmological term mimicked by the graviton mass. The St\"uckelberg fields are inhomogeneous, which could probably give rise to inhomogeneous perturbations of the homogeneous and isotropic backgrounds, although this effect should be suppressed by the smallness of the graviton mass.