Cosmic crystallography in a circle

TL;DR

This paper derives the expected histogram of pair separations for objects distributed on a circle, aiding cosmic crystallography by comparing replicated and non-replicated object distributions.

Contribution

It provides a theoretical model for pair separation histograms on a circle, enhancing analysis methods in cosmic crystallography.

Findings

Derived the expected pair separation histogram for objects on a circle.

Compared histograms for replicated and non-replicated objects.

Contributed to methods for detecting cosmic topology.

Abstract

In a circle (an S^1) with circumference 1 assume m objects distributed pseudo-randomly. In the universal covering R^1 assume the objects replicated accordingly, and take an interval L>1. In this interval, make the normalized histogram of the pair separations which are not an integer. The theoretical (expected) such histogram is obtained in this report, as well as its difference to a similar histogram for non-replicated objects. The whole study is of interest for the cosmic crystallography.