Weak Massive Gravity

We find a new class of theories of massive gravity with five propagating degrees of freedom where only rotations are preserved. Our results are based on a non-perturbative and background-independent Hamiltonian analysis. In these theories the weak field approximation is well behaved and the static gravitational potential is typically screened \`a la Yukawa at large distances, while at short distances no vDVZ discontinuity is found and there is no need to rely on nonlinear effects to pass the solar system tests. The effective field theory analysis shows that the ultraviolet cutoff is (m M_PL)^1/2 ~ 1/\mu m, the highest possible. Thus, these theories can be studied in weak-field regime at all the phenomenologically interesting scales, and are candidates for a calculable large-distance modified gravity.