4.1. Magnetic fields from primordial vorticity
The idea that primordial magnetic fields can be produced by plasma vortical motion during the radiation era of the early Universe has been first proposed by Harrison . Since this mechanism has been reviewed in several papers (see e.g. [1, 132]) we shall not discuss it in detail here. Harrison's mechanism is based on the consideration that the electron and ion rotational velocities should decrease differently in the expanding Universe in the pre-recombination era. The reason is that Thomson scattering is much more effective for electrons than ions. Therefore electrons remains tightly coupled for a longer time to the radiation and behave like relativistic matter whereas ions are already non-relativistic. It follows that during Universe expansion angular velocity decreases like a-1 for electrons and like a-2 for ions, where a is the Universe scale factor. The difference between these two velocities causes an electromotive force, hence an electric current which generate a magnetic field. Harrison  showed that if a primordial turbulence was present at the recombination this mechanism may lead to present time intergalactic magnetic fields as large as 10-8 G on a scale-length of 1 Mpc.
A problem, however, arises with this scenario. In fact, it was noted by Rees  that since rotational, or vector, density perturbations decay with cosmic expansion, in order to produce sizeable effects at recombination time this kind of fluctuations should has been dominant at the radiation-matter equality time. This seems to be incompatible with the standard scenario for galaxy formation.
Another related problem is that, in contrast to scalar or tensor perturbations, rotational perturbations cannot arise from small deviations from the isotropic Friedmann Universe near the initial singularity. This is a consequence of the Helmholtz-Kelvin circulation theorem which states that the circulation around a closed curve following the motion of matter is conserved. Such a problem, however, may be partially circumvented if collisionless matter was present during the big-bang (e.g. decoupled gravitons after the Planck era). Rebhan  showed that in this case the Helmholtz-Kelvin theorem does not apply and growing modes of vorticity on superhorizon scale can be obtained. In fact, a non perfect fluid can support anisotropic pressure which may generate nonzero vorticity even if it was zero at the singularity. It follows that the only constraint to the amount of primordial vorticity comes from the requirement that it does not produce too large anisotropies in the CMBR. Rebhan showed that this requirement implies the following upper limit to the strength of a present time intergalactic magnetic field, with coherence length L, produced by vortical plasma motion
Such a field might act as a seed for galactic dynamo.
If primordial vorticity is really not incompatible with standard cosmology, another interesting possibility is to generate primordial magnetic fields arises. It was noted by Vilenkin  that, as a consequence of parity violation in the Weinberg-Salam model of the electroweak interactions, macroscopic parity-violating currents may develop in a vortical thermal background. Vilenkin and Leavy  suggested that this currents may effectively give rise to strong magnetic fields. It was also recently noted by Brizard, Murayama and Wurtele  that in the presence of vorticity and a neutrino-antineutrino asymmetry, collective neutrino-plasma interactions may power astrophysical as well as cosmological magnetic fields.
Clearly, in order to implement these scenarios, a suitable mechanism to produce the required amount of primordial vorticity has to be found. Among other exotic possibilities, generation of vorticity and magnetic fields by the anisotropic collapse of conventional matter into the potential well of pre-existing dark matter condensations in a texture-seeded scenario of structure formation , and from rotating primordial black-holes  have been considered in the literature.
In our opinion, primordial phase transitions, may provide a more realistic source of vorticity. The generation of primeval magnetic fields during some of these transitions will be the subject of the next sections of this chapter.