Gravitating diBaryons in the $SU(3)$-Skyrme-Einstein theory, their flat limit and a novel finite-density transition

We construct the first analytic self-gravitating skyrmions with higher baryon charge in four dimensions for the $SU(3)$-Skyrme-Einstein-$\Lambda$ theory by combining the generalized hedgehog ansatz with the approach developed by Balachandran et al. to describe the first (numerical) example of a non-embedded solution. These are genuine $SU(3)$ analytic solutions instead of trivial embeddings of $SU(2)$ into $SU(3)$. The geometry is that of a Bianchi IX universe. The Skyrme ansatz is chosen in such a way that the Skyrme field equations are identically satisfied in the sector with baryon charge 4 (we call these configurations diBaryons anyway to emphasize the importance of the non-embedded ansatz). The field equations reduce to a dynamical system for the three Bianchi IX scale factors. Particular solutions are explicitly analyzed. Traversable wormholes with NUT-AdS asymptotics supported by a topologically non-trivial $SU(3)$-sigma soliton are also constructed. The self-gravitating solutions admit also a suitable flat limit giving rise to Skyrmions of charge 4 confined in a box of finite volume maintaining the integrability of the $SU(3)$ Skyrme field equations. This formalism discloses a novel transition at finite baryon density arising from the competition between embedded and non-embedded solutions in which the non-embedded solutions prevail at high density while are suppressed at low densities.