GR is the unique theory of a self interacting massless spin two field, the graviton. Here, we consider the case where the spin two field is instead massive. For a long time, it was believed that such theories were inconsistent due to a ghost instability that generically emerges at the nonlinear level. Only recently this problem has been overcome and several modifications that evade this instability have been found.

The simplest version of a massive gravity theory consists in adding to the Einstein-Hilbert action a potential built from non-derivative self-couplings of the metric. As a matter of fact, such modification needs additional structure besides the metric.

In this talk we present the canonical analysis for a general massive deformation of GR in a fully non-perturbative and background independent way and show the conditions that a potential has to satisfy in order to avoid such a instability. Among the possible candidates, we find that Lorentz invariance considerably restricts the viable potentials and, when Minkowski space is required as a background, they probably reduce to a single one.

For this potential we analyze the cosmology in the framework of bigravity theories, where the extra metric needed is taken to be dynamical. In particular, FRW solutions exist and are very close to the ones in GR at early times; at late times the universe flows to a dS phase. Unfortunately, scalar cosmological perturbations show an exponential instability that signals the unreliability of the standard perturbation theory in sharp contrast to GR.