Speculations on primordial magnetic helicity

We speculate that above or just below the electroweak phase transition magnetic fields are generated which have a net helicity (otherwise said, a Chern-Simons term) of the order of the magnitude NB+NL, where NB,L is the baryon or lepton number today. (To be more precise requires much more knowledge of B-, L-generating mechanisms than we currently have.) Electromagnetic helicity generation is associated (indirectly) with the generation of an electroweak Chern-Simons number through $B+L$ anomalies. This helicity, which in the early universe is some 30 orders of magnitude greater than what would be expected from fluctuations alone in the absence of $B+L$ violation, should be reasonably well conserved through the evolution of the universe to around the times of matter dominance and decoupling, because the early universe is an excellent conductor. Possible consequences include early structure formation, macroscopic manifestations of CP violation in the cosmic magnetic field (measurable at least in principle, if not in practice), and an inverse-cascade dynamo mechanism in which magnetic fields and helicity are unstable to transfer to larger and larger spatial scales. We give a quasilinear treatment of the general-relativistic magnetohydrodynamics inverse cascade instability, finding substantial growth for helicity of the assumed magnitude out to scales ~lMε^-1, where {ε} is roughly the $B+L$ to photon ratio and lM is the magnetic correlation length. We also elaborate further on an earlier proposal of the author for the generation of magnetic fields above the EW (electroweak) phase transition.