Spikes in Cosmic Crystallography

TL;DR

This paper analyzes the expected patterns in cosmic crystallography, showing that topological spikes are due to translations and that such spikes are insufficient to determine the universe's shape, especially in hyperbolic cases.

Contribution

It derives a general expression for the expected pair separation histogram and clarifies the limitations of cosmic crystallography in identifying universe topology.

Findings

Topological spikes in PSH are caused by translations.

Hyperbolic manifolds show no topological spikes in PSH.

Cosmic crystallography alone cannot conclusively determine universe topology.

Abstract

If the universe is multiply connected and small the sky shows multiple images of cosmic objects, correlated by the covering group of the 3-manifold used to model it. These correlations were originally thought to manifest as spikes in pair separation histograms (PSH) built from suitable catalogues. Using probability theory we derive an expression for the expected pair separation histogram (EPSH) in a rather general topological-geometrical-observational setting. As a major consequence we show that the spikes of topological origin in PSH's are due to translations, whereas other isometries manifest as tiny deformations of the PSH corresponding to the simply connected case. This result holds for all Robertson-Walker spacetimes and gives rise to two basic corollaries: (i) that PSH's of Euclidean manifolds that have the same translations in their covering groups exhibit identical spike spectra of topological origin, making clear that even if the universe is flat the topological spikes alone are not sufficient for determining its topology; and (ii) that PSH's of hyperbolic 3-manifolds exhibit no spikes of topological origin. These corollaries ensure that cosmic crystallography, as originally formulated, is not a conclusive method for unveiling the shape of the universe. We also present a method that reduces the statistical fluctuations in PSH's built from simulated catalogues.