Topological signatures in CMB temperature anisotropy maps

We propose an alternative formalism to simulate temperature anisotropy maps of CMB in $\Lambda$CDM universes with nontrivial spatial topologies. This formalism avoids the need to explicitly compute the eigenmodes of the Laplacian operator in the spatial sections. Instead, the covariance matrix of the coefficients of the spherical harmonic decomposition of the temperature anisotropies is expressed in terms of the elements of the covering group of the space. We obtain a decomposition of the correlation matrix that isolates the topological contribution to the CMB temperature anisotropies out of the ``noise'' due to the simply connected contribution. An additional decomposition of this topological signature of the correlation matrix for any complicated topology allows us to compute it in terms of correlation matrizes corresponding to simpler topologies, for which closed quadrature formulae might be derived. Another simplification in the calculations is obtained by using invariance properties of the correlation matrix due to some symmetries of the quocient manifolds. We also use the latter decomposition to show that temperature anisotropy maps of the CMB of (not too large) multiply connected universes must show ``patterns of alignement'', and propose a method to look for these patterns, thus opening the door to the development of new methods for detecting topology of our Universe even when the injectivity radius of space is slightly larger than the radius of the last scattering surface. We illustrate all these features with the simplest examples, those of flat homogeneous manifolds, i.e., torii, with special attention given to the cylinder, i.e., $T^1$ topology.