SU(2) Cosmological Solitons

TL;DR

This paper numerically constructs and analyzes SU(2) cosmological solitons in Einstein's gravity with a positive cosmological constant, revealing static regions, horizons, and black hole structures depending on parameters.

Contribution

It introduces a new class of SU(2) solutions with a static core in cosmological settings, extending understanding of nonlinear sigma models coupled to gravity.

Findings

Existence of static solutions with a regular center and horizons.

Identification of a maximum coupling constant where solutions become globally static.

Discovery of solutions resembling Einstein static universe and eternal cosmological black holes.

Abstract

We present a class of numerical solutions to the SU(2) nonlinear $\sigma$-model coupled to the Einstein equations with cosmological constant $\Lambda\geq 0$ in spherical symmetry. These solutions are characterized by the presence of a regular static region which includes a center of symmetry. They are parameterized by a dimensionless ``coupling constant'' $\beta$, the sign of the cosmological constant, and an integer ``excitation number'' $n$. The phenomenology we find is compared to the corresponding solutions found for the Einstein-Yang-Mills (EYM) equations with positive $\Lambda$ (EYM$\Lambda$). If we choose $\Lambda$ positive and fix $n$, we find a family of static spacetimes with a Killing horizon for $0 \leq \beta < \beta_{max}$. As a limiting solution for $\beta = \beta_{max}$ we find a {\em globally} static spacetime with $\Lambda=0$, the lowest excitation being the Einstein static universe. To interpret the physical significance of the Killing horizon in the cosmological context, we apply the concept of a trapping horizon as formulated by Hayward. For small values of $\beta$ an asymptotically de Sitter dynamic region contains the static region within a Killing horizon of cosmological type. For strong coupling the static region contains an ``eternal cosmological black hole''.