1754-0410-1-5 1754-0410 Research article <p>Semi-analytical approach to magnetized temperature autocorrelations</p> Giovannini Massimo massimo.giovannini@cern.ch

Centro "Enrico Fermi", Via Panisperna 89/A, 00184 Rome, Italy

Department of Physics, Theory Division, CERN, 1211 Geneva 23, Switzerland

 PMC Physics A 1754-0410 2007 1 1 5 http://www.physmathcentral.com/1754-0410/1/5 10.1186/1754-0410-1-5 18 10 2007 18 10 2007 18 10 2007 2007 Giovannini This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The cosmic microwave background (CMB) temperature autocorrelations, induced by a magnetized adiabatic mode of curvature inhomogeneities, are computed with semi-analytical methods. As suggested by the latest CMB data, a nearly scale-invariant spectrum for the adiabatic mode is consistently assumed. In this situation, the effects of a fully inhomogeneous magnetic field are scrutinized and constrained with particular attention to harmonics which are relevant for the region of Doppler oscillations. Depending on the parameters of the stochastic magnetic field a hump may replace the second peak of the angular power spectrum. Detectable effects on the Doppler region are then expected only if the magnetic power spectra have quasi-flat slopes and typical amplitude (smoothed over a comoving scale of Mpc size and redshifted to the epoch of gravitational collapse of the protogalaxy) exceeding 0.1 nG. If the magnetic energy spectra are bluer (i.e. steeper in frequency) the allowed value of the smoothed amplitude becomes, comparatively, larger (in the range of 20 nG). The implications of this investigation for the origin of large-scale magnetic fields in the Universe are discussed. Connections with forthcoming experimental observations of CMB temperature fluctuations are also suggested and partially explored.

1 Formulation of the problem

Since the Cosmic Microwave Background (CMB) is extremely isotropic in nearly all angular scales, it is rather plausible to infer that the Universe was quite homogeneous (and isotropic) at the moment when the ionization fraction dropped significantly and the photon mean free path became, almost suddenly, comparable with the present Hubble radius.

The inhomogeneities present for length-scales larger than the Hubble radius right before recombination are believed to be, ultimately, the seeds of structure formation and they can be studied by looking at the temperature autocorrelations which are customarily illustrated in terms of the angular power spectrum. The distinctive features of the angular power spectrum (like the Doppler peaks) can be phenomenologically reproduced by assuming the presence, before recombination, of a primordial adiabatic 2mode arising in a spatially flat Universe 12345. Possible deviations from this working hypothesis can also be bounded: they include, for instance, the plausible presence of non-adiabatic modes (see 678 and references therein), or even features in the power-spectrum that could be attributed either to the pre-inflationary stage of expansion or to the effective modification of the dispersion relations (see 9101112 and references therein). For a pedagogical introduction to the physics of CMB anisotropies see, for instance, Ref. 13. In short the purpose of the present paper is to show that CMB temperature autocorrelations may also be a source of valuable informations on large-scale magnetic fields whose possible presence prior to recombination sheds precious light on the origin of the largest magnetized structures we see today in the sky such as galaxies, clusters of galaxies and even some supercluster.

In fact, spiral galaxies and rich clusters possess a large-scale magnetic field that ranges from 500 nG 1415 (in the case of Abell clusters) to few μG in the case of spiral galaxies 16. Elliptical galaxies have also magnetic fields in the μG range but with correlation scales of the order of 10–100 pc (i.e. much smaller than in the spirals where typical correlation lengths are of the order of 30 kpc, as in the case of the Milky Way). The existence of large-scale magnetic fields in superclusters, still debatable because of ambiguities in the determination of the column density of electrons along the line of sight, would be rather intriguing. Recently plausible indications of the existence of magnetized structures in Hercules and Perseus-Pisces superclusters have been reported 17 (see also 18): the typical correlation scales of the fields would be 0.5 Mpc and the intensity 300 nG.

While there exist various ideas put forward throught the years, it is fair to say that the origin of these (pretty large) fields is still matter of debate 1519. Even if they are, roughly, one millionth of a typical planetary magnetic field (such as the one of the earth) these fields are pretty large for a cosmological standard since their energy density is comparable both with energy density of the CMB photons (i.e. TCMB4 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdsfaunaaDaaaleaacqqGdbWqcqqGnbqtcqqGcbGqaeaacqaI0aanaaaaaa@3082@) and with the cosmic ray pressure. The very presence of large scale magnetic fields in diffuse astrophysical plasmas and with large correlation scales (as large of, at least, 30 kpc) seems to point towards a possible primordial origin 15. At the same time, the efficiency of dynamo amplification can be questioned in different ways so that, at the onset of the gravitational collapse of the protogalaxy it seems rather plausible that only magnetic fields with intensities3 BL > 10-14 nG may be, eventually, amplified at an observable level 2021.

As emphasized many years ago by Harrison 222324, this situation is a bit reminiscent of what happened with the problem of justifying the presence of a flat spectrum of curvature perturbations that could eventually seed the structure formation paradigm. Today a possibility along this direction is provided by inflationary models in one of their various incarnations.

It seems therefore appropriate, especially in view of forthcoming satellite missions (like PLANCK Explorer 25), to discuss the effects of large-scale magnetic fields on CMB physics. In fact, all along the next decade dramatic improvements in the quality and quantity of CMB data can be expected. On the radio-astronomical side, the next generation of radio-telescopes such as Square Kilometre Array (SKA) 26 might be able to provide us with unprecedented accuracy in the full sky survey of Faraday Rotation measurements at frequencies that may be so large to be, roughly, comparable with 4 (even if always smaller than) the lower frequency channel of the PlANCK Explorer (i.e. about 30 GHz). The question before us today is, therefore, the following: is CMB itself able to provide compelling bounds on the strength of large-scale magnetic fields prior to hydrogen recombination? In fact, all the arguments connecting the present strength of magnetic field to their primordial value (say before recombination) suffer undeniable ambiguities. These ambiguities are related to the evolution of the Universe through the dark ages (i.e. approximately, between photon decoupling and galaxy formation). So, even if it is very reasonable to presume that during the stage of galaxy formation the magnetic flux and helicity are, according to Alfvén theorems, approximately conserved, the strengths of the fields prior to gravitational collapse is unknown and it is only predictable within a specific model for the origin of large-scale magnetic fields. In general terms, the magnetic fields produced in the early Universe may have different features. They may be helical or not, they may have different spectral slopes and different intensities. There are, however, aspects that are common to diverse mechanisms like the stochastic nature of the produced field. Furthermore, since as we go back in time the conductivity increases with the temperature, it can be expected that the flux freezing and the helicity conservation are better and better verified as the Universe heats up say from few eV to few MeV.

Along the past decade some studies addressed the analysis of vector and tensor modes induced by large-scale magnetic fields 28293031. There have been also investigations within a covariant approach to perturbation theory 3233. Only recently the analysis of the scalar modes has been undertaken 3435363738. The set-up of the aforementioned analyses is provided by an effective one-fluid description of the plasma which is essentially the curved space analog of magnetohydrodynamics (MHD). This approach is motivated since the typical length-scales of the problem are much larger of the Debye length. However, it should be borne in mind that the treatment of Faraday rotation is a typical two-fluid phenomenon. So if we would like to ask the question on how the polarization plane of the CMB is rotated by the presence of a uniform magnetic field a two-fluid description would be mandatory (see section 2 and references therein).

In the framework described in the previous paragraph, it has been shown that the magnetic fields affect the scalar modes in a threefold way. In the first place the magnetic energy density and pressure gravitate inducing a computable modification of the large-scale adiabatic solution. Moreover, the anisotropic stress and the divergence of the Lorentz force affect the evolution of the baryon-lepton fluid. Since, prior to decoupling, photons and baryons are tightly coupled the net effect will also be a modification of the temperature autocorrelations at angular scales smaller than the ones relevant for the ordinary SW contribution (i.e. ℓ > 30).

In the present paper, elaborating on the formalism developed in 343536, a semi-analtytical approach for the calculation of the temperature autocorrelations is proposed. Such a framework allows the estimate of the angular power spectrum also for angular scales compatible with the first Doppler peak. A gravitating magnetic field will be included from the very beginning and its effects discussed both at large angular scales and small angular scales. The main theme of the present paper can then be phrased by saying that large-scale magnetic fields affect the geometry and the evolution of the (scalar) sources. We ought to compute how all these effects combine in the final power spectra of the temperature autocorrelations. It should be remarked, incidentally, that the evolution of the density contrasts of the various species enter directly the scalar problem but neither the vector or the tensor modes are affected by their presence. As a consequence of this occurrence the self-consistent inclusion of the large-scale magnetic fields in the calculation is much more cumbersome than in the case of the tensor and vector modes.

The plan of the present paper will therefore be the following. In section 2 the typical scales of the problem will be discussed. In section 3 the attention will be focused on the large-scale evolution of the curvature perturbations with particular attention to the magnetized contribution, i.e. the contribution associated with the gravitating magnetic fields. In section 4 the evolution at smaller angular scales will be investigated accounting, in an approximate manner, for the finite thickness effects of the last-scattering surface. In section 5 the estimates of the angular power spectra of the temperature autocorrelations will be presented. Section 6 contains the concluding remarks. Some of the relevant theoretical tools needed for the discussion of the problem have been collected in the appendix with the sole aim to make the overall presentation more self-contained. The material presented in the appendix collects the main equations whose solutions are reported and discussed in section 3 and 4.

2 Typical scales of the problem

The analysis starts by defining all the relevant physical scales of the problem. These scales stem directly from the evolution equations of the gravitational perturbations in the presence of a stochastic magnetic field. The interested reader may also consult appendix A where some relevant technical aspects are briefly summarized.

2.1 Equality and recombination

According to the present understanding of the Doppler oscillations the space-time geometry is well described by a conformally flat line element of Friedmann-Robertson-Walker (FRW) type

d s 2 = a 2 ( τ ) [ d τ 2 − d x → 2 ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazcqWGZbWCdaahaaWcbeqaaiabikdaYaaakiabg2da9iabdggaHnaaCaaaleqabaGaeGOmaidaaOGaeiikaGccciGae8hXdqNaeiykaKIaei4waSLaemizaqMae8hXdq3aaWbaaSqabeaacqaIYaGmaaGccqGHsislcqWGKbazcuWG4baEgaWcamaaCaaaleqabaGaeGOmaidaaOGaeiyxa0LaeiilaWcaaa@41C1@

where t is the conformal time coordinate. In the present paper the general scheme will be to introduce the magnetic fields in the standard lore where the space-time geometry is spatially flat. This is the first important assumption which is supported by current experimental data including the joined analysis of, at least, three sets of data stemming, respectively from large-scale structure, from Type Ia supernovae and from the three year WMAP data (eventually combined with other CMB experiments). For the interpretation of the data a specific model must also be adopted. The framework of the present analysis will be the one provided by the ΛCDM model. This is probably the simplest case where the effects of magnetic fields can be included. Of course one may also ask the same question within a different underlying model (such as the open CDM model or the ΛCDM model with sizable contribution from the tensor modes and so on and so forth). While the calculational scheme will of course be a bit different, the main logic will remain the same. More details on the typical values of cosmological parameters inferred in the framework of the ΛCDM model can be found at the beginning of section 5.

In the geometry given by Eq. (2.1) the scale factor for the radiation-matter transition can be smoothly parametrized as

a ( τ ) = a e q [ ( τ τ 1 ) 2 + 2 ( τ τ 1 ) ] , τ 1 = 2 H 0 a eq Ω M 0 ≃ 288 ( h 0 2 Ω M 0 0.134 ) − 1 M p c . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemyyaeMaeiikaGccciGae8hXdqNaeiykaKIaeyypa0Jaemyyae2aaSbaaSqaaGqaaiab+vgaLjab+fhaXbqabaGcdaWadaqaamaabmaabaqcfa4aaSaaaeaacqWFepaDaeaacqWFepaDdaWgaaqaaiabigdaXaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaGccqGHRaWkcqaIYaGmdaqadaqaaKqbaoaalaaabaGae8hXdqhabaGae8hXdq3aaSbaaeaacqaIXaqmaeqaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaiabcYcaSaqaaiab=r8a0naaBaaaleaacqaIXaqmaeqaaOGaeyypa0tcfa4aaSaaaeaacqaIYaGmaeaacqWGibasdaWgaaqaaiabicdaWaqabaaaaOWaaOaaaeaajuaGdaWcaaqaaiabdggaHnaaBaaabaGaeeyzauMaeeyCaehabeaaaeaacqqHPoWvdaWgaaqaaiabb2eanjabicdaWaqabaaaaaWcbeaakiabloKi7iabikdaYiabiIda4iabiIda4maabmaabaqcfa4aaSaaaeaacqWGObaAdaqhaaqaaiabicdaWaqaaiabikdaYaaacqGHPoWvdaWgaaqaaiab+1eanjabicdaWaqabaaabaGaeGimaaJaeiOla4IaeGymaeJaeG4mamJaeGinaqdaaaGccaGLOaGaayzkaaaaamaaCaaaleqabaGaeyOeI0IaeGymaedaaOGae4xta0Kae4hCaaNae43yamMaeiOla4caaa@70ED@

Concerning Eqs. (2.1) and (2.2) few comments are in order:

• the conformal time coordinate is rather useful for the treatment of the evolution of magnetized curvature perturbations and is extensively employed in the appendix A;

• H0 is the present value of the Hubble constant and ΩM0 is the present critical fraction in non-relativistic matter, i.e. ΩM0 = Ωb0 + Ωc0, given by the sum of the CDM component and of the baryonic component;

• in the notation of Eq. (2.2) the equality time (i.e. the time at which the radiation contribution equals the contribution of dusty matter) is easily determined to be τeq = (2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaeGOmaidaleqaaaaa@2C02@ - 1)τ1, i.e. roughly, τeq ≃ τ1/2.

Equation (2.2) is a solution of the Friedmann-Lemaître equations whose specific form is

ℋ 2 = 8 π G 3 a 2 ρ t , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaahaaWcbeqaaGqaaiab+jdaYaaakiabg2da9KqbaoaalaaabaGaeGioaGdcciGae0hWdaNaem4raCeabaGaeG4mamdaaOGaemyyae2aaWbaaSqabeaacqaIYaGmaaGccqqFbpGCdaWgaaWcbaGaeeiDaqhabeaakiabcYcaSaaa@435D@

ℋ 2 − ℋ ′ = 4 π G a 2 ( ρ t + p t ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFlecsdaahaaWcbeqaaGqaaiab+jdaYaaakiabgkHiTiqb=TqiizaafaGaeyypa0JaeGinaqdcciGae0hWdaNaem4raCKaemyyae2aaWbaaSqabeaacqaIYaGmaaaccaGccqaFOaakcqqFbpGCdaWgaaWcbaGaeeiDaqhabeaakiabgUcaRiabdchaWnaaBaaaleaacqqG0baDaeqaaOGaeiykaKIaeiilaWcaaa@4951@

ρ ′ t + 3 ℋ ( ρ t + p t ) = 0 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFbpGCgaqbamaaBaaaleaacqqG0baDaeqaaOGaey4kaSIaeG4mamZenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae43cHGKaeiikaGIae8xWdi3aaSbaaSqaaiabbsha0bqabaGccqGHRaWkcqWGWbaCdaWgaaWcbaGaeeiDaqhabeaakiabcMcaPiabg2da9iabicdaWiabcYcaSaaa@462D@

where ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Tqiibaa@35AB@ = a'/a and the prime will denote, throughout the paper, a derivation with respect to τ. Equation (2.2) is indeed solution of Eqs. (2.3), (2.4) and (2.5) when the total energy density ρt is given by the sum of the matter density ρM and of the radiation density ρR (similarly pt = pR + pM).

Often, for notational convenience, the rescaled time coordinate x = τ/τ1 will be used. Within this x parametrization the critical fractions of radiation and dusty matter become

Ω R ( x ) = 1 α ( x ) + 1 = 1 ( x + 1 ) 2 , Ω M ( x ) = α ( x ) α ( x ) + 1 = x 2 + 2 x ( x + 1 ) 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeyyQdC1aaSbaaSqaaGqaaiab=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@64E6@

The redshift to equality is given, from Eq. (2.2), by

z e q + 1 ≃ ρ M 0 ρ R 0 = h 0 2 Ω M 0 h 0 2 Ω R 0 = 3228.91 ( h 0 2 Ω M 0 0.134 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEdaWgaaWcbaacbaGae8xzauMae8xCaehabeaakiabgUcaRiabigdaXiabloKi7KqbaoaalaaabaacciGae4xWdi3aaSbaaeaacqWFnbqtcqaIWaamaeqaaaqaaiab+f8aYnaaBaaabaGae8NuaiLaeGimaadabeaaaaGccqGH9aqpjuaGdaWcaaqaaiabdIgaOnaaDaaabaGaeGimaadabaGaeGOmaidaaiabgM6axnaaBaaabaGae8xta0KaeGimaadabeaaaeaacqWGObaAdaqhaaqaaiabicdaWaqaaiabikdaYaaacqGHPoWvdaWgaaqaaiab=jfasjabicdaWaqabaaaaOGaeyypa0JaeG4mamJaeGOmaiJaeGOmaiJaeGioaGJaeiOla4IaeGyoaKJaeGymaeZaaeWaaeaajuaGdaWcaaqaaiabdIgaOnaaDaaabaGaeGimaadabaGaeGOmaidaaiabgM6axnaaBaaabaGae8xta0KaeGimaadabeaaaeaacqaIWaamcqGGUaGlcqaIXaqmcqaIZaWmcqaI0aanaaaakiaawIcacaGLPaaacqGGUaGlaaa@60C8@

The redshift to recombination zrec is, approximately, between 1050 and 1150. From this hierarchy of scales, i.e. zdec > zrec, it appears that recombination takes place when the Universe is already dominated by matter. Furthermore, a decrease in the fraction of dusty matter delays the onset of the matter dominated epoch.

If the recombination happens suddenly, the ionization fraction xe drops abruptly from 1 to 10-5. Prior to recombination the photons interact with protons and electrons via Thompson scattering so that the relevant mean free path is, approximately,

λ T ( z r e c ) ≃ 1.8 x e ( 0.023 h 0 2 Ω b 0 ) ( 1100 1 + z r e c ) 2 ( 0.88 1 − Y p / 2 ) M p c , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6A33@

where Yp ≃ 0.24 is the abundance of 4He. Since mp = 0.938 GeV and me = 0.510 MeV, the mean free path of the photons will be essentially determined by the electrons because the Thompson cross section is smaller for protons than for electrons. Furthermore the protons and the electrons are even more tightly coupled, among them, by Coulomb scattering whose rate is larger than the Thompson rate of interaction. When the ionization fraction drops the photon mean free path gets as large as 104 Mpc. For the purposes of this investigation it will be also important to take into account, at least approximately, the finite thickness of the last scattering surface. This can be done by approximating the visibility function with a Gaussian profile 3940414243(see also 4445) with finite width. We recall that the visibility function simply gives the probability that a photon was last scattered between τ and τ + dτ (see section 4). The scale factor (2.2) can be used to express the ratios of two typical time-scales in terms of the ratio between the corresponding redshifts. So, for instance,

x rec = τ rec τ 1 = z eq + 1 z rec + 1 + 1 − 1 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaWgaaWcbaGaeeOCaiNaeeyzauMaee4yamgabeaakiabg2da9KqbaoaalaaabaacciGae8hXdq3aaSbaaeaacqqGYbGCcqqGLbqzcqqGJbWyaeqaaaqaaiab=r8a0naaBaaabaGaeGymaedabeaaaaGccqGH9aqpdaGcaaqaaKqbaoaalaaabaGaemOEaO3aaSbaaeaacqqGLbqzcqqGXbqCaeqaaiabgUcaRiabigdaXaqaaiabdQha6naaBaaabaGaeeOCaiNaeeyzauMaee4yamgabeaacqGHRaWkcqaIXaqmaaGccqGHRaWkcqaIXaqmaSqabaGccqGHsislcqaIXaqmcqGGSaalaaa@4E86@

which implies that, for zrec and h02ΩM0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdIgaOnaaDaaaleaacqaIWaamaeaacqaIYaGmaaGccqqHPoWvdaWgaaWcbaGaeeyta0KaeGimaadabeaaaaa@322E@ = 0.134, τrec = 1.01τ1.

There is another typical scale that plays an important role in the discussion of the Doppler oscillations. It is the baryon to photon ratio and it is defined as

R b ( z ) = 3 4 ρ b ρ γ = 0.664 ( h 0 2 Ω b 0 0.023 ) ( 1051 z + 1 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGsbGudaWgaaWcbaGaeeOyaigabeaakiabcIcaOiabdQha6jabcMcaPiabg2da9KqbaoaalaaabaGaeG4mamdabaGaeGinaqdaamaalaaabaacciGae8xWdi3aaSbaaeaacqqGIbGyaeqaaaqaaiab=f8aYnaaBaaabaGaeq4SdCgabeaaaaGaeyypa0JaeGimaaJaeiOla4IaeGOnayJaeGOnayJaeGinaqZaaeWaaeaadaWcaaqaaiabdIgaOnaaDaaabaGaeGimaadabaGaeGOmaidaaiabfM6axnaaBaaabaGaeeOyaiMaeGimaadabeaaaeaacqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaIZaWmaaaacaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiabigdaXiabicdaWiabiwda1iabigdaXaqaaiabdQha6jabgUcaRiabigdaXaaaaiaawIcacaGLPaaacqGGUaGlaaa@575B@

In the treatment of the angular power spectrum at intermediate angular scales Rb(z) appears ubiquitously either alone or in the expression of the sound speed of the photon-baryon system (see appendix A for further details)

c sb ( z ) = 1 3 ( R b ( z ) + 1 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGJbWydaWgaaWcbaGaee4CamNaeeOyaigabeaakiabcIcaOiabdQha6jabcMcaPiabg2da9KqbaoaalaaabaGaeGymaedabaWaaOaaaeaacqaIZaWmcqGGOaakcqWGsbGudaWgaaqaaiabbkgaIbqabaGaeiikaGIaemOEaONaeiykaKIaey4kaSIaeGymaeJaeiykaKcabeaaaaGaeiOla4caaa@3F88@

In the absence of a magnetized contribution, Rb(zrec) sets the height of the first Doppler peak as it can be easily argued by solving the evolution of the photon density contrast in the WKB approximation (see Eqs. (A.34) and (A.35)).

2.2 Plasma scales

The Debye scale and the plasma frequency of the electrons can be easily computed in terms of the cosmological parameters introduced so far. The results are, respectively:

λ D ( z ) = T e 8 π e 2 n e ≃ 4.26 x e ( 1050 z + 1 ) ( h 0 2 Ω b 0 0.023 ) − 1 / 2 c m , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6456@

ω p e ( z ) = 3.45 ( h 0 2 Ω b 0 0.023 ) 1 / 2 ( 1 + z 1050 ) 3 / 2 M H z . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@58E8@

By comparing Eqs. (2.8) and (2.12), λT ≫ λD both around equality and recombination. For typical scales comparable with the Hubble radius at recombination, therefore, the plasma will be, to an excellent approximation, globally neutral, i.e.

∇ → ⋅ E → = 4 π e ( n p − n e ) = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacuGHhis0gaWcaiabgwSixlqbdweafzaalaGaeyypa0JaeGinaqdcciGae8hWdaNaemyzauMaeiikaGIaemOBa42aaSbaaSqaaiabbchaWbqabaGccqGHsislcqWGUbGBdaWgaaWcbaGaeeyzaugabeaakiabcMcaPiabg2da9iabicdaWaaa@3EEF@

where E→(τ,x→)=a2(τ)ℰ→(τ,x→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdweafzaalaGaeiikaGccciGae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkcqGH9aqpcqWGHbqydaahaaWcbeqaaiabikdaYaaakiabcIcaOiab=r8a0jabcMcaPmrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiqb+btifzaalaGaeiikaGIae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkaaa@49B5@ denote the rescaled electric fields and where, by charge neutrality, the electron density equals the proton density, i.e.

n e ( z ) = n p ( z ) = x e η b n γ ( z ) , η b = 6.27 × 10 − 10 ( h 0 2 Ω b 0 0.023 ) ; MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaemOBa42aaSbaaSqaaiabbwgaLbqabaGccqGGOaakcqWG6bGEcqGGPaqkcqGH9aqpcqWGUbGBdaWgaaWcbaGaeeiCaahabeaakiabcIcaOiabdQha6jabcMcaPiabg2da9iabdIha4naaBaaaleaacqqGLbqzaeqaaGGacOGae83TdG2aaSbaaSqaaiabbkgaIbqabaGccqWGUbGBdaWgaaWcbaGae83SdCgabeaakiabcIcaOiabdQha6jabcMcaPiabcYcaSaqaaiab=D7aOnaaBaaaleaacqqGIbGyaeqaaOGaeyypa0JaeGOnayJaeiOla4IaeGOmaiJaeG4naCdaaiabgEna0kabigdaXiabicdaWmaaCaaaleqabaGaeyOeI0IaeGymaeJaeGimaadaaOWaaeWaaeaajuaGdaWcaaqaaiabdIgaOnaaDaaabaGaeGimaadabaGaeGOmaidaaiabgM6axnaaBaaabaGaeeOyaiMaeGimaadabeaaaeaacqaIWaamcqGGUaGlcqaIWaamcqaIYaGmcqaIZaWmaaaakiaawIcacaGLPaaacqGG7aWoaaa@6443@

ηb is the ratio between the baryonic charge density and the photon density. When the ionization fraction drops, the Debye scale is still the smallest length of the problem. From Eq. (2.13) the plasma frequency for the electrons is, around recombination, in the MHz range. The plasma frequency for the ions (essentially protons) will then be smaller (in the kHz range). Both these frequencies are smaller than the maximum of the CMB emission (which is, today, around 300 GHz and around 300 THz around recombination). Since the main focus of the present investigation will be on frequencies ω ≪ ωpe, the electromagnetic propagation of disturbances can be safely neglected and this implies, in terms of the rescaled electric and magnetic fields, that

∇ → × B → = 4 π J → , ∇ → ⋅ B → = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGafy4bIeTbaSaacqGHxdaTcuWGcbGqgaWcaiabg2da9iabisda0GGaciab=b8aWjqbdQeakzaalaGaeiilaWcabaGafy4bIeTbaSaacqGHflY1cuWGcbGqgaWcaaaacqGH9aqpcqaIWaamaaa@3BF3@

where B→(τ,x→)=a2ℬ→(τ,x→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaGaeiikaGccciGae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkcqGH9aqpcqWGHbqydaahaaWcbeqaaiabikdaYaaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaakiqb+XsiczaalaGaeiikaGIae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkaaa@4636@ and where

J → = σ c ( E → + v → × B → ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGkbGsgaWcaiabg2da9GGaciab=n8aZnaaBaaaleaacqqGJbWyaeqaaOGaeiikaGIafmyrauKbaSaacqGHRaWkcuWG2bGDgaWcaiabgEna0kqbdkeaczaalaGaeiykaKIaeiilaWcaaa@392C@

is the Ohmic current and σc = a(τ) σ¯c MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=n8aZzaaraWaaSbaaSqaaiabbogaJbqabaaaaa@2E50@ defined in terms of the rescaled conductivity. Since we are in the situation where T ≪ me, σ¯c=αem−1TT/me MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=n8aZzaaraWaaSbaaSqaaiabbogaJbqabaGccqGH9aqpcqWFXoqydaqhaaWcbaGaeeyzauMaeeyBa0gabaGaeyOeI0IaeGymaedaaOGaemivaq1aaOaaaeaacqWGubavcqGGVaWlcqWGTbqBdaWgaaWcbaGaeeyzaugabeaaaeqaaaaa@3BF8@. By now using the Ohmic electric field inside the remaining Maxwell equation, i.e.

∇ → × E → = − ∂ B → ∂ τ , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacuGHhis0gaWcaiabgEna0kqbdweafzaalaGaeyypa0JaeyOeI0scfa4aaSaaaeaaiiGacqWFciITcuWGcbGqgaWcaaqaaiab=jGi2kab=r8a0baakiabcYcaSaaa@384C@

the magnetic diffusivity equation can be obtained

∂ B → ∂ τ = ∇ → × ( v → × B → ) + 1 4 π σ c ∇ 2 B → . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaGGaciab=jGi2kqbdkeaczaalaaabaGae8NaIyRae8hXdqhaaOGaeyypa0Jafy4bIeTbaSaacqGHxdaTcqGGOaakcuWG2bGDgaWcaiabgEna0kqbdkeaczaalaGaeiykaKIaey4kaSscfa4aaSaaaeaacqaIXaqmaeaacqaI0aancqWFapaCcqWFdpWCdaWgaaWcbaGaee4yamgajuaGbeaaaaGccqGHhis0daahaaWcbeqaaiabikdaYaaakiqbdkeaczaalaGaeiOla4caaa@4968@

Equation (2.19) together with the previous equations introduced in the present subsection are the starting point of the magnetohydrodynamical (MHD) description adopted in the present paper. They hold for typical frequencies ω ≪ ωpe and for typical length scales much larger than the Debye scale. In this approximation (see Eq. (2.16)) the Ohmic current is solenoidal, i.e. ∇→⋅J→=0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbgEGirBaalaGaeyyXICTafmOsaOKbaSaacqGH9aqpcqaIWaamaaa@31FA@.

As in the flat-space case, the MHD equations can be obtained from a two-fluid description by combining the relevant equations and by using global variables. As a consequence of this derivation J→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdQeakzaalaaaaa@2C24@ will be the total current and v→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdAha2zaalaaaaa@2C7C@ will be the bulk velocity of the plasma, i.e. the centre-of-mass velocity of the electron-proton system 4647. It should be remembered that various phenomena involving the possible existence of a primordial magnetic field at recombination should not be treated within a single fluid approximation (as it will be done here) but rather within a two-fluid (or even kinetic) description. An example along this direction is Faraday rotation of the CMB polarization 48 or any other phenomenon where the electromagnetic branch of the plasma spectrum is relevant, i.e. ω > ωpe. In fact, the CMB is linearly polarized. So if a uniform magnetic field is present at recombination the polarization plane of the CMB can be rotated. From the appropriate dispersion relations (obtainable in the usual two-fluid description) the Faraday rotation rate can be computed bearing in mind that the Larmor frequency of electrons and ions at recombination, i.e.

ω Be = e B L ( τ rec ) m e c ≃ 18.08 ( B L ( τ rec ) 10 − 3 G ) kHz , ω Bi = e B L ( τ rec ) m i c ≃ 9.66 ( B L ( τ rec ) 10 − 3 G ) Hz , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqabeqacaaabaacciGae8xYdC3aaSbaaeaacqqGcbGqcqqGLbqzaeqaaiabg2da9maalaaabaGaemyzaugcbiGae4Nqai0aaSbaaeaacqqGmbataeqaaiabcIcaOiab=r8a0naaBaaabaGaeeOCaiNaeeyzauMaee4yamgabeaacqGGPaqkaeaacqWGTbqBdaWgaaqaaiabbwgaLbqabaGaem4yamgaaiabloKi7iabigdaXiabiIda4iabc6caUiabicdaWiabiIda4maabmaabaWaaSaaaeaacqGFcbGqdaWgaaqaaiabbYeambqabaGaeiikaGIae8hXdq3aaSbaaeaacqqGYbGCcqqGLbqzcqqGJbWyaeqaaiabcMcaPaqaaiabigdaXiabicdaWmaaCaaabeqaaiabgkHiTiabiodaZaaacqqGhbWraaaacaGLOaGaayzkaaGaee4AaSMaeeisaGKaeeOEaONaeiilaWcabaGae8xYdC3aaSbaaeaacqqGcbGqcqqGPbqAaeqaaiabg2da9maalaaabaGaemyzauMae4Nqai0aaSbaaeaacqqGmbataeqaaiabcIcaOiab=r8a0naaBaaabaGaeeOCaiNaeeyzauMaee4yamgabeaacqGGPaqkaeaacqWGTbqBdaWgaaqaaiabbMgaPbqabaGaem4yamgaaiabloKi7iabiMda5iabc6caUiabiAda2iabiAda2maabmaabaWaaSaaaeaacqGFcbGqdaWgaaqaaiabbYeambqabaGaeiikaGIae8hXdq3aaSbaaeaacqqGYbGCcqqGLbqzcqqGJbWyaeqaaiabcMcaPaqaaiabigdaXiabicdaWmaaCaaabeqaaiabgkHiTiabiodaZaaacqqGhbWraaaacaGLOaGaayzkaaGaeeisaGKaeeOEaONaeiilaWcaaaaa@88A1@

are both smaller than ωpe. In Eq. (2.20) BL(τrec) is the smoothed magnetic field strength at recombination.

It is the moment to spell out clearly two concepts that are central to the discussion of the evolution of large-scale magnetic fields in a FRW Universe with line element (2.1):

• the concept of comoving and physical magnetic fields;

• the concept of stochastic magnetic field.

The comoving magnetic field B→(τ,x→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaGaeiikaGccciGae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkaaa@31FD@ is related to the physical magnetic field ℬ→(τ,x→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiqb=XsiczaalaGaeiikaGccciGae4hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkaaa@3BC0@ as B→(τ,x→)=a2(τ)ℬ→(τ,x→) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaGaeiikaGccciGae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkcqGH9aqpcqWGHbqydaahaaWcbeqaaiabikdaYaaakiabcIcaOiab=r8a0jabcMcaPmrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiqb+XsiczaalaGaeiikaGIae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkaaa@49A8@. We will choose as the reference time the epoch of gravitational collapse of the protogalaxy. At this time the comoving and physical field coincide. So, for instance, a (physical) magnetic field of nG strength at the onset of gravitational collapse will be roughly of the order of the mG (i.e. 10-3 G) at the epoch of recombination. This conclusion stems directly from the fact that the physical magnetic field scales with a-2(τ), i.e. with z2 where z, as usual is the redshift. This implies, in turn, that B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaaaaa@2C14@ (i.e. the comoving field) is roughly constant (in time) if the plasma does not have sizable kinetic helicity5(i.e. 〈v→⋅∇→×v→〉=0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgMYiHlqbdAha2zaalaGaeyyXICTafy4bIeTbaSaacqGHxdaTcuWG2bGDgaWcaiabgQYiXlabg2da9iabicdaWaaa@3973@) (see, for instance, 152021). In this situation Eq. (2.19) dictates that B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaaaaa@2C14@ is constant for typical wave-numbers k <kσ (i.e. for sufficiently large comoving length-scales) where kσ sets the magnetic diffusivity scale whose value, at recombination, is