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Large-scale magnetic fields from hydromagnetic turbulence in the very early universe

We investigate hydromagnetic turbulence of primordial magnetic fields using magnetohydrodynamics (MHD) in an expanding universe. We present the basic, covariant MHD equations, find solutions for MHD waves in the early universe, and investigate the equations numerically for random magnetic fields in two spatial dimensions. We find the formation of magnetic structures at larger and larger scales as time goes on. In three dimensions we use a cascade (shell) model, that has been rather successful in the study of certain aspects of hydrodynamic turbulence. Using such a model we find that after ${\cal O}(10^9)$ times the initial time the scale of the magnetic field fluctuation (in the comoving frame) has increased by 4-5 orders of magnitude as a consequence of an inverse cascade effect (i.e. transfer of energy from smaller to larger scales). Thus {\it at large scales} primordial magnetic fields are considerably stronger than expected from considerations which do not take into account the effects of MHD turbulence.