Cosmic topology. Part Ic. Limits on lens spaces from circle searches

Cosmic microwave background (CMB) temperature and polarization observations indicate that in the best-fit Λ Cold Dark Matter model of the Universe, the local geometry is consistent with at most a small amount of positive or negative curvature, i.e., ΩK1. However, whether the geometry is flat (E³), positively curved (S³) or negatively curved (H³), there are many possible topologies. Among the topologies of S³ geometry, the lens spaces $L(p,q)$, where p and q ($p>1$ and $0<q<p$) are positive integers, are quotients of the covering space of S³ (the three-sphere) by Zₚ, the cyclic group of order p. We use the absence of any pair of circles on the CMB sky with matching patterns of temperature fluctuations to establish constraints on p and q as a function of the curvature scale that are considerably stronger than those previously asserted for most values of p and q. The smaller the value of ΩK, i.e., the larger the curvature radius, the larger the maximum allowed value of p. For example, if ΩK 0.05 then p≤ 9, while if ΩK 0.02, p can be as high as 24. Future work will extend these constraints to a wider set of S³ topologies.