It is well known that cosmic rays (CRs) contribute significantly to the pressure of the interstellar medium in our own Galaxy, suggesting that they may play an important role in regulating star formation during the formation and evolution of galaxies. We here discuss a novel numerical treatment of the physics of CRs and its implementation in the parallel smoothed particle hydrodynamics code GADGET-2. In our methodology, the non-thermal CR population of each gaseous fluid element is approximated by a simple power law spectrum in particle momentum, characterized by an amplitude, a cut-off, and a fixed slope. Adiabatic compression, and a number of physical source and sink terms are modelled which modify the CR pressure of each particle. The most important sources considered are injection by supernovae and diffusive shock acceleration, while the primary sinks are thermalization by Coulomb interactions, and catastrophic losses by hadronic interactions. We also include diffusion of CRs. Our scheme allows us to carry out the first cosmological structure formation simulations that self-consistently account for CR physics. In simulations of isolated galaxies, we find that CRs can significantly reduce the star formation efficiencies of small galaxies, with virial velocities below ~80 km/s, an effect that becomes progressively stronger towards low mass scales. In cosmological simulations at high redshift, the total mass-to-light ratio of small halos and the faint-end of the luminosity function are strongly affected. When CR acceleration in shocks is followed as well, up to ~40% of the energy dissipated at structure formation shocks can appear as CR pressure at z~3-6, but this fraction drops to ~10% at low redshifts when the shock distribution becomes increasingly dominated by lower Mach numbers. (abridged)
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