In this review we have analyzed a large variety of aspects of magnetic fields in the early Universe. Our exposition followed an inverse-chronological order. In the first part of Chap. 1 we discussed what observations tell us about recent time fields and their evolution in galaxies and clusters of galaxies. As we have seen, a final answer about the origin of these fields is not yet available. Several arguments, however, suggest that galactic and cluster fields were preexisting, or at least contemporary to, their hosts. The main reasons in favor of this thesis are: the ubiquity of the fields and the uniformity of their strength; the theoretical problems with the MHD amplification mechanisms, especially to explain the origin of cluster fields; the observation of µG magnetic fields in high-redshift galaxies. It is reassuring that new ideas continuously appear to determine these fields at all times. For example, a very recent one by Loeb and Waxman proposes to look for fluctuations in the radio background from intergalactic synchrotron emission of relativistic electrons interacting with CMBR .
A consistent and economical mechanism which may naturally explain the early origin of both galactic and cluster magnetic fields, is the adiabatic compression of a primeval field with strength in the range B0 ~ 10-9 - 10-10 Gauss (B0 is the intensity that the primordial field would have today under the assumption of adiabatic decay of the field due to the Hubble expansion). If this was the case, two other interesting effects may arise: a) magnetic fields may have affected structure formation perhaps helping to solve some of the problems of the CDM scenario; b) magnetic fields can have produced observable imprints in the CMBR. Given the current theoretical uncertainties about the MHD of galaxies and clusters, and the preliminary status of N-body simulations in the presence of magnetic fields, the most promising possibility to test the primordial origin hypothesis of cosmic magnetic fields comes from the forthcoming observations of the CMBR anisotropies.
In Chap. 2 we reviewed several possible effects of magnetic fields on the CMBR. In first place, we showed that magnetic fields may affect the isotropy of the CMBR. A magnetic field which is homogeneous through the entire Hubble volume, would spoil Universe isotropy giving rise to a dipole anisotropy in the CMBR. On the basis of this argument it was shown that COBE measurements provide an upper limit on the present time equivalent strength of a homogeneous cosmic magnetic field which is roughly 3 × 10-9 G. More plausibly, magnetic fields are tangled on scales much smaller than the Hubble radius. In this case the effect on the Universe geometry is negligible and much more interesting effects may be produced on small angular scales. Some of these effects arise as a consequence of MHD modes appearing in the magnetized photon-baryon plasma in place of the usual acoustic modes. The amplitude and velocity of the MHD modes depend on the magnetic field intensity and spatial direction. Some of these modes are quite different from standard scalar and tensor modes which are usually considered in the theoretical analysis of the CMBR distortions. For example, Alfvén waves have the peculiar property of not beeing depleted by the Universe expansion inspite of their vectorial nature. These modes are well suited to probe perturbations as those generated by cosmic defects and primordial phase transitions. Tangled magnetic fields, whose production is predicted by several models and, which are observed in the intercluster medium, are also expected to produce Alfvén waves. Another interesting aspect of this kind of isocurvature perturbations, is that they are not affected by Silk damping. The polarization power spectrum of CMBR can also be affected by primordial magnetic fields. This a consequence of the Faraday rotation produced by the field on the CMB photons on their way through the last scattering surface. Magnetic fields with strength B0 10-9 G may have produced a detectable level of depolarization. Furthermore, it was shown that because of the polarization dependence of the Compton scattering, the depolarization can feed-back into a temperature anisotropy. It was concluded that the best strategy to identify the imprint of primordial magnetic fields on the CMBR is probably to look for their signature in the temperature and polarization anisotropies cross-correlation. This method may probably reach a sensitivity of ~ 10-10 G for the present time equivalent magnetic field strength when the forthcoming balloon and satellite missions data is analyzed. Some results from the Boomerang and Maxima experiments  are already out with surprising results. It is too early to decide the reasons why, if experimentally confirmed, the second peak is low. We have verified, together with Edsjö , that magnetic fields can only decrease the ratio of amplitude of the second peak with respect to the first. Therefore the effect cannot be explained in terms of primordial magnetic fields. Interesting constraints on the strength of these fields will be available only when the amplitude of several peaks and the polarization of CMBR will be measured.
Another period of the Universe history when primordial magnetic fields may have produced observable consequences is the big-bang nucleosynthesis. This subject was treated in Chap. 3. Three main effects have been discussed: the effect of the magnetic field energy density on the Universe expansion; the modification produced by a strong magnetic field of the electron-positron gas thermodynamics; the modification of the weak processes keeping neutrons and protons in chemical equilibrium. All these effects produces a variation in the final neutron-to-proton ratio, hence in the relative abundances of light relic elements. The effect of the field on the Universe expansion rate was showed to be globally dominant, though the others cannot be neglected. Furthermore, the non-gravitational effects of the magnetic field can exceed that on the expansion rate in delimited regions where the magnetic field intensity may be larger than the Universe mean value. In this case these effects could have produced fluctuations in baryon to photon ratio and in the relic neutrino temperature. Apparently, the BBN upper bound on primordial magnetic fields, which is B0 7 × 10-5 G, looks less stringent than other limits which come from the Faraday rotation measurements (RMs) of distant quasars, or from the global isotropy of the CMBR. However, we showed in Sec. 3.4 that this conclusion is not correct if magnetic fields are tangled. The reason is that BBN probes length scales which are of the order of the Hubble horizon size at BBN time (which today corresponds approximately to 100 pc) whereas CMBR and the RMs probe much larger scales. Furthermore, constraints derived from the analysis of effects taking place at different times may not be directly comparable if the magnetic field evolution is not adiabatic.
In Chap. 4 we reviewed some of the models which predict the generation of magnetic fields in the early Universe. We first discussed those models which invoke a first order phase transition. This kind of transitions naturally provide some conditions, as charge separation, out-of-equilibrium condition, and high level of turbulence, which are known to be important ingredients of magnetogenesis. We discussed the cases of the QCD and the electroweak phase transitions (EWPT). In the case of the EWPT some extension of the particles physics standard model has to be invoked for the transition to be first order. Magnetic fields may be generated during the EWPT from the non-trivial dynamics of the gauge fields produced by the equilibration of the electrically charged components of the Higgs field. This effect resembles the Kibble mechanism for the formation of topological defects. It is interesting that such a mechanism gives rise to magnetic fields, though only on very small scales, even if the phase transition is second order. In general, since the production of magnetic fields during a phase transition is a causal phenomenon, the coherence length scale of these fields at the generation time cannot exceed the horizon radius at that time. Typically, once this length is adiabatically re-scaled to present time, one gets coherence cell sizes which are much smaller than those observed today in galaxies and the inter-cluster medium. This problem may be eased by the effect of the magnetic helicity which is expected to be produced during primordial phase transitions.Helicity may help the formation of large magnetic structures starting from small ones (inverse cascade). Indeed, this is a quite common phenomenon in MHD. Some estimates of the quantitative relevance of this effect have been given,for example, in Sec. 1.4. We have seen that the QCD phase transition might indeed give rise to phenomenological interesting values of the present time magnetic field strength and coherence size but only assuming quite optimistic conditions. Magnetic fields produced at the EWPT might have played a role in the generation of galactic magnetic fields only if they were amplified by a galactic dynamo. The problem with the small coherence scale of magnetic fields produced in the early Universe may be circumvented if the production mechanism was not-causal. This may be possible if magnetic fields were produced during inflation by the superadiabatic amplification of preexisting quantum fluctuations of the gauge fields. This phenomenon, however, can take place only if the conformal invariance of the electromagnetic field is broken. In Sec. 4.5 we have discussed several interesting mechanisms which have been proposed in the literature to avoid this obstacle. Unfortunately, although some results are encouraging, at the present status of art, none of these model seems to offer any firm prediction. Further work on the subject is, therefore, necessary.
Even if magnetic fields, produced during the electroweak phase transitions or before, are nor the progenitor of galactic magnetic fields, they may still have had other interesting cosmological consequences. Perhaps the most intriguing possibility is that magnetic fields played a role in the generation of the baryon asymmetry of the Universe (BAU). The magnetic fields may influence electroweak baryogenesis at two levels. At a first level, magnetic fields can play an indirect role on electroweak baryogenesis by modifying the free energy difference between the symmetric and broken phases, the Higgs effective potential, and the rate of sphaleron, baryon number violating, transitions. In Chap. 5 we showed, however, at this level, no significative modifications to arise with respect to the standard scenario. Magnetic fields, or better their hypermagnetic progenitors, may have played a much more direct role in the generation of the BAU if they possessed a net helicity. Indeed, it is well known from field theory that the hypermagnetic helicity coincides with the Chern-Simon number which can be converted into baryons and leptons numbers by the Abelian anomaly. The origin of the primordial magnetic helicity is still matter of speculation. Among other possibilities which we have review in Chap. 4, one of the more discussed in the literature is that a net hypermagnetic helicity of the Universe arise by an anomalous coupling of the gauge fields to an oscillating pseudoscalar field. The existence of pseudoscalar fields of this kind is required by several extensions of the particle physics standard model. However, it must be admitted that the mechanisms for generation of fields that are large and extended at the same time are far from being fully understood.
Large magnetic fields would also have a profound effect on chirality but it is also quite far from observability. As a rule of the thumb, particle physics effects will appear at the earliest at B = m2, the being the lightest hadron. This makes these effects difficult to test. Large fields are required. The only hope is the existence of superconductive strings.
Are the observed fields, so widespread, of early origin or some seeds fields were rapidly enhanced by a dynamo mechanism? This question remains unanswered but high precision CMBR acoustic peak measurements may very well provide a breakthrough.
The authors thank J. Adams, M. Bander, P. Coles, D. Comelli, U. Danielsson, J. Edsjö, F. De Felice, A. Dolgov, D. Harari, A. Loeb, M. Pietroni, G. Raffelt, A. Riotto, M. Shaposhnikov, A. Schwimmer, O. Törnkvist, T. Vachaspati and E. Waxman for several useful discussions.
This project was suggested to us by David Schramm. His untimely death left us without his advice.