Inflation in the closed FLRW model and the CMB

Abstract: Recent cosmic microwave background (CMB) observations put strong constraints on the spatial curvature via estimation of the parameter $\Omega_k$ assuming an almost scale invariant primordial power spectrum. We study the evolution of the background geometry and gauge-invariant scalar perturbations in an inflationary closed FLRW model and calculate the primordial power spectrum. We find that the inflationary dynamics is modified due to the presence of spatial curvature, leading to corrections to the nearly scale invariant power spectrum at the end of inflation. When evolved to the surface of last scattering, the resulting temperature anisotropy spectrum ($C_{\ell}^{TT}$) shows deficit of power at low multipoles ($\ell<20$). By comparing our results with the recent Planck data we discuss the role of spatial curvature in accounting for CMB anomalies and in the estimation of the parameter $\Omega_k$. Since the curvature effects are limited to low multipoles, the Planck estimation of cosmological parameters remains robust under inclusion of positive spatial curvature.