An analytical approach to the CMB Polarization in a Spatially Closed background

TL;DR

This paper derives analytical expressions for CMB polarization in a closed universe, revealing dependencies on cosmological parameters and comparing results with numerical simulations and observations.

Contribution

It provides the first analytical derivation of scalar mode polarization in a closed universe, explicitly showing parameter dependencies and validating with numerical and observational data.

Findings

Analytical EE and TE spectra agree with numerical results on large and moderate scales.

Peak positions match well with numerical results, peak heights within 20%.

Reveals properties of CMB polarization spectra through analytic expressions.

Abstract

The scalar mode polarization of the cosmic microwave background is derived in a spatially closed universe from the Boltzmann equation using the line of sight integral method. The EE and TE multipole coefficients have been extracted analytically by considering some tolerable approximations such as considering the evolution of perturbation hydrodynamically and sudden transition from opacity to transparency at the time of last scattering. As the major advantage of analytic expressions, EE and TE multipole coefficients explicitly show the dependencies on baryon density, matter density, curvature, primordial spectral index, primordial power spectrum amplitude, Optical depth, recombination width and recombination time. Using a realistic set of cosmological parameters taken from a fit to data from Planck, the closed universe EE and TE power spectrums in the scalar mode are compared with numerical results from the CAMB code and also latest observational data. The analytic results agree with the numerical ones on the big and moderate scales. The peak positions are in good agreement with the numerical result on these scales while the peak heights agree with that to within 20% due to the approximations have been considered for these derivations. Also, several interesting properties of CMB polarization are revealed by the analytic spectra.