 Research article
 Open Access
 Published:
Semianalytical approach to magnetized temperature autocorrelations
PMC Physics A volume 1, Article number: 5 (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
According to the present understanding of the Doppler oscillations the spacetime geometry is well described by a conformally flat line element of FriedmannRobertsonWalker (FRW) type
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.
In the geometry given by Eq. (2.1) the scale factor for the radiationmatter transition can be smoothly parametrized as
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.
Equation (2.2) is a solution of the FriedmannLemaître equations whose specific form is
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}).
Often, for notational convenience, the rescaled time coordinate x = τ/τ_{1} will be used. Within this x parametrization the critical fractions of radiation and dusty matter become
The redshift to equality is given, from Eq. (2.2), by
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,
where Y_{p} ≃ 0.24 is the abundance of ^{4}He. 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 10^{4} 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 [39–43](see also [44, 45]) 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 timescales in terms of the ratio between the corresponding redshifts. So, for instance,
which implies that, for z_{rec} and ${h}_{0}^{2}{\Omega}_{\text{M}0}$ = 0.134, τ_{rec} = 1.01τ_{1}.
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
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 photonbaryon system (see appendix A for further details)
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
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:
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.
where $\overrightarrow{E}(\tau ,\overrightarrow{x})={a}^{2}(\tau )\overrightarrow{\mathcal{E}}(\tau ,\overrightarrow{x})$ denote the rescaled electric fields and where, by charge neutrality, the electron density equals the proton density, i.e.
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
where $\overrightarrow{B}(\tau ,\overrightarrow{x})={a}^{2}\overrightarrow{\mathcal{B}}(\tau ,\overrightarrow{x})$ and where
is the Ohmic current and σ_{c} = a(τ) ${\overline{\sigma}}_{\text{c}}$ defined in terms of the rescaled conductivity. Since we are in the situation where T ≪ m_{e}, ${\overline{\sigma}}_{\text{c}}={\alpha}_{\text{em}}^{1}T\sqrt{T/{m}_{\text{e}}}$. By now using the Ohmic electric field inside the remaining Maxwell equation, i.e.
the magnetic diffusivity equation can be obtained
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. $\overrightarrow{\nabla}\cdot \overrightarrow{J}=0$.
As in the flatspace case, the MHD equations can be obtained from a twofluid description by combining the relevant equations and by using global variables. As a consequence of this derivation $\overrightarrow{J}$ will be the total current and $\overrightarrow{v}$ will be the bulk velocity of the plasma, i.e. the centreofmass velocity of the electronproton system [46, 47]. 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 twofluid (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 twofluid description) the Faraday rotation rate can be computed bearing in mind that the Larmor frequency of electrons and ions at recombination, i.e.
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.
The comoving magnetic field $\overrightarrow{B}(\tau ,\overrightarrow{x})$ is related to the physical magnetic field $\overrightarrow{\mathcal{B}}(\tau ,\overrightarrow{x})$ as $\overrightarrow{B}(\tau ,\overrightarrow{x})={a}^{2}(\tau )\overrightarrow{\mathcal{B}}(\tau ,\overrightarrow{x})$. 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 z^{2} where z, as usual is the redshift. This implies, in turn, that $\overrightarrow{B}$ (i.e. the comoving field) is roughly constant (in time) if the plasma does not have sizable kinetic helicity^{5}(i.e. $\u3008\overrightarrow{v}\cdot \overrightarrow{\nabla}\times \overrightarrow{v}\u3009=0$) (see, for instance, [15, 20, 21]). In this situation Eq. (2.19) dictates that $\overrightarrow{B}$ is constant for typical wavenumbers k <k_{ σ }(i.e. for sufficiently large comoving lengthscales) where k_{ σ }sets the magnetic diffusivity scale whose value, at recombination, is
Equation (2.21) can be compared with the estimate of the diffusive scale associated with Silk damping:
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.
Under the conditions of MHD, two (approximate) conservations laws may be derived, namely the magnetic flux conservation
and the magnetic helicity conservation
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.
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}
where
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.
It is often useful, in practical estimates, to regularize the twopoint function by using an appropriate "windowing". Two popular windows are, respectively, the Gaussian and the tophat functions, i.e.
For instance, the regularized magnetic energy density with Gaussian filter can be obtained from the previous expressions by shifting $\mathcal{Q}(k)\to \mathcal{Q}(k){W}_{\text{g}}^{2}(k,L)$. The result is
where F(a, b, x) ≡_{1} F_{1}(a, b, x) is the confluent hypergeometric function [52, 53]. Notice that the integral appearing in the trace converges for m > 3. The amplitude of the magnetic power spectrum ${\mathcal{Q}}_{0}$ can be traded for ${B}_{\text{L}}^{2}$ where ${B}_{\text{L}}^{2}$ is by definition the regularized twopoint function evaluated at coincident spatial points, i.e.
Combining Eq. (2.28) with Eq. (2.29) we have that ${\mathcal{Q}}_{0}$ becomes
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. $\u3008\overrightarrow{B}\cdot \overrightarrow{\nabla}\times \overrightarrow{B}\u3009=0$. There are situations where magnetic fields are produced in a state with nonvanishing 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)
but where now
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).
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. ${\overrightarrow{B}}^{2}$/(8π) and ${\overrightarrow{B}}^{2}$/(24π)) and the one related to the divergence of the MHD Lorentz force (i.e. $\overrightarrow{\nabla}\cdot [\overrightarrow{J}\times \overrightarrow{B}]$) 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 $4\pi \overrightarrow{J}=\overrightarrow{\nabla}\times \overrightarrow{B}$ the divergence of the Lorentz force will be proportional to $\overrightarrow{\nabla}\cdot [(\overrightarrow{\nabla}\times \overrightarrow{B})\times \overrightarrow{B}]$. The magnetic anisotropic stress ${\tilde{\Pi}}_{i}^{j}$ 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.
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 twopoint 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:
where
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.
It is practical, for the present purposes, to think the matterradiation fluid as a unique physical entity with timedependent barotropic index and timedependent sound speed:
where α = a/a_{eq}. According to Eq. (3.1), when a ≫ a_{eq} both ${c}_{\text{st}}^{2}$ and w_{t} go to zero (as appropriate when matter dominates) while in the opposite limit (i.e. α ≪ 1) ${c}_{\text{st}}^{2}$ ≃ 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 radiationmatter transition. This issue has been addressed in [34] by following, outside the Hubble radius, the evolution of the gaugeinvariant density contrast on uniform density hypersurfaces (customarily denoted by ζ):
where ψ is related to the fluctuation of the spatial component of the metric (i.e. δ_{s}g_{ ij }= 2a^{2}ψδ_{ ij }in the conformally Newtonian gauge) and
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 gaugeinvariant 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 $\mathcal{R}$. Since both $\mathcal{R}$ and ζ are gaugeinvariant, their mutual relation can be worked out in any gauge and, in particular, in the conformally Newtonian gauge where $\mathcal{R}$ can be expressed as [13]
where φ is defined as the spatial part of the perturbed metric in the conformally Newtonian gauge, i.e. δ_{s}g_{00} = 2a^{2}φ. In the same gauge the Hamiltonian constraint reads (see also appendix A and, in particular, Eq. (A.5))
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
implying that ^{7} for kτ ≪ 1, $\mathcal{R}$ (k) ~ ζ (k) + $\mathcal{O}$ (kτ^{2}). From the covariant conservation equation we can easily deduce the evolution for ζ:
In the case of a CDMradiation entropy mode we have that
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 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 $\mathcal{S}$_{*} since the present analysis can be easily extended, with some algebra, to the case of magnetized nonadiabatic modes. Recalling now the expression of the total sound speed ${c}_{\text{st}}^{2}$ given in Eq. (3.1) and noticing that
Eq. (3.7) can be recast in the following useful form ^{8}
whose solution is
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,
When ψ = φ we can also obtain the evolution of ψ for the large scales
Equation (3.14) can be easily solved by noticing that it can be rewritten as
implying that
where
By using the obvious change of variables y = β + 1 both integrals can be calculated with elementary methods with the result that
Inserting Eq. (3.18) into Eq. (3.16) the explicit result for ψ can be written as:
Equation (3.19) can be evaluated in the two limits mentioned above, i.e., respectively, well after and well before equality:
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 [35, 36] that the magnetic anisotropic stress can be neglected close to recombination but it is certainly relevant deep in the radiationdominated 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/τ_{1} ≪ 1 as
The Hamiltonian constraint (A.5) implies, always for x ≪ 1, that the following relation must hold among the various constants:
Going on along the same theme we have that Eq. (A.9) is automatically satisfied by Eq. (3.21) in the smallx limit. The solution of Eq. (A.4) can be obtained with similar methods and always well before equality:
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 = kτ_{1} and it is the wavenumber rescaled through τ_{1} which appears in Eq. (2.2). Notice that, as Ω_{B}(k) also σ_{B}(k) is approximately constant in time when the fluxfreezing 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:
where, as usual, x = τ/τ_{1} and where, according to Eqs. (2.2) and (2.6)
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,
where the constants ψ_{*} (k) and φ_{1}(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 $\mathscr{H}$^{2}Ω_{R}, Eq. (3.24) becomes
Let us now expand the right hand side of Eq. (3.27). We will have that, for x < 1
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}).
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:
Finally, from Eq. (A.25), the photonbaryon peculiar velocity field is determined to be:
By solving Eq. (A.19) (bearing in mind Eqs. (3.22) and (3.30)) the following relations can be obtained
allowing to determine, in conjunction with Eq. (3.22), the explicit form of φ_{1}(k) and of ψ_{1}(k):
If R_{ ν }= Ω_{B} = 0 we have that
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
The ordinary and integrated SachsWolfe contributions can now be computed. Recalling Eq. (A.45) the largescale limit of the brightness perturbation of the radiation field is (see also Eqs. (A.40) and (A.45) of the appendix A)
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, ${\delta}_{\gamma}^{(\text{f})}=4({\psi}^{(\text{f})}{\psi}^{(\text{i})})+{\delta}_{\gamma}^{(\text{i})}$ 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
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
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 $\mathcal{A}$_{ 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^{3}appears 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 $\mathcal{F}$(ε) and $\mathcal{G}$(ε) as well as the smoothed amplitude $\overline{\Omega}$_{BL} are defined as
From Eq. (3.43), recalling that T_{CMB} = 2.725K and that ${\overline{\rho}}_{\gamma}=({\pi}^{2}/15){T}_{\text{CMB}}^{4}$, we can also write, in more explicit terms:
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].
By performing the integration over the comoving wavenumber that appears in Eq. (3.37) the wanted result can be expressed as ^{10}
where
In Eq. (3.48) γ_{br} is the correlation angle that has been included to keep the expressions as general as possible. In what follows the main focus will however be on the case where the adiabatic mode of curvature perturbations is not correlated with the magnetized contribution (i.e. γ_{br} = π/2). The various pivot scales appearing in Eqs. (3.46), (3.47) and (3.48) will now be defined:
Let us now consider some simplified limits. The first one is to posit^{11} n_{ ζ }= 1 and ε < 1. We will have that the functions $\mathcal{Z}$ will be simplified. They become:
We now can enforce the normalization at large scales by assuming a dominant adiabatic mode. A preliminary manipulation is the following. We can write the previous expression as
We can now expand the relevant terms in powers of ε. We do get
where
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.
From Eqs. (A.32), (A.33) and (A.34) the photon density contrast can be determined under the assumption that the entropic contribution is absent. Thus, if only the magnetic fields and the adiabatic mode are present, Eqs. (A.34), (A.35) and (A.36) lead to the following solution
where ψ(k, τ) is assumed to be slowly varying in time and where, recalling Eq. (2.8)
The constant A_{1}(k) can be determined by matching the solution to the largescale (i.e. superHubble) behaviour of the fluctuations, i.e.
where ψ_{m} denotes the value of ψ(k) after equality and for kτ < 1. From the solution of the evolution equation of δ_{ γ }(k, τ) also θ_{γb}(k, τ) can be easily obtained (see, in particular, Eq. (A.24) of appendix A). The final result can be expressed, for the present purposes, as^{12}
The functions $\mathcal{L}$_{ ζ }(k, τ) and $\mathcal{M}$_{ ζ }(k, τ) are directly related to the curvature perturbations and can be determined by interpolating the largescale behaviour with the smallscale solutions. In the present case they can be written as
In Eqs. (4.6) and (4.7) the variable w = kτ_{0}/ℓ has been introduced. This way of writing may seem, at the moment, obscure. However, the variable w will appear as integration variable in the angular power spectrum, so it is practical, as early as possible, to express the integrands directly in terms of w. Finally the functions $\mathcal{L}$_{B}(k, τ) and $\mathcal{M}$_{B}(k, τ) are determined in similar terms and they can be written as
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.
The functions $\mathcal{D}$_{ ζ }(k) and $\mathcal{D}$_{B}(k) encode the informations related to the diffusivity wavenumber:
As introduced before k_{D} is the thermal diffusivity scale (i.e. shear viscosity). The quantity named k_{B} is the smallest momentum between the ones defined by magnetic diffusivity, by Alfvén diffusivity and by thermal diffusivity. The magnetic diffusivity has been already introduced in Eq. (2.21) and it arises because of the finite value of the conductivity. The Alfvén diffusivity arises when the magnetic field supports Alfvén waves that are subsequently damped for typical lengthscales that are a bit smaller than the Silk damping scale (see [30] and, in particular, [55]). Now, if the magnetic field is fully inhomogeneous (as in the present case) the dominant source of diffusivity is represented by the Silk length scale since it is larger than the magnetic diffusivity length and than the Alfvén diffusivity length [30]. For the purpose of simplifying the integrals to be evaluated numerically it is practical to introduce the following rescaled quantities:
after some algebra the angular power spectrum can be written as the sum of four integrals, i.e.
where:
where j_{ℓ}(y) denote the spherical Bessel functions of the first kind [52, 53] which are related to the ordinary Bessel functions of the first kind as ${j}_{\ell}=\sqrt{\pi /(2y)}{J}_{\ell +1/2}(y)$. The various functions appearing in Eqs. (4.13), (4.14), (4.15) and (4.16) are:
where ℓ_{S} and ℓ_{A} denote respectively the typical Silk multipole and the typical multipole associated with Alfvén diffusivity. They will be defined explicitly in a moment. In Eqs. (4.15) and (4.16) the oscillatory terms arising, originally, in the full expression of the angular power spectrum have been simplified. The two oscillatory contributions in Eqs. (4.15) and (4.16) go, respectively, as cos (2γℓw) and as cos (γℓw). The definition of γ can be easily deduced from the original parametrization of the oscillatory contribution in Eqs. (4.1) and (4.2). In fact we can write α(k, τ_{rec}) = γ(τ_{rec}) ℓw. Recalling that w = kτ_{0}/ℓ, and defining, for notational convenience, γ ≡ γ (τ_{rec}), the following expression for γ can be easily obtained
In Eq. (4.21) the first equality is simply the definition of γ while the second equality can be deduced by inserting in the definition the explicit expression of the scale factor of Eq. (2.2). By doing so the constant ν_{1} is just R_{b}z_{rec}/z_{eq}. The expression of γ can be made even more explicit by performing the integral appearing in Eq. (4.21):
It is now practical to recall that the ratio between τ_{1} and τ_{0} depends upon the critical fraction of the dark energy. So the scale factor (2.2) must be complemented, at late times, by the contribution of the dark energy. This standard calculation leads to the following estimate for a spatially flat Universe
where Ω_{Λ0} is the present critical fraction of dark energy (parametrized in terms of a cosmological constant in a ΛCDM framework) and where ν_{2} = 0.0858. Inserting Eq. (2.9) into Eq. (4.23) and recalling the explicit expression of ν_{1} we will have finally
At this point the spherical Bessel functions appearing in the above expressions can be evaluated in the limit of large ℓ with the result that the above expressions can be made more explicit. In particular, focussing the attention on j_{ℓ}(ℓw), we have that [52, 53]
Note that the expansion (4.25) has been used consistently by other authors (see, in particular, [44, 45] and also [39, 41–43]). The result expressed by Eq. (4.25) allows to write the integrals of Eqs. (4.13), (4.14), (4.15) and (4.16) as
where
where
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).
In the integrals (4.31), (4.32), (4.33) and (4.34) the scale ℓ_{t} stems from the finite thickness of the last scattering surface and it is defined as
Furthermore, within the present approximations,
To simplify further the obtained expressions we can also change variable in some of the integrals. Consider, as an example, the integrals appearing in the expression of C_{1}(ℓ) (see Eq. (4.27)). Changing the variable of integration as w = y^{2} + 1 we will have that
where, in explicit terms and after the change of variables,
and
In Eq. (4.39) the explicit dependence of the functions L_{B}(ℓ, y) and L_{ ζ }(ℓ, y) upon y can be simply deduced from the analog expressions in terms of w :
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.
The aforementioned list of statements refers to the case of a pure ΛCDM model. If, for instance, tensors are included, then the WMAP 3year data combined with CBI and VSA increase a bit the value of ${h}_{0}^{2}{\Omega}_{b0}$ which becomes, in this case closer to 0.023. While in the future it might be interesting to include prerecombination magnetic fields also in nonminimal ΛCDM scenarios, here the logic will be to take a best fit model to the WMAP data alone, compare it with the numerical scheme proposed in this paper, and, consequently, assess the accuracy of the semianalytical method. Once this step will be concluded the effects stemming from the presence of the magnetic fields will be carefully analyzed. Consider, therefore, the case when the magnetic field vanishes (i.e. B_{L} = 0) in a ΛCDM model with no tensors. In Fig. 1 the contribution of each of the integrals appearing in Eq. (4.26) is illustrated. The analytical form of these integrals has been derived in Eqs. (4.27), (4.28), (4.29) and (4.30). In Fig. 1 (plot at the left) the separate contributions of ℓ(ℓ + 1) C_{1}(ℓ)/(2π) and of ℓ(ℓ + 1)C_{2}(ℓ)/(2π) have been reported for a fiducial set of parameters (i.e. n_{ ζ }= 0.958, ${h}_{0}^{2}{\Omega}_{\text{M}0}$ = 0.1277 and ${h}_{0}^{2}{\Omega}_{b0}$ = 0.0229). This fiducial set of parameters corresponds to the best fit of the WMAP 3year data alone [1]. As mentioned in Eq. (3.49) the pivot wavenumber is k_{p} = 0.002 Mpc^{1}. This is also the choice made by WMAP team. In the plot at the right (always in Fig. 1) the separate contributions of ℓ(ℓ + 1)C_{3}(ℓ)/(2π) and of ℓ(ℓ + 1)C_{4}(ℓ)/(2π) is illustrated for the same fiducial set of parameters (which is also described at the top of the plot). The various contributions are expressed in units of (μK)^{2} (i.e. 1μK = 10^{6}K) which are the appropriate ones for the comparison with the data. The normalization of the calculation is set by evaluating (analytically) the largescale contribution for ℓ < 30 (see Eq. (3.45)) and by comparing it, in this region, with the WMAP 3year data release.
By summing up the four separate contributions illustrated in Fig. 1, Eq. (4.26) allows to determine, for a given choice of cosmological parameters, the full temperature autocorrelations. The results, always in the absence of magnetized contribution, are reported in Fig. 2. In the plot at the left of Fig. 2 the critical fractions of matter and baryons, as well as h_{0}, are all fixed. The only quantity allowed to vary from one curve to the other is the scalar spectral index of curvature perturbations, i.e. n. The full line denotes the pivot case n_{ ζ }= 0.958 (corresponding to the central value for the spectral index as determined according to the WMAP data alone). The dashed and dotdashed lines correspond, respectively, to n_{ ζ }= 0.974 and n_{ ζ }= 0.942 (which define the allowed range of n since n_{ ζ }= 0.958 ± 0.016 [1]).
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 contribution of the magnetic fields will now be included both in the SachsWolfe region (as discussed in section 3) and in the Doppler region (as discussed in section 4). In Fig. 3 the temperature autocorrelations are computed in the presence of a magnetized background. The values of the relevant magnetic parameters (i.e. the smoothed amplitude of the field B_{L} and the spectral slope ε) are reported at the top of each plot and in the legends. In the plot at the left of Fig. 3 the spectral slope is fixed as ε = 0.01 while B_{L} is allowed to vary. The other cosmological parameters are fixed to their concordance values stemming from the analysis of the WMAP 3year data and are essentially the ones already reported at the top of Fig. 2. The diamonds are the WMAP 3year data points. In the plot at the right of Fig. 3 the spectral slope is still reasonably flat but, this time, ε = 0.1. For a spectral slope ε = 0.01 the case B_{L} = 1nG is barely distinguishable (but not indistinguishable, as we shall see below) from the case B_{L} = 0. As soon as B_{L} increases from 1 to 5 nG three different phenomena take place:
• 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.
If the spectral slope increases a similar trend takes place as B_{L} increases. However, the formation of the hump takes place for values of B_{L} which are comparatively larger than in the case of nearly scaleinvariant magnetic energy spectrum. In Fig. 3 (plot at the right) the magnetic spectral slope is ε = 0.1 (while the adiabatic spectral slope is fixed to the concordance value, i.e. n_{ ζ }= 0.984). To observe the formation of the hump (which is of course excluded by experimental data) the values of B_{L} must be larger and in the range of 15 to 20 nG. As soon as ε increases towards 1 the minimal allowed B_{L} also increases. This is particularly evident from the two plots reported in Fig. 4 where the values of ε have been chosen to be 0.5 (plot at the left) and 0.9 (plot at the right).
In Fig. 4 the dashed curve in the plot at the right corresponds to B_{L} = 6 nG. For this value of B_{L} the hump is not yet present, while for ε = 0.01 already for B_{L} = 2nG the second peak is completely destroyed. These differences are related to the fact that an increase in ε implies, indirectly, that the amplitude of the power spectrum of the magnetized background decreases at large lengthscales, i.e. for small wavenumbers. From Fig. 3 it can be argued, for instance, that when the magnetic slope is nearly flat (i.e. ε ≃ 0.01), the allowed value of the smoothed field becomes B_{L} < 0.1 nG. It should be remarked, to avoid confusion, that the scale invariant limit for the curvature perturbations, according to the conventions of the present paper is n_{ ζ }→ 1 while the scale invariant limit for the magnetic energy density fluctuations is ε ≪ 1. Finally in Fig. 5 the effect of the variation of the magnetic pivot scale is illustrated. If k_{L} diminishes by one order of magnitude the temperature autocorrelations increase in a different way depending upon the value of ε. By diminishing k_{L} the magnetic field is smoothed over a larger lengthscale. The net effect of this choice will be to increase the temperature autocorrelations for the same values of B_{L} and ε.
For ℓ = 210 the experimental value of the temperature autocorrelations is [1–3] 5586 ± 106.25 (μK)^{2}, while for ℓ = 231 the experimental value is 5616.35 ± 99.94 (μK)^{2}. The next value, i.e. ℓ = 253 implies 5318.06 ± 86.19 (μK)^{2}. By requiring that the addition of the magnetic field does not shift appreciably the height of the first Doppler peak it is possible to find, for each value of the spectral slope ε a maximal magnetic field which approximately coincides, in the cases of Fig. 3 with the lowest curve of each plot. This argument is sharpened in Fig. 6 where the starred points represent the computed values of the temperature autocorrelations for two different values of B_{L} and for the interesting range of ε. The value of ℓ_{p}, i.e. the multipole corresponding to the first Doppler peak, has been taken, according to [1–3] to be 220. If, according to experimental data, the following condition
is enforced, then, the smoothed field intensity and the spectral slope will be bounded in terms of the position and height of the Doppler peak. This condition is indeed sufficient since, according to the numerical results reported in the previous figures, the distortion of the second and third peaks are always correlated with the increase of the first peak. Already at a superficial level, it is clear that if B_{L} ≤ 1nG the only spectral slopes compatible with the requirement of Eq. (5.1) are rather blue and, typically ε > 0.5. The numerical values obtained with the method described in Fig. 6 are well represented by the following interpolating formula