Matter dominated cosmology

Cosmology investigates questions about the origin, the evolution and the future development of the universe. Several scientific disciplines of physics contribute to these investigations. Restricting oneself to those aspects of cosmology that are purely governed by gravitation, then Einsteins equations are, according to current understanding, the system that describes the behaviour of the universe accurately. They provide the opportunity to consider models of the universe mathematically rigorous including those types of matter models that behave realistically in the case of low particle densities. Among those types of systems the Einstein-Vlasov system is the one that is mathematically best understood. It is this system we investigate rigorously in this research program. It is the aim of the discipline of Mathematical Cosmology to make rigorous statements about the structure of solutions of Einstein-matter systems by studying the Einstein equations using modern mathematical methods and to understand possible scenarios for the cosmological evolution. Particularly relevant are stability results of certain spacetimes, which confirm that those models may describe the generic behaviour of spacetime. In the envisioned research project we focus on one particular aspect of this general program, which we motivate in the following. Fundamental results in the area of Mathematical Cosmology concern exclusively those models of spacetime, whose long-time behaviour is similar to those of the pure vacuum equations. Consequently, in those models, matter has no fundamental influence on the geometry of spacetime on large scales. However, there exist certain solutions of Einsteins equations coupled to realistic matter models, whose long-time behaviour is determined by the matter. We call those solutions matter dominated. For the Einstein-Vlasov system there are two classes of such matter dominated models, the Einstein- deSitter solution and the Rendall class. Both solutions described expanding universes. It is the aim of the research program to investigate the asymptotic behaviour near those models. We study those system by different methods starting with numerical simulations in a symmetry reduced setting. Starting from those results we apply a series of analytical methods to establish the numerical findings rigorously. We will then use the numerical infrastructure to simulate those systems for more general classes of initial data and to obtain information on their long-time behaviour. Those investigations concern the case of large data, i.e. highly inhomogeneous perturbations of spacetime. This regime of Einsteins equations is also almost completely non-understood from an analytical point of view. Finally, we will study singularity formation of this systems, which is relevant for big-bang formation in those models, where we include extensions of our system to include electromagnetism and collisions of particles.