Cosmic topology. Part IId. Eigenmodes and correlation matrices of lens spaces
Abstract
The global topology of the Universe is a longstanding open question. In this work, we examine the statistical signatures of a positively curved universe with a Friedmann--Lemaitre--Robertson--Walker metric and the topology of a lens space L(p,q). Since these manifolds are generally statistically anisotropic and inhomogeneous, their cosmic microwave background (CMB) covariance matrices contain non-zero off-diagonal entries that depend on observer location. We compute these full harmonic-space covariance matrices for scalar perturbations for arbitrary lens spaces and observer position. Using the Kullback--Leibler divergence, we assess the distinguishability of these spaces from a simply connected three-sphere with the same curvature. The results show that topological signatures can still be significant in a cosmic-variance-limited regime even when the length, dNC, of the shortest loop around the Universe through the observer exceeds the diameter, dLSS, of the last-scattering surface by up to 10%. This distinguishability depends on the curvature, the lens space and the observer position mainly through the ratio dNC/dLSS, which is set not only by the curvature radius but also by p, q and the observer location. Therefore, for any value of the spatial curvature, however small, there are lens spaces and observer positions for which 0.985 dLSS<dNC<1.1 dLSS. Such observers would find no matched circles in the CMB large enough to have been detected to date, yet the topology would remain potentially discoverable through its statistical signatures.
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BibTeX
@misc{barandiaran2026cosmic,
title = {{Cosmic topology. Part IId. Eigenmodes and correlation matrices of lens spaces}},
author = {Mikel Martin Barandiaran and Catherine Petretti and Stefano Anselmi and Deyan P Mihaylov and Anna Negro and Glenn D Starkman and Andrius Tamosiunas and Yashar Akrami and George Alestas and Javier Carrón Duque and Craig J Copi and Fernando Cornet-Gomez and Linn Htat Lu and Andrew H Jaffe and Arthur Kosowsky and José Javier Ortega Gómez and Ricardo G Rodrigues and Amirhossein Samandar and Cynthia Trendafilova},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12418},
eprint = {2610.12418},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05842226v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cosmic topology. Part IId. Eigenmodes and correlation matrices of lens spaces
AU - Mikel Martin Barandiaran
AU - Catherine Petretti
AU - Stefano Anselmi
AU - Deyan P Mihaylov
AU - Anna Negro
AU - Glenn D Starkman
AU - Andrius Tamosiunas
AU - Yashar Akrami
AU - George Alestas
AU - Javier Carrón Duque
AU - Craig J Copi
AU - Fernando Cornet-Gomez
AU - Linn Htat Lu
AU - Andrew H Jaffe
AU - Arthur Kosowsky
AU - José Javier Ortega Gómez
AU - Ricardo G Rodrigues
AU - Amirhossein Samandar
AU - Cynthia Trendafilova
PY - 2026
DA - 2026/10/08
PB - HAL
DO - 10.48550/arxiv.2610.12418
UR - https://hal.science/hal-05842226v1
AB - The global topology of the Universe is a longstanding open question. In this work, we examine the statistical signatures of a positively curved universe with a Friedmann--Lemaitre--Robertson--Walker metric and the topology of a lens space L(p,q). Since these manifolds are generally statistically anisotropic and inhomogeneous, their cosmic microwave background (CMB) covariance matrices contain non-zero off-diagonal entries that depend on observer location. We compute these full harmonic-space covariance matrices for scalar perturbations for arbitrary lens spaces and observer position. Using the Kullback--Leibler divergence, we assess the distinguishability of these spaces from a simply connected three-sphere with the same curvature. The results show that topological signatures can still be significant in a cosmic-variance-limited regime even when the length, d_NC, of the shortest loop around the Universe through the observer exceeds the diameter, d_LSS, of the last-scattering surface by up to 10%. This distinguishability depends on the curvature, the lens space and the observer position mainly through the ratio d_NC/d_LSS, which is set not only by the curvature radius but also by p, q and the observer location. Therefore, for any value of the spatial curvature, however small, there are lens spaces and observer positions for which 0.985 d_LSS<d_NC<1.1 d_LSS. Such observers would find no matched circles in the CMB large enough to have been detected to date, yet the topology would remain potentially discoverable through its statistical signatures.
ER -