An approximate analytic formula for the polarization of cosmic microwave background radiation

In the decoupling of the early universe, photons interacted with electrons via Thompson scattering, whereby any anisotropy in the spatial distribution of the photon gas will lead to polarization of the cosmic microwave background (CMB) radiation which has recently been observed by WMAP. In this paper, starting from the Boltzmann equation of the photon gas and adopting the general expression of the optical depth function, we have derived separate, approximate analytic solutions for the polarization of the CMB radiation induced by primordial density perturbation and remnant gravitational waves, solutions that are valid for the general recombination process. For the scalar type of perturbations Fs, the derived approximate analytic expression of polarization is βs ⋍ −CFs(τd)Δτd, where τd and Δτd are the decoupling time and the decoupling duration, respectively, and C ⋍ (0.08 ∼ 0.12) depending on the recombination model. For the tensorial type of perturbations FT, the calculation is made in the long-wave approximation. By expanding the perturbation function in terms of the wave number and keeping the first two terms FT ⋍ F(1)T + F(2)T, we obtain the analytic expression of polarization βT ⋍ −[CF(1)T(τd) + DF(2)T(τd)]Δτd, where D ⋍ (0.22 ∼ 0.32), again depending on the recombination model. Our results may help to interpret the observed temperature-polarization cross correlation and to detect the contribution of remnant gravitational waves to the anisotropy of the CMB radiation.