Dynamical Equilibrium States of a Class of Irrotational Non-Orthogonally Transitive $G_{2}$ Cosmologies II: Models With One Hypersurface-Orthogonal Killing Vector Field

Abstract

We consider a class of inhomogeneous self-similar cosmological models in which the perfect fluid flow is tangential to the orbits of a three-parameter similarity group. We restrict the similarity group to possess both an Abelian $G_{2}$, and a single hypersurface orthogonal Killing vector field, and we restrict the fluid flow to be orthogonal to the orbits of the Abelian $G_{2}$. The temporal evolution of the models is forced to be power law, due to the similarity group, and the Einstein field equations reduce to a three-dimensional autonomous system of ordinary differential equations which is qualitatively analysed in order to determine the spatial structure of the models. The existence of two classes of well-behaved models is demonstrated. The first of these is asymptotically spatially homogeneous and matter dominated, and the second is vacuum dominated and either asymptotically spatially homogeneous or acceleration dominated, at large spatial distances.