Detectability of Cosmic Topology in Flat Universes

TL;DR

This paper investigates the observational detectability of the topology of flat universes, providing a classification of flat 3-manifolds, their geometric properties, and conditions under which their topology can be observed.

Contribution

It offers a complete classification of compact flat 3-manifolds, derives formulas for their injectivity radii and volumes, and establishes detectability conditions for flat universe topologies.

Findings

Derived expressions for injectivity radii and volumes of flat 3-manifolds.

Identified conditions under which flat universe topologies are detectable.

Constructed toy models illustrating topology detectability in flat universes.

Abstract

Recent observations seem to indicate that we live in a universe whose spatial sections are nearly or exactly flat. Motivated by this we study the problem of observational detection of the topology of universes with flat spatial sections. We first give a complete description of the diffeomorphic classification of compact flat 3-manifolds, and derive the expressions for the injectivity radii, and for the volume of each class of Euclidean 3-manifolds. There emerges from our calculations the undetectability conditions for each (topological) class of flat universes. To illustrate the detectability of flat topologies we construct toy models by using an assumption by Bernshtein and Shvartsman which permits to establish a relation between topological typical lengths to the dynamics of flat models.