Dynamics of matter in the decelerated epoch

In the early phases of cosmological evolution, the Universe expanded significantly more slowly than it does at present. The aim of this project is to understand how matter behaves during such a phase of slow expansion. This dynamics is particularly crucial for the emergence of structures in the present-day Universe. Specifically, we seek to determine the conditions under which matter homogenizes and when, by contrast, it condenses into structures. This is relevant for gaining an analytical understanding of the formation of cosmic structures such as galaxies and galaxy clusters, and for elucidating the mechanisms by which this process occurs. The starting point of our investigation is a recently discovered critical phenomenon in fluids that, depending on the expansion rate of the Universe, either homogenize or form shocks and thus structures. Using a hybrid approach that combines rigorous mathematical analysis with high-precision numerical simulations, we will investigate these phenomena under general conditions, with the goal of understanding the consequences of these effects for realistic cosmological scenarios. This hybrid methodology particularly reveals nonlinear effects that remain hidden to purely linear methods and provides a comprehensive picture of the actual behavior of matter. In particular, we will examine the conditions under which generic fluids in slowly expanding phases homogenize or form shocks. To this end, we will consider equations of state that have not previously been studied in the context of this phase transition between stability and instability, such as polytropic fluids, in order to understand how material properties influence the transition. In addition, we will analyze the influence of spacetime curvature on stability and shock formation in fluids, as this is especially relevant in the relativistic regime for applications in cosmology. Moreover, we will consider fluids with additional interactions, especially gravity and electromagnetism. From a heuristic perspective, gravity favors shock formation, while electric repulsion counteracts it. Because the fluid model breaks down at shocks, we will also consider the Vlasov model, in which the matter itself is collisionless. This model is used to describe matter on large scales, such as in galaxies or galaxy clusters. We will compare the dynamics of Vlasov matter with those of fluids to better understand how interactions within the fluid affect structure formation. Our goal is an integrated picture of matter dynamics that shows when matter in a slowly expanding Universe remains homogeneous and when it forms structures. This will fundamentally contribute to developing a comprehensive, definitive understanding of early structure formation in the cosmos.