  If the Universe has non-trivial spatial topology, observables depend on both
the parameters of the spatial manifold and the position and orientation of the
observer. In infinite Euclidean space, most cosmological observables arise from
the amplitudes of Fourier modes of primordial scalar curvature perturbations.
Topological boundary conditions replace the full set of Fourier modes with
specific linear combinations of selected Fourier modes as the eigenmodes of the
scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean
manifolds 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 observer
positions, generalizing previous treatments. Under the assumption that the
amplitudes of primordial scalar curvature eigenmodes are independent random
variables, for each topology we obtain the correlation matrices of Fourier-mode
amplitudes (of scalar fields linearly related to the scalar curvature) and the
correlation matrices of spherical-harmonic coefficients of such fields sampled
on a sphere, such as the temperature of the cosmic microwave background (CMB).
We evaluate the detectability of these correlations given the cosmic variance
of the observed CMB sky. We find that topologies where the distance to our
nearest clone is less than about 1.2 times the diameter of the last scattering
surface of the CMB give a correlation signal that is larger than cosmic
variance noise in the CMB. This implies that if cosmic topology is the
explanation of large-angle anomalies in the CMB, then the distance to our
nearest clone is not much larger than the diameter of the last scattering
surface. We argue that the topological information is likely to be better
preserved in three-dimensional data, such as will eventually be available from
large-scale structure surveys.
