$F(R)$ nonlinear massive theories of gravity and their cosmological
  implications
  We propose a nonlinear massive gravitational theory which includes $F(R)$
modifications. This construction inherits the benefits of the de
Rham-Gabadadze-Tolley model and is free of the Boulware-Deser ghost due to the
existence of a Hamiltonian constraint accompanied by a nontrivial secondary
one. The scalar perturbations in a cosmological background can be stabilized at
the linear level for a wide class of the $F(R)$ models. The linear scalar mode
arisen from the $F(R)$ sector can absorb the nonlinear longitudinal graviton,
and hence, our scenario demonstrates the possibility of a gravitational
Goldstone theorem. Finally, due to the combined contribution of the $F(R)$ and
graviton-mass sectors, the proposed theory allows for a large class of
cosmological evolutions, such as the simultaneous and unified description of
inflation and late-time acceleration.
