6.3. Inside the Hubble radius
The experimental evidence concerning large-scale magnetic fields suggests that magnetic fields should have similar strength over different length-scales. In this sense inflationary mechanisms seem to provide a rather natural explanation for the largeness of the correlation scale. At the same time, the typical amplitude of the obtained seeds is, in various models, rather minute.
Primordial magnetic fields can also be generated through physical mechanisms operating inside the Hubble radius at a given physical time. Particularly interesting moments in the life of the Universe are the epoch of the electroweak phase transition (EWPT) or the QCD phase transition where magnetic fields may be generated according to different physical ideas. In the following the different proposals emerged so far will be reviewed.
6.3.1. Phase transitions
At the time of the EWPT the typical size of the Hubble radius is of the order of 3 cm and the temperature is roughly 100 GeV. Before getting into the details of the possible electroweak origin of large-scale magnetic fields it is useful to present a kinematical argument based on the evolution of the correlation scale of the magnetic fields .
Suppose that, thanks to some mechanism, sufficiently large magnetic fields compatible with the critical density bound are generated inside the Hubble radius at the electroweak epoch. Assuming that the typical coherence length of the generated magnetic fields is maximal, the present correlation scale will certainly be much larger but, unfortunately, it does not seem to be as large as the Mpc scale at the epoch of the gravitational collapse. As already mentioned, the growth of the correlation scale may be enhanced, by various processes such as inverse cascade and helical inverse cascade. For instance, if the injection spectrum generated at the electroweak epoch is Gaussian and random a simple estimate shows that the present correlation scale is of the order of 100 AU which is already larger than what one would get only from the trivial expansion of length-scales (i.e. 1 AU) . If, in a complementary perspective, the injection spectrum is strongly helical, then the typical correlation scale can even be of the order of 100 pc but still too small than the typical scale of the gravitational collapse.
Large-scale magnetic fields can be generated at the electroweak epoch in various ways. Consider first the case when the phase transition is strongly first order.
Hogan  originally suggested the idea that magnetic fields can be generated during first-order phase transitions. Since during the phase transition there are gradients in the radiation temperature, similar thermoelectric source terms of MHD equations (which were discussed in the context of the Biermann mechanism) may arise. The magnetic fields, initially concentrated on the surface of the bubbles, are expelled when bubbles collide thanks to the finite value of the conductivity.
The idea that charge separation can be generated during first-order phase transitions has been exploited in . The suggestion is again that there are baryon number gradients at the phase boundaries leading to thermoelectric terms. In the process of bubble nucleation and collisions turbulence is then produced. In spite of the fact that the produced fields are sizable, the correlation scale, as previously pointed out, is constrained to be smaller than 100 pc.
In a first-order phase transition the phases of the complex order parameter of the nucleated bubbles are not correlated. When the bubbles undergo collisions a phase gradient arises leading to a source terms for the evolution equation of the gauge fields. Kibble and Vilenkin  proposed a gauge-invariant difference between the phases of the Higgs field in the two bubbles. This idea has been investigated in the context of the Abelian-Higgs model [244, 245, 246]. The collision of two spherical bubbles in the Abelian-Higgs model leads to a magnetic field which is localized in the region at the intersection of the two bubbles. The estimate of the strength of the field depends crucially upon the velocity of the bubble wall. The extension of this idea to the case of the standard model SU(2) × U(1) has been discussed in . A relevant aspect to be mentioned is that the photon field in the broken phase of the electroweak theory should be properly defined. In  it has been shown that the definition employed in  is equivalent to the one previously discussed in .
It has been argued by Vachaspati  that magnetic fields can be generated at the electroweak time even if the phase transition is of second order. The observat on is that, provided the Higgs field fluctuates, electromagnetic fields may be produced since the gradients of the Higgs field appear in the definition of the photon field in terms of the hypercharge and SU(2) fields. Two of the arguments proposed in  have been scrutinized in subsequent discussions. The first argument is related to the averaging which should be performed in order to get to the magnetic field relevant for the MHD seeds. Enqvist and Olesen  noticed that if line averaging is relevant the obtained magnetic field is rather strong. However  (see also ) volume averaging is the one relevant for MHD seeds.
The second point is related to the fact that the discussion of  was performed in terms of gauge-dependent quantities. The problem is then to give a gauge-invariant definition of the photon field in terms of the standard model fields. As already mentioned this problem has been addressed in  and the proposed gauge-invariant is equivalent  to the one proposed in .
In  a mechanism for the generation of magnetic fields at the electroweak epoch has been proposed in connection with the AMHD equations. The idea is to study the conversion of the right-electron chemical potential into hypercharge fields. In this context the baryon asymmetry is produced at some epoch prior to the electroweak phase. The obtained magnetic fields are rather strong (i.e. || ~ 1022 G at the EW epoch) but over a small scale , i.e. 10-6Hew-1 dangerously close to the diffusivity scale.
The final point to be mentioned is that, probably, the electroweak phase transition is neither first order nor second order but it is of even higher order at least in the context of the minimal standard model [253, 254]. This conclusion has been reached using non-perturbative techniques and the relevant point, in the present context, is that for Higgs boson masses larger than mW the phase transition seems to disappear and it is possible to pass from the symmetric to the broken phase without hitting any first or second order phase transition.
There have been also ideas concerniing the a possible generation of magnetic fields at the time of the QCD phase transition occurring roughly at T ~ 140 MeV, i.e. at the moment when free quarks combine to form colorless hadrons. The mechanism here is always related to the idea of Biermann with thermoelectric currents developed at the QCD time.Since the strange quark is heavier than the up and down quarks there may be the possibility that the quarks develop a net positive charge which is compensated by the electric charge in the leptonic sector. Again, invoking the dynamics of a first-order phase transition, it is argued that the shocks affect leptons and quarks in a different way so that electric currents are developed as the bubble wall moves in the quark-gluon plasma. In  the magnetic field has been estimated to be || ~ G at the time of the QCD phase transition and with typical scale of the order of the meter at the same epoch.
In [256, 257] it has been pointed out that, probably, the magnetic fields generated at the time of QCD phase transition may be much stronger than the ones estimated in . The authors of [256, 257] argue that strong magnetic fields may be generated when the broken and symmetric phase of the theory coexist. The magnetic fields generated at the boundaries between quark and hadron phases can be, according to the authors, as large as 106 G over scales of the order of the meter at the time of the QCD phase transition.
Recently, in a series of papers, Boyanovsky, de Vega and Simionato [258, 259, 260] studied the generation of large scale magnetic fields during a phase transition taking place in the radiation dominated epoch. The setting is a theory of N charged scalar fields coupled to an Abelian gauge field that undergoes a phase transition at a critical temperature much larger than the electroweak scale. Using non-equilibrium field theory techniques the authors argue that during the scaling regime (when the back-reaction effects are dominant) large scale magnetogenesis is possible. The claim is that the minimal dynamo requirement of Eq. (6.1) is achievable at the electroweak scale. Furthermore, much larger magnetic fields can be expected if the scaling regime can be extended below the QCD scale.