Circles-in-the-sky searches and observable cosmic topology in a flat universe

Abstract

[Abridged] In a Universe with a detectable nontrivial spatial topology the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations - the so-called circles-in-the-sky. Searches for nearly antipodal circles in maps of cosmic microwave background have so far been unsuccessful. This negative outcome along with recent theoretical results concerning the detectability of nearly flat compact topologies is sufficient to exclude a detectable nontrivial topology for most observers in very nearly flat positively and negatively curved Universes ($0<|\Omega_{tot}-1| \lesssim 10^{-5}$). Here we investigate the consequences of these searches for observable nontrivial topologies if the Universe turns out to be exactly flat ($\Omega_{tot}=1$). We demonstrate that in this case the conclusions deduced from such searches can be radically different. We show that for all multiply-connected orientable flat manifolds it is possible to directly study the action of the holonomies in order to obtain a general upper bound on the angle that characterizes the deviation from antipodicity of pairs of matching circles associated with the shortest closed geodesic. This bound is valid for all observers and all possible values of the compactification length parameters. We also show that in a flat Universe there are observers for whom the circles-in-the-sky searches already undertaken are insufficient to exclude the possibility of a detectable nontrivial spatial topology. It is remarkable how such small variations in the spatial curvature of the Universe, which are effectively indistinguishable geometrically, can have such a drastic effect on the detectability of cosmic topology.