CMB anisotropy power spectrum of 3-sphere universe for low $l$

Abstract

We calculate the CMB anisotropy power spectrum of a closed universe for large angle (i.e., low $l$) due to a scale invariant fluctuation of primordial universe by considering the spherical harmonics for 3-sphere. In particular, contrary to the wide belief, we show that this consideration affects the CMB anisotropy power spectrum; instead of constant $l(l+1)C_l$, our consideration results in the supression for $l=2$, currently explained by the cosmic variance. As a more concrete proof of our analysis, from the low $l$ CMB anisotropy data \emph{alone}, we obtained $\Omega_{\mathrm{tot}}=1.0018^{+0.0031}_{-0.0007}$, which agrees with $\Omega_{\mathrm{tot}}=1.0023^{+ 0.0056}_{- 0.0054}$ from the previous anlaysis of WMAP+BAO+$H_0$. Thus, we conclude that our Universe is closed.