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

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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:

[EQUATION]

[EQUATION]

where now (in contrast with the nomenclature above used) [FORMULA] and [FORMULA] are the unperturbed Faraday tensor (a function of the unperturbed mean electric and magnetic fields [FORMULA] and [FORMULA]) 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)

[EQUATION]

From this equation we obtain

[EQUATION]

[EQUATION]

[EQUATION]

(and other similar formulae)

[EQUATION]

[EQUATION]

With Eq. (A7) in (A4), (A5) and (A6) we have

[EQUATION]

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

[EQUATION]

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, [FORMULA] is the unperturbed magnetic field. For energy conservation we would have

[EQUATION]

where [FORMULA] is the photon energy density. If [FORMULA] is the direction of the magnetic field, we have for the conservation of momentum, taking into account the homogeneity of the Universe [FORMULA]

[EQUATION]

which does not imply [FORMULA] at all. This result is however, deduced from the Einstein field equations. We again benefit from the above calculations, with [FORMULA] now being the unperturbed magnetic field. It is obtained that

[EQUATION]

Components [FORMULA], [FORMULA] and [FORMULA] of this equation yields

[EQUATION]

[EQUATION]

[EQUATION]

This does not mean that [FORMULA]. One of them may be non-vanishing, for instance [FORMULA]. But then components [FORMULA] and [FORMULA] yield

[EQUATION]

[EQUATION]

and subtracting we obtain [FORMULA], and therefore

[EQUATION]

in a Robertson-Walker metric.

We conclude that in a universe with a Robertson-Walker metric, the existence of [FORMULA], 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

[EQUATION]

[EQUATION]

Eliminating [FORMULA] in these two equations it is obtained

[EQUATION]

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 [FORMULA], with [FORMULA] 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

[EQUATION]

As the mean field vanishes, let us again term [FORMULA]. After the introduction of perturbations, this becomes

[EQUATION]

where [FORMULA] is the mean velocity. We would have in general [FORMULA], with [FORMULA] being the fluctuative velocity, but it would be present only in second order terms. Therefore [FORMULA] obeys

[EQUATION]

where [FORMULA] is the position vector. With

[EQUATION]

[EQUATION]

we obtain

[EQUATION]

The solution would be of the form

[EQUATION]

where [FORMULA] is the amplitude that only depends on time, and [FORMULA] is a function only of [FORMULA]:

[EQUATION]

where [FORMULA] is constant. Substituting in the induction equation, we have

[EQUATION]

hence

[EQUATION]

and

[EQUATION]

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).

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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