Evolution of magnetic fields through cosmological perturbation theory

Abstract

The origin of galactic and extra-galactic magnetic fields is an unsolved problem in modern cosmology. A possible scenario comes from the idea of these fields emerged from a small field, a seed, which was produced in the early universe (phase transitions, inflation, ...) and it evolves in time. Cosmological perturbation theory offers a natural way to study the evolution of primordial magnetic fields. The dynamics for this field in the cosmological context is described by a cosmic dynamo like equation, through the dynamo term. In this paper we get the perturbed Maxwell's equations and compute the energy momentum tensor to second order in perturbation theory in terms of gauge invariant quantities. Two posible scenarios are discussed, first we consider a FLRW background without magnetic field and we study the perturbation theory introducing the magnetic field as a perturbation. The second scenario, we consider a magnetized FLRW and build up the perturbation theory from this background. We compare the cosmological dynamo like equation in both scenarios. PACS numbers: 98.80.-k, 95.30.Qd.

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

“…It is more natural to compare the magnetised Bianchi I model that we have examined with previous work in [8,27,28,29,30], where (besides considering a similar theoretical framework based on Satistical Mechanics) the magnetic interaction in the infinite conductivity regime and the cosmic fluid are both studied in a purely FLRW context, in which the frozen magnetic field is introduced as part of the background (the "weakly magnetised FLRW" spacetime in [30]) as a relativistic scalar correction to the energy density and the isotropic pressure of a perfect fluid particle mixture considered as source of an FLRW metric (see equations (3.26)-(3.29) of [8] and section IX of [30]). 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]).…”

“…Besides considering observational bounds on the magnetic field, we also discuss how a comparison of our approach and results can be made with those based on cosmological perturbations discussed and summarised in [9,10,11] and in previous work based on the weak field approximation [8,27,28,29,30]. 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.…”

“…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 weak field approximation was used also in earlier literature [8,27,28] (see update in [29,30]) in a purely FLRW context to examine the magnetic interaction at early cosmic times.…”

“…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.…”

Contrasting formulations of cosmological perturbations in a magnetic FLRW cosmology

“…Cosmological perturbations models permeated by a large-scale primordial magnetic field has been widely worked by Tsagas [25,26,27] and Ellis [28], where they found the complete equations system which shows a direct coupling between the Maxwell and the Einstein fields and also, gauge invariant for magnetic fields were built in the frame of 1+3 covariant approach. Furthermore, in previous works, we have obtained a set of equations which describe the evolution of cosmological magnetic fields up to second order in the gauge invariant approach, with their respective gauge transformations for the fields, important for building the gauge invariant magnetic variables [29]. Therefore, studying in detail the magnetic gauge invariant quantities in each one of the formalisms, we can find equivalences between themselves.…”