CMB Polarization and Theories of Gravitation with Massive Gravitons

We study in this paper three different theories of gravitation with massive gravitons - the modified Fierz-Pauli (FP) model, Massive Gravity and the bimetric theory proposed by Visser - in linear perturbation theory around a Minkowski and a flat FRW background. For the TT tensor perturbations we show that the three theories give rise to the same dynamical equations and to the same form of the Boltzmann equations for the radiative transfer in General Relativity (GR). 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 FP theory the vector gravitational-wave (GW) polarization modes ($\Psi_{3}$ amplitudes in the Newman-Penrose (NP) formalism) do not decay too fast as it 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 $Y_{2,\pm 1}(\theta,\varphi)$ on the CMB 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.