SU(2) cosmological solitons

Abstract

We present a class of numerical solutions to the SU(2) nonlinear $\ensuremath{\sigma}$ model coupled to the Einstein equations with a cosmological constant $\ensuremath{\Lambda}>~0$ in spherical symmetry. These solutions are characterized by the presence of a regular static region which includes a center of symmetry. They are parametrized by a dimensionless ``coupling constant'' $\ensuremath{\beta},$ 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 a positive $\ensuremath{\Lambda}$ $(\mathrm{EYM}\ensuremath{\Lambda}).$ If we choose $\ensuremath{\Lambda}$ positive and fix n, we find a family of static spacetimes with a Killing horizon for $0<~\ensuremath{\beta}<{\ensuremath{\beta}}_{\mathrm{max}}.$ As a limiting solution for $\ensuremath{\beta}={\ensuremath{\beta}}_{\mathrm{max}}$ we find a globally static spacetime with $\ensuremath{\Lambda}=0,$ 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 $\ensuremath{\beta}$ 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.''