The geodesic rule and the spectrum of the vacuum

TL;DR

This paper examines the geodesic rule's validity in predicting topological defect densities during phase transitions, deriving a formula for string defect estimates and analyzing its accuracy for spherical vacua.

Contribution

It introduces a new formula for estimating the number of string-like defects formed, based on the geodesic rule, and discusses its application and limitations for spherical vacuum spaces.

Findings

Derived a formula for defect density estimation.

Applied the formula to spherical vacua.

Discussed deviations from expected defect counts.

Abstract

We analyze the consequences of a recent argument justifying the validity of the "geodesic rule" which can be used to determine the density of global topological defects. We derive a formula that provides a rough estimate of the number of string-like defects formed in a phase transition. We apply this formula to vacua which are spheres. We provide some reasons for the deviation of our predictions from the corresponding accepted values.