Dynamical behavior and Jacobi stability analysis of wound strings

TL;DR

This paper investigates the dynamical stability of wound cosmic strings with internal windings using numerical solutions and Jacobi stability analysis, revealing that internal space topology influences microscopic behavior but not macroscopic dynamics.

Contribution

It introduces a combined numerical and KCC theory approach to analyze the stability of wound strings in simple internal geometries, providing new insights into their long-term behavior.

Findings

Internal space topology affects microscopic string behavior.

Macroscopic string dynamics are insensitive to internal windings.

Stability bounds depend on internal space curvature and topology.

Abstract

We numerically solve the equations of motion (EOM) for two models of circular cosmic string loops with windings in a simply connected internal space. Since the windings cannot be topologically stabilized, stability must be achieved (if at all) dynamically. As toy models for realistic compactifications, we consider windings on a small section of $\mathbb{R}^2$, which is valid as an approximation to any simply connected internal manifold if the winding radius is sufficiently small, and windings on an $S^2$ of constant radius $\mathcal{R}$. We then use Kosambi-Cartan-Chern (KCC) theory to analyze the Jacobi stability of the string equations and determine bounds on the physical parameters that ensure dynamical stability of the windings. We find that, for the same initial conditions, the curvature and topology of the internal space have nontrivial effects on the microscopic behavior of the string in the higher dimensions, but that the macroscopic behavior is remarkably insensitive to the details of the motion in the compact space. This suggests that higher-dimensional signatures may be extremely difficult to detect in the effective $(3+1)$-dimensional dynamics of strings compactified on an internal space, even if configurations with nontrivial windings persist over long time periods.