# Cosmological Einstein-Skyrme solutions with non-vanishing topological charge

## TL;DR

This paper investigates time-dependent Einstein-Skyrme solutions with topological charge, revealing static spherically symmetric solutions, their stability, and the spontaneous breaking of symmetry due to gravitational coupling.

## Contribution

It provides an explicit integrable static solution and analyzes the stability and symmetry-breaking behavior of gravitating Skyrmions in a cosmological setting.

## Key findings

- Existence of a static, spherically symmetric solution described by the Ermakov-Pinney system.
- For positive cosmological constant, the solution is neutrally stable under small deformations.
- Spacetime becomes locally flat at late times despite anisotropy in the Skyrmion.

## Abstract

Time-dependent analytic solutions of the Einstein-Skyrme system --gravitating Skyrmions--, with topological charge one are analyzed in detail. In particular, the question of whether these Skyrmions reach a spherically symmetric configuration for $t\rightarrow+\infty$ is discussed. It is shown that there is a static, spherically symmetric solution described by the Ermakov-Pinney system, which is fully integrable by algebraic methods. For $\Lambda>0$ this spherically symmetric solution is found to be in a "neutral equilibrium" under small deformations, in the sense that under a small squashing it would neither blow up nor dissapear after a long time, but it would remain finite forever (plastic deformation). Thus, in a sense, the coupling with Einstein gravity spontaneously breaks the spherical symmetry of the solution. However, in spite of the lack of isotropy, for $t \to\infty$ (and $\Lambda>0$) the space time is locally flat and the anisotropy of the Skyrmion only reflects the squashing of spacetime.