Lower bound on the magnetic field strength in the hot universe

It is assumed that long range coherent magnetic fields in the universe were spontaneously generated at high temperature due to vacuum polarization of non-Abelian gauge fields, and resulted in the present intergalaxy magnetic field. The zero value of the screening mass for fields of this type was discovered recently. Here, a procedure to estimate the field strengths at different temperatures is developed and the lower bound on the magnetic field strength $B \sim 10^{14} G$, at the electroweak phase transition temperature, is derived. As a particular case, the standard model is considered. Some model dependent peculiarities of the phenomena under investigation are briefly discussed.