Cosmic topology. Part IIb. Eigenmodes, correlation matrices, and detectability of non-orientable Euclidean manifolds

Abstract 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. In an earlier work we provided a comprehensive treatment of orientable Euclidean three-manifolds; but a thorough exploration of cosmic topology must include non-orientable three-manifolds as candidates for the geometry of space. In this paper we consider the non-orientable Euclidean topologies E 7 – E 10 , E 13 – E 15 , and E 17 , encompassing 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 CMB sky. As for orientable three-manifolds, we find that in manifolds where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB, we expect a correlation signal that is larger than cosmic variance noise in the CMB. The parameter space of the non-orientable Euclidean manifolds is quite rich, supporting, for example, complex dependencies of clone distances on those parameters. Our limited selection of manifold parameters — both the values of those we fix, and the choices of which to vary — are therefore exemplary of interesting behaviors (e.g., of how well the manifold can be distinguished from the covering space), but not necessarily representative. Future searches for topology will certainly require a much more thorough exploration of the parameter space to determine what values of the parameters predict statistical correlations that are convincingly attributable to topology.