An Analytic Approach to the CMB Polarization Generated by Relic Gravitational Waves

We develop Polnarev’s analytic method to calculate of the polarization power spectrum of the cosmic microwave background radiation (CMB) generated by cosmic relic gravitational waves. In this analytic approach the physics involved in this process is more transparent. Consequently, the effects due to various elements of physics can be isolated easily. In numerically calculating the evolution of the gravitational waves during the radiation-matter transition, the WKB approximation for the scale factor has 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 effects on the power spectrum at small scale up to the second order of the tight coupling due to the collisions. An analytic polarization spectra CXX l has been obtained with 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 relic gravitational waves from the sudden-change approximation, we have demonstrated the dependence of CXX l on the dark energy and the baryon. 2. Our analytic half-gaussian approximation of the visibility function fits analytically better than the usual Gaussian model, and its time integration yields a parameter-dependent damping factor. This improves the spectrum ∆CXX l /C XX l ∼ 30% around the second and third peaks. 3. The second order of tight coupling limit reduces the amplitude of CXX l by 58%, comparing with the first order. 4. The influence of the power spectrum index of relic GW is such that a larger value of nT produces higher polarization spectra CXX l . PACS numbers: 98.80.-k, 98.80.Es, 04.30.-w, 04.62.+v,