ou rnal of Cosmology and Astroparticle Physics

We analyze the theories of gravity modied by a generic nonderivative potential built from the metric, under the minimal requirement of unbroken spatial rotations. Using the canonical analysis, we classify the potentialsV according to the number of degrees of freedom (DoF) that propagate at the nonperturbative level. We then compare the nonperturbative results with the perturbative DoF propagating around Minkowski and FRW backgrounds. A genericV implies 6 propagating DoF at the non-perturbative level, with a ghost on Minkowski background. There exist potentials which propagate 5 DoF, as already studied in previous works. Here, no V with unbroken rotational invariance admitting 4 DoF is found. Theories with 3 DoF turn out to be strongly coupled on Minkowski background. Finally, potentials with only the 2 DoF of a massive graviton exist. Their eect on cosmology is simply equivalent