A gauge-invariant analysis of magnetic fields in general-relativistic cosmology

Abstract

We provide a fully general-relativistic treatment of cosmological perturbations in a universe permeated by a large-scale primordial magnetic field using the Ellis-Bruni gauge-invariant formalism. The exact non-linear equations for general relativistic magnetohydrodynamic evolution are derived. A number of applications are made: the behaviour of small perturbations to Friedmann universes are studied; a comparison is made with earlier Newtonian treatments of cosmological perturbations and some effects of inflationary expansion are examined.

A non-perturbative study of the evolution of cosmic magnetised sources

“…This approach represents a further simplification of the weak field limit that is applicable to an early radiation dominated era (well before structure formation) in which matter and radiation perturbations are also neglected, and can be justified if the magnetic field is "not too tangled on scales smaller than the magnetic dissipation scale" (see discussion in [53] and [8]). This simplification of the weak field regime is not gauge invariant, but for the cosmic times under considera-tion it yields the same solutions as the weak field (and thus as gauge invariant perturbations) in the infinite conductivity regime [25,26].…”

“…Such a perturbative approach of a frozen magnetic field is fully justified if the latter is tangled on scales smaller than the Hubble horizon [21] and has lead to a comprehensive literature [20,21,22,23,24,25] in which the perturbations are covariant and gauge invariant.…”

“…However, a classical Maxwellian electromagnetic interaction (not necessarily restricted to early cosmic times) can also be incorporated into the full dynamics of General Relativity through an elegant first order system based on covariant objects defined in a 4-velocity frame (the 1+3 formalism [9,20,21]). In particular, this formalism is useful to study (following a perturbative or non-perturbative approach) the dynamics of a "frozen" magnetic field without electric currents through the limit of infinite conductivity [20,21,22,23,24,25]. Whenever a spatially flat FLRW background is assumed in a perturbative treatment, a "weak field approximation" can be defined in which the anisotropic stresses of the the magnetic field (associated with vector perturbations) are neglected and the latter field is described (at first order) as a sort of scalar source that dilutes as B ∼ a −2 [20,21,22,23,24,25,26].…”

“…In particular, this formalism is useful to study (following a perturbative or non-perturbative approach) the dynamics of a "frozen" magnetic field without electric currents through the limit of infinite conductivity [20,21,22,23,24,25]. Whenever a spatially flat FLRW background is assumed in a perturbative treatment, a "weak field approximation" can be defined in which the anisotropic stresses of the the magnetic field (associated with vector perturbations) are neglected and the latter field is described (at first order) as a sort of scalar source that dilutes as B ∼ a −2 [20,21,22,23,24,25,26]. While this approximation is not gauge invariant, it leads to the same results (see proof in [26]) as gauge invariant perturbations when infinite conductivity is assumed.…”

“…This comparison is facilitated by our choice of a Bianchi I model that becomes a spatially flat FLRW model when the magnetic field vanishes (i.e. the magnetic field is the sole cause of anisotropy of the source), as in this case a sufficiently weak field can be regarded approximately as a sort of supra-horizon magnetic perturbation on a spatially flat FLRW background (see comments on this issue in [37]), but it lends naturally with a comparison in the weak field regime studied in [25,26,30] and in [8]. Without neglecting the importance of observational bounds, the obtained results may be useful in potential situations in which strong cosmic magnetic field need to be examined in a non-perturbative manner.…”