Astron. Astrophys. 326, 13-22 (1997)

1. Introduction

Recent measurements of intergalactic magnetic fields (see the review by Kronberg, 1994, and references therein) have provided evidence of the following facts:

1. Magnetic fields of the order of are not uncommon. Values larger than have been found in all reported measurements (see also Feretti et al. 1995). Fields of this strength are quantitatively important because they correspond to an energy density equal to that of CMBR. Kronberg (1994) even suggests that fields are ubiquitous. In this case, a "background magnetism" would be in equipartition of energy with the CMBR. This possibility, even whilst theoretically attractive, is still based on a limited number of measurements and therefore will not be assumed here.
2. Large magnetic fields have also been found in protogalactic clouds (e.g. Wolfe, Lanzetta, & Oren 1992; Welter, Perry, & Kronberg, 1984). Probably, primordial magnetic fields to a large degree contributed to the present intergalactic ones. Kronberg suggested that magnetic fields play a role in the formation of structures in the Universe.

These facts, even if we do not know exactly how ubiquitous and how persistent in time intergalactic magnetic fields are, stimulate the analysis of their evolution and interrelation with density inhomogeneities.

In this paper we consider a universe dominated by relativistic particles. More precisely we have in mind a universe dominated by photons before Equality. The equations however are valid for any kind of dominant relativistic particles including hot dark matter.

The study of the evolution of density inhomogeneities for a universe with photons and barions, when no magnetic fields are present, is a classical topic (Weinberg 1972; Peebles 1980; Kolb & Turner 1990; Börner 1988; Battaner 1996). It may be divided into three periods:

i) Post-Recombination era. During this era a Newtonian analysis is appropriate, but nonlinear effects require rather sophisticated numerical techniques. Inhomogeneities grow as R first, becoming proportional to and when nonlinear effects become more and more important. The inclusion of magnetic fields in the study of this epoch has been carried out by Wasserman (1978), Coles (1992) and Kim, Olinto, & Rosner (1994), this last work including nonlinear effects. Classical treatments of Birkeland currents in the plasma Universe have been carried out by Peratt (1988)

ii) Acoustic era. A relativistic treatment is necessary; viscosity and heat conduction must be included (Field 1969; Weinberg 1972; and others) as these effects explain the Silk mass. Inhomogeneities do not increase during this era, which ends at Recombination, its beginning being dependent on the rest mass of the primordial cloud, around . As far as we know, no attempt at introducing magnetic fields into this analysis has been made.

iii) Radiation dominated era. This era ends when the acoustic one begins, and is therefore not perfectly defined, roughly at , so it corresponds to a photon dominated universe. The beginning of this era is also rather indeterminate. During this epoch non-magnetic inhomogeneities increase as .

The inclusion of magnetic fields in the study of this third era is the objective of this paper. Basically, our objective in this paper is to extend the work by Wasserman (1978) and Kim, Olinto and Rosner (1994) to the radiation dominated era. We will deal with the evolution of magnetic fields and their influence on density inhomogeneities in a radiation dominated Universe. The mathematical procedure must be relativistic, but the inclusion of nonlinear and imperfect fluid effects is not necessary, which greatly simplifies the problem. The upper time boundary will be placed at approximately , before the Acoustic epoch, and near Equality. The lower time boundary is undefined, but in particular we consider a Post-Annihilation era, in order to avoid sudden jumps in the temperature of photons and because positron-electron and quark-gluon plasmas, which constitute the plasma state at earlier epochs, require another analytical formulation. Therefore, the period under study broadly extends from Annihilation to Equality.

We consider that the evolution of magnetic fields is not perturbed by creation and loss processes. Some mechanisms have been invoked for later stages (Rees 1987; Lesch & Chiba 1995; Ruzmaikin, Sokoloff, & Shukurov 1989; and others) but these probably do not affect the epoch studied here. Some mechanisms producing primordial fields, prior to Annihilation, are implicitly assumed (see Turner & Widrow 1988; Quashnock, Loeb, & Spergel 1989; Vachaspati 1991; Ratra 1992; Enqvist & Olesen 1993, 1994; Davis & Dimopoulos 1996) but no assumption about their order of magnitude is here adopted.

Some important works have recently dealt with MHD in an expanding universe (Holcomb 1989, 1990; Dettmann, Frankel, & Kowalenko 1993; Gailis et al. 1994; Gailis, Frankel, & Dettmann 1995; Brandenburg, Enqvist, & Olesen 1996). However our objective is not MHD, but the influence of magnetic fields on the formation of large scale structure. In these papers the metric is unperturbed. Here magnetic fields themselves are responsible for perturbations in the metric, which induce motions and density inhomogeneities, which in turn affect the perturbed metric, and possibly the magnetic fields.

We have not included either protons or electrons in the system of equations. In a first attempt to solve this problem, this omission can be accepted, especially when we are considering an epoch of the Universe that is dominated by photons in which charges are considered to play a minor role. Nevertheless, exists, which creates a charge current (see Eq. (44)) and has an influence on the remaining equations. The existence of charges is implicitly assumed to support magnetic fields and electric currents, but equations of this third component are not explicitly considered. The influence of magnetic fields on the photon inhomogeneities lies in the fact that curvature is decided by the energy-momentum tensor, which is modified taking into account the magnetic contribution. This contribution is probably too small to affect the expansion itself, but not so as to have a large influence on the internal structure. The contribution of magnetic fields to the energy-momentum tensor is very different with respect to the contribution of photon and baryon densities alone. This inclusion is not, therefore, trivial and indeed the results are clearly different. In the presence of magnetic fields, inhomogeneities evolve in a completely different way, as is demonstrated here in this particular epoch in the lifetime of the Universe.

In agreement with the Cosmological Principle, we consider than no mean magnetic field exists at cosmological scales, so (or rather, we demonstrate in Appendix A that in a Robertson-Walker metric). Classically, this condition is equivalently reached from , Gauss theorem and the Cosmological Principle. However random magnetic fields do exist at lower scales in characteristic cells. These fluctuative magnetic fields, even with random orientations, are present everywhere, so that is non-vanishing. There is no mean magnetic field in the Universe, but there is a mean magnetic energy. No assumption about its value, such as that of equipartition with the CMBR energy density or any other hypothesis, is made "a priori".

In this paper we obtain the equations and derive the basic conclusions. In forthcoming papers we will deal with inhomogeneities affected by selected particular magnetic configurations, and with the influence of a large scale magnetic field distribution on the large scale density distribution. Paper II deals with the influence of magnetic flux tubes on the distribution of the density, showing that primordial magnetic fields can be responsible for the observed present filamentary structure

The curvature has been set equal to zero, which is, in any case, a good assumption for this epoch.

© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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