Models of the Universe : An analysis of the asymptotic behaviour of non-linear dynamical systems

Sammanfattning: In this thesis we present some relevant theory, and then we rigorously investigate the existence intervals and the asymptotic behaviors of three cosmological models. The first model we investigate is based on the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which is consistent with the cosmological principle. This is a common assumption which asserts that the universe is spatially homogeneous and isotropic. The second and third models are of Bianchi type I and Bianchi type II respectively, which are both anisotropic, but spatially homogeneous models. For all models we find that the existence interval is (0,∞), meaning that they all predict an origin of the universe for some past time, while guaranteeing the existence of the universe for all future times. Furthermore we prove that in all models the universe expands exponentially for times far in the future and that the non-isotropic solutions tend towards isotropic solutions forward in time. Differences were found in the asymptotic behavior backward in time, as the FLRW-model was shown to behave like the square root for times close to t=0, while the anisotropy in the Bianchi type I and Bianchi type II models became unbounded close to t=0. It was found that there were no differences in the asymptotic behavior between the two anisotropic models. Finally we investigated some interesting aspects specific for each model. For instance the behaviour of light-like curves were analysed in the FLRW-solutions and vacuum solutions were investigated in the Bianchi type I and Bianchi type II models.