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Large-scale magnetic fields, curvature fluctuations and the thermal history of the Universe

It is shown that gravitating magnetic fields affect the evolution of curvature perturbations in a way that is reminiscent of a pristine non-adiabatic pressure fluctuation. The gauge-invariant evolution of curvature perturbations is used to constrain the magnetic power spectrum. Depending on the essential features of the thermodynamic history of the Universe, the explicit derivation of the bound is modified. The theoretical uncertainty in the constraints on the magnetic energy spectrum is assessed by comparing the results obtained in the case of the conventional thermal history with the estimates stemming from less conventional (but phenomenologically allowed) post-inflationary evolutions.


Introduction
It is tempting to speculate that large-scale magnetic fields are generated in the early Universe [1,2,3] during a phase where the Weyl invariance of their evolution equations is broken either spontaneously of explicitly (see, for instance, [4] and references therein). As a result the obtained magnetic fields will be necessarily tangled over typical wavelengths larger than the Hubble radius [5,6,7,8,9]. In this situation, the presence of magnetic fields may affect the evolution of curvature inhomogeneities in a way that is reminiscent of what happens in the presence of a non-adiabatic pressure fluctuations.
Suppose, indeed, that large-scale magnetic fields are generated thanks to the amplification of the quantum fluctuations of an Abelian gauge field which can be identified with the hypercharge field whose modes will not be screened by thermal effects. In spite of the specific model of parametric amplification, the result of the process will be a stochastically distributed field whose two point function can be written as where P B (k) is the magnetic power spectrum (which may change depending upon the specific way the Weyl invariance is broken) and P j i (k) = (δ j i − k i k j /k 2 ) is the traceless projector. In Eq. (1.1) the momenta appearing in the correlator are the comoving wave-numbers. The homogeneous component of this configuration vanishes and, as a result, the magnetic fields will not break spatial isotropy 2 of the background geometry whose line element can be written, in the spatially flat case and in terms of the conformal time coordinate τ , as (1. 2) The typical wavelengths that set the initial conditions of the CMB anisotropies 3 are still larger than the Hubble radius (i.e. kτ < 1 ) at recombination, taking place for an approximate redshift z rec ∼ 1050 when the visibility function is peaked. When kτ < 1 the evolution equations of the perturbations of the spatial curvature will receive contributions from three independent sources. The adiabatic mode typically provides the first contribution arising from the quantum fluctuations of a (single) inflaton field. The second contribution may come from a collection of non-adiabatic modes that may be present either because the inflationary evolution is driven by more than one scalar degree of freedom, or because the primordial plasma contains a number of spectator fields which do not drive the evolution of the background but whose inhomogeneities contribute to the overall fluctuations of the spatial curvature. Finally, the third contribution to the evolution of curvature fluctuations can be attributed to gravitating magnetic fields. Up to now various studies addressed the interplay between large-scale magnetic fields and the scalar [12,13,14] (see also [15,16]), vector [17,18,19] and tensor [19] modes of the geometry (see [20] and [21] for two recent reviews). As far as the scalar modes are concerned, it was assumed that large-scale magnetic fields are present prior to recombination (but after equality) and the resulting corrections to the Sachs-Wolfe plateau have been computed [12,13]. The analysis has been also recently extended to cover the Doppler region, i.e. the first, second and third peaks of the temperature autocorrelations [14]. The main theme of the present paper will be 2 If a background magnetic field exist, it must necessarily be oriented along a specific direction [10]. The effective spatial isotropy is broken and the angular power spectrum depends on the direction of the magnetic field. As a consequence a number of constraints, stemming directly from the breaking of spatial isotropy, can be derived [11]. 3 By this expression we mean the wavelengths that set the initial conditions of the lowest multipoles of the Boltzmann hierarchy.
to discuss possible (further) numerical constraints arising from the coupled evolution of large-scale magnetic fields and curvature perturbations. This analysis, when correctly performed, allows to assess the theoretical uncertainty of the computational scheme. In fact, the results obtained in the case of the conventional thermal history can be compared with the estimates stemming from less conventional (but phenomenologically allowed) post-inflationary evolutions.
In the most simplistic 4 scenario a conventional inflationary phase is followed, after a sudden reheating, by a radiation-dominated phase which is replaced, at equality by the standard matterdominated phase. This evolutionary history allows to compute the curvature perturbations that are induced, for instance, by a single primordial adiabatic mode. It is also possible, in the same framework, to include the peculiar effect of large-scale magnetic fields whose inhomogeneities may indeed affect the curvature perturbations [12,13,15,16]. The result of this calculation allows the estimate of the Sachs-Wolfe plateau and, ultimately, the estimate of the Doppler oscillations [14].
If the evolutionary history of the background geometry changes in its early phases, also the effect of large-scale magnetic fields on the induced curvature perturbations will be different. The evolution of magnetized curvature perturbations is sensitive, both, to the total barotropic index and to the total sound speed, i.e.
where the prime denotes a derivation with respect to τ and the subscript t reminds that the barotropic index and the sound speed are the ones pertaining to the total fluid.
If the rate of expansion changes in the early phases of the thermal history of the Universe (typically prior to big bang nucleosynthesis) also w t and c 2 st will be different and the curvature perturbations will have a slightly different evolution. This qualitative argument will now be corroborated by a detailed calculation and this is one of the themes of the present investigation. The problem then becomes to estimate the total curvature perturbation that will be sensitive both to the adiabatic and to the magnetized contributions.
The strategy adopted in the present investigation will then be to scrutinize different thermodynamic histories of the Universe and to compute, in each of these cases, the resulting curvature perturbations for wavelengths that are still larger than the Hubble radius after matter-radiation equality. It will be shown that possible variations in the evolution of the total barotropic index will modify the interplay between the adiabatic and the magnetized components of curvature perturbations. This procedure will also allow to determine the theoretical error caused by selecting the most simplistic thermal history in comparison with its non-minimal counterparts.
The plan of the present analysis will then be the following. In section 2 the essential theoretical 4 Notice that the most simplistic scenario is also the one contemplated by the minimal paradigm compatible with the three data sets that are used to infer the values of the cosmological parameters, i.e. the ΛCDM paradigm. These three data include, in general terms, the large-scale structure observations, by the CMB observations and by the type Ia supernovae observations. 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 data [22,23,24,25,26] have been combined, so far, with various sets of data. These data sets include the 2dF Galaxy Redshift Survey [27], the combination of Boomerang and ACBAR data [28,29], the combination of CBI and VSA data [30,31]. Furthermore the WMAP 3-year data have been also combined with the Hubble Space Telescope Key Project (HSTKP) data [32] as well as with the Sloan Digital Sky Survey (SDSS) [33,34] data. Finally, the WMAP 3-year data can be also usefully combined with the weak lensing data [35,36] and with the observations of type Ia supernovae (in particular the data of the Supernova Legacy Survey (SNLS) [37] and the so-called Supernova "Gold Sample" (SNGS) [38,39]). tools will be introduced. A consistent gauge-invariant description will allow to follow, in one shot, the evolution of the curvature fluctuations and of the density contrasts on comoving orthogonal hypersurfaces. In section 3 different models for the evolution of the barotropic index will be scrutinized and motivated. The resulting (magnetized) curvature perturbations will be computed. In section 4 a set of bounds on the magnetic field intensity will be derived and compared. Section 5 contains the concluding remarks.

Evolution equations
The notations ubiquitously employed in the present script imply that the Friedmann-Lemaître equations in the spatially flat case (with line element given in Eq. (1.2)) are given by: where H = a ′ /a and the prime denotes, as in Eq. (1.3) a derivation with respect to τ . Notice that H = aH where H =ȧ/a is the Hubble expansion rate and the dot denotes a derivation with respect to the cosmic time coordinate t (recall that the connection between t and τ is given by the differential relation dt = a(τ )dτ ). By virtue of the connection between H and H, a given wavelength is larger than the Hubble radius provided the corresponding comoving wave-number k satisfies the condition k/(aH) ≃ kτ < 1. Equations (1.3) and (2.3) imply so that c 2 st = w t iff the (total) barotropic index is constant in time. For the purposes of the present investigation it is practical to start with the evolution equations of the fluctuations of the geometry expressed in fully gauge-invariant terms. The Hamiltonian and momentum constraints can be written as where δρ t is the total (and gauge-invariant) density contrast of the fluid sources and δρ B (τ, x) = B 2 ( x)/(8πa 4 ). The fluctuations Φ and Ψ are the gauge-invariant Bardeen potentials [40,41] that coincide with the longitudinal fluctuations of the metric in the conformally Newtonian gauge. In Eq. (2.6) θ t is the three-divergence of the total peculiar velocity. At the right-hand side of Eq. (2.6) the three-divergence of the Poyniting vector leads to a term going as ∇ · [ E × B]/a 4 , where E ≃ J/σ ≃ ( ∇ × B)/σ is the Ohmic electric field. The conductivity σ is large during most of the thermodynamic history of the Universe, since it is proportional to the temperature when the temperature is much larger than the mass of the correponding species. The contribution of the Ohmic Poynting vector will therefore be neglected in comparison with the other components of the magnetic energy-momentum tensor. Introducing the density contrast on comoving orthogonal hypersurfaces, the Hamiltonian constraint (2.5) takes the peculiar form which is (just formally) analog to the Poisson equation typical of the non-relativistic treatment of gravitational inhomogeneities. Another relevant pair of gauge-invariant quantities is represented by . (2.10) The variable ζ [41,42] where Eq. (2.12) follows from Eq. (2.11) making use of the generalized Poisson equation (2.8). It is relevant to remark that when the wavelengths are larger than the Hubble radius, at a given epoch, (2.13) In spite of the physical differences between ζ, R and ǫ m the explicit solution of the whole system in terms of one of these variables allows to compute the others. From the fluctuation of the covariant conservation equation of the total plasma the following equation can be easily obtained (2.14) Using now Eq. (2.9) inside Eq. (2.14) a first-order differential equation for ζ emerges and it is given by: The term δp nad in Eq. (2.15) accounts for the non-adiabatic pressure inhomogeneities; the following decomposition of the total pressure fluctuation has been tacitly assumed in Eqs. (2.14) and (2.15). While the first (adiabatic) contribution gives the pressure fluctuation produced by the inhomogeneous density fluctuation when the specific entropy is constant (i.e. δς = 0), the second (non-adiabatic) contribution arises even if the density is unperturbed (i.e. δρ t = 0) but the plasma possesses many different components (for instance, in the pre-recombinantion plasma, photons, baryons, neutrinos and dark matter).
The evolution of R can be obtained, in similar terms, from the equations of Φ and Ψ, i.e.
where Π j i is the anisotropic stress of the fluid and is the anisotropic stress of the magnetic field. Inserting Eq. (2.10) into Eq. (2.17) and recalling Eq.
(2.16) the evolution equation of R can be written as where the notation has been used. If there are collisionless particles in the plasma (like neutrinos, after weak interactions have fallen out of thermal equilibrium) we will have that Concerning Eq. (2.23) two comments are in order. We referred the magnetic anisotropic stress to the photon background. This is natural since, for typical length-scales larger than the magnetic diffusivity scale, both quantities redshift at the same rate with the expansion of the Universe. The second comment involves the relevance of the anisotropic stress. As discussed in relation with the estimate of the CMB autocorrelation induced by magnetic fields, the magnetic anisotropic stress is of upmost importance at intermediate scales.
In particular the interplay with the neutrino anisotropic stress leads to the correct initial conditions for the magnetized CMB anisotropies. However, in the present paper we will be mostly concerned with the largest scales and we will try to assess what could be the influence, on those scales, of slightly different thermal histories of the Universe. For typical wavelengths larger than the Hubble radius at recombination the anisotropic stress can be neglected in the first approximation. If needed, however, it can be included with the standard iterative procedure [44,45] (see also [46]) where, to lowest order, the solution is the one where Φ ≃ Ψ.

Different thermal histories
By looking at Eq. (2.15) it is clear that, to leading order in k 2 τ 2 , the evolution of ζ (and of R) is determined by three independent pieces of information: • the presence (or absence) of non-adiabatic pressure fluctuations; • presence (or absence) of large-scale magnetic fields; • the specific time dependence of w t and c 2 st The simplest possible situation compatible with the presence of super-Hubble magnetic fields arises when the Universe becomes suddenly dominated by radiation at the end of inflation. This is, incidentally, also the underlying assumption in the standard ΛCDM scenario. The Universe will then become dominated by dusty matter at a redshift 5 z eq ≃ 3200. If inflation was driven by a single inflaton and if other spectator fields with scale-invariant spectrum were absent, then it is also rather plausible to enforce the condition δp nad = 0. In this situation, the exact solution of Eqs. (2.1), (2.2) and (2.3) implies that Using Eqs. (3.1) and (3.2) inside Eq. (2.15) and changing the variable from τ to α the following expression can be obtained: In Eq. (3.3) the magnetic energy density has been expressed in units of the photon energy density and R γ denotes the photon fraction in the post-inflationary radiation background. After neutrinos decouple for temperatures of the order of the MeV the photon fraction will be related with the neutrino fraction as R γ = 1 − R ν where R ν = 0.405 (for three neutrino families). Direct integration of Eq. (3.3) gives the sought result, namely The constant ζ * (k) stands for the adiabatic mode produced (for instance during a phase of conventional inflation). After matter-radiation equality (but before recombination) to leading order in k 2 τ 2 we will have, from Eq. (2.11), that This result can be easily interpreted from the physical point of view. When the rate of expansion increases from radiation to matter the barotropic index (and the sound speed) are both decreasing from 1/3 to 0. Thus the overall effect on the source term for the evolution of curvature perturbations implies that the magnetic contribution tends to cancel the contribution of the adiabatic mode. 5 When not otherwise stated it will be assumed that h 2 0 Ω M0 ≃ 0.1326, where Ω M0 denotes the present fraction in dusty matter. This value for the total critical fraction of matter emerges, in the context of the ΛCDM paradigm, when the WMAP data [22,23] are combined with all the remaining cosmological data sets of different origin (excluding weak lensing data).
In the context of this reference scenario when the Universe only passes, after inflation, from radiation to matter also ǫ m and θ t can be computed, respectively, from Eqs. (2.8) and (2.6). It suffices to extract Ψ from ζ and then use the aforementioned constraints. Going to Fourier space result of this manipulation will be where Thanks to the success of big-bang nucleosynthesis, it is rather plausible to imagine that the Universe was already dominated by radiation for temperatures as large and few MeV. However, prior to that time there are no direct probes of the expansion rate of the Universe. In a somehow indirect way this could be achieved through the study of the cosmic background of gravitational waves. At the moment, for instance, there is no compelling evidence of the fact that, after inflation, the Universe became suddenly dominated by radiation. It is, on the contrary, rather reasonable that, before settling on a radiation-dominated stage of expansion the Universe passed through a phase where, for instance, the rate of expansion was smaller than the Hubble rate during radiation. This occurrence is realized when the total barotropic index of the sources driving the geometry satisfies 1/3 < w t ≤ 1. In this case the sources are said to be stiffer than radiation [47]. The stiffest equation of state we can imagine is the one where the barotropic index is 1 which implies that the sound speed coincides with the speed of light. The case of speed of sound equal to speed of light is the one contemplated by the so-called Zeldovich model (see [50] and references therein). A similar kind of evolution arises in quintessential inflationary models where the inflaton and the quintessence field are unified in a single degree of freedom. In this case (as in the more general case of stiffer phases [47]) the spectrum of relic gravitons presents a sharp increase [51] and a peak [52] which are potentially accessible to direct observations. In the context of quintessential inflation the stiff epoch is effectively dominated by the kinetic energy of the inflaton. Stiff phases also arise in brane-world scenarios [53] where the calculation of the graviton spectrum mirrors the four-dimensional case [51,52].
The transition from a stiff epoch to a radiation-dominated phase of expansion can be understood in terms of the following exact solution: parametrize the physical situation where, at τ * some (small) amount of radiation is present in comparison with the stiff contribution (i.e. ρ R * ≪ ρ s * ). In the context of quintessential inflationary models [54] the initial amount of radiation comes indeed from quantum fluctuations [55] and, consequently, ρ R * ≃ H 4 * . This occurence implies, in turn, that the hierarchy between the initial scale factor at τ * and the scale factor at the onset of radiation (i.e. a r is determined by M P /H * ). Since H * < 10 −6 M P , the duration of the stiff phase is correctly bounded because H r > H bbn . According to Eq. (2.4), the barotropic index and the sound speed are: . (3.11) Using therefore Eq. (3.11) inside Eq. (2.15) and changing variable from τ to α: Integrating Eq. (3.12) between α = 0 and α f we do get where α f ≫ 1 and it is defined deep in the radiation epoch. This result implies that the magnetic contribution enhances (rather than canceling) the adiabatic term ζ * (k) of inflationary origin.
As in the case of the radiation-matter transition the total density contrast and the total peculiar velocity can be computed. In Fourier space the result is: where This result can be generalized when the stiff phase is parametrized in terms of a source p s = δρ s with 1 < δ < 1/3. Repeating the same steps outlined above we will have that the dependence of the barotropic index and of the sound speed upon the scale factor are given, in this case, by: The equation for ζ and its related solution can the be written as As in the other two cases ǫ m and θ t can be easily computed: (3.22) The upper limit of integration in Eqs. (3.22) corresponds to s(α) = α 3δ−1 . For each value of δ the above integrals can be evaluated in explicit terms.
Suppose now to investigate a three stage model where the Universe expanded, initially, at a rate that was slower than radiation then got trapped in a radiation phase which turned subsequently into an epoch dominated by dusty matter. In this case we will have that the barotropic index and the sound speed are given, respectively, by: In this case the evolution equation of curvature perturbations is given by (3.25) The (approximate) solution of Eq. (3.25) in the matter-dominated epoch reads: • if the Universe is dominated by radiation from the end of inflation (as it happens in the case of the standard evolution) the magnetized contribution and the adiabatic mode have opposite sign; • if, prior to the electroweak epoch, the thermodynamic history of the Universe deviates from a radiation epoch the magnetized contribution is more suppressed in comparison with the standard case.
In the framework of conventional inflationary models this type of deviation from the standard scenario is pretty general. By general we mean that, within the current bounds on the expansion of the Universe arising from big-bang nucleosynthesis we can just imagine drastic deviation from the radiation-dominated evolution between the end of inflation and, say, neutrino decoupling. In fact, deviation from a radiation-dominated evolution after (or right before) big bang nucleosyntheis will necessarily jeopardize the production of the abundances of the light elements.
Up to now it has been assumed that the fluids of the primeval plasma do not exchange energy. Let us now address this interesting situation that could arise, for instance, in the course of reheating. Let us therefore imagine that a matter fluid (which can model the coherent oscillations of the inflaton) decays into massless particles. This dynamics can be parametrized by the following system 6 : The Universe, initially dominated (right after inflation) by dusty matter, becomes rapidly dominated by radiation at a rate that is controlled by Γ. It should be immediately noticed that, by summing up Eqs.
(3.28) and (3.29) the total energy density ρ t = ρ r +ρ m is covariantly conserved, i.e.ρ t +3H(ρ t +p t ) = 0. Equations (3.28) and (3.29) can be approximately solved: where t d ≃ Γ −1 . As a consequence of the relations of Eq. (3.30) we can also say that In this situation the total barotropic index and the total sound speed are slightly modified thanks to the physical differences of the system. In particular we will have that From Eq. (3.32) is clear that c 2 st → 1/3 when Γ ≫ H. The evolution equations of curvature perturbations to be solved read: where g = H/(Γ + 3H). Let us now concentrate, as discussed above, on the conventional situation where the non-adiabatic modes are totally absent. In this case, initially, δp nad ∝ (ζ r − ζ m ) ≃ 0 and ζ r = ζ m ≃ ζ * . Using Eq. (3.32) inside Eq. (3.33) we will simply have thaṫ Recalling that, after t d , Γ ≫ H and ρ m is exponentially suppressed, Eq. (3.35) can be written as Integrating once Eq. (3.36) the result will be The integral mentioned above can be done introducing the error function, i.e. (3.38) Taking the limit y → ∞, we then have (3.40)

Bounds on different thermal histories
The two-point correlation function of curvature perturbations will receive contributions, in general, from the adiabatic mode, from one (or more) non-adiabatic modes, and from the magnetic field. As argued in the previous sections, the simplest situation is the one where only the adiabatic mode is present together with the magnetized contribution that we ought to constrain. This choice is motivated by the experimental evidence that the Doppler peak structure of the temperature autocorrelations strongly suggests that, after equality, the large-scale curvature perturbations were, predominantly adiabatic [22,23,24,25,26].
Consider therefore, the curvature perturbations present at recombination in the framework of the different thermal histories discussed in the previous section. The curvature perturbations can be written, in Fourier space, as 7 where the subscript h refers to each different thermal history and where ζ * (k) represents the adiabatic contribution normalized to the large-scale value of the (ordinary) Sachs-Wolfe contribution to the temperature autocorrelations. The total (scalar) power spectrum P h (k), i.e. the Fourier transform of the two-point function computed at different spatial locations but at the same time is defined as From Eq. (4.1) and within the conventions summarized by Eq. (4.2) we then have that where γ is the correlation angle between the adiabatic mode and the magnetized contribution. In Eq. (4.4) P ζ is the power spectrum of the adiabatic contribution defined as In Eq. (4.5) n ζ is the spectral index of the adiabatic mode and A ζ is the amplitude referred to the pivot scale k p . Following the WMAP conventions the value of the pivot scale will be fixed as k p = 0.002 Mpc −1 . The numerical factor appearing in the expression of A ζ has been obtained by taking consistently T cmb = 2.725 × 10 6 (expressed in units of µK). In the absence of any other contributions the WMAP 3-year data imply 8 n ζ ≃ 0.947 and A(k p ) ≃ 0.815 when combined with the remaining cosmological data sets, i.e. supernovae and large scale structure 9 . In Eq. (4.5) the magnetic part of the correlation function is expressed as [13,14] P Ω (k) = F (ǫ)Ω where 2ǫ the spectral index of the magnetic energy density fluctuations and k L is the magnetic pivot scale that will be defined in a moment. In Eq. (4.6) we have that (4.7) In Eq. (4.7) B L is the value of the magnetic field smoothed, through a Gaussian window function, over a typical comoving length L which is related to the magnetic pivot scale as L = 2π/k L . In what follows the fiducial value k L = Mpc −1 will be consistently adopted. It is relevant to appreciate that B L represents the smoothed magnetic field red-shifted to the present epoch. This convention is a bit confusing but we will follow it since it is a common practice in the field. The source of the possible confusion is, in short, the following. The field B L is the value of the magnetic field at recombination redshifted to the present epoch and assuming magnetic flux conservation which is well justified since the value of the conductivity is always rather large. However, B L is not the present value of the magnetic field intensity observed in galaxies and galaxy clusters. The rationale for this 8 In this analysis the values of the cosmological parameters will be fixed to the best fit values of the WMAP data combined with the other sets of cosmological observables. This choice is not crucial since the values of the cosmological parameters are purely illustrative. The essential features of the present analysis are unchanged if the best fit values of the WMAP data alone (or partially combined with the other data sets) are consistently assumed. 9 The data [32] and [35,36] are not included in the joined analysis. statement is that during the gravitational collapse of protogalaxies the magnetic field intensity B L will be amplified by compressional amplification and, probably, also by some dynamo action. Assuming just compressional amplification (which is the most certain aspect of the dynamics) the amplification factor may be of the order 10 4 or even 10 5 .
Notice that, from Eq. (4.6) the nearly scale-invariant limit is achieved for ǫ < 1. Furthermore, as discussed elsewhere, the power spectrum of the magnetic energy density, being quartic in the field intensities, is computed in terms of the appropriate convolutions that have been evaluated, to get to Eq. (4.6), for ǫ < 1. It should be borne in mind that, in the opposite case (i.e. ǫ > 1) the magnetic energy density has a violet spectrum. This implies that, at very large length-scales the gravitational effect will be negligible and that, furthermore, the most significant bounds on the intensity of the magnetic field will come from comoving momenta close to the diffusion scale. Notice that the nearly scaleinvariant spectrum is rather suggestive also because it would imply a nearly scale-invariant spectrum of large-scale magnetic fields from galaxies, to clusters, to super-clusters. This observation is, today, beyond our observational capabilities. However, by looking at recent determinations of magnetic fields in clusters and in some typical supercluster, it is indeed tempting to speculate that the resulting power spectrum of the magnetic energy density is nearly scale-invariant.
The correlation function of curvature perturbations enters directly the determination of the ordinary Sachs-Wolfe contribution which is the dominant source of temperature anisotropies at large angular scales. Let us require that the magnetized contribution is always much smaller than the adiabatic contribution. So, we will assume, in a rather conservative approach, that the magnetized contribution is always smaller (by a factor η) than the adiabatic contribution. The parameter η can be O(10 −3 ), in a conservative approach. Following this strategy the magnetic field intensity can be bounded in each thermal history of the Universe. Consider, for instance, the plots appearing in Fig.  (1) where, on the vertical axis, the base 10 logarithm of the smoothed magnetic field is reported in units of nG. In the plot at the left the three different lines correspond to the cases of the three different histories discussed in the previous sections. These three different histories will imply three different functions T h with h = 1, 2, 3 and with, in general T 1 = T 2 = T 3 . At late times the functions T h can be approximated as 12α rec + 4 3α rec + 4 . (4.10) The function T 1 arises in the simplest case, i.e. when the Universe, after a phase of conventional inflation, passes from a radiation-dominated epoch to the matter stage. In T 1 the reheating is assumed to be a sudden process. The function T 2 describes the situation when there is an intermediate stiff phase expanding at a rate slower than radiation. Finally the function T 3 arises when there is a prolonged reheating and then the Universe is, subsequently, dominated by radiation and matter. In Fig. 1 (plot at the left) we illustrate the bounds as a function of the magnetic spectral index. The region below each of the curves implies that the contribution of the magnetic field to the two-point function is smaller than 10 −3 .
The less restrictive case is, surprisingly enough, T 2 . The rationale for this occurrence can be simply understood in terms of the results reported in the previous section. In the transition from the stiff phase to the radiation phase the magnetized component gets reduced while in the subsequent transition from radiation to matter the magnetized contribution gets enhanced. Since the two effects tend to cancel, the net result will be an overall suppression of the magnetized contribution. This dynamical suppression allows the magnetic field amplitude to be larger than in the other two cases (i.e. T 1 and T 3 ) where there is no "destructive" interference between the two subsequent effects.
In the plot at the right of Fig. 1 this pattern is confirmed with an extra piece of information: if the matter fraction increases the bound become looser in the case of T 3 and a bit tighter in the case of T 1 and T 2 . In fact, in Fig. 1 (plot at the right) the magnetic spectral index has been kept fixed while the matter fraction is allowed to move from the fiducial value h 2 0 Ω M0 = 0.1326 which is the one assumed in the left plot of Fig. 1. The increase (or decrease) of the critical fraction of dusty matter simply means, physically, that the recombination epoch is either delayed or anticipated.
It should also be remarked that the case of prolonged reheating is the most constraining. This occurrence can be also simply understood from the considerations of the previous section. In the case of prolonged reheating c 2 st = 1/3. So during this phase the curvature perturbations decrease and this decrease will interfere constructively with the usual radiation matter transition since both contribution have the same sign. In Fig. 1 the value of the correlation angle has been fixed to π/2 implying that the adiabatic and the magnetized mode are uncorrelated. Again we may relax this assumption and allow γ to vary while all the other parameters are fixed to their fiducial values. This has been done in Fig. 2. As expected the bounds is tighter in the case when the two components are correlated and looser when the two components are anti-correlated. In Fig. 2 (plot at the right) the variation of the adiabatic spectral index is illustrated always in the case where the two components are uncorrelated. It is now relevant to remark that while P ζ (k) slightly decreases as k increases, the opposite is true for P Ω (k). It is therefore clear that, given this situation, the most constraining wavenumbers are the largest, i.e. the small-scale behaviour of the spectrum is the one that leads to the most stringent bounds. This consideration has been taken into account in deriving the bounds expressed by Figs. 1 and 2. In particular, it has been assumed that the diffusion scale is the one roughly associated with    Fig. 1 (plot at the left). In the plot at the right the two classes of curves have illustrate the same phenomenon but in terms of the dependence on n ζ and should be compared with the right plot appearing in Fig. 2.
the Silk wavenumber, i.e. There might be slightly different choices for the diffusion scale. For instance it has been noticed in the past that the diffusion scale of the magnetic fields should be related with the induced velocity of Alfvén waves. In this case the diffusion length-scale will be slightly smaller than the Silk scale by a factor which is essentially proportional, in our notations, to (B L /nG). In this case the diffusion wavenumber will be given by [19]: Mpc −1 (4.12) Figure 3 illustrates the situation described by Eq. (4.12). It is clear that the patterns of the different thermal histories remain the same. However, the bound is improved by roughly one order of magnitude.

Concluding remarks
There are various lessons that can be drawn from the exercise reported in this paper. The main question we addressed has been the possible influence of slight variations in the thermal history of the Universe on the curvature perturbations induced by a magnetic field present for typical scales that are larger than the Hubble radius at recombination. Various inflationary mechanisms for the generation of cosmic magnetic field fall in this situation. It has been found that, indeed, the thermal history of the Universe may either suppress or increase the relative weight of magnetized curvature perturbations. The suppression occurs when, during the dynamical evolution, there is a sort of destructive interference. This phenomenon takes place, amusingly enough, when there was a phase, in the life of the Universe, when the rate of expansion was slower than the one of radiation. This kind of evolution must occur prior to the onset of big-bang nucleosynthesis and explicit examples have been provided. Also the opposite phenomenon can be realized, i.e. a sort of constructive interference when the overall contribution of the magnetized curvature perturbations gets enhanced in comparison with the adiabatic contribution present at the end of inflation. This situation takes place if the reheating phase is sufficiently prolonged in comparison with the sudden reheating approximation.
These considerations can certainly be developed along different directions. It is interesting to remark that the nature of the bounds on the magnetic field intensity may change quantitatively and qualitatively. Different thermal histories imply that the constraints on the magnetic field intensities may vary, generally speaking, of one order of magnitude. In the case when the Universe contains after inflation (but before radiation) a stiff phase the present value of the smoothed magnetic field must be qualitatively 10 B L < 1.5 × 10 −7 G while in the context of prolonged reheating it must be as small as 1.6 × 10 −8 G.
The presented considerations are not totally insensitive on the estimate of the diffusive wavenumber. In particular, it occurs that while experimental data favour a slightly red spectrum of curvature perturbations (i.e. an adiabatic spectral index slightly smaller than 1), the magnetized contribution is typically nearly scale-invariant but slightly blue. In this situation the most constraining wavenumbers are the ones close to diffusion scale. If the diffusivity length-scale is slightly smaller than the Silk scale, then the overall bounds on the smoothed magnetic field intensity range, depending on the details of the specific thermal history, from B L < nG to B L < 0.1 nG. It can be discussed if the Alfvén velocity should enter the thermal diffusivity scale when the magnetic field is fully inhomogeneous. Indeed Alfvén waves are only excited when there is a background magnetic field in the game. This is not the case when the magnetic field does not break spatial isotropy which the case investigated in the present paper and which is also, in our opinion, the most realistic one (see the introduction). With these caveats, however, it is important to notice that different thermal histories may either relax or improve the bounds on the magnetic field intensity by one order of magnitude.