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

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2. The mean magnetic field

Before considering the interrelation between fluctuative magnetic fields and density inhomogeneities, let us consider the mean magnetic field and its influence on the motion of the Universe as a whole. It is intuitive that an isotropic universe cannot possess a mean magnetic field. Nevertheless, it is demonstrated in Appendix A that the existence of a mean magnetic field is incompatible with the Robertson-Walker metric, by examining the form adopted by Maxwell equations, the equation of motion and the Einstein field equations in a magnetized universe. Therefore, we consider [FORMULA]. However, magnetic fields may be present (and actually they are) in smaller cells, with the direction of the field being random at larger scales, so that we assume [FORMULA]. There is no mean magnetic field but there is a mean magnetic energy.

In Einstein field equations we must include the mean magnetic energy-momentum tensor, deduced from:

[EQUATION]

where [FORMULA] is the Faraday tensor. We adopt for the contravariant-covariant form of the Faraday tensor in a Minkowskian frame

[EQUATION]

or in brief

[EQUATION]

where

[EQUATION]

and [FORMULA] and [FORMULA] are the electric and magnetic three-vectors, as seen by an inertial observer.

The Robertson-Walker metric for [FORMULA] can be written as

[EQUATION]

where (as usual) Latin indexes denote only spatial coordinates, and R is the cosmological scale factor. Throughout the paper we will consider that R is measured taking its present value [FORMULA] as unity. Therefore, R is dimensionless, [FORMULA] at present, and we have approximately [FORMULA], with z being the redshift, taking into account that we are dealing with times long before Recombination. Now we take into account the transformation of coordinates that transforms [FORMULA] into [FORMULA]. This transformation is achieved by

[EQUATION]

We then obtain

[EQUATION]

Using the Robertson-Walker metric we obtain the other forms of the Faraday tensor

[EQUATION]

We then assume infinite conductivity, so that electrical fields in the rest frame of the charged particles vanish. Magnetic fields are assumed to be tied to charges. Cheng & Olinto (1994) showed that the effects of finite conductivity in the early universe may be neglected. The electromagnetic momentum-energy tensor becomes

[EQUATION]

where

[EQUATION]

and

[EQUATION]

We also have

[EQUATION]

[EQUATION]

The subindex M denotes "magnetic"

[EQUATION]

with

[EQUATION]

This is easily checked. For instance [FORMULA]. Hence [FORMULA]. As

[EQUATION]

is the magnetic energy density, we have as the equation of state

[EQUATION]

The form of the magnetic energy-momentum tensor is identical to the form of the radiative energy-momentum tensor. For photons we consider

[EQUATION]

where [FORMULA] is the radiative hydrostatic pressure, [FORMULA] the radiative energy density, and we consider [FORMULA] and [FORMULA]. Therefore

[EQUATION]

Therefore the total energy momentum is

[EQUATION]

where

[EQUATION]

and

[EQUATION]

are the total density and hydrostatic pressure. Given the similarity of form between [FORMULA] and [FORMULA] the Einstein field equations must provide familiar results, i.e. the same expansion and cooling laws that hold for a purely radiative universe.

It is straightforwardly obtained for the Robertson-Walker metric that

[EQUATION]

[EQUATION]

[EQUATION]

with [FORMULA], as usual, and therefore

[EQUATION]

and

[EQUATION]

If [FORMULA] we find the familiar result [FORMULA], but in general, we must specify a relation between [FORMULA] and [FORMULA] to deduce both. For instance, if there was at some epoch equipartition of radiative and magnetic energy [FORMULA], both would decrease at the same rate and the condition of equipartition would be conserved. If this condition were confirmed at present (as suggested by Kronberg 1994) the epoch of the Universe dominated by photons would become co-dominated equally by photons and by magnetic fields. However, we emphasize that this equipartition condition is not assumed (or deduced) in this paper. If it held, the relation [FORMULA] would be maintained, but now the constant of proportionality would be higher, by a factor of less than [FORMULA], and the expansion would be faster.

If we assume that the magnetic energy density is much less than the radiative one, as is usually done, and indeed this is what our results suggest, then we would obtain the obvious result that the expansion is unaffected by magnetic fields.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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