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If the Universe has non-trivial spatial topology, observables depend on boththe parameters of the spatial manifold and the position and orientation of theobserver. In infinite Euclidean space, most cosmological observables arise fromthe amplitudes of Fourier modes of primordial scalar curvature perturbations.Topological boundary conditions replace the full set of Fourier modes withspecific linear combinations of selected Fourier modes as the eigenmodes of thescalar Laplacian. We present formulas for eigenmodes in orientable Euclideanmanifolds with the topologies $E_1$ - $E_6$, $E_{11}$, $E_{12}$, $E_{16}$, and$E_{18}$ that encompass the full range of manifold parameters and observerpositions, generalizing previous treatments. Under the assumption that theamplitudes of primordial scalar curvature eigenmodes are independent randomvariables, for each topology we obtain the correlation matrices of Fourier-modeamplitudes (of scalar fields linearly related to the scalar curvature) and thecorrelation matrices of spherical-harmonic coefficients of such fields sampledon a sphere, such as the temperature of the cosmic microwave background (CMB).We evaluate the detectability of these correlations given the cosmic varianceof the observed CMB sky. We find that topologies where the distance to ournearest clone is less than about 1.2 times the diameter of the last scatteringsurface of the CMB give a correlation signal that is larger than cosmicvariance noise in the CMB. This implies that if cosmic topology is theexplanation of large-angle anomalies in the CMB, then the distance to ournearest clone is not much larger than the diameter of the last scatteringsurface. We argue that the topological information is likely to be betterpreserved in three-dimensional data, such as will eventually be available fromlarge-scale structure surveys.

Cosmic topology. Part II. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds