Cosmic microwave background anisotropies in multiconnected flat spaces

This article investigates the signature of the seventeen multiconnected flat spaces in cosmic microwave background (CMB) maps. For each such space it recalls a fundamental domain and a set of generating matrices, and then goes on to find an orthonormal basis for the set of eigenmodes of the Laplace operator on that space. The basis eigenmodes are expressed as linear combinations of eigenmodes of the simply connected Euclidean space. A preceding work, which provides a general method for implementing multiconnected topologies in standard CMB codes, is then applied to simulate CMB maps and angular power spectra for each space. Unlike in the 3-torus, the results in most multiconnected flat spaces depend on the location of the observer. This effect is discussed in detail. In particular, it is shown that the correlated circles on a CMB map are generically not back to back, so that negative search of back-to-back circles in the Wilkinson Microwave Anisotropy Probe data does not exclude a vast majority of flat or nearly flat topologies.