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
$\mathrm{\Lambda}$ 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.

PREPRINT

# Cosmic Topology I: Limits on Orientable Euclidean Manifolds from Circle Searches

Pip Petersen, Yashar Akrami, Craig J. Copi, Andrew H. Jaffe, Arthur Kosowsky, Glenn D. Starkman, Andrius Tamosiunas, Johannes R. Eskilt, Özenç Güngör, Samanta Saha, Quinn Taylor

Submitted on 4 November 2022, last revised on 22 November 2022

## Abstract

## Preprint

Comment: 21 pages, 7 figures. v2: formatting changes

Subjects: Astrophysics - Cosmology and Nongalactic Astrophysics; General Relativity and Quantum Cosmology; High Energy Physics - Phenomenology; High Energy Physics - Theory