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

An important observable signature of a detectable nontrivial spatial topology of the Universe is the presence in the cosmic microwave background sky of pairs of matching circles with the same distributions of temperature fluctuations---the so-called circles in the sky. Most of the recent attempts to find these circles, including the ones undertaken by the Planck Collaboration, were restricted to antipodal or nearly antipodal circles with radii $\ensuremath{\lambda}\ensuremath{\ge}15\ifmmode^\circ\else\textdegree\fi{}$. In the most general search, pairs of circles with deviation from antipodality angles $0\ifmmode^\circ\else\textdegree\fi{}\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}169\ifmmode^\circ\else\textdegree\fi{}$ and radii $10\ifmmode^\circ\else\textdegree\fi{}\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}90\ifmmode^\circ\else\textdegree\fi{}$ were investigated. No statistically significant pairs of matching circles were found in the searches so far undertaken. Assuming that the negative result of general search can be confirmed through analysis made with data from Planck and future cosmic microwave background experiments, we examine the question as to whether there are nearly flat universes with compact topology, satisfying Planck constraints on cosmological parameters, that would give rise to circles in the sky whose observable parameters $\ensuremath{\lambda}$ and $\ensuremath{\theta}$ fall outside the parameter ranges covered by this general search. We derive the expressions for the deviation from antipodality and for the radius of the circles associated to a pair of elements ($\ensuremath{\gamma}$, ${\ensuremath{\gamma}}^{\ensuremath{-}1}$) of the holonomy group $\mathrm{\ensuremath{\Gamma}}$ which define the spatial section of any positively curved universe with a nontrivial compact 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 $\ensuremath{\theta}$ is larger than 169\ifmmode^\circ\else\textdegree\fi{} and whose radii of the circles $\ensuremath{\lambda}$ are smaller than 10\ifmmode^\circ\else\textdegree\fi{} for some observers's positions. This makes it apparent that slightly positively curved nearly flat universes with cosmological parameters within Planck bounds can be endowed with a nontrivial spatial topology with values of the observable parameters $\ensuremath{\lambda}$ and $\ensuremath{\theta}$ outside the ranges covered by the searches for circles carried out so far with either WMAP or Planck data. Thus, these circles-in-the-sky searches carried out so far are not sufficient to exclude the possibility of a universe with a detectable nontrivial cosmic topology. We present concrete examples of lens spaces universes whose associated circles have both, or at least one, value of the observable parameters ($\ensuremath{\lambda}$, $\ensuremath{\theta}$) outside the ranges covered by these searches. We also present a brief discussion of the implications of our results in view of unavoidable practical limits of the circles-in-the-sky method.