Relativistic magnetohydrodynamics in the early Universe

We review the conservation laws of magnetohydrodynamics (MHD) in an expanding homogeneous and isotropic Universe that can be applied to the study of early Universe physics during the epoch of radiation domination. The conservation laws for a conducting perfect fluid with relativistic bulk velocities in an expanding background are presented (for the first time in their non-conservation form, i.e., as dynamical equations for the velocity and energy density fluid variables), and extending previous results that apply in the limit of subrelativistic bulk motion. Furthermore, it is shown that the subrelativistic limit presents new corrections that have not been considered in previous work. We discuss the conformal invariance of the MHD equations for a radiation-dominated fluid and different types of scaling of the fluid variables that are relevant for other equations of state when the bulk velocity is subrelativistic. In particular, we review the super-comoving coordinates that have been proposed for matter-dominated fluids and present this choice of coordinates for any equation of state. First-order fluid dynamics to include imperfect relativistic fluids and the scaling of the transport coefficients with temperature in the early Universe are presented. We review the propagation of sound waves, Alfv\'en waves, and magnetosonic waves in the early Universe plasma. The Boris correction for relativistic Alfv\'en speeds is presented and adapted for early Universe applications. This review is an extension, including new results, of part of the lectures presented at the minicourse "Simulations of Early Universe Magnetohydrodynamics" lectured by A. Roper Pol and J. Schober at EPFL, as part of the six-week program "Generation, evolution, and observations of cosmological magnetic fields" at the Bernoulli Center in May 2024.