A Topological Signature in Cosmic Topology

TL;DR

This paper introduces two methods for extracting the topological signature of multiply connected Robertson-Walker universes, demonstrating their effectiveness across different curvature types through simulations and theoretical analysis.

Contribution

It presents novel procedures for obtaining the topological signature of RW universes and validates their consistency and applicability across flat, elliptic, and hyperbolic models.

Findings

Both approaches yield consistent topological signatures.

The topological signature can be detected even with few images per object.

The mean pair separation histogram technique is effective for studying topological signatures.

Abstract

Two procedures for obtaining (extracting and constructing) the topological signature of any multiply connected Robertson-Walker (RW) universe are presented. It is shown through computer-aided simulations that both approaches give rise to the same topological signature for a multiply connected flat RW universe. The strength of these approaches is illustrated by extracting the topological signatures of a flat ($k=0$), an elliptic ($k=1$), and a hyperbolic ($k=-1$) multiply connected RW universes. We also show how separated contributions of the covering isometries add up to form the topological signature of a RW flat universe. There emerges from our theoretical results and simulations that the topological signature arises (in the mean) even when there are just a few images for each object. It is also shown that the mean pair separation histogram technique works, and that it is a suitable approach for studying the topological signatures of RW universes as well as the role of non-translational isometries.