Cosmological analogues of the Bartnik-McKinnon solutions

We present a numerical classification of the spherically symmetric, static solutions to the Einstein-Yang-Mills equations with a cosmological constant $\ensuremath{\Lambda}$. We find three qualitatively different classes of configurations, where the solutions in each class are characterized by the value of $\ensuremath{\Lambda}$ and the number of nodes, $n$, of the Yang-Mills amplitude. For sufficiently small, positive values of the cosmological constant, $\ensuremath{\Lambda}<{\ensuremath{\Lambda}}_{\mathrm{crit}}(n)$, the solutions generalize the Bartnik-McKinnon solitons, which are now surrounded by a cosmological horizon and approach the de Sitter geometry in the asymptotic region. For a discrete set of values ${\ensuremath{\Lambda}}_{\mathrm{reg}}(n)>{\ensuremath{\Lambda}}_{\mathrm{crit}}(n)$, the solutions are topologically three-spheres, the ground state ($n=1$) being the Einstein universe. In the intermediate region, that is, for ${\ensuremath{\Lambda}}_{\mathrm{crit}}(n)<\ensuremath{\Lambda}<{\ensuremath{\Lambda}}_{\mathrm{reg}}(n)$, there exists a discrete family of global solutions with an horizon and "finite size."