Cosmic topology. Part I. Limits on orientable Euclidean manifolds from circle searches

TL;DR

This paper investigates the limits on orientable Euclidean universe topologies using circle searches in the cosmic microwave background, revealing that many complex topologies remain observationally viable beyond previous constraints.

Contribution

It translates circle search constraints into limits on parameters of all nine orientable Euclidean manifolds and provides a computational tool for these analyses.

Findings

No circle pairs detected above noise, constraining topology.

Most non-trivial topologies can have loops shorter than the last scattering surface.

A broader class of manifolds remains compatible with observations.

Abstract

The Einstein field equations of general relativity constrain the local curvature at every point in spacetime, but say nothing about the global topology of the Universe. Cosmic microwave background anisotropies have proven to be the most powerful probe of non-trivial topology since, within $\Lambda$CDM, these anisotropies have well-characterized statistical properties, the signal is principally from a thin spherical shell centered on the observer (the last scattering surface), and space-based observations nearly cover the full sky. The most generic signature of cosmic topology in the microwave background is pairs of circles with matching temperature and polarization patterns. No such circle pairs have been seen above noise in the WMAP or Planck temperature data, implying that the shortest non-contractible loop around the Universe through our location is longer than 98.5% of the comoving diameter of the last scattering surface. We translate this generic constraint into limits on the parameters that characterize manifolds with each of the nine possible non-trivial orientable Euclidean topologies, and provide a code which computes these constraints. In all but the simplest cases, the shortest non-contractible loop in the space can avoid us, and be shorter than the diameter of the last scattering surface by a factor ranging from 2 to at least 6. This result implies that a broader range of manifolds is observationally allowed than widely appreciated. Probing these manifolds will require more subtle statistical signatures than matched circles, such as off-diagonal correlations of harmonic coefficients.