CMB polarization in theories of gravitation with massive gravitons

We study in this paper three different theories of gravitation with massive gravitons-the modified Fierz-Pauli model, massive gravity and the bimetric theory proposed by Visser-in linear perturbation theory around a Minkowski and a flat Friedmann-Robertson-Walker background. For the transverse-traceless tensor perturbations we show that the three theories give rise to the same dynamical equations, to the same form of the tensor Sachs-Wolfe effect, and consequently to the same form of the Boltzmann equations for the radiative transfer in general relativity. We then analyze vector perturbations in these theories and show that they do not give the same results as in the previous case. We first show that vector perturbations in massive gravity present the same form as found in general relativity, whereas in the modified Fierz-Pauli theory the vector gravitational-wave polarization modes (psi(3) amplitudes in the Newman-Penrose formalism) do not decay too fast as happens in the former case. Rather, we show that such psi(3) polarization modes give rise to an unusual vector Sachs-Wolfe effect, leaving a signature in the quadrupole form Y-2 +/- 1(theta,phi) on the cosmic microwave background radiation polarization. We then derive the details for the Thomson scattering of CMB photons for these psi(3) modes, and then construct the correspondent Boltzmann equations. Based upon these results we then qualitatively show that psi(3)-mode vector signatures-if they do exist - could clearly be distinguished on the CMB polarization from the usual psi(4) tensor modes. We also estimate that the graviton mass limit for the vector modes is m = 10(-66) g-10(-29) cm(-1), so that vector modes with masses below this limit exhibit the same dynamical evolution as the massless gravitons. We argue at the end of this paper that CMB polarization experiments can be decisive to test alternative theories of gravitation by measuring CMB polarization in the E-mode.