Evolution of the large-scale tail of primordial magnetic fields

Cosmic magnetic fields may be generated during early cosmic phase transition, such as the QCD or electroweak transitions. The magnitude of the remainder of such fields at the present epoch crucially depends on the exponent n of their (initially super-Hubble) large-scale tail, i.e., B-lambda similar to lambda(-n). It has been claimed that causality requires n = 5/2, contrary to much earlier claims of n = 3/2. Here we analyze this question in detail. First, we note that contrary to current belief, the large-scale magnetic field tail is not established at the phase transition itself, but rather continuously evolves up to the present epoch. Neglecting turbulent flows we find n = 7/2, i.e., very strongly suppressed large-scale fields. However, in the inevitable presence of turbulent flows we find that the large-scale magnetic field tail has sufficient time to evolve to that of the fluid turbulence. For white noise fluid turbulence this yields n = 3/2 up to a certain scale and n = 5/2 beyond for the magnetic field spectrum. This picture is also not changed when primordial viscosity and fluid flow dissipation is taken into account. Appreciable primordial magnetic fields originating from cosmic phase transitions thus seem possible.