It is now well accepted that the observed structure of our universe is best reproduced in the presence of cold dark matter and dark energy. Within this framework of a LCDM cosmology, structures form in a hierarchical bottom up fashion. The size, range and completeness of observational data increased recently using the latest generation of astronomical instruments, and opened the so-called period of precision cosmology. Hence, the basic parameters describing such a LCDM universe can be in principle determined with a precision of ten per cent or better. In this hierarchical picture of structure formation, small objects collapse first and then merge in a complex manner to form larger and larger structures. To a first approximation, one can study the formation of cosmic structures using N-body simulations which follow the gravitational collapse of collision-less particles. However, with the possible exception of gravitational lensing, observations mainly reflect the state of the ordinary (baryonic) matter. Therefore, their interpretation in the framework of cosmic evolution requires that we understand the complex, non-gravitational, physical processes which determine the evolution of the cosmic baryons. The evolution of each of the underlying building blocks within the hierarchical formation scenario will contribute to the state and composition of the intra-cluster medium by being seed for metal pollution, magnetic fields, high energy particles and energy. Depending on their origin, these components will be blown out by jets, winds or ram pressure effects and finally mixed with the surrounding intra-cluster medium. Some of these effects will be naturally followed within hydrodynamic simulations (like ram pressure effects), others have to be improved via effective models (like star formation and related feedback and chemical pollution by supernova). Further components like magnetic fields and high energy particles need additional modeling of their injection processes and evolution, and also have to be self consistently coupled with hydrodynamics. Here, cosmological simulations are a valuable probe of our understanding of the underlying physical processes and harbor enormous potential for the interpretation of observational data.
Magnetic fields on large scales are observed within galaxies, along galactic outflows and even within the intra-cluster medium (ICM), filling the space inside galaxy clusters. It is expected that also the diffuse baryons within filaments and maybe even in voids -- which are the largest structures observed in the Universe -- are magnetized. However, despite their importance for various transport processes, our understanding of the origin, evolution and interplay of magnetic fields with structure formation is still very limited. Therefore, numerical experiments are a key tool for understanding the build up of large scale magnetic fields. They also enable the prediction of the expected properties of the magnetic field among different classes of objects (voids, filaments or galaxy clusters). Numerical simulations of the formation of cosmological structures and predictions of their magnetic field properties allow to put constraints on different models for the origin of cosmic magnetism. This also puts constraints onto the various mechanisms prop
osed for the magnetic seed fields, and relates the evolution of the magnetic field structure to the dynamical state of the cosmic structures. Particularly in case of
galaxy clusters this allows to put additional constraints on the presence of cosmic ray protons by comparing in detail the induced radio emission in such simulations with observations. Additionally, such simulations can be used to constrain the transport and the deflection of ultra high energy particles (e.g. protons, photons) within the large scale structures of the universe.