An analytical approach to the CMB anisotropies in a spatially closed background

Abstract

The scalar mode temperature fluctuations of the cosmic microwave background has been derived in a spatially closed universe from two different methods. First, by following the photon trajectory after the last scattering and then from the Boltzmann equation in a closed background and the line of sight integral method. An analytic expression for the temperature multipole coefficient has been extracted at the hydrodynamical limit, where we have considered some tolerable approximations. By considering a realistic set of cosmological parameters taken from a fit to data from Planck, the TT power spectrum in the scalar mode for the closed universe has been compared with numerical one by using the CAMB code and also latest observational data. The analytic result agrees with the numerical one on almost all scales. The peak positions are in very good agreement with numerical result while the peak heights agree with that to within 10% due to the approximations have been considered for this derivation. arXiv:1709.06549v1 [gr-qc]

Suppression of the long-wavelength CMB spectrum from the Hartle–Hawking wave function in the Starobinsky-type inflation model

“…On the other hand, the eigenvalues and eigenfunctions on S 3 sphere for the Laplace-Beltrami operator can be found as follows [30]:…”

An analytical approach to the CMB polarization in a spatially closed background

“…In this section we give a brief overview of the perturbation theory in a spatially closed universe and then review the procedures we have used in our previous work [28] to extract the scalar mode temperature fluctuation from the Boltzmann equation.…”

“…Inserting all above relations in Eq. ( 7), one can show that the Boltzmann equation for the matrix J ij (x, p, t) yields two coupled Boltzmann equations for ∆ T n (µ, t) and ∆ P n (µ, t) as [28] ∆T…”

“…By Assuming sudden transition from opacity to transparency at the time of last scattering, considering the evolution of perturbation hydrodynamically and also neglecting the ISW effect that is important only for relatively small values of , one can derive the final formula for the scalar mode temperature fluctuations as [28] ∆T (n)…”

“…[18] also gave a unifying framework for all spectra in both tensor and scalar modes. In our previous work [28] we performed a detail analytic calculation of the scalar mode (in synchronous gauge) temperature power spectrum C S T T, in a spatially closed background and now we continue that way to extract analytical expressions for EE and TE polarization multipole coefficients. Like what we have done for temperature power spectrum derivation, we will apply some of the results and techniques developed in the study of the CMB anisotropies in a flat spatial geometry by extending them into the closed geometry case.…”