An Analytic Approach to CMB Polarizations Generated by Relic Gravitational Waves

By Polnarev’s method we analytically calculate the polarization spectra of the cosmic microwave background radiation (CMB) generated by cosmic relic gravitational waves (RGW). In this analytic approach the physics involved in this generating process is more transparent. Consequently, the effects due to various elements of physics can be isolated easily. In solving the equation for evolution of RGW in the expanding universe, both the sudden transition and the WKB approximation for the scale factor during the radiationmatter transition have been taken. To describe more precisely the decoupling process, we have introduced an analytic expression for the visibility function, consisting of two pieces of half-Gaussian curves. We also include the damping on polarizations due to the photon diffusion up to the second order of the tight coupling. Analytic polarization spectra CXX l have been obtained with the following several improvements over the previous results. 1. The approximate analytic result is quite close to the numerical one evaluated from the cmbfast code, especially, for the first three peaks of the spectrum that are observable. By using the analytic exact solution of RGW in the sudden transition approximation, we have demonstrated the dependence of CXX l on the dark energy and the baryons. 2. Our analytic half-Gaussian approximation of the visibility function fits better than the usual Gaussian model, and its time integration yields an analytic damping factor, which is parameter-dependent. This improves the spectra by ∼ 30% around the second and third peaks. 3. The second order of tight coupling reduces the overall amplitude of CXX l by 58%, comparing with the tight coupling limit. 4. The influences of inflation on RGW and on CMB polarization are explicitly demonstrated. PACS numbers: 98.70Vc, 04.30.-w, 98.80.-k, 98.80.Es,