Damping of cosmic magnetic fields

We examine the evolution of magnetic fields in an expanding fluid composed of matter and radiation with particular interest in the evolution of cosmic magnetic fields. We derive the propagation velocities and damping rates for relativistic and non-relativistic fast and slow magnetosonic and Alfv\'en waves in the presence of viscous and heat conducting processes. The analysis covers all magnetohydrodynamics modes in the radiation diffusion and the free-streaming regimes. When our results are applied to the evolution of magnetic fields in the early universe, we find that cosmic magnetic fields are damped from prior to the epoch of neutrino decoupling up to recombination. Similar to the case of sound waves propagating in a demagnetized plasma, fast magnetosonic waves are damped by radiation diffusion on all scales smaller than the radiation diffusion length. The characteristic damping scales are the horizon scale at neutrino decoupling (M_10^-4M_ in baryons) and the Silk mass at recombination (M_10^13M_ in baryons). In contrast, the oscillations of slow magnetosonic and Alfv\'en waves get overdamped in the radiation diffusion regime, resulting in frozen-in magnetic field perturbations. Further damping of these perturbations is possible only if before recombination the wave enters a regime in which radiation free-streams on the scale of the perturbation. The maximum damping scale of slow magnetosonic and Alfv\'en modes is always smaller than or equal to the damping scale of fast magnetosonic waves, and depends on the magnetic field strength and its direction relative to the wave vector. Our findings have multifold implications for cosmology. The dissipation of magnetic field energy into heat during the epoch of neutrino decoupling ensures that most magnetic field configurations generated in the very early universe satisfy big bang nucleosynthesis constraints. Further dissipation before recombination constrains models in which primordial magnetic fields give rise to galactic magnetic fields or density perturbations. Finally, the survival of Alfv\'en and slow magnetosonic modes on scales well below the Silk mass may be of significance for the formation of structure on small scales.