Comparing different closure relations for cosmic ray hydrodynamics

Cosmic ray (CR) hydrodynamics is a (re-)emerging field of high interest due to the importance of CRs for the dynamical evolution of the interstellar, the circumgalactic, and the intracluster medium. In these environments, CRs with GeV energies can influence large-scale dynamics by regulating star formation, driving galactic winds or by altering the pressure balance of galactic halos. Recent efforts have moved the focus of the community from a one-moment description of CR transport towards a two-moment model as this allows for a more accurate description of the microphysics impacting the CR population. Like all hydrodynamical theories, these two-moment methods require a closure relation for a consistent and closed set of evolution equations. The goal of this paper is to quantify the impact of different closure relations on the resulting solutions. To this end, we review the common P1 and M1 closure relations, derive a new four-moment H1 description for CR transport and describe how to incorporate CR scattering by Alfv\'en waves into these three hydrodynamical models. While there are significant differences in the transport properties of radiation in the P1 and M1 approximations in comparison to more accurate radiative transfer simulations using the discrete ordinates approximation, we only find small differences between the three hydrodynamical CR transport models in the free streaming limit when we neglect CR scattering. Most importantly, for realistic applications in the interstellar, circumgalactic or intracluster medium where CR scattering is frequent, these differences vanish and all presented hydrodynamical models produce the same results.