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Magnetic fields are pervading. Planets, stars, galaxies and clusters of galaxies have been observed that carry fields that are large and extensive. Though strong homogeneous fields are ruled out by the uniformity of the cosmic background radiation, large domains with uniform fields are possible.

A crucial ingredient for the survival of magnetic fields on astrophysical scales is for them to live in a medium with a high electrical conductivity. As we shall see in Chap. 1, this condition is comfortably fulfilled for the cosmic medium during most of the evolution of the Universe. As a consequence, it is possible for magnetic fields generated during the big-bang or later to have survived until today as a relic.

To establish the existence and properties of primeval magnetic fields would be of extreme importance for cosmology. Magnetic fields may have affected a number of relevant processes which took place in the early Universe as well as the Universe geometry itself. Because of the Universe high conductivity, two important quantities are almost conserved during Universe evolution: the magnetic flux and the magnetic helicity (see Sec. 1.4). As we will see, valuable information about fundamental physics which took place before the recombination time may be encoded in these quantities.

In the past years a considerable amount of work has been done about cosmic magnetic fields both on the astrophysical and the particle physics point of view. The main motivations of such wide interest are the following.

The origin of the magnetic fields observed in the galaxies and in the clusters of galaxies is unknown. This is an outstanding problem in modern cosmology and, historically, it was the first motivation to look for a primordial origin of magnetic fields. Some elaborated magnetohydrodynamical (MHD) mechanisms have been proposed to amplify very weak magnetic fields into the µG fields generally observed in galaxies (see Sec. 1.1). These mechanisms, known as galactic dynamo, are based on the conversion of the kinetic energy of the turbulent motion of the conductive interstellar medium into magnetic energy. Today, the efficiency of such a kind of MHD engines has been put in question both by improved theoretical work and new observations of magnetic fields in high redshift galaxies (see Sec. 1.2). As a consequence, the mechanism responsible for the origin of galactic magnetic fields has probably to be looked back in the remote past, at least at a time comparable to that of galaxy formation. Furthermore, even if the galactic dynamo was effective, the origin of the seed fields which initiated the processes has still to be identified.

Even more mysterious is the origin of magnetic fields in galaxy clusters. These fields have been observed to have strength and coherence size comparable to, and in some cases larger than, galactic fields. In the standard cold dark matter (CDM) scenario of structure formation clusters form by aggregation of galaxies. It is now understood that magnetic fields in the inter-cluster medium (ICM) cannot form from ejection of the galactic fields (see Sec. 1.2). Therefore, a common astrophysical origin of both types of fields seems to be excluded. Although, independent astrophysical mechanisms have been proposed for the generation of magnetic fields in galaxies and clusters, a more economical, and conceptually satisfying solution would be to look for a common cosmological origin.

Magnetic fields could have played a significant role in structure formation. It may be not a coincidence that primordial magnetic fields as those required to explain galactic fields, without having to appeal to a MHD amplification, would also produce pre-recombination density perturbations on protogalactic scales. These effects go in the right direction to solve one of the major problems of the CDM scenario of structure formation (see Sec. 1.3). Furthermore, if primordial magnetic fields affected structure formation they also probably left detectable imprints in the temperature and polarization anisotropies, or the thermal spectrum, of the cosmic microwave background radiation (CMBR) (see Chap. 2).

Field theory provides several clues about the physical mechanisms which may have produced magnetic fields in the early Universe. Typically, magnetogenesis requires an out-of-thermal equilibrium condition and a macroscopic parity violation. These conditions could be naturally provided by those phase transitions which presumably took place during the big-bang. Well known examples are the QCD (or quark confinement) phase transition, the electroweak (EW) phase transition, the GUT phase transition. During these transitions magnetic fields can be either generated by the turbulent motion induced in the ambient plasma by the rapid variation of some thermodynamic quantities (if the transition is first order) or by the dynamics of the Higgs and gauge fields. In the latter case the mechanism leading to magnetogenesis shares some interesting common aspects with the mechanism which have been proposed for the formation of topological defects. On the other hand, if cosmic strings were produced in the early Universe they could also generate cosmic magnetic fields in several ways. Inflation, which provides a consistent solution to many cosmological puzzles, has also several features which make it interesting in the present context (see Sec. 4.5). Although to implement an inflationary scenario of magnetogenesis requires some nontrivial extensions of the particle physics standard model, recent independent developments in field theory may provide the required ingredients. Magnetic fields might also be produced by a preexisting lepton asymmetry by means of the Abelian anomaly (see Sec. 4.4). Since the predictions about the strength and the spatial distribution of the magnetic fields are different for different models, the possible detection of primeval magnetic fields may shed light on fundamental physical processes which could, otherwise, be unaccessible.

Even if primordial magnetic fields did not produce any relevant effect after recombination, they may still have played a significant role in several fundamental processes which occurred in the first 100,000 years. For example, we shall show that magnetic fields may have affected the big-bang nucleosynthesis, the dynamics of some phase transitions, and baryogenesis. Since big-bang nucleosynthesis (BBN) has been often used to derive constraints on cosmological and particle physics parameters, the reader may be not surprised to learn here that BBN also provides interesting limits on the strength of primordial magnetic fields (see Chap. 3). Even more interesting is the interplay which may exist between baryogenesis and magnetogenesis. Magnetic fields might have influenced baryogenesis either by affecting the dynamics of the electroweak phase transition or by changing the rate of baryon number violating sphaleron processes (see Chap. 5). Another intriguing possibility is that the hypercharge component of primeval magnetic fields possessed a net helicity (Chern-Simon number) which may have been converted into baryons and leptons by the Abelian anomaly (see Chap. 4). In other words, primordial magnetic fields may provide a novel scenario for the production of the observed matter-antimatter asymmetry of the Universe.

An interesting aspect of magnetic fields is their effect on the constituents of matter. This in turn is of importance on many aspects of the processes that took place in the early times. Masses of hadrons get changed so that protons are heavier than neutrons. The very nature of chirality could get changed see (Chap. 5). However the characteristic field for this to happen is H = mpi2 which is about 1018 G. These fields cannot exist at times when hadrons are already existing and therefore are probably not relevant. Near cosmic superconductive strings the story may be different.

Clearly, this is a quite rich and interdisciplinary subject and we will not be able to cover with the same accuracy all its different aspects. Our review is manly focused on the production mechanism and the effects of magnetic fields before, or during, the photon decoupling from matter.

In Chap. 1 we shortly review the current status of the observations. In order to establish some relation between recent time and primeval magnetic fields we also provide a short description of some of the mechanisms which are supposed to control the evolution of magnetic fields in the galaxies and in the intergalactic medium. We only give a very short and incomplete description of the effect of magnetic fields on structure formation. Some basic aspects of this subject are, anyhow, presented in Chap. 2 where we discuss the effect of magnetic fields on the anisotropies of the cosmic microwave background radiation. From a phenomenological point of view Chap. 2 is certainly the most interesting of our review. The rapid determination of the CMBR acoustic peaks at the level of a few percent will constrain these fields significantly. We briefly touch upon the recent determination of the second acoustic peak. In Chap. 3 we describe several effects of strong magnetic fields on the BBN and present some constraints which can be derived by comparing the theoretical predictions of the light elements relic abundances with observations. Since it can be of some relevance for BBN, propagation of neutrinos in magnetized media is also shortly discussed at the end of that chapter. In Chap. 4 we review several models which predict the generation of magnetic fields in the early Universe. In the same chapter some possible mutual effects of magnetogenesis and baryogenesis are also discussed. Some aspects of the effects which are described in Chapts. 3 and 4, which concern the stability of strong magnetic fields and the effect that they may produce on matter and gauge fields, are discussed in more details in Chap. 5. At the end we report our conclusions.

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