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b"We take a multi-faceted approach to study the relativistic cosmic ray (CR) proton population in galaxy clusters. CR protons may be accelerated by structure formation shock waves, injected from radio galaxies into the intra-cluster medium, or result from supernova driven galactic winds. This\\nthesis addresses the following questions: do CR protons exist in galaxy clusters? What is the dynamic and cosmological impact of CRs? How can we observe them? How can we describe CRs and their interactions? The first major\\npart of this thesis investigates the question of the dynamic influence of CRs on the intra-cluster medium and searches for unbiased tracers of their existence using multi-frequency observational results. To this end, I develop an analytical framework to describe the hadronic interactions of CR protons with the ambient thermal plasma. In the second part, a description of CR gas for cosmological applications is presented that is especially suited for\\nhydrodynamical simulations. During the course of this work, I focus on developing a formalism for instantaneously identifying and estimating the strength of structure formation shocks during cosmological simulations to\\naccelerate CRs through diffusive shock acceleration.\\n\\nSince the energetically dominant CR population is trapped by cluster magnetic fields, it can only be observed indirectly through non-thermal radiative\\nprocesses. CR protons interact hadronically with the ambient plasma and produce mainly neutral and charged pions that successively decay into gamma-rays, secondary electrons, and neutrinos. I develop an analytic formalism which\\ndescribes the induced radio synchrotron, inverse Compton, and gamma-ray emission. Comparing the expected gamma-ray flux to the upper limits obtained by the gamma-ray observatory EGRET, I am able to constrain the CR proton energy density in nearby cooling core clusters to < 20% relative to the thermal energy density. In this context, I study the hypothesis that the diffuse radio synchrotron emission of galaxy clusters is produced by hadronically originating relativistic electrons and I develop a non-parametric criterion to obtain the minimum energy state for an observed radio synchrotron emission: the excellent\\nagreement between the observed and theoretically expected radio surface brightness profile of the Perseus mini-halo and the small amount of energy density in CR protons needed to account for the observed radio emission makes this hadronic model an attractive explanation of radio mini-halos found in cooling core clusters. To explain the giant radio halo of Coma within the hadronic model of secondary electrons, the CR proton-to-thermal energy density profile has to increase radially up to moderate CR energy densities.\\nCosmological simulations that self-consistently follow CR acceleration at shock waves predict such an energy density profile: strong shock waves, that occur predominantly in low density regions, are able to efficiently accelerate\\nhigh-energetic CRs, whereas weak central flow shocks inject only a low-energetic CR population which is strongly diminished by Coulomb interactions. This implies that the dynamic importance of the shock-injected CR energy density is largest in the low-density halo infall regions, but is\\nless important for the weaker shocks occurring in central high-density cluster regions.\\n\\nAs an extension of this work, I propose a new method in order to elucidate the content of the radio plasma bubbles located at cool cores of galaxy clusters. Using the Sunyaev-Zel'dovich (SZ) effect, the Atacama Large Millimeter Array and the Green Bank Telescope should be able to infer the dynamically dominant CR component of the plasma bubbles in suitable galaxy clusters within short observation times. Future high-sensitivity multi-frequency SZ observations\\nwill be able to infer the energy spectrum of the dynamically dominant electron population. This knowledge can yield indirect indications for an underlying composition of relativistic outflows of radio galaxies because plasma bubbles represent the relic fluid of jets.\\n\\nIn the second major part of my thesis, I address the problem of constructing an accurate and self-consistent model for the description of CRs that aims at studying the dynamic influence of CRs on structure formation and galaxy\\nevolution. This will not only allow the production of realistic non-thermal emission signatures of galaxies and clusters of galaxies, but also allow in-vivo studies of dynamic effects driven by relativistic particles and the\\nstar formation history. The developed model self-consistently traces relativistic protons originating from various kinds of sources, such as structure formation shock waves and supernovae driven galactic winds, and also\\naccounts for dissipative processes in the relativistic gas component. To this end, I develop a formalism for the identification and accurate estimation of the strength of structure formation shocks during cosmological smoothed\\nparticle hydrodynamics simulations. Shocks not only play a decisive role for the thermalization of gas in virializing structures but also for the acceleration of CRs through diffusive shock acceleration. The formalism is\\napplicable both to ordinary non-relativistic thermal gas and to plasmas composed of CRs and thermal gas. I apply these methods to studying the properties of structure formation shocks in high-resolution hydrodynamic simulations of the LambdaCDM model and find that most of the energy is\\ndissipated in weak internal shocks which are predominantly central flow shocks or merger shock waves traversing halo centers. Collapsed cosmological structures are surrounded by external shocks with a much higher Mach number, but they play only a minor role in the energy balance of thermalization. I show that after the epoch of cosmic reionization, the Mach number distribution is significantly modified by an efficient suppression of strong external shock waves due to the associated increase of the sound speed of the diffuse gas."