Cosmic topology. Part IIIb. Eigenmodes and correlation matrices of spin-2 perturbations in orientable Euclidean manifolds

Abstract: We study the eigenmodes of the spin-2 Laplacian in orientable Euclidean manifolds and their implications for the tensor-induced part of the cosmic microwave background (CMB) temperature and polarization anisotropies. We provide analytic expressions for the correlation matrices of Fourier-mode amplitudes and of spherical harmonic coefficients. We demonstrate that non-trivial spatial topology alters the statistical properties of CMB tensor anisotropies, inducing correlations between harmonic coefficients of differing $\ell$ and $m$ and across every possible pair of temperature and $E$- and $B$-modes of polarization. This includes normally forbidden $TB$ and $EB$ correlations. We compute the Kullback-Leibler (KL) divergence between the pure tensor-induced CMB fluctuations in the usual infinite covering space and those in each of the non-trivial manifolds under consideration, varying both the size of the manifolds and the location of the observer. We find that the amount of information about the topology of the Universe contained in tensor-induced anisotropies does not saturate as fast as its scalar counterpart; indeed, the KL divergence continues to grow with the inclusion of higher multipoles up to the largest $\ell$ we have computed. Our results suggest that CMB polarization measurements from upcoming experiments can provide new avenues for detecting signatures of cosmic topology, motivating a full analysis where scalar and tensor perturbations are combined and noise is included.

## Overview of Spin-2 Perturbations in Cosmic Topology

The research presented in "Cosmic topology. Part IIIb. Eigenmodes and correlation matrices of spin-2 perturbations in orientable Euclidean manifolds" explores the intricate relationship between cosmic topology and the tensor-induced components of the Cosmic Microwave Background (CMB) anisotropies. Utilizing eigenmodes of the spin-2 Laplacian within orientable Euclidean manifolds, this study illustrates how non-trivial topology can significantly alter CMB temperature and polarization signals.

### Key Findings and Methods

The authors calculate analytic expressions for the correlation matrices of Fourier-mode amplitudes and spherical harmonic coefficients. It is demonstrated that non-trivial spatial topology, which breaks statistical isotropy, results in unusual correlations in CMB anisotropies. Intriguingly, these include normally forbidden correlations such as $TB$ and $EB$.

The study employs the Kullback-Leibler (KL) divergence, a statistical measure of difference between probability distributions, to quantify how much information about the Universe's topology can be inferred from tensor-induced CMB fluctuations. The findings suggest that the information content in tensor-induced anisotropies increases with the inclusion of higher multipole expansions, without saturating as quickly as that of scalar-induced anisotropies.

### Implications and Future Directions

Practically, this work implies that future CMB experiments focusing on polarization could unravel new signatures of cosmic topology that remain invisible in scalar field studies. Theoretically, the research supports the notion that manifold topology has measurable implications on cosmological observables, offering crucial insights into the early universe's structure.

Future work could extend these findings by integrating noise and foreground effects into the analysis. Additionally, combining scalar and tensor perturbations in a cohesive model could enhance our understanding of large-scale cosmic phenomena. Examination across wider multipole ranges and other cosmic geometries could further underpin the universality or particularity of the observed phenomena.

In conclusion, the paper advances our comprehension of the universe’s topology, highlighting that even subtle topological features bear significant imprints on observable cosmological phenomena, meriting continued investigation in the emergent field of cosmic topology detection.