Time Evolution of the Large-Scale Tail of Non-Helical Primordial Magnetic Fields with Back-Reaction of the Turbulent Medium

We present a derivation of the time evolution equations for the energy content of non-helical magnetic fields and the accompanying turbulent flows from first principles of magnetohydrodynamics in the general framework of homogeneous and isotropic turbulence. This is then applied to the Early Universe, i.e. the evolution of primordial magnetic fields. Numerically integrating the equations we find that most of the energy is concentrated at an integral wave number scale k_I where the turbulence turn over time equals the Hubble time. At larger length scales L, or smaller wavenumbers q = 2 π / L << k_I, independent of the assumed turbulent flow power spectrum, mode-mode coupling tends to develop a small q magnetic field tail from zero with a violet noise slope proportional to the fourth inverse power of L and therefore a scaling for the magnetic field of B ~ L^(-5/2).