Dust-filled axially symmetric universes with a cosmological constant

Following the recent recognition of a positive value for the vacuum energy density and the realization that a simple Kantowski-Sachs model might fit classical tests of cosmology, we study the qualitative behavior of three anisotropic and homogeneous models, Kantowski-Sachs, Bianchi type-I and Bianchi type-III universes, with dust and a cosmological constant, in order to find out which are physically permitted. We find that these models undergo isotropization up to the point that the observations will not be able to distinguish between them and the standard model, except for the Kantowski-Sachs model $({\Omega }_{k_0}<0)$ and for the Bianchi type-III model $({\Omega }_{k_0}>0)$ with ${\Omega }_{{\Lambda }_0}$ smaller than some critical value ${\Omega }_{{\Lambda }_M}.$ Even if one imposes that the Universe should be nearly isotropic since the last scattering epoch $\left(z\approx 1000\right),$ meaning that the Universe should have approximately the same Hubble parameter in all directions (considering the COBE 4-year data), there is still a large range for the matter density parameter compatible with Kantowski-Sachs and Bianchi type-III models if $|{\Omega }_0+{\Omega }_{{\Lambda }_0}-1|<~\delta ,$ for a very small $\delta .$ The Bianchi type-I model becomes exactly isotropic owing to our restrictions and we have ${\Omega }_0+{\Omega }_{{\Lambda }_0}=1\mathrm{}$ in this case. Of course, all these models approach locally an exponential expanding state provided the cosmological constant ${\Omega }_{\Lambda }>{\Omega }_{{\Lambda }_M}.$