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An important, and potentially detectable, signature of a non-trivial topology for the universe is the presence of so called circles-in-the-sky in the cosmic microwave background (CMB). Recent searches, confined to antipodal and nearly antipodal circles, have however failed to detect any. This outcome, coupled with recent theoretical results concerning the detectability of very nearly flat universes, is sufficient to exclude a detectable non-trivial cosmic topology for most observers in the inflationary limit ($0< |\Omega_{tot}-1| \lesssim 10^{-5}$). In a recent paper we have studied the consequences of these searches for circles if the Universe turns out to be exactly flat ($\Omega_{tot} = 1 $) as is often assumed. More specifically, we have derived the maximum angles of deviation possible from antipodicity of pairs of matching circles associated with the shortest closed geodesic for all multiply-connected flat orientable $3$-manifolds. These upper bounds on the deviation from antipodicity demonstrate that in a flat universe for some classes of topology there remains a substantial fraction of observers for whom the deviation from antipodicity of the matching circles is considerably larger than zero, which implies that the searches for circles-in-the-sky undertaken so far are not enough to exclude the possibility of a detectable non-trivial flat topology. Here we briefly review these results and discuss their consequences in the search for circles-in-the-sky in a flat universes.

OBSERVABLE CIRCLES-IN-THE-SKY IN FLAT UNIVERSES