Cosmological Analogues of the Bartnik--McKinnon Solutions

Abstract

We present a numerical classification of the spherically symmetric, static solutions to the Einstein--Yang--Mills equations with cosmological constant $\Lambda$. We find three qualitatively different classes of configurations, where the solutions in each class are characterized by the value of $\Lambda$ and the number of nodes, $n$, of the Yang--Mills amplitude. For sufficiently small, positive values of the cosmological constant, $\Lambda <\text{Llow}(n)$, the solutions generalize the Bartnik--McKinnon solitons, which are now surrounded by a cosmological horizon and approach the deSitter geometry in the asymptotic region. For a discrete set of values ${\Lambda }_{\mathrm{reg}}(n)>{\Lambda }_{\mathrm{crit}}(n)$, the solutions are topologically $3$--spheres, the ground state $(n=1)$ being the Einstein Universe. In the intermediate region, that is for $\text{Llow}(n)<\Lambda <\text{Lhig}(n)$, there exists a discrete family of global solutions with horizon and ``finite size''.

Comments: 16 pages, LaTeX, 9 Postscript figures, uses epsf.sty