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 Ω_M0 is the present critical fraction in non-relativistic matter, i.e. Ω_M0 = Ω_b0 + Ω_c0, 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 τ_eq = (2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamaakaaabaGaeGOmaidaleqaaaaa@2C02@ - 1)τ₁, i.e. roughly, τ_eq ≃ 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 ρ_t is given by the sum of the matter density ρ_M and of the radiation density ρ_R (similarly p_t = p_R + p_M).

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 z_rec is, approximately, between 1050 and 1150. From this hierarchy of scales, i.e. z_dec > z_rec, 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 x_e 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 Y_p ≃ 0.24 is the abundance of ⁴He. Since m_p = 0.938 GeV and m_e = 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 z_rec and h02ΩM0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdIgaOnaaDaaaleaacqaIWaamaeaacqaIYaGmaaGccqqHPoWvdaWgaaWcbaGaeeyta0KaeGimaadabeaaaaa@322E@ = 0.134, τ_rec = 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 R_b(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, R_b(z_rec) 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)).

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), λ_T ≫ λ_D 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@

η_b 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 ≪ ω_pe, 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 σ_c = 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 ≪ m_e, 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 ≪ ω_pe 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 > ω_pe. 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 ω_pe. In Eq. (2.20) B_L(τ_rec) 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 <k_σ (i.e. for sufficiently large comoving length-scales) where k_σ sets the magnetic diffusivity scale whose value, at recombination, is

1 k σ ≃ 1.24 × 10 − 14 ( h 0 2 Ω M 0 0.134 ) − 3 / 4 Mpc . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabigdaXaqaaiabdUgaRnaaBaaabaacciGae83WdmhabeaaaaGccqWIdjYocqaIXaqmcqGGUaGlcqaIYaGmcqaI0aancqGHxdaTcqaIXaqmcqaIWaamdaahaaWcbeqaaiabgkHiTiabigdaXiabisda0aaakmaabmaabaqcfa4aaSaaaeaacqWGObaAdaqhaaqaaiabicdaWaqaaiabikdaYaaacqGHPoWvdaWgaaqaaiabb2eanjabicdaWaqabaaabaGaeGimaaJaeiOla4IaeGymaeJaeG4mamJaeGinaqdaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcqaIZaWmcqGGVaWlcqaI0aanaaGccqqGnbqtcqqGWbaCcqqGJbWycqGGUaGlaaa@51C6@

Equation (2.21) can be compared with the estimate of the diffusive scale associated with Silk damping:

1 k D τ rec = 9.63 × 10 − 3 ( h 0 2 Ω b 0.023 ) − 1 / 2 ( h 0 2 Ω M 0.134 ) 1 / 4 ( 1050 z rec ) 3 / 4 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabigdaXaqaaiabdUgaRnaaBaaabaGaeeiraqeabeaaiiGacqWFepaDdaWgaaqaaiabbkhaYjabbwgaLjabbogaJbqabaaaaOGaeyypa0JaeGyoaKJaeiOla4IaeGOnayJaeG4mamJaey41aqRaeGymaeJaeGimaaZaaWbaaSqabeaacqGHsislcqaIZaWmaaGcdaqadaqaaKqbaoaalaaabaGaemiAaG2aa0baaeaacqaIWaamaeaacqaIYaGmaaGaeyyQdC1aaSbaaeaacqqGIbGyaeqaaaqaaiabicdaWiabc6caUiabicdaWiabikdaYiabiodaZaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaeGymaeJaei4la8IaeGOmaidaaOWaaeWaaeaajuaGdaWcaaqaaiabdIgaOnaaDaaabaGaeGimaadabaGaeGOmaidaaiabgM6axnaaBaaabaGaeeyta0eabeaaaeaacqaIWaamcqGGUaGlcqaIXaqmcqaIZaWmcqaI0aanaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabigdaXiabc+caViabisda0aaakmaabmaabaqcfa4aaSaaaeaacqaIXaqmcqaIWaamcqaI1aqncqaIWaamaeaacqWG6bGEdaWgaaqaaiabbkhaYjabbwgaLjabbogaJbqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIZaWmcqGGVaWlcqaI0aanaaGccqGGUaGlaaa@7002@

Hence, for the typical value of the matter fraction appearing in Eq. (2.21), τ_rec ≃ τ₁ and, consequently k_σ ≫ k_D. While finite conductivity effects are rather efficient in washing out the magnetic fields for large wave-numbers, the thermal diffusivity effects (related to shear viscosity and, ultimately, to Silk damping) affect typical wave-numbers that are much smaller than the ones affected by conductivity.

Under the conditions of MHD, two (approximate) conservations laws may be derived, namely the magnetic flux conservation

d d τ ∫ Σ B → ⋅ d Σ → = − 1 4 π σ c ∫ Σ ∇ → × ∇ → × B → ⋅ d Σ → , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKbqaaiabdsgaKHGaciab=r8a0baadaWdraqaaiqbdkeaczaalaGaeyyXICTaemizaqMafu4OdmLbaSaaaSqaaiabfo6atbqab0Gaey4kIipakiabg2da9iabgkHiTKqbaoaalaaabaGaeGymaedabaGaeGinaqJae8hWdaNae83Wdm3aaSbaaeaacqqGJbWyaeqaaaaakmaapebabaGafy4bIeTbaSaacqGHxdaTcuGHhis0gaWcaiabgEna0kqbdkeaczaalaGaeyyXICTaemizaqMafu4OdmLbaSaaaSqaaiabfo6atbqab0Gaey4kIipakiabcYcaSaaa@5445@

and the magnetic helicity conservation

d d τ ( ∫ V d 3 x A → ⋅ B → ) = − 1 4 π σ c ∫ V d 3 x   B → ⋅ ∇ → × B → . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKbqaaiabdsgaKHGaciab=r8a0baakmaabmaabaWaa8qeaeaacqWGKbazdaahaaWcbeqaaiabiodaZaaakiabdIha4jqbdgeabzaalaGaeyyXICTafmOqaiKbaSaaaSqaaiabdAfawbqab0Gaey4kIipaaOGaayjkaiaawMcaaiabg2da9iabgkHiTKqbaoaalaaabaGaeGymaedabaGaeGinaqJae8hWdaNae83Wdm3aaSbaaeaacqqGJbWyaeqaaaaakmaapebabaGaemizaq2aaWbaaSqabeaacqaIZaWmaaGccqWG4baEcqqGGaaicuWGcbGqgaWcaiabgwSixlqbgEGirBaalaGaey41aqRafmOqaiKbaSaaaSqaaiabdAfawbqab0Gaey4kIipakiabc6caUaaa@573C@

In Eq. (2.23) Σ is an arbitrary closed surface that moves with the plasma. In Eq. (2.24) A→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdgeabzaalaaaaa@2C12@ is the vector potential. According to Eq. (2.23), in MHD the magnetic field has to be always solenoidal (i.e. ∇→⋅B→=0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbgEGirBaalaGaeyyXICTafmOqaiKbaSaacqGH9aqpcqaIWaamaaa@31EA@). Thus, the magnetic flux conservation implies that, in the ideal MHD limit (i.e. σ_c → ∞) the magnetic flux lines, closed because of the transverse nature of the field, evolve always glued together with the plasma element. In this approximation, as far as the magnetic field evolution is concerned, the plasma is a collection of (closed) flux tubes. The theorem of flux conservation states then that the energetical properties of large-scale magnetic fields are conserved throughout the plasma evolution.

While the flux conservation concerns the energetic properties of the magnetic flux lines, the magnetic helicity, i.e. Eq. (2.24), concerns chiefly the topological properties of the magnetic flux lines. In the simplest situation, the magnetic flux lines will be closed loops evolving independently in the plasma and the helicity will vanish. There could be, however, more complicated topological situations 51 where a single magnetic loop is twisted (like some kind of Möbius stripe) or the case where the magnetic loops are connected like the rings of a chain: now the non-vanishing magnetic helicity measures, essentially, the number of links and twists in the magnetic flux lines 47. Furthermore, in the superconducting limit, the helicity will not change throughout the time evolution. The conservation of the magnetic flux and of the magnetic helicity is a consequence of the fact that, in ideal MHD, the Ohmic electric field is always orthogonal both to the bulk velocity field and to the magnetic field. In the resistive MHD approximation this conclusion may not apply. The quantity at the right-hand-side of Eq. (2.24), i.e. B→⋅∇→×B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaGaeyyXICTafy4bIeTbaSaacqGHxdaTcuWGcbGqgaWcaaaa@332C@ is called magnetic gyrotropy and it is a gauge-invariant measure of the number of contact points in the magnetic flux lines. As we shall see in a moment, the only stochastic fields contributing to the scalar fluctuations of the goemetry are the ones for which the magnetic gyrotropy vanishes.

Nearly all mechanisms able to generate large scale magnetic fields imply the existence of a stochastic background of magnetic disturbances 15 that could be written, in Fourier space, as ⁶

〈 B i ( k → , τ ) B j ( p → , τ ) 〉 = P i j ( k ) δ ( 3 ) ( k → + p → ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadaqaaiabdkeacnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIafm4AaSMbaSaacqGGSaaliiGacqWFepaDcqGGPaqkcqWGcbGqdaWgaaWcbaGaemOAaOgabeaakiabcIcaOiqbdchaWzaalaGaeiilaWIae8hXdqNaeiykaKcacaGLPmIaayPkJaGaeyypa0Jaemiuaa1aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGOaakcqWGRbWAcqGGPaqkcqWF0oazdaahaaWcbeqaaiabcIcaOiabiodaZiabcMcaPaaakiabcIcaOiqbdUgaRzaalaGaey4kaSIafmiCaaNbaSaacqGGPaqkcqGGSaalaaa@5009@

where

P i j ( k ) = Q ( k ) ( δ i j − k i k j k 2 ) , Q ( k ) = Q 0 k m . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5EBC@

From Eq. (2.26) the magnetic field configuration of Eq. (2.25) depends on the amplitude of the field Q0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=br8rnaaBaaaleaacqaIWaamaeqaaaaa@37A3@ and on the spectral index m.

It is often useful, in practical estimates, to regularize the two-point function by using an appropriate "windowing". Two popular windows are, respectively, the Gaussian and the top-hat functions, i.e.

W g ( k , L ) = e − k 2 L 2 2 , W t h ( k , L ) = 3 k L j 1 ( k L ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5D47@

For instance, the regularized magnetic energy density with Gaussian filter can be obtained from the previous expressions by shifting Q(k)→Q(k)Wg2(k,L) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=br8rjabcIcaOiabdUgaRjabcMcaPiabgkziUkab=br8rjabcIcaOiabdUgaRjabcMcaPiabdEfaxnaaDaaaleaacqqGNbWzaeaacqaIYaGmaaGccqGGOaakcqWGRbWAcqGGSaalcqWGmbatcqGGPaqkaaa@4936@. The result is

〈 B i ( τ , x → ) B i ( τ , x → + r → ) 〉 = 2 Q 0 ( 2 π ) 2 Γ ( m + 3 2 ) L 3 + m F ( m + 3 2 , 3 2 , − r 2 4 L 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadaqaaiabdkeacnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGccciGae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkcqWGcbGqdaahaaWcbeqaaiabdMgaPbaakiabcIcaOiab=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@733A@

where F(a, b, x) ≡₁ F₁(a, b, x) is the confluent hypergeometric function 5253. Notice that the integral appearing in the trace converges for m > -3. The amplitude of the magnetic power spectrum Q0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=br8rnaaBaaaleaacqaIWaamaeqaaaaa@37A3@ can be traded for BL2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkeacnaaDaaaleaacqqGmbataeaacqaIYaGmaaaaaa@2E40@ where BL2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabdkeacnaaDaaaleaacqqGmbataeaacqaIYaGmaaaaaa@2E40@ is by definition the regularized two-point function evaluated at coincident spatial points, i.e.

B L 2 = lim ⁡ r → 0 〈 B i ( τ , x → ) B i ( τ , x → + r → ) 〉 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGcbGqdaqhaaWcbaGaeeitaWeabaGaeGOmaidaaOGaeyypa0ZaaCbeaeaacyGGSbaBcqGGPbqAcqGGTbqBaSqaaiabdkhaYjabgkziUkabicdaWaqabaGcdaaadaqaaiabdkeacnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGccciGae8hXdqNaeiilaWIafmiEaGNbaSaacqGGPaqkcqWGcbGqdaahaaWcbeqaaiabdMgaPbaakiabcIcaOiab=r8a0jabcYcaSiqbdIha4zaalaGaey4kaSIafmOCaiNbaSaacqGGPaqkaiaawMYicaGLQmcaaaa@4C8C@

Combining Eq. (2.28) with Eq. (2.29) we have that Q0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=br8rnaaBaaaleaacqaIWaamaeqaaaaa@37A3@ becomes

Q 0 = ( 2 π ) m + 5 2 k L − ( 3 + m ) Γ ( m + 3 2 ) B L 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGimaadabeaakiabg2da9KqbaoaalaaabaGaeiikaGIaeGOmaidcciGae4hWdaNaeiykaKYaaWbaaeqabaGaemyBa0Maey4kaSIaeGynaudaaaqaaiabikdaYaaadaWcaaqaaiabdUgaRnaaDaaabaacbaGae0htaWeabaGaeyOeI0IaeiikaGIaeG4mamJaey4kaSIaemyBa0MaeiykaKcaaaqaaiabfo5ahnaabmaabaWaaSaaaeaacqWGTbqBcqGHRaWkcqaIZaWmaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaOGaemOqai0aa0baaSqaaiab9XeambqaaiabikdaYaaakiabcYcaSaaa@555C@

where k_L = 2π/L. The two main parameters that will therefore characterize the magnetic background will be the smoothed amplitude B_L and the spectral slope. For reasons related to the way power spectra are assigned for curvature perturbations, it will be practical to define the magnetic spectral index as ε = m + 3 (see Eqs. (3.40)-(3.41) and comments therein).

In the case of the configuration (2.25) the magnetic gyrotropy is vanishing, i.e. 〈B→⋅∇→×B→〉=0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabgMYiHlqbdkeaczaalaGaeyyXICTafy4bIeTbaSaacqGHxdaTcuWGcbGqgaWcaiabgQYiXlabg2da9iabicdaWaaa@38A3@. There are situations where magnetic fields are produced in a state with non-vanishing gyrotropy (or helicity) (see for instance 54 and references therein). In the latter case, the two point function can be written in the same form given in Eq. (2.25)

〈 B i ( k → , τ ) B j ( p → , τ ) 〉 = P ˜ i j ( k ) δ ( 3 ) ( k → + p → ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadaqaaiabdkeacnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIafm4AaSMbaSaacqGGSaaliiGacqWFepaDcqGGPaqkcqWGcbGqdaWgaaWcbaGaemOAaOgabeaakiabcIcaOiqbdchaWzaalaGaeiilaWIae8hXdqNaeiykaKcacaGLPmIaayPkJaGaeyypa0JafmiuaaLbaGaadaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcIcaOiabdUgaRjabcMcaPiab=r7aKnaaCaaaleqabaGaeiikaGIaeG4mamJaeiykaKcaaOGaeiikaGIafm4AaSMbaSaacqGHRaWkcuWGWbaCgaWcaiabcMcaPiabcYcaSaaa@5018@

but where now

P ˜ i j ( k ) = Q ( k ) ( δ i j − k i k j k 2 ) + i Q ˜ ( k ) ε i j ℓ k ℓ k , Q ˜ ( k ) = Q ˜ 0 k m ˜ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGafmiuaaLbaGaadaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcIcaOiabdUgaRjabcMcaPiabg2da9mrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=br8rjabcIcaOiabdUgaRjabcMcaPmaabmaabaacciGae4hTdq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGHsisljuaGdaWcaaqaaiabdUgaRnaaBaaabaGaemyAaKgabeaacqWGRbWAdaWgaaqaaiabdQgaQbqabaaabaGaem4AaS2aaWbaaeqabaGaeGOmaidaaaaaaOGaayjkaiaawMcaaiabgUcaRiabdMgaPjqb=br8rzaaiaGaeiikaGIaem4AaSMaeiykaKIae4xTdu2aaSbaaSqaaiabdMgaPjabdQgaQjabloriSbqabaqcfa4aaSaaaeaacqWGRbWAdaahaaqabeaacqWItecBaaaabaGaem4AaSgaaOGaeiilaWcabaGaf8heXhLbaGaacqGGOaakcqWGRbWAcqGGPaqkcqGH9aqpcuWFqeFugaacamaaBaaaleaacqaIWaamaeqaaOGaem4AaS2aaWbaaSqabeaacuWGTbqBgaacaaaaaaGccqGGUaGlaaa@709B@

From Eq. (2.32) we can appreciate that, on top of the parity-invariant contribution (already defined in Eqs. (2.25) and (2.26)), there is a second term proportional to the Levi-Civita ε_ijℓ . In Fourier space, the introduction of gyrotropic configurations implies also the presence of a second function of the momentum Q˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiqb=br8rzaaiaaaaa@3698@(k). In the case of scalar fluctuations of the geometry this second power spectrum will not give any contribution (but it does contribute to the vector modes of the geometry as well as in the case of the tensor modes).

The correlators that contribute to the evolution of the scalar fluctuations of the geometry will be essentially the ones of magnetic energy density and pressure (i.e. B→2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaWaaWbaaSqabeaacqaIYaGmaaaaaa@2D33@/(8π) and B→2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbdkeaczaalaWaaWbaaSqabeaacqaIYaGmaaaaaa@2D33@/(24π)) and the one related to the divergence of the MHD Lorentz force (i.e. ∇→⋅[J→×B→] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbgEGirBaalaGaeyyXICTaei4waSLafmOsaOKbaSaacqGHxdaTcuWGcbGqgaWcaiabc2faDbaa@35BC@) which appears as source term in the evolution equation of the divergence of the peculiar velocity of the baryons (see Eqs. (A.23) and (A.25) of the appendix A). Since in MHD 4πJ→=∇→×B→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabisda0GGaciab=b8aWjqbdQeakzaalaGaeyypa0Jafy4bIeTbaSaacqGHxdaTcuWGcbGqgaWcaaaa@34B2@ the divergence of the Lorentz force will be proportional to ∇→⋅[(∇→×B→)×B→] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbgEGirBaalaGaeyyXICTaei4waSLaeiikaGIafy4bIeTbaSaacqGHxdaTcuWGcbGqgaWcaiabcMcaPiabgEna0kqbdkeaczaalaGaeiyxa0faaa@3B0D@. The magnetic anisotropic stress Π˜ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbfc6aqzaaiaWaa0baaSqaaiabdMgaPbqaaiabdQgaQbaaaaa@2F67@ does also contribute to the scalar problem but it can be related, through simple vector identities, to the magnetic energy density and to the divergence of the Lorentz force (see Eqs. (A.28) and (A.29)). To specify the effect of the stochastic background of magnetic fields on the scalar modes of the geometry we shall therefore need the correlation functions of two dimensionless quantities denoted, in what follows, by Ω_B and σ_B, i.e.

Ω B ( τ , x → ) = 1 a 4 ( τ ) ρ γ ( τ ) B → 2 ( τ , x → ) 8 π , ∂ j ∂ i Π ˜ j i = ( p γ + ρ γ ) ∇ 2 σ B MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeyyQdC1aaSbaaSqaaiabbkeacbqabaGccqGGOaakiiGacqWFepaDcqGGSaalcuWG4baEgaWcaiabcMcaPiabg2da9KqbaoaalaaabaGaeGymaedabaGaemyyae2aaWbaaeqabaGaeGinaqdaaiabcIcaOiab=r8a0jabcMcaPiab=f8aYnaaBaaabaGae83SdCgabeaacqGGOaakcqWFepaDcqGGPaqkaaWaaSaaaeaacuWGcbGqgaWcamaaCaaabeqaaiabikdaYaaacqGGOaakcqWFepaDcqGGSaalcuWG4baEgaWcaiabcMcaPaqaaiabiIda4iab=b8aWbaakiabcYcaSaqaaiab=jGi2oaaBaaaleaacqWGQbGAaeqaaOGae8NaIy7aaWbaaSqabeaacqWGPbqAaaGccuqHGoaugaacamaaDaaaleaacqWGQbGAaeaacqWGPbqAaaGccqGH9aqpcqGGOaakcqWGWbaCdaWgaaWcbaGae83SdCgabeaakiabgUcaRiab=f8aYnaaBaaaleaacqWFZoWzaeqaaOGaeiykaKIaey4bIe9aaWbaaSqabeaacqaIYaGmaaGccqWFdpWCdaWgaaWcbaGaeeOqaieabeaaaaaaaa@68A4@

where ρ_γ is the energy density of the photons. Since Ω_B and σ_B are both quadratic in the magnetic field intensity, their corresponding two-point functions will be quartic in the magnetic field intensities. Consequently Ω_B and σ_B will have Fourier transforms that are defined as convolutions of the original magnetic fields and, more precisely:

Ω B ( x → , τ ) = δ ρ B ( x → , τ ) ρ γ ( τ ) = 1 ( 2 π ) 3 / 2 ∫ d 3 q   Ω B ( q → , τ ) e − i q → ⋅ x → , δ p B = δ ρ B 3 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaGaeyyQdC1aaSbaaSqaaiabbkeacbqabaGccqGGOaakcuWG4baEgaWcaiabcYcaSGGaciab=r8a0jabcMcaPiabg2da9KqbaoaalaaabaGae8hTdqMae8xWdi3aaSbaaeaacqqGcbGqaeqaaiabcIcaOiqbdIha4zaalaGaeiilaWIae8hXdqNaeiykaKcabaGae8xWdi3aaSbaaeaacqWFZoWzaeqaaiabcIcaOiab=r8a0jabcMcaPaaakiabg2da9KqbaoaalaaabaGaeGymaedabaGaeiikaGIaeGOmaiJae8hWdaNaeiykaKYaaWbaaeqabaGaeG4mamJaei4la8IaeGOmaidaaaaakmaapeaabaGaemizaq2aaWbaaSqabeaacqaIZaWmaaGccqWGXbqCcqqGGaaicqGHPoWvdaWgaaWcbaGaeeOqaieabeaaaeqabeqdcqGHRiI8aOGaeiikaGIafmyCaeNbaSaacqGGSaalcqWFepaDcqGGPaqkcqWGLbqzdaahaaWcbeqaaiabgkHiTiabdMgaPjqbdghaXzaalaGaeyyXICTafmiEaGNbaSaaaaGccqGGSaalaeaacqWF0oazcqWGWbaCdaWgaaWcbaGaeeOqaieabeaaaaGccqGH9aqpjuaGdaWcaaqaaiab=r7aKjab=f8aYnaaBaaabaGaeeOqaieabeaaaeaacqaIZaWmaaGccqGGSaalaaa@7598@

σ B ( x → , τ ) = 1 ( 2 π ) 3 / 2 ∫ d 3 q e − i q → ⋅ x → σ B ( q → , τ ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaWgaaWcbaacbaGae4NqaieabeaakiabcIcaOiqbdIha4zaalaGaeiilaWIae8hXdqNaeiykaKIaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqGGOaakcqaIYaGmcqWFapaCcqGGPaqkdaahaaqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaaaaOWaa8qaaeaacqWGKbazdaahaaWcbeqaaiabiodaZaaakiabdghaXbWcbeqab0Gaey4kIipakiabdwgaLnaaCaaaleqabaGaeyOeI0IaemyAaKMafmyCaeNbaSaacqGHflY1cuWG4baEgaWcaaaakiab=n8aZnaaBaaaleaacqGFcbGqaeqaaOGaeiikaGIafmyCaeNbaSaacqGGSaalcqWFepaDcqGGPaqkcqGGSaalaaa@55E5@

where

Ω B ( q → , τ ) = 1 ( 2 π ) 3 / 2 1 8 π ρ ¯ γ ∫ d 3 p B i ( p → ) B i ( q → − p → ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHPoWvdaWgaaWcbaGaeeOqaieabeaakiabcIcaOiqbdghaXzaalaGaeiilaWccciGae8hXdqNaeiykaKIaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqGGOaakcqaIYaGmcqWFapaCcqGGPaqkdaahaaqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaaaamaalaaabaGaeGymaedabaGaeGioaGJae8hWdaNaf8xWdiNbaebadaWgaaqaaiab=n7aNbqabaaaaOWaa8qaaeaacqWGKbazdaahaaWcbeqaaiabiodaZaaakiabdchaWjabdkeacnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIafmiCaaNbaSaacqGGPaqkcqWGcbGqdaahaaWcbeqaaiabdMgaPbaakiabcIcaOiqbdghaXzaalaGaeyOeI0IafmiCaaNbaSaacqGGPaqkcqGGSaalaSqabeqaniabgUIiYdaaaa@58F4@

σ B ( q → , τ ) = 1 ( 2 π ) 3 / 2 1 16 π ρ ¯ γ 1 q 2 ∫ d 3 p [ 3 ( q j − p j ) p i B j ( p → ) B i ( q → − p → ) − q 2 B i ( q → − p → ) B i ( p → ) ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaWgaaWcbaGaeeOqaieabeaakiabcIcaOiqbdghaXzaalaGaeiilaWIae8hXdqNaeiykaKIaeyypa0tcfa4aaSaaaeaacqaIXaqmaeaacqGGOaakcqaIYaGmcqWFapaCcqGGPaqkdaahaaqabeaacqaIZaWmcqGGVaWlcqaIYaGmaaaaamaalaaabaGaeGymaedabaGaeGymaeJaeGOnayJae8hWdaNaf8xWdiNbaebadaWgaaqaaiab=n7aNbqabaaaamaalaaabaGaeGymaedabaGaemyCae3aaWbaaeqabaGaeGOmaidaaaaakmaapeaabaGaemizaq2aaWbaaSqabeaacqaIZaWmaaGccqWGWbaCdaWadaqaaiabiodaZiabcIcaOiabdghaXnaaCaaaleqabaGaemOAaOgaaOGaeyOeI0IaemiCaa3aaWbaaSqabeaacqWGQbGAaaGccqGGPaqkcqWGWbaCdaahaaWcbeqaaiabdMgaPbaakiabdkeacnaaBaaaleaacqWGQbGAaeqaaOGaeiikaGIafmiCaaNbaSaacqGGPaqkcqWGcbGqdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiqbdghaXzaalaGaeyOeI0IafmiCaaNbaSaacqGGPaqkcqGHsislcqWGXbqCdaahaaWcbeqaaiabikdaYaaakiabdkeacnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIafmyCaeNbaSaacqGHsislcuWGWbaCgaWcaiabcMcaPiabdkeacnaaCaaaleqabaGaemyAaKgaaOGaeiikaGIafmiCaaNbaSaacqGGPaqkaiaawUfacaGLDbaacqGGUaGlaSqabeqaniabgUIiYdaaaa@7D91@

having defined, for notational convenience, ρ¯γ=ργ(τ)a4(τ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f8aYzaaraWaaSbaaSqaaiab=n7aNbqabaGccqGH9aqpcqWFbpGCdaWgaaWcbaGae83SdCgabeaakiabcIcaOiab=r8a0jabcMcaPiabdggaHnaaCaaaleqabaGaeGinaqdaaOGaeiikaGIae8hXdqNaeiykaKcaaa@3CA1@.

Consider the large angular scales that were outside the horizon at recombination. While smaller angular scales (compatible with the first Doppler peak) necessarily demand the inclusion of finite thickness effects of the last scattering surface, the largest angular scales (corresponding to harmonics ℓ ≤ 25) can be safely treated in the approximation that the visibility function is a Dirac delta function centered around τ_rec. Moreover, for the modes satisfying the condition kτ_rec < 1 the radiation-matter transition takes place when the relevant modes have wavelengths still larger than the Hubble radius.

It is practical, for the present purposes, to think the matter-radiation fluid as a unique physical entity with time-dependent barotropic index and time-dependent sound speed:

w t ( α ) = p t ρ t = 1 3 ( α + 1 ) , c st 2 = p ′ t ρ ′ t = 4 3 ( 4 + 3 α ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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f7aHjabcMcaPaaacqGGSaalaaaaaa@5A3B@

where α = a/a_eq. According to Eq. (3.1), when a ≫ a_eq both cst2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGqaciab=ngaJnaaDaaaleaacqqGZbWCcqqG0baDaeaacqaIYaGmaaaaaa@3046@ and w_t go to zero (as appropriate when matter dominates) while in the opposite limit (i.e. α ≪ 1) cst2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGqaciab=ngaJnaaDaaaleaacqqGZbWCcqqG0baDaeaacqaIYaGmaaaaaa@3046@ ≃ w_t → 1/3 which is the usual result of the radiation epoch. Since recombination takes place after equality it will be crucial, for the present purposes, to determine the perturbations of the spatial curvature at this moment. The presence of fully inhomogeneous magnetic fields affects the evolution of the curvature perturbations across the radiation-matter transition. This issue has been addressed in 34 by following, outside the Hubble radius, the evolution of the gauge-invariant density contrast on uniform density hypersurfaces (customarily denoted by ζ):

ζ = − ψ + ℋ ( δ ρ t + δ ρ B ) ρ ′ t . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF2oGEcqGH9aqpcqGHsislcqWFipqEcqGHRaWkjuaGdaWcaaqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab+TqiijabcIcaOiab=r7aKjab=f8aYnaaBaaabaacbaGae0hDaqhabeaacqGHRaWkcqWF0oazcqWFbpGCdaWgaaqaaiabbkeacbqabaGaeiykaKcabaGaf8xWdiNbauaadaWgaaqaaiabbsha0bqabaaaaOGaeiOla4caaa@4C59@

where ψ is related to the fluctuation of the spatial component of the metric (i.e. δ_sg_ij = 2a²ψδ_ij in the conformally Newtonian gauge) and

δ ρ t = δ ρ γ + δ ρ ν + δ ρ c + δ ρ b , δ ρ B = B 2 ( x → ) 8 π a 4 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaacciGae8hTdqMae8xWdi3aaSbaaSqaaiabbsha0bqabaGccqGH9aqpcqWF0oazcqWFbpGCdaWgaaWcbaGae83SdCgabeaakiabgUcaRiab=r7aKjab=f8aYnaaBaaaleaacqWF9oGBaeqaaOGaey4kaSIae8hTdqMae8xWdi3aaSbaaSqaaiabbogaJbqabaGccqGHRaWkcqWF0oazcqWFbpGCdaWgaaWcbaGaeeOyaigabeaakiabcYcaSaqaaiab=r7aKjab=f8aYnaaBaaaleaacqqGcbGqaeqaaOGaeyypa0tcfa4aaSaaaeaacqWGcbGqdaahaaqabeaacqaIYaGmaaGaeiikaGIafmiEaGNbaSaacqGGPaqkaeaacqaI4aaocqWFapaCcqWGHbqydaahaaqabeaacqaI0aanaaaaaaaakiabcYcaSaaa@59CB@

are, respectively, the total density fluctuation of the fluid sources (i.e. photons, neutrinos, CDM and baryons) and the density fluctuations induced by a fully inhomogeneous magnetic field. The gauge-invariant density contrast on uniform curvature hypersurfaces is related, via the Hamiltonian constraint (see Eq. (A.5)), to the curvature perturbations on comoving orthogonal hypersufaces customarily denoted by ℛ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Trisbaa@35C5@. Since both ℛ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Trisbaa@35C5@ and ζ are gauge-invariant, their mutual relation can be worked out in any gauge and, in particular, in the conformally Newtonian gauge where ℛ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Trisbaa@35C5@ can be expressed as 13

ℛ = − ψ − 2 ρ t 3 ( ρ t + p t ) ( φ + ψ ′ ℋ ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFBeIucqWFGaaicqGH9aqpcqGHsisliiGacqGFipqEcqGHsisljuaGdaWcaaqaaiabikdaYiab+f8aYnaaBaaabaGaeeiDaqhabeaaaeaacqaIZaWmcqGGOaakcqGFbpGCdaWgaaqaaiabbsha0bqabaGaey4kaSIaemiCaa3aaSbaaeaacqqG0baDaeqaaiabcMcaPaaakmaabmaabaGae4NXdyMaey4kaSscfa4aaSaaaeaacuGFipqEgaqbaaqaaiab=TqiibaaaOGaayjkaiaawMcaaiabcYcaSaaa@51BF@

where φ is defined as the spatial part of the perturbed metric in the conformally Newtonian gauge, i.e. δ_sg₀₀ = 2a²φ. In the same gauge the Hamiltonian constraint reads (see also appendix A and, in particular, Eq. (A.5))

∇ 2 ψ − 3 ℋ ( ℋ φ + ψ ′ ) = 4 π G a 2 ( δ ρ t + δ ρ B ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHhis0daahaaWcbeqaaiabikdaYaaaiiGakiab=H8a5jabgkHiTiabiodaZmrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab+TqiijabcIcaOiab+Tqiijab=z8aMjabgUcaRiqb=H8a5zaafaGaeiykaKIaeyypa0JaeGinaqJae8hWdaNaem4raCKaemyyae2aaWbaaSqabeaacqaIYaGmaaGccqGGOaakcqWF0oazcqWFbpGCdaWgaaWcbaGaeeiDaqhabeaakiabgUcaRiab=r7aKjab=f8aYnaaBaaaleaacqqGcbGqaeqaaOGaeiykaKIaeiOla4caaa@56D6@

Using Eq. (2.5) inside Eq. (3.2) and inserting the obtained equation into Eq. (3.5) we obtain, through Eq. (3.4) the following relation

ζ = ℛ + ∇ 2 ψ 12 π G a 2 ( ρ t + p t ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF2oGEcqGH9aqpt0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFBeIucqGHRaWkjuaGdaWcaaqaaiabgEGirpaaCaaabeqaaiabikdaYaaacqWFipqEaeaacqaIXaqmcqaIYaGmcqWFapaCcqWGhbWrcqWGHbqydaahaaqabeaacqaIYaGmaaGaeiikaGIae8xWdi3aaSbaaeaacqqG0baDaeqaaiabgUcaRiabdchaWnaaBaaabaGaeeiDaqhabeaacqGGPaqkaaGccqGGUaGlaaa@4EA6@

implying that ⁷ for kτ ≪ 1, ℛ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Trisbaa@35C5@ (k) ~ ζ (k) + O MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=5q8pbaa@3685@ (|kτ|²). From the covariant conservation equation we can easily deduce the evolution for ζ:

ζ ′ = − ℋ p t + ρ t δ p nad + ℋ p t + ρ t ( c st 2 − 1 3 ) δ ρ B − θ t 3 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWF2oGEgaqbaKqbakabg2da9iabgkHiTmaalaaabaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae43cHGeabaGaemiCaa3aaSbaaeaacqqG0baDaeqaaiabgUcaRiab=f8aYnaaBaaabaGaeeiDaqhabeaaaaGae8hTdqMaemiCaa3aaSbaaeaacqqGUbGBcqqGHbqycqqGKbazaeqaaiabgUcaRmaalaaabaGae43cHGeabaGaemiCaa3aaSbaaeaacqqG0baDaeqaaiabgUcaRiab=f8aYnaaBaaabaGaeeiDaqhabeaaaaWaaeWaaeaacqWGJbWydaqhaaqaaiabbohaZjabbsha0bqaaiabikdaYaaacqGHsisldaWcaaqaaiabigdaXaqaaiabiodaZaaaaiaawIcacaGLPaaacqWF0oazcqWFbpGCdaWgaaqaaiabbkeacbqabaGaeyOeI0YaaSaaaeaacqWF4oqCdaWgaaqaaiabbsha0bqabaaabaGaeG4mamdaaiabc6caUaaa@650B@

In the case of a CDM-radiation entropy mode we have that

δ p nad = ρ M c s 2 S ∗ , S ∗ = δ ζ ζ , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaacciGae8hTdqMaemiCaa3aaSbaaSqaaiabb6gaUjabbggaHjabbsgaKbqabaGccqGH9aqpcqWFbpGCdaWgaaWcbaGaeeyta0eabeaakiabdogaJnaaDaaaleaacqqGZbWCaeaacqaIYaGmaaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaqcfaOae4NeXpLcdaWgaaWcbaGaey4fIOcabeaakiabcYcaSaqaaKqbakab+jr8tPWaaSbaaSqaaiabgEHiQaqabaGccqGH9aqpjuaGdaWcaaqaaiab=r7aKjab=z7a6bqaaiab=z7a6baacqGGSaalaaaaaa@52F9@

where S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaKqbakab=jr8tbaa@371B@_* is the relative fulctuation of the specific entropy ζ = T³/n_CDM defined in terms of the temperature T and in terms of the CDM concentration n_CDM.

The possible presence of entropic contributions will be neglected since the attention will now be focused on the simplest situation which implies solely the presence of an adiabatic mode. It is however useful to keep, for a moment, the dependence of the curvature perturbations also upon S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaKqbakab=jr8tbaa@371B@_* since the present analysis can be easily extended, with some algebra, to the case of magnetized non-adiabatic modes. Recalling now the expression of the total sound speed cst2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGqaciab=ngaJnaaDaaaleaacqqGZbWCcqqG0baDaeaacqaIYaGmaaaaaa@3046@ given in Eq. (3.1) and noticing that

ρ M ρ t + p t = 3 α 3 α + 4 , ρ R ρ t + p t = 3 3 α + 4 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqabeqacaaabaWaaSaaaeaaiiGacqWFbpGCdaWgaaqaaiabb2eanbqabaaabaGaeqyWdi3aaSbaaeaacqqG0baDaeqaaiabgUcaRiabdchaWnaaBaaabaGaeeiDaqhabeaaaaGaeyypa0ZaaSaaaeaacqaIZaWmcqaHXoqyaeaacqaIZaWmcqaHXoqycqGHRaWkcqaI0aanaaGaeiilaWcabaWaaSaaaeaacqWFbpGCdaWgaaqaaiabbkfasbqabaaabaGae8xWdi3aaSbaaeaacqqG0baDaeqaaiabgUcaRiabdchaWnaaBaaabaGaeeiDaqhabeaaaaGaeyypa0ZaaSaaaeaacqaIZaWmaeaacqaIZaWmcqWFXoqycqGHRaWkcqaI0aanaaGaeiilaWcaaaaa@4FB4@

Eq. (3.7) can be recast in the following useful form ⁸

d ζ d α = − 4 S ∗ ( 3 α + 4 ) 2 − 3 R γ Ω B ( 3 α + 4 ) 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKHGaciab=z7a6bqaaiabdsgaKjab=f7aHbaacqGH9aqpcqGHsisldaWcaaqaaiabisda0mrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab+jr8tnaaBaaabaGaey4fIOcabeaaaeaacqGGOaakcqaIZaWmcqWFXoqycqGHRaWkcqaI0aancqGGPaqkdaahaaqabeaacqaIYaGmaaaaaiabgkHiTmaalaaabaGaeG4mamJaemOuai1aaSbaaeaacqWFZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaaaeaacqGGOaakcqaIZaWmcqWFXoqycqGHRaWkcqaI0aancqGGPaqkdaahaaqabeaacqaIYaGmaaaaaiabcYcaSaaa@5765@

whose solution is

ζ ( k , τ ) = ζ ∗ ( k ) − α [ 4 S ∗ ( k ) + 3 R γ Ω B ( k ) ] 4 ( 3 α + 4 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF2oGEcqGGOaakcqWGRbWAcqGGSaalcqWFepaDcqGGPaqkcqGH9aqpcqWF2oGEdaWgaaWcbaGaey4fIOcabeaakiabcIcaOiabdUgaRjabcMcaPiabgkHiTKqbaoaalaaabaGae8xSdeMaei4waSLaeGinaqZenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae4NeXp1aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRiabiodaZiabdkfasnaaBaaabaGae83SdCgabeaacqqHPoWvdaWgaaqaaiabbkeacbqabaGaeiikaGIaem4AaSMaeiykaKIaeiyxa0fabaGaeGinaqJaeiikaGIaeG4mamJae8xSdeMaey4kaSIaeGinaqJaeiykaKcaaiabcYcaSaaa@618F@

where ζ_*(k) is the constant value of curvature perturbations implied by the presence of the adiabatic mode; Ω_B(k) has been introduced in Eq. (2.36). The dependence upon the Fourier mode k has been explicitly written to remind that ζ_*(k) is constant in time but not in space. In the two relevant physical limits, i.e. well before and well after equality, Eq. (3.11) implies, respectively,

lim ⁡ α ≫ 1 ζ ( k ) = ζ ∗ ( k ) − S ∗ ( k ) 3 − R γ Ω B ( k ) 4 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaadaWfqaqaaiGbcYgaSjabcMgaPjabc2gaTbWcbaacciGae8xSdeMaeS4AI8JaeGymaedabeaakiab=z7a6jabcIcaOiabdUgaRjabcMcaPiabg2da9iab=z7a6naaBaaaleaacqGHxiIkaeqaaOGaeiikaGIaem4AaSMaeiykaKIaeyOeI0scfa4aaSaaaeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFse=udaWgaaqaaiabgEHiQaqabaGaeiikaGIaem4AaSMaeiykaKcabaGaeG4mamdaaiabgkHiTmaalaaabaGaemOuai1aaSbaaeaacqWFZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkaeaacqaI0aanaaGaeiilaWcaaa@5C2A@

lim ⁡ α ≪ 1 ζ = ζ ∗ ( k ) S ∗ ( k ) 4 α − 3 R γ Ω B ( k ) 16 α . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaadaWfqaqaaiGbcYgaSjabcMgaPjabc2gaTbWcbaacciGae8xSdeMaeSOAI0JaeGymaedabeaakiab=z7a6jabg2da9iab=z7a6naaBaaaleaacqGHxiIkaeqaaOGaeiikaGIaem4AaSMaeiykaKscfa4aaSaaaeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFse=udaWgaaqaaiabgEHiQaqabaGaeiikaGIaem4AaSMaeiykaKcabaGaeGinaqdaaiab=f7aHjabgkHiTmaalaaabaGaeG4mamJaemOuai1aaSbaaeaacqWFZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkaeaacqaIXaqmcqaI2aGnaaGaeqySdeMaeiOla4caaa@5D50@

When ψ = φ we can also obtain the evolution of ψ for the large scales

d ψ d α + 5 α + 6 2 α ( α + 1 ) ψ = − 3 α + 4 2 α ( α + 1 ) ζ ∗ + 4 S ∗ + 3 R γ Ω B 8 ( α + 1 ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabdsgaKHGaciab=H8a5bqaaiabdsgaKjab=f7aHbaacqGHRaWkdaWcaaqaaiabiwda1iab=f7aHjabgUcaRiabiAda2aqaaiabikdaYiab=f7aHjabcIcaOiab=f7aHjabgUcaRiabigdaXiabcMcaPaaacqWFipqEcqGH9aqpcqGHsisldaWcaaqaaiabiodaZiab=f7aHjabgUcaRiabisda0aqaaiabikdaYiab=f7aHjabcIcaOiab=f7aHjabgUcaRiabigdaXiabcMcaPaaacqWF2oGEdaWgaaqaaiabgEHiQaqabaGaey4kaSYaaSaaaeaacqaI0aant0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFse=udaWgaaqaaiabgEHiQaqabaGaey4kaSIaeG4mamJaemOuai1aaSbaaeaacqaHZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaaaeaacqaI4aaocqGGOaakcqWFXoqycqGHRaWkcqaIXaqmcqGGPaqkaaGaeiOla4caaa@6DC5@

Equation (3.14) can be easily solved by noticing that it can be rewritten as

∂ ∂ α ( α 3 α + 1 ψ ) α + 1 α 3 = − 3 α + 4 2 α ( α + 1 ) ζ ∗ + 4 S ∗ + 3 R γ Ω B 8 ( α + 1 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaiabgkGi2cqaaiabgkGi2IGaciab=f7aHbaadaqadaqaamaalaaabaGae8xSde2aaWbaaeqabaGaeG4mamdaaaqaamaakaaabaGae8xSdeMaey4kaSIaeGymaedabeaaaaGae8hYdKhacaGLOaGaayzkaaWaaSaaaeaadaGcaaqaaGGaaiab+f7aHjabgUcaRiabigdaXaqabaaabaGae8xSde2aaWbaaeqabaGaeG4mamdaaaaacqGH9aqpcqGHsisldaWcaaqaaiabiodaZiab=f7aHjabgUcaRiabisda0aqaaiabikdaYiab=f7aHjabcIcaOiab=f7aHjabgUcaRiabigdaXiabcMcaPaaacqWF2oGEdaWgaaqaaiabgEHiQaqabaGaey4kaSYaaSaaaeaacqaI0aant0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqqFse=udaWgaaqaaiabgEHiQaqabaGaey4kaSIaeG4mamJaemOuai1aaSbaaeaacqWFZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaaaeaacqaI4aaocqGGOaakcqWFXoqycqGHRaWkcqaIXaqmcqGGPaqkaaGaeiilaWcaaa@6D1C@

implying that

α 3 α + 1 ψ = − ζ ∗ 2 ℐ 1 ( α ) + 4 S ∗ + 3 R γ Ω B 8 ℐ 2 ( α ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGdaWcaaqaaGGaciab=f7aHnaaCaaabeqaaiabiodaZaaaaeaadaGcaaqaaiab=f7aHjabgUcaRiabigdaXaqabaaaaiab=H8a5jabg2da9iabgkHiTmaalaaabaGae8NTdO3aaSbaaeaacqGHxiIkaeqaaaqaaiabikdaYaaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFqessdaWgaaqaaiabigdaXaqabaGaeiikaGIae8xSdeMaeiykaKIaey4kaSYaaSaaaeaacqaI0aancqGFse=udaWgaaqaaiabgEHiQaqabaGaey4kaSIaeG4mamJaemOuai1aaSbaaeaacqWFZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaaaeaacqaI4aaoaaGae4heHK0aaSbaaeaacqaIYaGmaeqaaiabcIcaOiab=f7aHjabcMcaPiabcYcaSaaa@5B55@

where

ℐ 1 ( α ) = ∫ 0 α β 2 ( 3 β + 4 ) ( β + 1 ) 3 / 2 d β , ℐ 2 ( α ) = ∫ 0 α β 3 ( β + 1 ) 3 / 2 d β . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6DFB@

By using the obvious change of variables y = β + 1 both integrals can be calculated with elementary methods with the result that

ℐ 1 ( α ) = 2 { 16 [ α + 1 − 1 ] + α [ α ( 9 α + 2 ) − 8 ] } 15 α + 1 , ℐ 2 ( α ) = 2 { 16 [ 1 − α + 1 ] + α [ 8 + α ( α − 2 ) ] } 5 α + 1 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@86DB@

Inserting Eq. (3.18) into Eq. (3.16) the explicit result for ψ can be written as:

ψ ( k , τ ) = − ζ ∗ ( k ) 15 α 3 { 16 [ α + 1 − 1 ] + α [ α ( 9 α + 2 ) − 8 ] } + 4 S ∗ ( k ) + 3 R γ Ω B ( k ) 20 α 3 { 16 [ 1 − α + 1 ] + α [ 8 + α ( α − 2 ) ] } . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqaaeGabaaabaacciGae8hYdKNaeiikaGIaem4AaSMaeiilaWIae8hXdqNaeiykaKIaeyypa0JaeyOeI0YaaSaaaeaacqWF2oGEdaWgaaqaaiabgEHiQaqabaGaeiikaGIaem4AaSMaeiykaKcabaGaeGymaeJaeGynauJae8xSde2aaWbaaeqabaGaeG4mamdaaaaacqGG7bWEcqaIXaqmcqaI2aGncqGGBbWwdaGcaaqaaiab=f7aHjabgUcaRiabigdaXaqabaGaeyOeI0IaeGymaeJaeiyxa0Laey4kaSIae8xSdeMaei4waSLae8xSdeMaeiikaGIaeGyoaKJae8xSdeMaey4kaSIaeGOmaiJaeiykaKIaeyOeI0IaeGioaGJaeiyxa0LaeiyFa0habaGaey4kaSYaaSaaaeaacqaI0aant0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqGFse=udaWgaaqaaiabgEHiQaqabaGaeiikaGIaem4AaSMaeiykaKIaey4kaSIaeG4mamJaemOuai1aaSbaaeaacqWFZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkaeaacqaIYaGmcqaIWaamcqWFXoqydaahaaqabeaacqaIZaWmaaaaaiabcUha7jabigdaXiabiAda2iabcUfaBjabigdaXiabgkHiTmaakaaabaGae8xSdeMaey4kaSIaeGymaedabeaacqGGDbqxcqGHRaWkcqWFXoqycqGGBbWwcqaI4aaocqGHRaWkcqWFXoqycqGGOaakcqWFXoqycqGHsislcqaIYaGmcqGGPaqkcqGGDbqxcqGG9bqFcqGGUaGlaaaaaa@9655@

Equation (3.19) can be evaluated in the two limits mentioned above, i.e., respectively, well after and well before equality:

lim ⁡ α ≫ 1 ψ ( k , τ ) = − 3 5 ζ ∗ ( k ) + 4 S ∗ ( k ) + 3 R γ Ω B ( k ) 20 , lim ⁡ α ≪ 1 ψ ( k , τ ) = − 2 3 ζ ∗ ( k ) + α 32 [ 4 3 ζ ∗ ( k ) + 4 S ∗ ( k ) + 3 R γ Ω B ( k ) ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqaaeGabaaabaWaaCbeaeaacyGGSbaBcqGGPbqAcqGGTbqBaeaaiiGacqWFXoqycqWIRjYpcqaIXaqmaeqaaiab=H8a5jabcIcaOiabdUgaRjabcYcaSiab=r8a0jabcMcaPiabg2da9iabgkHiTmaalaaabaGaeG4mamdabaGaeGynaudaaiab=z7a6naaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkdaWcaaqaaiabisda0mrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab+jr8tnaaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkcqaIZaWmcqWGsbGudaWgaaqaaiab=n7aNbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPaqaaiabikdaYiabicdaWaaacqGGSaalaeaadaWfqaqaaiGbcYgaSjabcMgaPjabc2gaTbqaaiab=f7aHjablQMi9iabigdaXaqabaGae8hYdKNaeiikaGIaem4AaSMaeiilaWIae8hXdqNaeiykaKIaeyypa0JaeyOeI0YaaSaaaeaacqaIYaGmaeaacqaIZaWmaaGae8NTdO3aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRmaalaaabaGae8xSdegabaGaeG4mamJaeGOmaidaamaadmaabaWaaSaaaeaacqaI0aanaeaacqaIZaWmaaGae8NTdO3aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRiabisda0iab+jr8tnaaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkcqaIZaWmcqWGsbGudaWgaaqaaiab=n7aNbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPaGaay5waiaaw2faaiabc6caUaaaaaa@9E4B@

Notice that ζ_*(k) appears also in the correction which goes as α = a/a_eq. In this derivation the role of the anisotropic stress has been neglected. As full numerical solutions of the problem (in the tight coupling approximation) shows 3536 that the magnetic anisotropic stress can be neglected close to recombination but it is certainly relevant deep in the radiation-dominated regime. To address this issue let us solve directly the system provided by the evolution equations of the longitudinal fluctuations of the geometry (i.e. Eqs. (A.4), (A.5) and (A.6)-(A.9))coupled with the evolution equations of the matter sources which are reported in appendix A. The evolution of the background will be the one dictated by Eq. (2.2) and by Eq. (2.6). The solution of the Hamiltonian constraint (A.5) and of the evolution equations for various density contrasts (i.e. δ_ν, δ_γ, δ_b and δ_c) can be written, in the limit x = τ/τ₁ ≪ 1 as

δ γ ( k , τ ) = δ ν ( k , τ ) = − 2 φ ∗ ( k ) − R γ Ω B ( k ) + 4 ψ 1 ( k ) x , δ c ( k , τ ) = δ c ( k , τ ) = 3 4 δ γ ( k , τ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqaaeGabaaabaacciGae8hTdq2aaSbaaeaacqWFZoWzaeqaaiabcIcaOiabdUgaRjabcYcaSiab=r8a0jabcMcaPiabg2da9iab=r7aKnaaBaaabaGae8xVd4gabeaacqGGOaakcqWGRbWAcqGGSaalcqWFepaDcqGGPaqkcqGH9aqpcqGHsislcqaIYaGmcqWFgpGzdaWgaaqaaiabgEHiQaqabaGaeiikaGIaem4AaSMaeiykaKIaeyOeI0IaemOuai1aaSbaaeaacqWFZoWzaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkcqaI0aancqWFipqEdaWgaaqaaiabigdaXaqabaGaeiikaGIaem4AaSMaeiykaKIaemiEaGNaeiilaWcabaGae8hTdq2aaSbaaeaacqqGJbWyaeqaaiabcIcaOiabdUgaRjabcYcaSiab=r8a0jabcMcaPiabg2da9iab=r7aKnaaBaaabaGaee4yamgabeaacqGGOaakcqWGRbWAcqGGSaalcqWFepaDcqGGPaqkcqGH9aqpdaWcaaqaaiabiodaZaqaaiabisda0aaacqWF0oazdaWgaaqaaiab=n7aNbqabaGaeiikaGIaem4AaSMaeiilaWIae8hXdqNaeiykaKIaeiOla4caaaaa@7A33@

The Hamiltonian constraint (A.5) implies, always for x ≪ 1, that the following relation must hold among the various constants:

φ 1 ( k ) + 3 ψ 1 ( k ) + φ ∗ ( k ) 2 = 3 4 R γ Ω B ( k ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGajuaGcqWFgpGzdaWgaaqaaiabigdaXaqabaGaeiikaGIaem4AaSMaeiykaKIaey4kaSIaeG4mamJae8hYdK3aaSbaaeaacqaIXaqmaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRmaalaaabaGae8NXdy2aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPaqaaiabikdaYaaacqGH9aqpdaWcaaqaaiabiodaZaqaaiabisda0aaacqWGsbGudaWgaaqaaiab=n7aNbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPiabc6caUaaa@4CDD@

Going on along the same theme we have that Eq. (A.9) is automatically satisfied by Eq. (3.21) in the small-x limit. The solution of Eq. (A.4) can be obtained with similar methods and always well before equality:

σ ν ( k , τ ) = R γ R ν σ B ( k ) + κ 2 x 2 6 R ν ( ψ ∗ ( k ) − φ ∗ ( k ) ) + κ 2 x 3 6 R ν [ ( ψ ∗ ( k ) − φ ∗ ( k ) ) + ( ψ 1 ( k ) − φ 1 ( k ) ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGajuaGcqWFdpWCdaWgaaqaaiab=17aUbqabaGaeiikaGIaem4AaSMaeiilaWIae8hXdqNaeiykaKIaeyypa0ZaaSaaaeaacqWGsbGudaWgaaqaaiab=n7aNbqabaaabaGaemOuai1aaSbaaeaacqWF9oGBaeqaaaaacqWFdpWCdaWgaaqaaiabbkeacbqabaGaeiikaGIaem4AaSMaeiykaKIaey4kaSYaaSaaaeaacqWF6oWAdaahaaqabeaacqaIYaGmaaGaemiEaG3aaWbaaeqabaGaeGOmaidaaaqaaiabiAda2iabdkfasnaaBaaabaGae8xVd4gabeaaaaGaeiikaGIae8hYdK3aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabgkHiTiab=z8aMnaaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGGPaqkcqGHRaWkdaWcaaqaaiab=P7aRnaaCaaabeqaaiabikdaYaaacqWG4baEdaahaaqabeaacqaIZaWmaaaabaGaeGOnayJaemOuai1aaSbaaeaacqWF9oGBaeqaaaaacqGGBbWwcqGGOaakcqWFipqEdaWgaaqaaiabgEHiQaqabaGaeiikaGIaem4AaSMaeiykaKIaeyOeI0Iae8NXdy2aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabcMcaPiabgUcaRiabcIcaOiab=H8a5naaBaaabaGaeGymaedabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHsislcqWFgpGzdaWgaaqaaiabigdaXaqabaGaeiikaGIaem4AaSMaeiykaKIaeiykaKIaeiyxa0LaeiilaWcaaa@8509@

where σ_ν (k, τ) is the neutrino anisotropic stress and σ_B(k, τ) has been already introduced in Eq. (2.37); in Eq. (3.23) κ = kτ₁ and it is the wave-number rescaled through τ₁ which appears in Eq. (2.2). Notice that, as Ω_B(k) also σ_B(k) is approximately constant in time when the flux-freezing condition is verified. To derive Eq. (3.23) we take the (i ≠ j) component of the perturbed Einstein equation, i.e. Eq. (A.4) of the Appendix. From this equation we can write that:

σ ν = − R γ R ν σ B + k 2 6 ℋ 2 Ω R R ν ( ψ − φ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGajuaGcqWFdpWCdaWgaaqaaiab=17aUbqabaGaeyypa0JaeyOeI0YaaSaaaeaacqWGsbGudaWgaaqaaiab=n7aNbqabaaabaGaemOuai1aaSbaaeaacqWF9oGBaeqaaaaacqWFdpWCdaWgaaqaaiabbkeacbqabaGaey4kaSYaaSaaaeaacqWGRbWAdaahaaqabeaacqaIYaGmaaaabaGaeGOnayZenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae43cHG0aaWbaaeqabaGaeGOmaidaaiabfM6axnaaBaaabaGaeeOuaifabeaacqWGsbGudaWgaaqaaiab=17aUbqabaaaaiabcIcaOiab=H8a5jabgkHiTiab=z8aMjabcMcaPiabc6caUaaa@5661@

where, as usual, x = τ/τ₁ and where, according to Eqs. (2.2) and (2.6)

ℋ = 2 ( x + 1 ) τ 1 x ( x + 2 ) , Ω R = 1 ( x + 1 ) 2 , Ω M = x ( x + 2 ) ( x + 1 ) 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@6404@

The solution for ψ and φ is parametrized as

ψ(k, τ) = ψ_*(k) + ψ₁(k)x,   φ(k, τ) = φ_* + φ₁(k)x,

where the constants ψ_* (k) and φ₁(k) will be determined by consistency with the other equations. Now we are interested in the solution valid for x ≪ 1. So we have to expand all the terms of Eq. (3.24) for x ≪ 1. Taking into account the exact form of ℋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=Tqiibaa@35AB@²Ω_R, Eq. (3.24) becomes

σ ν = − R γ R ν σ B + k 2 τ 1 2 6 R ν x 2 ( 1 + x 2 ) 2 [ ψ − φ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGajuaGcqWFdpWCdaWgaaqaaiab=17aUbqabaGaeyypa0JaeyOeI0YaaSaaaeaacqWGsbGudaWgaaqaaiab=n7aNbqabaaabaGaemOuai1aaSbaaeaacqWF9oGBaeqaaaaacqWFdpWCdaWgaaqaaiabbkeacbqabaGaey4kaSYaaSaaaeaacqWGRbWAdaahaaqabeaacqaIYaGmaaGae8hXdq3aa0baaeaacqaIXaqmaeaacqaIYaGmaaaabaGaeGOnayJaemOuai1aaSbaaeaacqWF9oGBaeqaaaaacqWG4baEdaahaaqabeaacqaIYaGmaaWaaeWaaeaacqaIXaqmcqGHRaWkdaWcaaqaaiabdIha4bqaaiabikdaYaaaaiaawIcacaGLPaaadaahaaqabeaacqaIYaGmaaGaei4waSLae8hYdKNaeyOeI0Iae8NXdyMaeiykaKIaeiOla4caaa@555F@

Let us now expand the right hand side of Eq. (3.27). We will have that, for x < 1

σ ν = − R γ R ν σ B + k 2 τ 1 2 6 R ν x 2 ( 1 + x ) [ ψ ∗ − φ ∗ + ( ψ 1 − φ 1 ) x ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGajuaGcqWFdpWCdaWgaaqaaiab=17aUbqabaGaeyypa0JaeyOeI0YaaSaaaeaacqWGsbGudaWgaaqaaiab=n7aNbqabaaabaGaemOuai1aaSbaaeaacqWF9oGBaeqaaaaacqWFdpWCdaWgaaqaaiabbkeacbqabaGaey4kaSYaaSaaaeaacqWGRbWAdaahaaqabeaacqaIYaGmaaGae8hXdq3aa0baaeaacqaIXaqmaeaacqaIYaGmaaaabaGaeGOnayJaemOuai1aaSbaaeaacqWF9oGBaeqaaaaacqWG4baEdaahaaqabeaacqaIYaGmaaGaeiikaGIaeGymaeJaey4kaSIaemiEaGNaeiykaKIaei4waSLae8hYdK3aaSbaaeaacqGHxiIkaeqaaiabgkHiTiab=z8aMnaaBaaabaGaey4fIOcabeaacqGHRaWkcqGGOaakcqWFipqEdaWgaaqaaiabigdaXaqabaGaeyOeI0Iae8NXdy2aaSbaaeaacqaIXaqmaeqaaiabcMcaPiabdIha4jabc2faDjabc6caUaaa@6093@

Rearranging the terms of Eq. (3.28) and keeping the terms O MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=5q8pbaa@3685@(x³), Eq. (3.23) can be immediately reproduced (recall, as previously posited, that κ = kτ₁).

Using Eq. (3.21) into the evolution equations of the peculiar velocities (i.e. Eqs. (A.13), (A.18) and (A.25)), the explicit expressions for θ_c, θ_ν and θ_γb can be easily obtained. In particular, for θ_c and θ_ν we have:

θ c ( k , τ ) = κ 2 2 [ φ ∗ ( k ) x + x 2 6 ( 4 φ 1 ( k ) − φ ∗ ( k ) ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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@5660@

θ ν ( k , τ ) = κ 2 x 4 [ 2 φ ∗ ( k ) + R γ R ν ( 4 σ B ( k ) − R ν Ω B ( k ) ) ] + κ 2 x 2 2 [ ψ 1 ( k ) + φ 1 ( k ) ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF4oqCdaWgaaWcbaGae8xVd4gabeaakiabcIcaOiabdUgaRjabcYcaSiab=r8a0jabcMcaPiabg2da9KqbaoaalaaabaGae8NUdS2aaWbaaeqabaGaeGOmaidaaiabdIha4bqaaiabisda0aaadaWadaqaaiabikdaYiab=z8aMnaaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkdaWcaaqaaiabdkfasnaaBaaabaGae83SdCgabeaaaeaacqWGsbGudaWgaaqaaiab=17aUbqabaaaaiabcIcaOiabisda0iab=n8aZnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHsislcqWGsbGudaWgaaqaaiab=17aUbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPiabcMcaPaGaay5waiaaw2faaiabgUcaRmaalaaabaGae8NUdS2aaWbaaeqabaGaeGOmaidaaiabdIha4naaCaaabeqaaiabikdaYaaaaeaacqaIYaGmaaGaei4waSLae8hYdK3aaSbaaeaacqaIXaqmaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRiab=z8aMnaaBaaabaGaeGymaedabeaacqGGOaakcqWGRbWAcqGGPaqkcqGGDbqxcqGGUaGlaaa@73B0@

Finally, from Eq. (A.25), the photon-baryon peculiar velocity field is determined to be:

θ γ b ( k , τ ) = κ 2 x 8 [ R ν Ω B ( k ) − 4 σ B ( k ) + 2 φ ∗ ( k ) ] + κ 2 x 2 3 [ ψ 1 ( k ) + φ 1 ( k ) ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGajuaGcqWF4oqCdaWgaaqaaiab=n7aNjabbkgaIbqabaGaeiikaGIaem4AaSMaeiilaWIae8hXdqNaeiykaKIaeyypa0ZaaSaaaeaacqWF6oWAdaahaaqabeaacqaIYaGmaaGaemiEaGhabaGaeGioaGdaaiabcUfaBjabdkfasnaaBaaabaGae8xVd4gabeaacqqHPoWvdaWgaaqaaiabbkeacbqabaGaeiikaGIaem4AaSMaeiykaKIaeyOeI0IaeGinaqJae83Wdm3aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRiabikdaYiab=z8aMnaaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGGDbqxcqGHRaWkdaWcaaqaaiab=P7aRnaaCaaabeqaaiabikdaYaaacqWG4baEdaahaaqabeaacqaIYaGmaaaabaGaeG4mamdaaiabcUfaBjab=H8a5naaBaaabaGaeGymaedabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkcqWFgpGzdaWgaaqaaiabigdaXaqabaGaeiikaGIaem4AaSMaeiykaKIaeiyxa0LaeiOla4caaa@6DBA@

By solving Eq. (A.19) (bearing in mind Eqs. (3.22) and (3.30)) the following relations can be obtained

ψ ∗ ( k ) = ( 1 + 2 5 R ν ) φ ∗ ( k ) + R γ 5 [ 4 σ B ( k ) − R ν Ω B ( k ) ] , ψ 1 ( k ) − φ 1 ( k ) = 4 15 R ν ( ψ 1 ( k ) + φ 1 ( k ) ) − 2 5 R ν φ ∗ ( k ) − R γ 5 [ 4 σ B ( k ) − R ν Ω B ( k ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqaaeGabaaabaacciGae8hYdK3aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabg2da9maabmaabaGaeGymaeJaey4kaSYaaSaaaeaacqaIYaGmaeaacqaI1aqnaaGaemOuai1aaSbaaeaacqWF9oGBaeqaaaGaayjkaiaawMcaaiab=z8aMnaaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkdaWcaaqaaiabdkfasnaaBaaabaGae83SdCgabeaaaeaacqaI1aqnaaGaei4waSLaeGinaqJae83Wdm3aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPiabgkHiTiabdkfasnaaBaaabaGae8xVd4gabeaacqqHPoWvdaWgaaqaaiabbkeacbqabaGaeiikaGIaem4AaSMaeiykaKIaeiyxa0LaeiilaWcabaGae8hYdK3aaSbaaeaacqaIXaqmaeqaaiabcIcaOiabdUgaRjabcMcaPiabgkHiTiab=z8aMnaaBaaabaGaeGymaedabeaacqGGOaakcqWGRbWAcqGGPaqkcqGH9aqpdaWcaaqaaiabisda0aqaaiabigdaXiabiwda1aaacqWGsbGudaWgaaqaaiab=17aUbqabaGaeiikaGIae8hYdK3aaSbaaeaacqaIXaqmaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRiab=z8aMnaaBaaabaGaeGymaedabeaacqGGOaakcqWGRbWAcqGGPaqkcqGGPaqkcqGHsisldaWcaaqaaiabikdaYaqaaiabiwda1aaacqWGsbGudaWgaaqaaiab=17aUbqabaGae8NXdy2aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabgkHiTmaalaaabaGaemOuai1aaSbaaeaacqWFZoWzaeqaaaqaaiabiwda1aaacqGGBbWwcqaI0aancqWFdpWCdaWgaaqaaiabbkeacbqabaGaeiikaGIaem4AaSMaeiykaKIaeyOeI0IaemOuai1aaSbaaeaacqWF9oGBaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkcqGGDbqxcqGGSaalaaaaaa@9F9D@

allowing to determine, in conjunction with Eq. (3.22), the explicit form of φ₁(k) and of ψ₁(k):

ψ 1 ( k ) = 15 + 4 R ν 60 + 8 R ν [ 3 4 R γ Ω B ( k ) − φ ∗ ( k ) 2 ] − 1 60 + 8 R ν [ 2 5 R ν φ ∗ ( k ) + R γ 5 ( 4 σ B ( k ) − R ν Ω B ( k ) ) ] , φ 1 ( k ) = 15 − 4 R ν 60 + 8 R ν [ 3 4 R γ Ω B ( k ) − φ ∗ ( k ) 2 ] + 3 60 + 8 R ν [ 2 5 R ν φ ∗ ( k ) + R γ 5 ( 4 σ B ( k ) − R ν Ω B ( k ) ) ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqaaeGabaaabaacciGae8hYdK3aaSbaaeaacqaIXaqmaeqaaiabcIcaOiabdUgaRjabcMcaPiabg2da9maalaaabaGaeGymaeJaeGynauJaey4kaSIaeGinaqJaemOuai1aaSbaaeaacqWF9oGBaeqaaaqaaiabiAda2iabicdaWiabgUcaRiabiIda4iabdkfasnaaBaaabaGae8xVd4gabeaaaaWaamWaaeaadaWcaaqaaiabiodaZaqaaiabisda0aaacqWGsbGudaWgaaqaaiab=n7aNbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPiabgkHiTmaalaaabaGae8NXdy2aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPaqaaiabikdaYaaaaiaawUfacaGLDbaacqGHsisldaWcaaqaaiabigdaXaqaaiabiAda2iabicdaWiabgUcaRiabiIda4iabdkfasnaaBaaabaGae8xVd4gabeaaaaWaamWaaeaadaWcaaqaaiabikdaYaqaaiabiwda1aaacqWGsbGudaWgaaqaaiab=17aUbqabaGae8NXdy2aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRmaalaaabaGaemOuai1aaSbaaeaacqWFZoWzaeqaaaqaaiabiwda1aaacqGGOaakcqaI0aancqWFdpWCdaWgaaqaaiabbkeacbqabaGaeiikaGIaem4AaSMaeiykaKIaeyOeI0IaemOuai1aaSbaaeaacqWF9oGBaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkcqGGPaqkaiaawUfacaGLDbaacqGGSaalaeaacqWFgpGzdaWgaaqaaiabigdaXaqabaGaeiikaGIaem4AaSMaeiykaKIaeyypa0ZaaSaaaeaacqaIXaqmcqaI1aqncqGHsislcqaI0aancqWGsbGudaWgaaqaaiab=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n7aNbqabaaabaGaeGynaudaaiabcIcaOiabisda0iab=n8aZnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHsislcqWGsbGudaWgaaqaaiab=17aUbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPiabcMcaPaGaay5waiaaw2faaiabc6caUaaaaaa@D7F5@

If R_ν = Ω_B = 0 we have that

φ 1 ( k ) = ψ 1 ( k ) = − φ ∗ ( k ) 8 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGajuaGcqWFgpGzdaWgaaqaaiabigdaXaqabaGaeiikaGIaem4AaSMaeiykaKIaeyypa0Jae8hYdK3aaSbaaeaacqaIXaqmaeqaaiabcIcaOiabdUgaRjabcMcaPiabg2da9iabgkHiTmaalaaabaGae8NXdy2aaSbaaeaacqGHxiIkaeqaaiabcIcaOiabdUgaRjabcMcaPaqaaiabiIda4aaacqGGSaalaaa@416B@

and this result coincides precisely with the result already obtained in Eq. (3.13). In fact, recalling that α(x) = x² + 2x, we have that, in the small-x region ψ(k, τ) ≃ -(2/3) ζ_*(k) + (x/12) ζ_*(k). But recalling now that, in the limit R_ν → 0 and Ω_B → 0, ζ_*(k) = - (3/2)ψ_* (k), Eq. (3.34) is recovered. The obtained large-scale solutions will be important both for the explicit evaluation of the Sachs-Wolfe plateau as well as for the normalization of the solution at smaller k that will be discussed in the forthcoming section.

It is useful to add that, in the limit (R_γσ_B + R_νσ_ν) → 0 and R_ν → 0 the result reported in Eq.(3.11) is also recovered. Infact, in this limit, ψ_* = φ_* and ζ = ζ_* + O MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=5q8pbaa@3685@(α).

The ordinary and integrated Sachs-Wolfe contributions can now be computed. Recalling Eq. (A.45) the large-scale limit of the brightness perturbation of the radiation field is (see also Eqs. (A.40) and (A.45) of the appendix A)

Δ I ( SW ) ( k → , τ 0 ) = − ζ ∗ ( k ) 5 + R γ Ω B ( k ) 20 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGcqqHuoardaqhaaqaaiabbMeajbqaaiabcIcaOiabbofatjabbEfaxjabcMcaPaaacqGGOaakcuWGRbWAgaWcaiabcYcaSGGaciab=r8a0naaBaaabaGaeGimaadabeaacqGGPaqkcqGH9aqpcqGHsisldaWcaaqaaiab=z7a6naaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkaeaacqaI1aqnaaGaey4kaSYaaSaaaeaacqWGsbGudaWgaaqaaiab=n7aNbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPaqaaiabikdaYiabicdaWaaacqGGSaalaaa@4DB6@

Δ I ( ISW ) ( k → , τ 0 ) = 10 − 4 R ν 5 ( 15 + 4 R ν ) ζ ∗ ( k ) + 3 10 R γ Ω B ( k ) + 4 5 R γ ( R ν + 5 ) ( 4 σ B ( k ) − R ν Ω B ( k ) ) 5 ( 15 + 4 R ν ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGcqqHuoardaqhaaqaaiabbMeajbqaaiabcIcaOiabbMeajjabbofatjabbEfaxjabcMcaPaaacqGGOaakcuWGRbWAgaWcaiabcYcaSGGaciab=r8a0naaBaaabaGaeGimaadabeaacqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXiabicdaWiabgkHiTiabisda0iabdkfasnaaBaaabaGae8xVd4gabeaaaeaacqaI1aqncqGGOaakcqaIXaqmcqaI1aqncqGHRaWkcqaI0aancqWGsbGudaWgaaqaaiab=17aUbqabaGaeiykaKcaaiab=z7a6naaBaaabaGaey4fIOcabeaacqGGOaakcqWGRbWAcqGGPaqkcqGHRaWkdaWcaaqaaiabiodaZaqaaiabigdaXiabicdaWaaacqWGsbGudaWgaaqaaiab=n7aNbqabaGaeuyQdC1aaSbaaeaacqqGcbGqaeqaaiabcIcaOiabdUgaRjabcMcaPiabgUcaRmaalaaabaGaeGinaqdabaGaeGynaudaamaalaaabaGaemOuai1aaSbaaeaacqWFZoWzaeqaaiabcIcaOiabdkfasnaaBaaabaGae8xVd4gabeaacqGHRaWkcqaI1aqncqGGPaqkcqGGOaakcqaI0aancqWFdpWCdaWgaaqaaiabbkeacbqabaGaeiikaGIaem4AaSMaeiykaKIaeyOeI0IaemOuai1aaSbaaeaacqWF9oGBaeqaaiabfM6axnaaBaaabaGaeeOqaieabeaacqGGOaakcqWGRbWAcqGGPaqkcqGGPaqkaeaacqaI1aqncqGGOaakcqaIXaqmcqaI1aqncqGHRaWkcqaI0aancqWGsbGudaWgaaqaaiab=17aUbqabaGaeiykaKcaaiabc6caUaaa@8641@

As in the standard case, the ISW effect mimics the ordinary SW effect and it actually cancels partially the SW contribution at large angular scales. Notice that, in order to derive the explicit form of the ordinary SW it is practical to observe that, for wavelengths larger than the Hubble radius at recombination (δ_γ - 4ψ)' ≃ 0. This observation implies that, clearly, δγ(f)=4(ψ(f)−ψ(i))+δγ(i) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGacKqbakab=r7aKnaaDaaabaGae83SdCgabaGaeiikaGIaeeOzayMaeiykaKcaaiabg2da9iabisda0iabcIcaOiab=H8a5naaCaaabeqaaiabcIcaOiabbAgaMjabcMcaPaaacqGHsislcqWFipqEdaahaaqabeaacqGGOaakcqqGPbqAcqGGPaqkaaGaeiykaKIaey4kaSIae8hTdq2aa0baaeaacqWFZoWzaeaacqGGOaakcqqGPbqAcqGGPaqkaaaaaa@47CA@ where the superscripts f (for final) and i (for initial) indicate that the values of the corresponding quantities are taken, respectively, well after and well before equality. The large angular scale expression of the temperature autocorrelations are defined as

C ℓ ( SW ) = 2 π ∫ k 3 d ln ⁡ k | Δ I ( SW ) ( k , τ 0 ) | 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGcqWGdbWqdaqhaaqaaiabloriSbqaaiabcIcaOiabbofatjabbEfaxjabcMcaPaaacqGH9aqpdaWcaaqaaiabikdaYaqaaGGaciab=b8aWbaadaWdbaqaaiabdUgaRnaaCaaabeqaaiabiodaZaaacqWGKbazcyGGSbaBcqGGUbGBcqWGRbWAcqGG8baFcqqHuoardaqhaaqaaiabbMeajbqaaiabcIcaOiabbofatjabbEfaxjabcMcaPaaacqGGOaakcqWGRbWAcqGGSaalcqWFepaDdaWgaaqaaiabicdaWaqabaGaeiykaKIaeiiFaW3aaWbaaeqabaGaeGOmaidaaaqabeqacqGHRiI8aiabc6caUaaa@515F@

To evaluate Eq. (3.37) in explicit terms we have to mention the conventions for the curvature and for the magnetic power spectra. The correlators of ζ_*(k), Ω_B(k) and σ_B(k) are defined, respectively, as

〈 ζ ∗ ( k → ) ζ ∗ ( p → ) 〉 = | ζ ∗ ( k ) | 2 δ ( 3 ) ( k → + p → ) , 〈 Ω B ( k → ) Ω B ( p → ) 〉 = | Ω B ( k ) | 2 δ ( 3 ) ( k → + p → ) , 〈 σ B ( k → ) σ B ( p → ) 〉 = | σ B ( k ) | 2 δ ( 3 ) ( k → + p → ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=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r7aKnaaCaaabeqaaiabcIcaOiabiodaZiabcMcaPaaacqGGOaakcuWGRbWAgaWcaiabgUcaRiqbdchaWzaalaGaeiykaKIaeiOla4caaaaa@9A9B@

In the case of the curvature perturbations we will have that

where k_p denotes the pivot scale at which the spectrum of curvature fluctuations is computed and A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaKqbakab=bq8bbaa@36F7@_ζ is, by definition, the amplitude of the spectrum at the pivot scale. In similar terms the magnetized contributions can be written as

where k_L (defined in Eq. (2.30)) denotes, in some sense, the magnetic pivot scale. The spectral index of the magnetic correlator defined in Eq. (2.32) is related to ε as m + 3 = ε. Notice also that in defining the correlators of Ω_B and of σ_B the same conventions used for the curvature perturbations have been adopted. These conventions imply that a factor k^-3appears at the right hand side of the first relation of Eq. (3.39).

Since the spectrum of the magnetic energy density implies the calculation of a convolution k_L is also related to the smoothing scale of the magnetic energy density (see, for instance, 35). In Eqs. (3.40) and (3.41) the functions ℱ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaajuaGcqWFXeIraaa@35BC@(ε) and G MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaajuaGcqWFge=raaa@3662@(ε) as well as the smoothed amplitude Ω¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiqbfM6axzaaraaaaa@2C9B@_BL are defined as

ℱ ( ε ) = 4 ( 6 − ε ) ( 2 π ) 2 ε ε ( 3 − 2 ε ) Γ 2 ( ε / 2 ) , G ( ε ) = 4 ( 188 − 4 ε 2 − 66 ε ) ( 2 π ) 2 ε 3 ε ( 3 − ε ) ( 2 ε + 1 ) Γ 2 ( ε / 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqabeqacaaabaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8xmHyKaeiikaGccciGae4xTduMaeiykaKIaeyypa0ZaaSaaaeaacqaI0aancqGGOaakcqaI2aGncqGHsislcqGF1oqzcqGGPaqkcqGGOaakcqaIYaGmcqGFapaCcqGGPaqkdaahaaqabeaacqaIYaGmcqGF1oqzaaaabaGae4xTduMaeiikaGIaeG4mamJaeyOeI0IaeGOmaiJae4xTduMaeiykaKIaeu4KdC0aaWbaaeqabaGaeGOmaidaaiabcIcaOiab+v7aLjabc+caViabikdaYiabcMcaPaaacqGGSaalaeaacqWFge=rcqGGOaakcqGF1oqzcqGGPaqkcqGH9aqpdaWcaaqaaiabisda0iabcIcaOiabigdaXiabiIda4iabiIda4iabgkHiTiabisda0iab+v7aLnaaCaaabeqaaiabikdaYaaacqGHsislcqaI2aGncqaI2aGncqGF1oqzcqGGPaqkcqGGOaakcqaIYaGmcqGFapaCcqGGPaqkdaahaaqabeaacqaIYaGmcqGF1oqzaaaabaGaeG4mamJae4xTduMaeiikaGIaeG4mamJaeyOeI0Iae4xTduMaeiykaKIaeiikaGIaeGOmaiJae4xTduMaey4kaSIaeGymaeJaeiykaKIaeu4KdC0aaWbaaeqabaGaeGOmaidaaiabcIcaOiab+v7aLjabc+caViabikdaYiabcMcaPaaacqGGSaalaaaaaa@8A1D@

Ω ¯ BL = ρ BL ρ ¯ γ , ρ BL = B L 2 8 π , ρ ¯ γ = a 4 ( τ ) ρ γ ( τ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaajuaGfaqabeqadaaabaGafuyQdCLbaebadaWgaaqaaiabbkeacjabbYeambqabaGaeyypa0ZaaSaaaeaaiiGacqWFbpGCdaWgaaqaaiabbkeacjabbYeambqabaaabaGaf8xWdiNbaebadaWgaaqaaiab=n7aNbqabaaaaiabcYcaSaqaaiab=f8aYnaaBaaabaGaeeOqaiKaeeitaWeabeaacqGH9aqpdaWcaaqaaiabdkeacnaaDaaabaGaeeitaWeabaGaeGOmaidaaaqaaiabiIda4iab=b8aWbaacqGGSaalaeaacuWFbpGCgaqeamaaBaaabaGae83SdCgabeaacqGH9aqpcqWGHbqydaahaaqabeaacqaI0aanaaGaeiikaGIae8hXdqNaeiykaKIae8xWdi3aaSbaaeaacqWFZoWzaeqaaiabcIcaOiab=r8a0jabcMcaPiabc6caUaaaaaa@56B7@

From Eq. (3.43), recalling that T_CMB = 2.725K and that ρ¯γ=(π2/15)TCMB4 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=wiFfYlOipiY=Hhbbf9v8qqaqFr0xc9vqpe0di9q8qqpG0dHiVcFbIOFHK8Feei0lXdar=Jb9qqFfeaYRXxe9vr0=vr0=LqpWqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaGGaciqb=f8aYzaaraWaaSbaaSqaaiab=n7aNbqabaGccqGH9aqpcqGGOaakcqWFapaCdaahaaWcbeqaaiabikdaYaaakiabc+caViabigdaXiabiwda1iabcMcaPiabdsfaunaaDaaaleaacqqGdbWqcqqGnbqtcqqGcbGqaeaacqaI0aanaaaaaa@3CA0@, we can also write, in more explicit terms:

Ω ¯ BL = 7.565 × 10 − 9 ( B L nG ) 2 . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8bkY=MipeYlH8Hipec8Eeeu0xXdbba9frFj0xb9Lqpepeea0xd9q8qiYRWxGi6xij=hbbc9s8aq0=yqpe0xbbG8A8frFve9Fve9Fj0dmeaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHPoWvgaqeamaaBaaaleaacqqGcbGqcqqGmbataeqaaOGaeyypa0JaeG4naCJaeiOla4IaeGynauJaeGOnayJaeGynauJaey41aqRaeGymaeJaeGimaaZaaWbaaSqabeaacqGHsislcqaI5aqoaaGcdaqadaqaaKqbaoaalaaabaGaemOqai0aaSbaaeaacqqGmbataeqaaaqaaiabb6gaUjabbEeahbaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaOGaeiOla4caaa@4346@

It should finally be appreciated that the power spectra of the magnetic energy density and of the anisotropic stress are proportional since we focus our attention to magnetic spectral slopes ε < 1 which are the most relevant at large length-scales ⁹. In principle, the present analysis can be also extended to the case when the magnetic power spectra are very steep in k (i.e. ε > 1). In the latter case the power spectra are often said to be violet and they are severely constrained by thermal diffusivity effects 30.

By performing the integration over the comoving wave-number that appears in Eq. (3.37) the wanted result can be expressed as ¹⁰

C ℓ ( S W ) = [