Relativistic fluids in cosmology

In cosmology there is a fundamental desire to understand how the universe reached its current state in the course of its evolution and what we can learn from current observations about its development. A basic questions in this direction is: How did the present structures and arrangements of matter on large scales emerge? This area is usually referred to as structure formation. From an initial state of high density shortly after the Big Bang the universe expanded to its present state, where matter is mainly concentrated in the form of galaxies. Which mechanisms precisely led to the formation of those structure and how they occurred are questions of cosmological research. The starting point of the present research project is to investigate how the pure interaction of general relativistic gravitation in the form of Einsteins theory with matter moving in the expanding spacetime geometry, can explain this formation of structures. The methods we use are the mathematical-analytical study of Einsteins equations coupled to the Euler equations for relativistic fluids, which are a model for matter on large scales. A fundamental feature of fluids is the formation of shocks in finite time. This means that from arbitrarily small initial fluctuations in a fluid arbitrary large fluctuations can form, which become infinite in a suitable sense in finite time. This behaviour is used to describe the process of structure formation. From a relatively homogeneous initial distribution of matter, regions with highly concentrated matter formed in finite time, for instance pre-stadia of present day galaxies. Studying this phenomenon carefully one finds out that there is a relation between the speed of expansion and this type of structure formation in an expanding universe. The faster a universe expands the stronger is the suppression of shock formation in the fluid contained in that spacetime. A consequence for our universe is then that the epoch of structure formation had to coincide with a phase of sufficiently slow expansion to allow for the effect of shock formation in fluids. From a precise analysis of the expanding spacetime geometry and the matter contained in it we will determine in the project at hand what conditions on the speed of expansion are sufficient for structure formation to occur. This research has the aim to lay profound theoretical foundations for the understanding of this critical phase of the development of our universe and will in this form provide an important contribution for the complete picture of cosmology.

The temporal evolution of the structure of the universe is determined on large scales by the gravitational interaction of accumulations of matter and the geometry of space-time. This behavior is precisely described by the Einstein equations for gravity and, in the case of matter, by the Euler equations. In general, so-called shocks can form in fluids, which can be an indication of extreme events in the evolution of the universe and thus explain how matter structures have formed in the course of its evolution. However, the formation of such shocks, and therefore of structures within the matter distributions, is influenced by the expansion speed of the universe. If the universe expands fast enough, shocks do not occur. The central question that could be answered as part of the Relativistic Fluids in Cosmology project is that of the minimum necessary propagation speed that suppresses shock formation and the conditions under which structures can form in matter. Through a combination of analytical and numerical methods, a precise quantitative relationship between the speed of sound of the fluid and the speed of expansion of the universe was discovered, which marks the phase transition between shock formation and its suppression.