Semianalytical approach to magnetized temperature autocorrelations
 Massimo Giovannini^{1, 2}Email author
Received: 18 October 2007
Accepted: 18 October 2007
Published: 18 October 2007
Abstract
The cosmic microwave background (CMB) temperature autocorrelations, induced by a magnetized adiabatic mode of curvature inhomogeneities, are computed with semianalytical methods. As suggested by the latest CMB data, a nearly scaleinvariant spectrum for the adiabatic mode is consistently assumed. In this situation, the effects of a fully inhomogeneous magnetic field are scrutinized and constrained with particular attention to harmonics which are relevant for the region of Doppler oscillations. Depending on the parameters of the stochastic magnetic field a hump may replace the second peak of the angular power spectrum. Detectable effects on the Doppler region are then expected only if the magnetic power spectra have quasiflat slopes and typical amplitude (smoothed over a comoving scale of Mpc size and redshifted to the epoch of gravitational collapse of the protogalaxy) exceeding 0.1 nG. If the magnetic energy spectra are bluer (i.e. steeper in frequency) the allowed value of the smoothed amplitude becomes, comparatively, larger (in the range of 20 nG). The implications of this investigation for the origin of largescale magnetic fields in the Universe are discussed. Connections with forthcoming experimental observations of CMB temperature fluctuations are also suggested and partially explored.
1 Formulation of the problem
Since the Cosmic Microwave Background (CMB) is extremely isotropic in nearly all angular scales, it is rather plausible to infer that the Universe was quite homogeneous (and isotropic) at the moment when the ionization fraction dropped significantly and the photon mean free path became, almost suddenly, comparable with the present Hubble radius.
The inhomogeneities present for lengthscales larger than the Hubble radius right before recombination are believed to be, ultimately, the seeds of structure formation and they can be studied by looking at the temperature autocorrelations which are customarily illustrated in terms of the angular power spectrum. The distinctive features of the angular power spectrum (like the Doppler peaks) can be phenomenologically reproduced by assuming the presence, before recombination, of a primordial adiabatic ^{2}mode arising in a spatially flat Universe [1–5]. Possible deviations from this working hypothesis can also be bounded: they include, for instance, the plausible presence of nonadiabatic modes (see [6–8] and references therein), or even features in the powerspectrum that could be attributed either to the preinflationary stage of expansion or to the effective modification of the dispersion relations (see [9–12] and references therein). For a pedagogical introduction to the physics of CMB anisotropies see, for instance, Ref. [13]. In short the purpose of the present paper is to show that CMB temperature autocorrelations may also be a source of valuable informations on largescale magnetic fields whose possible presence prior to recombination sheds precious light on the origin of the largest magnetized structures we see today in the sky such as galaxies, clusters of galaxies and even some supercluster.
In fact, spiral galaxies and rich clusters possess a largescale magnetic field that ranges from 500 nG [14, 15] (in the case of Abell clusters) to few μG in the case of spiral galaxies [16]. Elliptical galaxies have also magnetic fields in the μG range but with correlation scales of the order of 10–100 pc (i.e. much smaller than in the spirals where typical correlation lengths are of the order of 30 kpc, as in the case of the Milky Way). The existence of largescale magnetic fields in superclusters, still debatable because of ambiguities in the determination of the column density of electrons along the line of sight, would be rather intriguing. Recently plausible indications of the existence of magnetized structures in Hercules and PerseusPisces superclusters have been reported [17] (see also [18]): the typical correlation scales of the fields would be 0.5 Mpc and the intensity 300 nG.
While there exist various ideas put forward throught the years, it is fair to say that the origin of these (pretty large) fields is still matter of debate [15, 19]. Even if they are, roughly, one millionth of a typical planetary magnetic field (such as the one of the earth) these fields are pretty large for a cosmological standard since their energy density is comparable both with energy density of the CMB photons (i.e. ${T}_{\text{CMB}}^{4}$) and with the cosmic ray pressure. The very presence of large scale magnetic fields in diffuse astrophysical plasmas and with large correlation scales (as large of, at least, 30 kpc) seems to point towards a possible primordial origin [15]. At the same time, the efficiency of dynamo amplification can be questioned in different ways so that, at the onset of the gravitational collapse of the protogalaxy it seems rather plausible that only magnetic fields with intensities^{3} B_{L} > 10^{14} nG may be, eventually, amplified at an observable level [20, 21].
As emphasized many years ago by Harrison [22–24], this situation is a bit reminiscent of what happened with the problem of justifying the presence of a flat spectrum of curvature perturbations that could eventually seed the structure formation paradigm. Today a possibility along this direction is provided by inflationary models in one of their various incarnations.
It seems therefore appropriate, especially in view of forthcoming satellite missions (like PLANCK Explorer [25]), to discuss the effects of largescale magnetic fields on CMB physics. In fact, all along the next decade dramatic improvements in the quality and quantity of CMB data can be expected. On the radioastronomical side, the next generation of radiotelescopes such as Square Kilometre Array (SKA) [26] might be able to provide us with unprecedented accuracy in the full sky survey of Faraday Rotation measurements at frequencies that may be so large to be, roughly, comparable with ^{4} (even if always smaller than) the lower frequency channel of the PlANCK Explorer (i.e. about 30 GHz). The question before us today is, therefore, the following: is CMB itself able to provide compelling bounds on the strength of largescale magnetic fields prior to hydrogen recombination? In fact, all the arguments connecting the present strength of magnetic field to their primordial value (say before recombination) suffer undeniable ambiguities. These ambiguities are related to the evolution of the Universe through the dark ages (i.e. approximately, between photon decoupling and galaxy formation). So, even if it is very reasonable to presume that during the stage of galaxy formation the magnetic flux and helicity are, according to Alfvén theorems, approximately conserved, the strengths of the fields prior to gravitational collapse is unknown and it is only predictable within a specific model for the origin of largescale magnetic fields. In general terms, the magnetic fields produced in the early Universe may have different features. They may be helical or not, they may have different spectral slopes and different intensities. There are, however, aspects that are common to diverse mechanisms like the stochastic nature of the produced field. Furthermore, since as we go back in time the conductivity increases with the temperature, it can be expected that the flux freezing and the helicity conservation are better and better verified as the Universe heats up say from few eV to few MeV.
Along the past decade some studies addressed the analysis of vector and tensor modes induced by largescale magnetic fields [28–31]. There have been also investigations within a covariant approach to perturbation theory [32, 33]. Only recently the analysis of the scalar modes has been undertaken [34–38]. The setup of the aforementioned analyses is provided by an effective onefluid description of the plasma which is essentially the curved space analog of magnetohydrodynamics (MHD). This approach is motivated since the typical lengthscales of the problem are much larger of the Debye length. However, it should be borne in mind that the treatment of Faraday rotation is a typical twofluid phenomenon. So if we would like to ask the question on how the polarization plane of the CMB is rotated by the presence of a uniform magnetic field a twofluid description would be mandatory (see section 2 and references therein).
In the framework described in the previous paragraph, it has been shown that the magnetic fields affect the scalar modes in a threefold way. In the first place the magnetic energy density and pressure gravitate inducing a computable modification of the largescale adiabatic solution. Moreover, the anisotropic stress and the divergence of the Lorentz force affect the evolution of the baryonlepton fluid. Since, prior to decoupling, photons and baryons are tightly coupled the net effect will also be a modification of the temperature autocorrelations at angular scales smaller than the ones relevant for the ordinary SW contribution (i.e. ℓ > 30).
In the present paper, elaborating on the formalism developed in [34–36], a semianaltytical approach for the calculation of the temperature autocorrelations is proposed. Such a framework allows the estimate of the angular power spectrum also for angular scales compatible with the first Doppler peak. A gravitating magnetic field will be included from the very beginning and its effects discussed both at large angular scales and small angular scales. The main theme of the present paper can then be phrased by saying that largescale magnetic fields affect the geometry and the evolution of the (scalar) sources. We ought to compute how all these effects combine in the final power spectra of the temperature autocorrelations. It should be remarked, incidentally, that the evolution of the density contrasts of the various species enter directly the scalar problem but neither the vector or the tensor modes are affected by their presence. As a consequence of this occurrence the selfconsistent inclusion of the largescale magnetic fields in the calculation is much more cumbersome than in the case of the tensor and vector modes.
The plan of the present paper will therefore be the following. In section 2 the typical scales of the problem will be discussed. In section 3 the attention will be focused on the largescale evolution of the curvature perturbations with particular attention to the magnetized contribution, i.e. the contribution associated with the gravitating magnetic fields. In section 4 the evolution at smaller angular scales will be investigated accounting, in an approximate manner, for the finite thickness effects of the lastscattering surface. In section 5 the estimates of the angular power spectra of the temperature autocorrelations will be presented. Section 6 contains the concluding remarks. Some of the relevant theoretical tools needed for the discussion of the problem have been collected in the appendix with the sole aim to make the overall presentation more selfcontained. The material presented in the appendix collects the main equations whose solutions are reported and discussed in section 3 and 4.
2 Typical scales of the problem
The analysis starts by defining all the relevant physical scales of the problem. These scales stem directly from the evolution equations of the gravitational perturbations in the presence of a stochastic magnetic field. The interested reader may also consult appendix A where some relevant technical aspects are briefly summarized.
2.1 Equality and recombination
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 spacetime 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 largescale 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.
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;
• H_{0} is the present value of the Hubble constant and Ω_{M0} is the present critical fraction in nonrelativistic 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} = ($\sqrt{2}$  1)τ_{1}, i.e. roughly, τ_{eq} ≃ τ_{1}/2.
where $\mathscr{H}$ = 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}).
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.
which implies that, for z_{rec} and ${h}_{0}^{2}{\Omega}_{\text{M}0}$ = 0.134, τ_{rec} = 1.01τ_{1}.
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)).
2.2 Plasma scales
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. $\overrightarrow{\nabla}\cdot \overrightarrow{J}=0$.
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 largescale 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.
Hence, for the typical value of the matter fraction appearing in Eq. (2.21), τ_{rec} ≃ τ_{1} and, consequently k_{ σ }≫ k_{D}. While finite conductivity effects are rather efficient in washing out the magnetic fields for large wavenumbers, the thermal diffusivity effects (related to shear viscosity and, ultimately, to Silk damping) affect typical wavenumbers that are much smaller than the ones affected by conductivity.
In Eq. (2.23) Σ is an arbitrary closed surface that moves with the plasma. In Eq. (2.24) $\overrightarrow{A}$ is the vector potential. According to Eq. (2.23), in MHD the magnetic field has to be always solenoidal (i.e. $\overrightarrow{\nabla}\cdot \overrightarrow{B}=0$). 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 largescale 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 nonvanishing 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 righthandside of Eq. (2.24), i.e. $\overrightarrow{B}\cdot \overrightarrow{\nabla}\times \overrightarrow{B}$ is called magnetic gyrotropy and it is a gaugeinvariant 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.
From Eq. (2.26) the magnetic field configuration of Eq. (2.25) depends on the amplitude of the field ${\mathcal{Q}}_{0}$ and on the spectral index m.
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).
From Eq. (2.32) we can appreciate that, on top of the parityinvariant contribution (already defined in Eqs. (2.25) and (2.26)), there is a second term proportional to the LeviCivita ε_{ijℓ }. In Fourier space, the introduction of gyrotropic configurations implies also the presence of a second function of the momentum $\tilde{\mathcal{Q}}$(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).
having defined, for notational convenience, ${\overline{\rho}}_{\gamma}={\rho}_{\gamma}(\tau ){a}^{4}(\tau )$.
3 Largescale solutions
After equality but before recombination the fluctuations of the geometry evolve coupled with the fluctuations of the plasma. The plasma contains four species: photons, neutrinos (that will be taken to be effectively massless at recombination), baryons and cold dark matter (CDM) particles. The evolution equations go under the name of EinsteinBoltzmann system since they are formed by the perturbed Einstein equations and by the evolution equations of the brightness perturbations. In the case of temperature autocorrelations, the relevant Boltzmann hierarchy will be the one associate with the I stokes parameter giving the intensity of the Thompson scattered radiation field. Furthermore, since neutrinos are collisionless after 1 MeV, the Boltzmann hierarchy for neutrinos has also to be consistently included. In practice, however, the lowest multipoles (i.e. the density contrast, the velocity and the anisotropic stress) will be the most important ones for the problem of setting the prerecombination initial conditions.
Since stochastic magnetic fields are present prior to recombination, the EinsteinBoltzmann system has to be appropriately modified. This system has been already derived in the literature (see Ref. [34, 35]) but since it will be heavily used in the present and in the following sections the main equations have been collected and discussed in appendix A. It is also appropriate to remark, on a more technical ground, that the treatment of the curvature perturbations demands the analysis of quantities that are invariant under infinitesimal coordinate transformations (or, for short, gauge invariant). The strategy adopted in the appendix has been to pick up a specific gauge (i.e. the conformally Newtonian gauge) and to derive, in this gauge, the relevant evolution equations for the appropriate gaugeinvariant quantities such as the density contrast on uniform density hypersurfaces (denoted, in what follows, by ζ) and the curvature perturbations on comoving orthogonal hypersurfaces (denoted, in what follows, by $\mathcal{R}$). Defining as k the comoving wavenumber of the fluctuations, the magnetized EinsteinBoltzmann system can be discussed in three complementary regimes:
• the wavelengths that are larger than the Hubble radius at recombination, i.e. kτ_{rec} < 1;
• the wavelengths that crossed the Hubble radius before recombination but that were still larger than the Hubble radius at equality, i.e. kτ_{eq} < 1;
• the wavelengths that crossed the Hubble radius prior to equality and that are, consequently, inside the Hubble radius already at equality (i.e. kτ_{eq} > 1).
The wavelengths that are larger than the Hubble radius at recombination determine the largescale features of temperature autocorrelations and, in particular, the socalled SachsWolfe plateau. The wavelengths that crossed the Hubble radius around τ_{rec} determine the features of the temperature autocorrelations in the region of the Doppler oscillations.
The initial conditions of the EinsteinBoltzmann system are set in the regime when the relevant wavelengths are larger than the Hubble radius before equality (i.e. deep in the radiation epoch). The standard unknown is represented, in this context, by the primordial spectrum of the metric fluctuations whose amplitude and slope are two essential parameters of the ΛCDM model. To this unknown we shall also add the possible presence of a stochastically distributed magnetized background. In the conventional case, where magnetic fields are not contemplated, the system of metric fluctuations admits various (physically different) solutions that are customarily classified in adiabatic and nonadiabatic modes (see, for instance, [6, 7] and also [13]). For the adiabatic modes the fluctuations of the specific entropy vanish at large scales. Conversely, for nonadiabatic (also sometimes named isocurvature) solutions the fluctuations of the specific entropy do not vanish. The WMAP 3year data [1–3] suggest that the temperature autocorrelations are well fitted by assuming a primordial adiabatic mode of curvature perturbations with nearly scaleinvariant power spectrum. Therefore, the idea will be now to assume the presence of an adiabatic mode of curvature perturbations and to scrutinize the effects of fully inhomogeneous magnetic fields. It should be again stressed that this is the minimal assumption compatible with the standard ΛCDM paradigm. As it will be briefly discussed later on, all the nonadiabatic solutions in the preequality regime can be generalized to include a magnetized background [35]. However, for making the discussion both more cogent and simpler, the attention will be focussed on the physical system with the fewer number of extraparameters, i.e. the case of a magnetized adiabatic mode.
3.1 Curvature perturbations
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 radiationmatter transition takes place when the relevant modes have wavelengths still larger than the Hubble radius.
where $\mathcal{S}$_{*} is the relative fulctuation of the specific entropy ζ = T^{3}/n_{CDM} defined in terms of the temperature T and in terms of the CDM concentration n_{CDM}.
3.2 Magnetized adiabatic mode
The solution for ψ and φ is parametrized as
ψ(k, τ) = ψ_{*}(k) + ψ_{1}(k)x, φ(k, τ) = φ_{*} + φ_{1}(k)x,
Rearranging the terms of Eq. (3.28) and keeping the terms $\mathcal{O}$(x^{3}), Eq. (3.23) can be immediately reproduced (recall, as previously posited, that κ = kτ_{1}).
and this result coincides precisely with the result already obtained in Eq. (3.13). In fact, recalling that α(x) = x^{2} + 2x, we have that, in the smallx 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 largescale solutions will be important both for the explicit evaluation of the SachsWolfe 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_{ γ }σ_{B} + R_{ ν }σ_{ 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_{*} + $\mathcal{O}$(α).
3.3 Estimate of the ordinary SachsWolfe contribution
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^{3}appears at the right hand side of the first relation of Eq. (3.39).
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 lengthscales ^{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].
We can then compute the various pieces. They will set the scale of the numerical results. In particular, it is easy to argue that the presence of the cross correlation enhances the results at smaller scales. As a final comment it is relevant to remark that the largescale solutions are not only important per se but they will be used to deduce the appropriate normalization for the results arising at smaller angular scales.
4 Intermediate scales
As discussed in the previous section the attention will now be focussed on the situation of a dominant adiabatic mode. This is just because we want to delicately extend the ΛCDM paradigm and make contact with the standard scenario. In fact, it should be clear that our attempts are not alternative to the standard lore but, more modestly, complementary.
It should be stressed that Eqs. (4.6) and (4.7) can be obtained from the standard form of the ΛCDM transfer function in the case when Ω_{b0} ≪ Ω_{M0} and for a spatially flat Universe. Under these assumptions (which are the ones clearly spelled out at the beginning of the present paper) the transfer function (conventionally denoted by T(q)) will depend solely upon q = k/(14k_{eq}). When k_{eq} <k < 11k_{eq}(which is the interesting range if we want to study the first two Doppler peaks), then the full transfer function T(q) given in [57] can be approximated by T(k) ≃ (1/4) ln [14k_{eq}/k], which leads to Eq. (4.6) once we recall the definition of w and once we normalize, at largescales, the SW contribution to the adiabatic initial conditions in the presence of the magnetized contribution derived before in the long wavelength limit.
Concerning Eqs. (4.27)–(4.30) and (4.31)–(4.34) the following comments are in order:
• the lower limit of integration over w is 1 in Eqs. (4.27)–(4.30) since the asymptotic expansion of Bessel functions implies that k_{τ0 }≥ ℓ, i.e. w ≥ 1;
• the obtained expressions will be valid for the angular power spectrum will be applicable for sufficiently large ℓ; in practice, as we shall see the obtained results are in good agreement with the data in the Doppler region;
• the function $\beta (w,\ell )=\ell \sqrt{{w}^{2}1}\ell \phantom{\rule{0.5em}{0ex}}\mathrm{arccos}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}({w}^{1})\frac{\pi}{4}$ leads to a rapidly oscillating argument whose effect will be to slow down the convergence of the numerical integration; it is practical, for the present purposes, to replace cos^{2} [β (w, ℓ)] by its average (i.e. 1/2).
With similar manipulations it is possible to transform also all the other integrands appearing in Eqs. (4.28), (4.29) and (4.30).
5 Calculation of the temperature autocorrelations
So far the necessary ingredients for the estimate of the magnetized temperature autocorrelations have been sorted out. In particular the angular power spectrum has been computed semianalytically in the two relevant regions, i.e. the SachsWolfe regime (corresponding to large angular scales and ℓ ≤ 30) and the Doppler region, i.e. ℓ > 100. Furthermore, for the nature of the approximations made we do not expect the greatest accuracy of the algorithm in the intermediate region (i.e. 30 < ℓ < 100). Indeed, it was recognized already in the absence of magnetic fields that it is somehow necessary to smooth the joining of the two regimes by assuming an interpolating form of the metric fluctuations that depends upon two fitting parameters [42, 43]. We prefer here to stress that this method is inaccurate in the matching regime since the spherical Bessel functions have been approximated for large ℓ. Therefore, the comparison with experimental data should be preferentially conducted, for the present purposes, in the Doppler region. The strategy adopted in the present section is, therefore, the following:
• by taking a concordance model as a starting point, the shape and amplitude of the Doppler oscillations will be analyzed when the amplitude and spectral slope of the stochastic field are allowed to vary;
• constraints can then be derived from the temperature autocorrelations induced by the simultaneous presence of the standard adiabatic mode and of the stochastic magnetic field.
Before plunging into the discussion, it is appropriate to comment on the choice of the cosmological parameters that will be employed throughout this section. The WMAP 3year [1] data have been combined, so far, with various sets of data. These data sets include the 2dF Galaxy Redshift Survey [58], the combination of Boomerang and ACBAR data [59, 60], the combination of CBI and VSA data [61, 62]. Furthermore the WMAP 3year data can be also combined with the Hubble Space Telescope Key Project (HSTKP) data [63] as well as with the Sloan Digital Sky Survey (SDSS) [64, 65] data. Finally, the WMAP 3year data can be also usefully combined with the weak lensing data [66, 67] and with the observations of type Ia supernovae ^{13}(SNIa). Each of the data sets mentioned in the previous paragraph can be analyzed within different frameworks. The minimal ΛCDM model with no cutoff in the primordial spectrum of the adiabatic mode and with vanishing contribution of tensor modes is the simplest concordance framework. This is the one that has been adopted in this paper. Diverse completions of this minimal model are possible: they include the addition of the tensor modes, a sharp cutoff in the spectrum and so on and so forth. One of the conclusions of the present study is that the observational cosmologists may also want to include, in their analyses, the possibility of prerecombination largescale magnetic fields.
All these sets of data (combined with different theoretical models) lead necessarily to slightly different determinations of the relevant cosmological parameters To have an idea of the range of variations of the parameters the following examples are useful^{14}:

the WMAP 3year data alone [1] (in a ΛCDM framework) seem to favour a slightly smaller value ${h}_{0}^{2}{\Omega}_{\text{M}0}$ = 0.127;

if the WMAP 3year data are combined with the "gold" sample of SNIa [69] (see also [70]) the favoured value is ${h}_{0}^{2}{\Omega}_{\text{M}0}$ is of the order of 0.134; if the WMAP 3year data are combined with all the data sets ${h}_{0}^{2}{\Omega}_{\text{M}0}$ = 0.1324.

similarly, if the WMAP data alone are considered, the preferred value of ${h}_{0}^{2}{\Omega}_{b0}$ is 0.02229 while this value decreases to 0.02186 if the WMAP data are combined with all the other data sets.
As already stressed, the regime ℓ < 100 is only reasonably reproduced while the most interesting region, for the present purposes, is rather accurate (as the comparison with the WMAP data shows). The region of very large ℓ (i.e. ℓ > 1200) is also beyond the treatment of diffusive effects adopted in the present paper. In Figure 2 (plot at the right) the adiabatic spectral index is fixed (i.e. n_{ ζ }= 0.958) while the total (present) fraction of nonrelativistic matter is allowed to vary (h_{0} and ${h}_{0}^{2}{\Omega}_{b0}$ are, again, kept fixed). It can be observed that, according to Fig. 2, the amplitude of the first peak increases as the total (dusty) matter fraction decreases.
• the first Doppler peak increases dramatically and it reaches a value of the order of 1.2 × 10^{4} (μK)^{2} when B_{L} = 2 nG;
• already for 0.1 nG <B_{L} < 2 nG the third peak increases while the second peak becomes less pronounced;
• as soon as B_{L} ≥ 2 nG the second peak practically disappears and it is replaced by a sort of hump.
Thus, according to the results described so far it is possible to say that to avoid gross distortion of the temperature autocorrelations attributed to largescale magnetic fields we have to demand that the stochastic field satisfies
B_{L} ≤ 0.08nG, 0.001 ≤ ε < 1
If a magnetic field with smoothed amplitude B_{L} ≤ 0.1 nG is present before recombination the implication for the formation of magnetized structures are manifold. We recall that the value B_{L} is the smoothed magnetic field redshifted at the epoch of the gravitational collapse of the protogalaxy. We know that, during collapse, the freezing of magnetic flux justifies the compressional amplification of the preexisting field that will be boosted by roughly four orders of magnitude during the collapse [15]. This will bring the amplitude of the field to the μG level. It is however premature to speculate on these issues. There are, at the moment, two important steps to be undertaken:
• the forthcoming PLANCK explorer data will allow to strengthen the constraints derived in the present paper and, in particular, the formulae derived in the present section will allow to constraint directly the possible magnetized distortions stemming from the possible presence of largescale magnetic fields;
• another precious set of informations may come from the analysis of the magnetic fields in clusters and superclusters; it would be interesting to know, for instance, which is the spectral slope of the magnetic fields in galaxies, clusters and superclusters.
The other interesting suggestion of the present analysis is that the inclusion of a largescale magnetic field as a fit parameter in an extended ΛCDM model is definitely plausible. The ΛCDM model has been extended to include, after all, different possibilities like the ones arising in the darkenergy sector. Here we have the possibility of adding the parameters of a magnetized background which are rather well justified on the physical ground. Notice, in particular, that interesting degeneracies can be foreseen. For instance, the increase of the first peak caused by a decrease in the darkmatter fraction can be combined with the presence of a magnetic field whose effect, as we demonstrated, is to shift the first Doppler peak upwards. These issues are beyond the scopes of the present paper.
6 Concluding remarks
There are no compelling reasons why largescale magnetic fields should not be present prior to recombination. In this paper, via a semianalytical approach, the temperature autocorrelations induced by largescale magnetic fields have been computed and confronted with the available experimental data. Of course the data analysis can be enriched by combining the WMAP data also with other data sets and by checking the corresponding effects of largescale magnetic fields. The main spirit of this investigation was, however, not to discuss the analysis of data but to show that the effects of largescale magnetic fields on the temperature autocorrelations can be brought at the same theoretical standard of the calculations that are usually performed in the absence of magnetic fields.
According to this perspective it is interesting to notice that, at the level of the preequality initial conditions, the presence of magnetic field induces a quasiadiabatic mode. Depending on the features of the magnetic spectrum (i.e. its smoothed amplitude B_{L} and its spectral slope ε), possible distortions of the first and second peaks can jeopardize the shape of the observed temperature autocorrelations. In particular, for sufficiently strong magnetic backgrounds (i.e. B_{L} > 10 nG and ε ≤ 0.3), the second peak turns into a hump. From the analysis of these distortions it was possible to derive a bound that depends solely upon measurable quantities such as the location of the first peak and its height. The derived formulae will allow a swifter comparison of the possible effects of largescale magnetic fields with the forthcoming experimental data such as the ones of PLANCK explorer. The available WMAP data suggest that B_{L} ≤ 0.08 nG for 0.001 ≤ ε < 1. This range of parameters does not exclude that magnetic fields present prior to recombination could be the seeds of magnetized structures in the sky such as galaxies, clusters and superclusters. It is also interesting to remark that the allowed range of parameters does not exclude the possibility that the magnetic field of galaxies is produced from the prerecombination field even without a strong dynamo action whose possible drawbacks and virtues are, at present, a subject of very interesting debates.
In recent years CMB data have been confronted with a variety of cosmological scenarios that take as a pivotal model the ΛCDM paradigm. Some of the parameters usually added encode informations stemming from effects that, even if extremely interesting, arise at very high energy and curvature scales. While it is certainly important to test any predictive cosmological scenario, we would like to stress that the purpose of the present work is, in some sense, more modest. We hope to learn from CMB not only what was the initial state of the Universe when the Hubble rate was only one millionth times smaller than the Planck (or string) mass scale; if possible we would like to learn from CMB how and why the largest magnetized structures arose in the sky. Since we do see magnetic fields today it is definitely a well posed scientific question to know what were their effects prior to recombination. It would be desirable, for instance, to find clear evidence of the absence of prerecombination magnetic fields. It would be equally exciting to determine the possible presence of this natural component. It is therefore opinion of the author that the inclusion of a magnetized component in future experimental studies of CMB observables represents a physically motivated option which we do hope will be seriously considered by the various collaborations which are today active in experimental cosmology.
A Magnetized gravitational perturbations
where δ_{s} signifies the scalar nature of the fluctuation. While these equations are available in the literature [34, 35], it seems appropriate to give here an explicit and reasonably selfcontained treatment of some technical tools that constitute the basis of the results reported in the bulk of the paper.
The magnetic fields are here treated in the magnetohydrodynamical (MHD) approximation where the displacement current is neglected and where the three dynamical fields of the problem (i.e., respectively, the magnetic field, the Ohmic electric field and the total Ohmic current) are all solenoidal. The bulk velocity field, in this approach, is given by the centre of mass velocity of the electronproton system. This is physically justified since electrons and protons are strongly coupled by Coulomb scattering. Photons and baryon are also strongly coupled by Thompson scattering, at least up to recombination which is the relevant timescale for the effects of magnetic fields on temperature autocorrelations. The bulk velocity of the plasma can be separated into an irrotational part and into a rotational part which contributes to the evolution of the vector modes of the geometry [27]. In the present investigation only the scalar modes are treated and, therefore, only the irrotational part of the velocity field will be relevant. In the MHD approach the magnetic fields enter, both, the perturbed Einstein equations and the Boltzmann hierarchy.
A.1 Perturbed Einstein equations
is the magnetic anisotropic stress. Using the practical notation^{15} is ${\partial}_{i}{\partial}^{j}{\tilde{\Pi}}_{j}^{i}=({p}_{\gamma}+{\rho}_{\gamma}){\nabla}^{2}{\sigma}_{B}$ the spatial (and traceless) components of the perturbed Einstein equations imply
∇^{4}(φ  ψ) = 12πGa^{2}[p_{ ν }+ ρ_{ ν })∇^{2}σ_{ ν }+ (p_{ γ }+ ρ_{ γ })∇^{2}σ_{B}],
where δ_{ρt }= ∑_{ a }δ_{ ρa }is the total density fluctuation (with the sum running over the four species of the plasma, i.e. photons, baryons, neutrinos and CDM particles). Equations (A.5) and (A.6) are simply derived from the perturbed components of the (00) and (0i) Einstein equations [34, 35] (see also [13] for a comparison with the conventional situation where magnetic fields are absent). Notice that the MHD Pointying vector has not been included in the momentum constraint. The rationale for this approximation stems from the fact that this contribution is proportional to $\overrightarrow{\nabla}\cdot (\overrightarrow{E}\times \overrightarrow{B})$ and it contains one electric field which is suppressed, in MHD, by one power of σ_{c}, i.e. the Ohmic conductivity.
where η denotes the shear viscosity coefficient which is particularly relevant for the baryonphoton system and which is related to the photon mean free path (see below in this Appendix).