We investigate the impact of cosmic rays (CR) and different modes of CR transport on the properties of Milky Way-mass galaxies in cosmological magneto-hydrodynamical simulations in the context of the AURIGA project. We systematically study how advection, anisotropic diffusion and additional Alfven-wave cooling affect the galactic disc and the circum-galactic medium (CGM). Global properties such as stellar mass and star formation rate vary little between simulations with and without various CR transport physics, whereas structural properties such as disc sizes, CGM densities or temperatures can be strongly affected. In our simulations, CRs affect the accretion of gas onto galaxies by modifying the CGM flow structure. This alters the angular momentum distribution which manifests itself as a difference in stellar and gaseous disc size. The strength of this effect depends on the CR transport model: CR advection results in the most compact discs while the Alfven-wave model resembles more the AURIGA model. The advection and diffusion models exhibit large ($rsim50$ kpc) CR pressure-dominated gas haloes causing a smoother and partly cooler CGM. The additional CR pressure smoothes small-scale density peaks and compensates for the missing thermal pressure support at lower CGM temperatures. In contrast, the Alfven-wave model is only CR pressure dominated at the disc-halo interface and only in this model the gamma-ray emission from hadronic interactions agrees with observations. In contrast to previous findings, we conclude that details of CR transport are critical for accurately predicting the impact of CR feedback on galaxy formation.
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