Approximate Analytic Spectra of Reionized CMB Anisotropies and Polarization generated by Relic Gravitational Waves

Abstract

We present an approximate, analytical calculation of the reionized spectra $C_l^{XX}$ of cosmic microwave background radiation (CMB) anisotropies and polarizations generated by relic gravitational waves (RGWs). Three simple models of reionization are explored, whose visibility functions are fitted by gaussian type of functions as approximations. We have derived the analytical polarization $β_l$ and temperature anisotropies $α_l$, both consisting of two terms proportional to RGWs at the decoupling and at the reionization as well. The explicit dependence of $β_l$ and $α_l$ upon the reionization time $η_r$, the duration $Δη_r$, and the optical depth $κ_r$ are demonstrated. Moreover, $β_l$ and $α_l$ contain $κ_r$ in different coefficients, and the polarization spectra $C_l^{EE}$ are $C_l^{BB}$ are more sensitive probes of reionization than $C_l^{TT}$. These results facilitate examination of the reionization effects, in particular, the degeneracies of $κ_r$ with the normalization amplitude and with the initial spectral index of RGWs. It is also found that reionization also causes a $κ_r$-dependent shift $Δl\sim 20$ of the zero multipole $l_0$ of $C_l^{TE}$, an effect that should be included in order to detect the traces of RGWs. Compared with numerical results, the analytical $C_l^{XX}$ as approximation have the limitation. For the primary peaks in the range $l\simeq (30, 600)$, the error is $\le 3%$ in three models. In the range $l < 20$ for the reionization bumps, the error is $\le 15%$ for $C_l^{EE}$ and $C_l^{BB}$ in the two extended reionization models, and $C_l^{TT}$ and $C_l^{TE}$ have much larger departures for $l<10$. The bumps in the sudden reionization model are too low.