Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant

We examine the late-time behavior of intially expanding homogeneous cosmological models satisfying Einstein's equation with a positive cosmological constant $\Lambda$. It is shown that such models of all Bianchi types except IX exponentially evolve toward the de Sitter solution, with time scale ${\left(\frac{3}{\Lambda }\right)}^{\frac{1}{2}}$. The behavior of Bianchi type-IX universes is similar, provided that $\Lambda$ is sufficiently large compared with spatial-curvature terms. Thus, a positive cosmological constant provides an effective means of isotropizing homogeneous universes.