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

Abstract

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., |ΩK| ≪ 1. 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.