7.2. Big-bang nucleosynthesis epoch
Large scale magnetic fields possibly present at the BBN epoch can have an impact on the light nuclei formation. By reversing the argument, the success of BBN can be used in order to bound the magnetic energy density possibly present at the time of formation of light nuclei. Magnetic fields, at the BBN epoch may have four distinct effects:
they can enhance the rate of the Universe expansion at the corresponding epoch in a way proportional to B;
they can affect the reaction rate in a way proportional to em B;
they can affect the phase space of the electrons;
they can induce, prior to the formation of the light nuclei, matter-antimatter fluctuations.
While the first three effects are direct, the fourth effect is mainly caused by hypermagnetic fields present even before BBN.
The idea that magnetic fields may increase the Universe expansion and, consequently, affect directly the abundance of 4He (which is directly sensitive to the expansion rate) has been pointed out long ago by Greenstein  and by Matese and O'Connel . From a qualitative point of view the effects over the expansion should be leading in comparison with the effects over the rate of interactions (which are suppressed by em).
Thanks to detailed numerical simulations performed indipendently by different groups (i.e. Cheng et al. [266, 267]; Kernan, Starkman and Vachaspati [268, 269]; Grasso and Rubinstein [270, 271]), the common opinion is that the Universe expansion is the leading effect even if this conclusion has been reached only recently and not without some disagreements on the details of the calculations (see, for instance, ).
In order to prevent the Universe from expanding too fast at the BBN epoch B < 0.27 where is the energy density contributed by the standard three light neutrinos with masses much smaller than the MeV. The BBN bound is physically relevant in many situations since it is a bit more constraining than the critical density bound.
The BBN bounds discussed so far are derived assuming a stochastic magnetic field at the BBN epoch. This means that the isotropy of the geometry is not affected. However, it is not unreasonable to think of the possibility that a magnetic field with a preferred direction may break the isotropy of the background. Thus bounds on the isotropy of the expansion at the BBN epoch may be turned into bounds on the shear induced by the presence of a magnetic field (see, for instance, [272, 273] and [274, 275]).
Let us now come to the last point. The matter-antimatter fluctuations created at the electroweak epoch thanks to the presence of hypermagnetic fields may survive down to the epoch of BBN. Since the fluctuations in the baryon to entropy ratio are, in general, not positive definite, they will induce fluctuations in the baryon to photon ratio, at the BBN epoch. The possible effect of matter-antimatter fluctuations on BBN depends on the typical scale of of the baryon to entropy ratio at the electroweak epoch. Recalling that for T ~ Tc ~ 100 GeV the size of the electrowek horizon is approximately 3 cm, fluctuations whose scale is well inside the EW horizon at Tc have dissipated by the BBN time through (anti)neutron diffusion. The neutron diffusion scale at Tc is
The neutron diffusion scale at T = 1 keV is 105 m, while, today it is 10-5 pc, i.e. of the order of the astronomical unit. Matter-antimatter fluctuations smaller than 105 m annihilate before neutrino decoupling and have no effect on BBN. Two possibilities can then be envisaged. We could require that the matter-antimatter fluctuations (for scales r rn) are small. This will then imply a bound, in the (, ) plane on the strength of the hypermagnetic background. In Fig. 8 some typical baryon number fluctuations have been reported for different choices of the parameters and . In Fig. 9 such an exclusion plot is reported with the full line. With the dashed line the bound implied by the increase in the expansion rate (i.e. H < 0.2) is reported. Finally with the dot-dashed line the critical density bound is illustrated for the same hypermagnetic background.
Figure 8. The value of the baryon number fluctuations for different parameters of the hypermagnetic background and is reported.
Figure 9. The parameter space of the hypermagnetic background in the case (r, tc) < nB / s for r > rn (full line).
The second possibility is to study the effects of large matter-matter domains. These studies led to a slightly different scenario of BBN , namely BBN with matter-antimatter regions. This analysis has been performed in a series of papers by Rehm and Jedamzik [277, 282] and by Kurki-Suonio and Sihvola [278, 279] (see also [280, 281]). The idea is to discuss BBN in the presence of spherically symmetric regions of anti-matter characterized by their radius rA and by the parameter R, i.e. the matter/antimatter ratio. Furthermore, in this scenario the net baryon-to-photon ratio, , is positive definite and non zero. Antimatter domains larger than 105 m at 1 keV may survive until BBN and their dissipation has been analyzed in detail . Antimatter domains in the range
at 1 keV annihilate before BBN for temperatures between 70 keV and 1 MeV. Since the antineutrons annihilate on neutrons, the neutron to proton ratio gets smaller. As a consequence, the 4He abundance gets reduced if compared to the standard BBN scenario. The maximal scale of matter-antimatter fluctuations is determined by the constraints following from possible distortions of the CMB spectrum. The largest scale is of the order of 100 pc (today), corresponding to 1012 m at 1 keV.
Figure 10. From  the 4He yield is illustrated in the (R, rA) plane for = 6 × 10-10. As the matter/antimatter ratio decreases, we recover the standard 4He yield.
Suppose that matter-antimatter regions are present in the range of Eq. (7.7). Then the abundance of 4He get reduced. The yield of 4He are reported as a function of R, the matter-antimatter ratio and rA. Now, we do know that by adding extra-relativistic species the 4He can be increased since the Universe expansion gets larger. Then the conclusion is that BBN with matter-antimatter domains allows for a larger number of extra-relativistic species if compared to the standard BBN scenario. This observation may have implications for the upper bounds on the stochastic GW backgrounds of cosmological origin  since the extra-relativistic species present at the BBN epoch can indeed be interpreted as relic gravitons.