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 ${\beta }_l$ and temperature anisotropies ${\alpha }_l$, both consisting of two terms proportional to RGWs at the decoupling and at the reionization as well. The explicit dependence of ${\beta }_l$ and ${\alpha }_l$ upon the reionization time ${\eta }_r$, the duration $\Delta {\eta }_r$, and the optical depth ${\kappa }_r$ are demonstrated. Moreover, ${\beta }_l$ and ${\alpha }_l$ contain ${\kappa }_r$ 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 ${\kappa }_r$ with the normalization amplitude and with the initial spectral index of RGWs. It is also found that reionization also causes a ${\kappa }_r$-dependent shift $\Delta l\sim 20$ 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.

33 pages, 15 figures, to appear in PRD

References in corpus (20)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data
• Three Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Polarization Analysis
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectra
• Improved Calculation of the Primordial Gravitational Wave Spectrum in the Standard Model
• Relic Gravitational Waves in the Accelerating Universe
• Imprints of Relic Gravitational Waves in Cosmic Microwave Background Radiation
• Analytic spectrum of relic gravitational waves modified by neutrino free streaming and dark energy
• An exact analytic spectrum of relic gravitational waves in an accelerating universe
• The Maximum B-mode Polarization of the Cosmic Microwave Background from Inhomogeneous Reionization
• Modifications by QCD transition and $e^{+}{e}^{-}$ annihilation on analytic spectrum of relic gravitational waves in accelerating universe
• The Polarization of the Cosmic Microwave Background Due to Primordial Gravitational Waves
• A Linear Perturbation Theory of Inhomogeneous Reionization
• Impact of reionization on CMB polarization tests of slow-roll inflation
• CMB Temperature Polarization Correlation and Primordial Gravitational Waves
• The effect of reionization on the CMB-density correlation
• Relic Gravitational Waves And CMB Polarization In The Accelerating Universe
• Planck and reionization history: a model selection view
• Model independent approaches to reionization in the analysis of upcoming CMB data

Cited by in corpus (1)

• Relic Gravitational Waves with A Running Spectral Index and Its Constraints at High Frequencies