Amplification of hypercharge electromagnetic fields by a cosmological pseudoscalar

If, in addition to the standard model fields, a new pseudoscalar field exists and couples to hypercharge topological number density, it can exponentially amplify hyperelectric and hypermagnetic fields in the symmetric phase of the electroweak plasma, while coherently rolling or oscillating. We present the equations describing the coupled system of a pseudoscalar field and hypercharge electromagnetic fields in the electroweak plasma at temperatures above the electroweak phase transition, discuss approximations to the equations, and their validity. We then solve the approximate equations using assorted analytical and numerical methods, and determine the parameters for which hypercharge electromagnetic fields can be exponentially amplified.