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

## Appendix A: mean magnetic field in the Robertson-Walker metric

The purpose here is to show that the mean magnetic field is null in a universe obeying the metric of Robertson-Walker, in order to justify this assumption in the paper.

First consider the Maxwell equations:  where now (in contrast with the nomenclature above used) and are the unperturbed Faraday tensor (a function of the unperturbed mean electric and magnetic fields and ) and the unperturbed charge density electric current vector. All these quantities have been assumed to be null throughout the paper. We would now have, from (A2) From this equation we obtain   (and other similar formulae)  With Eq. (A7) in (A4), (A5) and (A6) we have Therefore the Maxwell equations under a Robertson-Walker metric are not incompatible with a mean magnetic field. The other set of Maxwell equations simply tells us that the Universe should be macroscopically neutral and that electric currents given by exist. Neither is the law of conservation of momentum-energy very restrictive with respect to the existence of a cosmological magnetic field. We can similarly benefit from the derivation carried out in section (5). Now, again, is the unperturbed magnetic field. For energy conservation we would have where is the photon energy density. If is the direction of the magnetic field, we have for the conservation of momentum, taking into account the homogeneity of the Universe  which does not imply at all. This result is however, deduced from the Einstein field equations. We again benefit from the above calculations, with now being the unperturbed magnetic field. It is obtained that Components , and of this equation yields   This does not mean that . One of them may be non-vanishing, for instance . But then components and yield  and subtracting we obtain , and therefore in a Robertson-Walker metric.

We conclude that in a universe with a Robertson-Walker metric, the existence of , even if compatible with the Maxwell equations and with the equations of conservation of energy-momentum, is incompatible with the Einstein field equations.

From (A13) it is then obtained  Eliminating in these two equations it is obtained which is the generalization of the Einstein-Friedmann equation, and which was implicitly used when obtaining (29).

## Appendix B: linear Newtonian evolution of magnetic fields

It will be shown that in the linear Post-Recombination epoch, the magnetic structures evolve according to the law , with being the amplitude of the magnetic propagating inhomogeneity (we are not now using comoving coordinates) in a similar way to their evolution in the Radiation dominated epoch.

Let us consider the induction equation As the mean field vanishes, let us again term . After the introduction of perturbations, this becomes where is the mean velocity. We would have in general , with being the fluctuative velocity, but it would be present only in second order terms. Therefore obeys where is the position vector. With  we obtain The solution would be of the form where is the amplitude that only depends on time, and is a function only of : where is constant. Substituting in the induction equation, we have hence and It can therefore be concluded that the magnetic spatial primordial pattern has been preserved until very recently when the back-reaction of the velocity field onto the magnetic field became important. The importance of the back-reaction has been pointed out by Kim et al. (1994).    © European Southern Observatory (ESO) 1997

Online publication: April 20, 1998 