Approximate analytic spectra of reionized CMB anisotropies and polarization generated by relic gravitational waves

We present an approximate, analytical calculation of the reionized spectra CₗXX of cosmic microwave background radiation anisotropies and polarizations generated by relic gravitational waves (RGWs). Three simple models of reionization are explored, whose visibility functions are fitted by Gaussian types of functions as approximations. We have derived the analytical polarization βₗ and temperature anisotropies αₗ, which both consist of two terms proportional to RGWs at the decoupling and at the reionization as well. The explicit dependence of βₗ and αₗ upon the reionization time ηᵣ, the duration Δηᵣ, and the optical depth κᵣ is demonstrated. Moreover, βₗ and αₗ contain κᵣ in different coefficients, and the polarization spectra CₗEE and CₗBB are more sensitive probes of reionization than CₗTT. These results facilitate examination of the reionization effects, in particular, the degeneracies of κᵣ with the normalization amplitude and with the initial spectral index of RGWs. It is also found that reionization causes a κᵣ-dependent shift Δl~20 of the zero multipole l₀ of CₗTE, an effect that should be included in order to detect the traces of RGWs. Compared with numerical results, the analytical CₗXX are approximate and have the limitation. For the primary peaks in the range l(30,600), the error is ≤3% in three different models. In the range $l<20$ for the reionization bumps, the error is ≤15% for CₗEE and CₗBB in the two extended reionization models, and CₗTT and CₗTE have much larger departures for $l<10$. The bumps in the sudden reionization model are too low.