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

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.