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 $\mathrm{\Lambda }$ 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., $\mathrm{\mid }{\mathrm{\Omega }}_K\mathrm{\mid }\ll 1$. However, whether the geometry is flat ($E^3$), positively curved ($S^3$) or negatively curved ($H^3$), there are many possible topologies. Among the topologies of $S^3$ 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 $S^3$ (the three-sphere) by ${\mathbb{Z}}_p$, 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 $\mathrm{\mid }{\mathrm{\Omega }}_K\mathrm{\mid }$, i.e., the larger the curvature radius, the larger the maximum allowed value of $p$. For example, if $\mathrm{\mid }{\mathrm{\Omega }}_K\mathrm{\mid }\simeq 0.05$ then $p\le 9$, while if $\mathrm{\mid }{\mathrm{\Omega }}_K\mathrm{\mid }\simeq 0.02$, $p$ can be as high as 24. Future work will extend these constraints to a wider set of $S^3$ topologies.