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

We present a derivation of the time evolution equations for the energy content of nonhelical magnetic fields and the accompanying turbulent flows from first principles of incompressible 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 wavenumber scale k_I where the turbulence turn over time equals the Hubble time. At larger length scales L, i.e., smaller wavenumbers q = 2 \pi / L << k_I, independent of the assumed turbulent flow power spectrum, mode-mode coupling tends to develop a small q magnetic field tail with a Batchelor spectrum proportional to the fourth inverse power of L and therefore a scaling for the magnetic field of B ~ L^(-5/2).