Limits of the circles-in-the-sky searches in the determination of cosmic topology of nearly flat universes

Abstract

[Abridged] An observable signature of a detectable nontrivial spatial topology of the Universe is the circles-in-the-sky in the CMB sky. In the most general search, pairs of circles with deviation from antipodality $0^\circ \leq θ\leq 169^\circ$ and radii $10^\circ \leq λ\leq 90^\circ$ were investigated, but no matching circles were found. Assuming this negative result, we examine the question as to whether there are nearly flat universes with compact topology that would give rise to circles whose observable parameters $λ$ and $θ$ fall o outside the ranges covered by this search. We derive the expressions for the deviation from antipodality and for the radius of the circles associated to a pair elements ($γ\,$,$γ^{-1}$) of the holonomy group $Γ$ which define the spatial section of any positively curved universe with a nontrivial topology. We show that there is a critical position that maximizes the deviation from antipodality, and prove that no matter how nearly flat the Universe is, it can always have a nontrivial spatial topology that gives rise to circles whose deviation from antipodality $θ$ is larger than $169^\circ$, and whose radii of the circles $λ$ are smaller than $10^\circ$ for some observers. This makes apparent that slightly positively curved universes with cosmological parameters within Planck bounds can be endowed with a nontrivial spatial topology with values of the parameters $λ$ and $θ$ outside the ranges covered by the searches for circles carried out so far. Thus, these circles searches so far undertaken are not sufficient to exclude the possibility of a universe with a detectable nontrivial cosmic topology. We present concrete examples of such nearly flat universes, and discuss the implications of our results in view of unavoidable practical limits of the circles-in-the-sky method.