On the CMB circular polarization. I. The Cotton-Mouton effect

Generation of cosmic microwave background (CMB) elliptic polarization due to the Cotton-Mouton (CM) effect in a cosmic magnetic field is studied. We concentrate on the generation of CMB circular polarization and on the rotation angle of the CMB polarization plane from the decoupling time until at present. For the first time, a rather detailed analysis of the CM effect for an arbitrary direction of the cosmic magnetic field with respect to photon direction of propagation is done. Considering the CMB linearly polarized at the decoupling time, it is shown that the CM effect is one of the most substantial effects in generating circular polarization especially in the low part of the CMB spectrum. It is shown that in the frequency range $10^8$ Hz $\leq \nu_0\leq 10^9$ Hz, the degree of circular polarization of the CMB at present for perpendicular propagation with respect to the cosmic magnetic field is in the range $ 10^{-13}\lesssim P_C(t_0)\lesssim 7.65\times 10^{-7}$ or Stokes circular polarization parameter $2.7 \times 10^{-13}$ K $\lesssim |V(t_0)|\lesssim 2 \times 10^{-6}$ K for values of the cosmic magnetic field amplitude at present in the range $10^{-9}$ G $\lesssim B\lesssim 8\times 10^{-8}$ G. On the other hand, for not perpendicular propagation with respect to the cosmic magnetic field we find $10^{-15}\lesssim P_C(t_0)\lesssim 6\times 10^{-12}$ or $2.72 \times 10^{-15}$ K $\lesssim |V(t_0)| \lesssim 10^{-11}$ K, for the same values of the cosmic magnetic field amplitude and same frequency range. Estimates on the rotation angle of the CMB polarization plane $\delta\psi_0$ due to the CM effect and constraints on the cosmic magnetic field amplitude from current constraints on $\delta\psi_0$ due to a combination of the CM and Faraday effects are found.