Cosmology investigates questions about the origin, evolution, and future development of the universe, drawing on various scientific disciplines of physics. When focusing on aspects governed purely by gravitation, Einstein’s equations are currently understood to describe the Universe's behavior accurately. These equations allow for the consideration of mathematically rigorous universe models, especially those with low particle densities. Among these systems, the Einstein-Vlasov system is the best understood mathematically, and it is the focus of this research program.
Associate Professor David Fajman from the research group Gravitational Physics is leading this project, bringing his expertise in the fields of Geometric Analysis and Mathematical Physics on problems in General Relativity and Cosmology. As an expert in the Einstein-Vlasov system, which poses several mathematically challenging problems, he will spearhead this scientific program in Mathematical Cosmology.
The discipline of Mathematical Cosmology aims to make rigorous statements about the solutions of Einstein-matter systems by studying Einstein’s equations with modern mathematical methods to understand possible scenarios for cosmological evolution. Stability results of certain spacetimes are particularly relevant, as they confirm that these models may describe the generic behavior of spacetime.
This research project focuses on a specific aspect of this broader program. Fundamental results in Mathematical Cosmology generally concern models of spacetime whose long-term behavior resembles that of pure vacuum equations, implying that matter does not fundamentally influence the geometry of spacetime on large scales. However, there are solutions to Einstein’s equations coupled with realistic matter models where long-term behavior is matter-dominated.
For the Einstein-Vlasov system, two classes of matter-dominated models exist: the Einstein-deSitter solution and the Rendall class. Both describe expanding universes. The research program aims to investigate the dynamics of cosmology near these models. The project will employ numerical simulations in a symmetry-reduced setting and then use analytical methods to rigorously establish the numerical findings. Subsequently, utilizing the numerical infrastructure to simulate these systems for more general initial data to obtain information on their long-term behavior. These investigations address the case of large data, i.e., highly inhomogeneous perturbations of spacetime, which remains largely unexplored analytically.
Finally, the project will study singularity formation in these systems, which is relevant for understanding big-bang formation. This includes extending the system to incorporate electromagnetism and particle collisions.
