Cosmic topology. Part IId. Eigenmodes and correlation matrices of lens spaces

Abstract

The global topology of the Universe is a longstanding open question. In this work, we examine the statistical signatures of a positively curved universe with a Friedmann--Lemaitre--Robertson--Walker metric and the topology of a lens space $L(p,q)$. Since these manifolds are generally statistically anisotropic and inhomogeneous, their cosmic microwave background (CMB) covariance matrices contain non-zero off-diagonal entries that depend on observer location. We compute these full harmonic-space covariance matrices for scalar perturbations for arbitrary lens spaces and observer position. Using the Kullback--Leibler divergence, we assess the distinguishability of these spaces from a simply connected three-sphere with the same curvature. The results show that topological signatures can still be significant in a cosmic-variance-limited regime even when the length, $d_{\rm NC}$, of the shortest loop around the Universe through the observer exceeds the diameter, $d_{\rm LSS}$, of the last-scattering surface by up to $10\%$. This distinguishability depends on the curvature, the lens space and the observer position mainly through the ratio $d_{\rm NC}/d_{\rm LSS}$, which is set not only by the curvature radius but also by $p$, $q$ and the observer location. Therefore, for any value of the spatial curvature, however small, there are lens spaces and observer positions for which $0.985\,d_{\rm LSS}<d_{\rm NC}<1.1\,d_{\rm LSS}$. Such observers would find no matched circles in the CMB large enough to have been detected to date, yet the topology would remain potentially discoverable through its statistical signatures.