Lorentz-violating inflationary magnetogenesis

A non-conformally invariant coupling between the inflaton and the photon in the minimal Lorentz-violating standard model extension is analyzed. For specific forms of the Lorentz-violating background tensor, the strong-coupling and backreaction problems of magnetogenesis in de Sitter inflation with scale $\sim 10^{16}$GeV are evaded, the electromagnetic-induced primordial spectra of (Gaussian and non-Gaussian) scalar and tensor curvature perturbations are compatible with cosmic microwave background observations, and the inflation-produced magnetic field directly accounts for cosmic magnetic fields.


I. INTRODUCTION
Coherent magnetic fields as strong as B ∼ 10 −6 G have been detected in any type of galaxies and in galaxy clusters (for reviews on cosmic magnetic fields, see [1][2][3][4][5][6][7][8][9]). Their origin is still an open issue and is puzzling to the point that "cosmic magnetism" should be considered one of the biggest mysteries in cosmology.
Nowadays, what it is clear enough is that seed magnetic fields present prior to galaxy formation can be amplified by protogalaxy collapse and magnetohydrodynamic turbulence effects and then, at least in principle, they can reproduce the properties of presently-observed galactic fields.
Indeed, it has been recently pointed out [10] (see also references therein) that a small-scale dynamo could exponentially amplify small-scale seed magnetic fields during the process of galactic disk formation. Successively, differential rotation of the newly formed galactic disk would order the chaotic field resulting from the small-scale dynamo in such a way to reproduce the main features of the observed galactic magnetic fields. This mechanism would explain galactic magnetism if a sufficient strong seed field is present prior to galaxy formation but leave substantially unanswered the question of the presence of strong magnetic fields in clusters of galaxies.
A plethora of mechanism acting in the early Universe have been proposed to produce seed fields since Fermi's proposal of existence of cosmic magnetic fields back in 1949 [11].
Promising candidates are those mechanism operating during inflation since inflation-generated fields can be correlated on super-horizon scales, and then their comoving correlation length can be as large as the galactic one. If magnetic fields are created after inflation, instead, their correlation length cannot exceed the dimension of the horizon at the time of generation, so that they are correlated on scales generally much smaller than the characteristic scale of the observed cosmic fields.
Since standard Maxwell electromagnetism in a Friedmann-Robertson-Walker universe is invariant under conformal transformations, magnetic fields cannot be generated during inflation, as a consequence of the well-known "Parker theorem" [12,13]. For this reason, all inflationary models proposed in the literature repose on the breaking of conformal invariance of (standard) electrodynamics.
Turner and Widrow [14] analyzed the consequences of adding, to the Maxwell Lagrangian, nonstandard conformal-breaking gravitational couplings of the photon.
Ratra [15], instead, introduced a nonstandard conformal-breaking coupling between the scalar field φ responsible for inflation (the inflaton) and the electromagnetic field.
After these two seminal papers on the generation of large-scale magnetic fields at inflation, many others mechanisms have been proposed, most of which introduces nonstandard photon couplings to break conformal invariance (see, e.g., ).
There are, however, three mechanisms proposed in the literature that work without resorting to nonstandard physics.
Dolgov [51] argued that the well-know conformal anomaly in quantum field theory in curved spacetime induces a breaking of conformal invariance of standard electrodynamics, which in turn stimulates the generation of strong, large-scale magnetic field at inflation. Barrow and Tsagas [52] (see, also, [53] and [54]) showed that, within the framework of conventional electromagnetism, astrophysically interesting magnetic fields can be generated if one assumes, contrarily to what previously assumed in the literature of cosmic magnetic fields, that the spatial curvature of the Universe is nonzero and compatible with astrophysical observations.
The author pointed out in [55] (see, also, [56]) that the process of renormalization of inflationary quantum magnetic fluctuations naturally breaks conformal invariance giving, as a result, a strong, scale-independent today magnetic field.
Recently enough, however, a potential problem for inflationary mechanisms of magnetogenesis has been pinpointed by Demozzi, Mukhanov, and Rubinstein [57], and it is now know as the "strong coupling problem". The problem consists in the fact that the full electrodynamics theory including both the nonstandard cou-plings of the photon with other fields (as the inflaton) and the standard one with conserved external currents must be always in a weak-coupling regime, in order to trust the results. This problem, when combined with the so-called "backreaction problem", which appears in the theory when the inflation-produced electromagnetic field appreciably back-reacts on the inflationary dynamics, excludes all the models of inflationary magnetogenesis based on nonstandard physics.
After the work [57], only three scenarios for inflationary magnetogenesis have been suggested in which both the strong coupling and the backreaction problems are avoided.
Ferreira, Jain, and Sloth [58] considered a magnetogenesis scenarioà la Ratra where the inflaton is kinetically coupled to the photon and where a low scale inflation is followed by a prolonged reheating phase dominated by a stiff fluid.
Caprini and Sorbo [59] proposed a generalization of Ratra-like model, where both a kinetic and an axion-like coupling are present.
Very interesting is the scenario recently proposed by Tasinato [60]. A (nonstandard) derivative interaction between fermion fields (which give rise to the external currents) and a scalar field (which is kinetically coupled to the photon and amplifies electromagnetic vacuum fluctuations) "renormalizes" the electric charge during inflation in such a way that the theory is always in the weak-coupling regime.
Beside the strong coupling and the backreaction problems, there is another possible problem, hereinafter referred to as the "curvature perturbation problem", firstly pointed out by Barnaby, Namba, and Peloso [61]. It consists in the fact that inflationary electromagnetic fields generate both scalar (Gaussian and non-Gaussian) and tensor curvature perturbations that could be in conflict with recent observations of cosmic microwave background (CMB) anisotropies.
In this paper, we discuss a generalization of the Ratra model where the inflaton φ is kinetically coupled to the photon through a Lorentz-violating coupling of the form where f is a generic positive-defined function, L M is the standard Maxwell Lagrangian, while L LV contains all Lorentz-violating terms that involve the photon field, and that are implemented by external background tensors. The motivation behind the investigation of possible effects of Lorentz-violating in inflationary magnetogenesis is that in some theories of quantum gravity, such as loop quantum gravity [62] and string theory [63], the breakdown of Lorentz symmetry is expected to take place around the Planck scale, and so before the beginning of inflation.
Working in the weak-coupling regime, we will show that strong, scaling-invariant, magnetic fields can be cre-ated without back-reacting on the inflationary dynamics and without generating curvature perturbations in conflict with CMB results. This is possible if the external tensors, which represent new degrees of freedom with respect to the Ratra model, assume specific (fine-tuned) forms that assure that the electric part of the electromagnetic energy-momentum tensor (that would give rise to the the three aforementioned problems) is vanishing during inflation.
The paper is organized as follows. In section II, we discuss the characteristics (intensity and correlation length), that a comoving cosmic magnetic field must have in order to explain the magnetic fields detected in galaxies and clusters of galaxies. In section III, we briefly review, in the context of the Ratra-like model, the strong coupling and backreaction problems in inflationary magnetogenesis. In sections IV and V, we introduce and quantize our model of magnetogenesis based on Lorentzviolating couplings between the inflaton and the photon. In section VI, we derive the conditions under which the inflation-produced electromagnetic field does not appreciably back-react on the inflationary dynamics. In section VII, we evolve the produced magnetic field from the end of inflation until today. In section VIII, we discuss additional constraints that could be eventually imposed on the inflation-produced electromagnetic energymomentum tensor. In section IX, we calculate the spectrum, bispectrum, and trispectrum of the scalar curvature perturbations and estimate the spectrum of the tensor modes generated by the electromagnetic field, and compare them to the current bounds derived by the Planck mission. In section X, we discuss our results. Finally, in section XI, we draw our conclusions.

II. SEED MAGNETIC FIELDS
Magnetic fields have been detected in all type of galaxies with intensities of order µG. Galaxies at high redshift (still in the process of being formed) and irregular galaxies do not possess structured magnetic fields, while magnetic fields in fully formed galaxies, such as spiral or barred galaxies, typically trace the large-scale structure of galaxies [1,5].
These observations could be explained if a sufficiently intense large-scale magnetic field were present prior to galaxy formation. In this case, and due to the high conductivity of the protogalactic plasma, the magnetic field would remain frozen in the plasma and its final spatial configuration would reflect that of the galaxy. This rearrangement of the structure (not amplification) of the magnetic field could be easily realized by a galactic dynamo action, whose efficiency in reorganizing the primordial field is subjected to the only condition that the comoving magnetic correlation length be greater than about 100pc [1]. Moreover, due to the Alfvén theo-rem (see, e.g., [3]), a frozen-in magnetic field is amplified by a factor of [ρ gal /ρ m (t)] 2/3 and its correlation decreased by [ρ gal /ρ m (t)] 1/3 during protogalactic collapse [14]. Here, ρ gal and ρ m (z) ∝ (1 + z) 3 are, respectively, the galactic and cosmic matter densities, and for typical galaxies ρ gal /ρ m (t 0 ) ∼ 10 6 at the present cosmic time t 0 4 × 10 17 s. Therefore, a comoving seed field B 0 ∼ 10 −10 [(1+z gal )/(1+z ta )] 2 G, correlated on a comoving scale greater than λ B ∼ 10[(1 + z ta )/(1 + z gal )] kpc, could explain the galactic magnetism. The redshiftdependent factors come from the fact that between the turn-around redshift z ta (when the protogalactic collapse begins) and the galaxy redshift z gal , a frozen-in primordial magnetic field is decoupled from the Hubble flow and does not evolve adiabatically. Typically, z ta ∼ f ew tens, while z gal ranges from 0 to f ew [2].
The observation of galaxy clusters reveals the presence of intracluster large-scale-correlated µG magnetic fields. The intensity of such fields rise to tens of µG in the cluster cores, but this can be probably ascribed to fast-acting dynamo mechanisms due to cluster cooling flows [1].
Numerical simulations [64] of cluster formation starting at redshift z ta = 15 have shown that a f ew × 10 −10 G seed field is processed by magnetohydrodynamic effects in such a way to reproduce the observed magnetic Faraday rotation maps of clusters at low redshifts (z cl 0). It has also be found that the initial magnetic field correlation properties are inessential to the final result, although the scale of the initial magnetic field fluctuations was limited by the resolution length of order 100kpc [65]. The overall amplification of a factor f ew × 10 3 is explained as a Alfvén frozen flux effect of [ρ cl /ρ m (t 0 )] 2/3 ∼ 10 2 during cluster collapse [since, typically, ρ cl /ρ m (t 0 ) ∼ 10 3 ], plus an amplification of a factor of f ew tens, probably due to a Kelvin-Helmholtz instability of the intracluster plasma flows [64]. Therefore, a comoving seed field B 0 ∼ 10 −10 (1 + z ta ) −2 G, correlated on a comoving scale greater than λ B ∼ 100 (1 + z ta ) kpc, could explain cluster magnetic fields.
Roughly speaking, then, in order to explain both galactic magnetism and galaxy cluster magnetic fields, it suffices to have a comoving seed magnetic field such that [55] Limits on primordial magnetic fields. -If cosmic magnetic fields are relics from inflation, they could modify the standard evolution of the universe in radiation and matter eras. However, this is not the case, since the present limits on primordial magnetic fields do not exclude the existence of large-scale magnetic fields as strong as those in Eq. (2). Indeed, the most significant limits on large-scale cosmic magnetic fields come from big bang nucleosynthesis analyses, B 0 1 × 10 −6 G [ [66][67][68], data on large scale structures, B 0 f ew × 10 −9 G [68,69], studies of CMB radiation, B 0 f ew × 10 −9 G [68, [70][71][72][73], studies of the ionization history of our Universe, B 0 10 −9 G [74], Faraday rotation maps of distant quasars, B 0 10 −11 G [75,76], and blazars observations, B 0 7 × 10 −14 G [77][78][79], where the latter lower limit refers to the less conservative bound from the Blazar 1ES 0229+200 [79]. It is interesting to observe that the upper limit from Faraday rotation maps and the lower limit from blazars observations are just few times outside the interval of B 0 in Eq. (2). Narrowing the above limits could then eventually reveal the primordial nature of cosmic magnetic fields.

III. STRONG COUPLING AND BACKREACTION IN INFLATIONARY MAGNETOGENESIS
Let us now discuss, in some details, two requirements that have to be imposed on any magnetogenesis mechanism operating during inflation. We focus our attention to the "standard" kinetically coupled scenario for magnetogenesis, where the inflaton field φ is coupled to the standard kinetic Maxwell term via a generic coupling f (φ). This represents an extended version of the model proposed by Ratra [15], where f (φ) ∝ e αφ , with α being a constant.
The first requirement, that the full theory, namely when including conserved external currents, must be in a weak-coupling regime, has been discussed only recently in [57].
The second requirement, namely that the inflationproduced electromagnetic field must not appreciably back-react on the inflationary dynamics, has been instead firstly discussed by Ratra [15], but ignored in the seminal paper [14] by Turner and Widrow on inflationary magnetogenesis.

IIIa. Strong coupling
Let us consider the action for the electromagnetic field A µ , where g is the determinant of the metric tensor g µν , and L em the electromagnetic Lagrangian density, and let us assume that the electromagnetic field is coupled to the (homogeneous) inflaton field φ through a general coupling of the form Here, f (φ) is a generic, positive-defined function of the inflaton, L M = − 1 4 F µν F µν is the standard free Maxwell Lagrangian density, with F µν = ∂ µ A ν − ∂ ν A µ , while L int = j µ A µ is the standard interaction term with conserved external current. If, for the sake of simplicity, we assume that j µ is provided just by a charged massless fermion fields ψ, we than have j µ = eψγ µ ψ, where e is the electric charge, and γ µ are the Dirac matrices in curved spacetime. The latter are related to the standard Dirac matrices in Minkowski spacetime through γ µ = e µ a γ a , with e µ a being the vierbein [12]. 1 (In this paper, indices in Minkowski spacetime are indicated with the first letters of the Latin alphabet and run from 0 to 3. Indices in curved spacetimes are indicated with Greek letters and run from 0 to 3. Latin indices from the middle of the alphabet run from 1 to 3 and indicates spatial components of a given tensor.) Re-writing the Lagrangian density (5) as we see that the quantity e/f (φ) plays the role of an effective, time-dependent electric charge. The case f (φ) 1 would then correspond to a strong coupling between the fermion and the electromagnetic fields, and the theory would be in a (unmanageable) strong-coupling regime, as firstly pointed out in [57]. For this reason, we assume that f (φ) 1 during inflation and, obviously, f (φ) 1 at the end of inflation in order to recover the standard electrodynamics. Accordingly, we will consistently neglect, in the following, the interaction term L int in Eq. (5).

IIIb. Backreaction
It is tacitely assumed in the literature that inflationary magnetogenesis takes place in a fixed curved spacetime background. Therefore, we must consistently check that the vacuum expectation value (VEV) of the electromagnetic energy-momentum tensor is always negligible with respect to the energy-momentum tensor of the inflaton.
The electromagnetic energy-momentum tensor can be found by varying the action with respect to the metric tensor, We obtain (T em ) µν = f (φ)(T M ) µν , where (T M ) µν = F αµ F α ν + 1 4 F αβ F αβ g µν is the standard Maxwell energymomentum tensor. 1 The vierbein satisfies the condition e µ a e bµ = η ab , and is such that gµν = e a µ e b ν η ab , where η ab is the metric tensor in Minkowski spacetime.
Let us restrict our analysis to the case of a spatially flat, Friedmann-Robertson-Walker universe, described by the line element where η is the conformal time and a(η) is the expansion parameter [the latter is normalized to unity at the present conformal time η 0 , a(η 0 ) = 1]. Moreover, we assume, for the sake of simplicity, that inflation is described by a de Sitter phase. In this case, the conformal time is inversely proportional to the expansion parameter, η = −1/Ha, and the Hubble parameter H is a constant. Moreover, the energy-momentum tensor of the inflaton is Here, M is the scale of inflation, related to the energy density of inflation, ρ inf , through M 4 = ρ inf = 3H 2 /(8πG), where G = 1/m 2 Pl is the Newton constant and m Pl ∼ 10 19 GeV is the Planck mass.
Since both the background spacetime and coupling function f (φ) are homogeneous and isotropic, the VEV of the electromagnetic energy-momentum tensor takes on the simple form where ρ em = (T em ) 0 0 is the VEV of the electromagnetic energy density. (For the quantization of the theory and a formal definition of the vacuum, see section V.) Consequently, the condition that the electromagnetic backreaction on inflation is negligible can be expressed as ρ em ρ inf . The electromagnetic energy density is made up of an electric contribution and a magnetic part, where the electric and magnetic fields are defined as usual as a 2 E = −Ȧ and a 2 B = ∇ × A, with A µ = (0, A).
Here, and in the following, we work the Coulomb gauge, A 0 = ∂ i A i = 0, we denote the differentiation with respect to the conformal time with a dot, and we use the symbol ∇ for indicating the nabla operator in comoving coordinates.
The two-point correlators E 2 and B 2 are formally infinite due to the ultraviolet divergence of the corresponding spectra. This kind of divergence, typical in quantum theory in curved spacetime, can be cured by the standard techniques of renormalization, such as adiabatic renormalization. Nevertheless, we are principally interested to large-scale electromagnetic modes which are outside the horizon. These modes, which are expected to behave classically, belong to the non-divergent, infrared part of the spectra. Therefore, it is convenient to work in Fourier space and introduce the the so-called electric and magnetic power spectra, P E (k, η) and P B (k, η), through The electromagnetic energy density stored on the mode k is then where ρ em (k, η) is the electromagnetic energy spectrum defined by ρ em = ∞ 0 dk k ρ em (k). The condition that the electromagnetic backreaction on inflation is negligible can be then defined, mode-by-mode, by A particularly interesting class of models is that for which the coupling function f (φ) scales in time as f (φ) ∝ η 6 . This gives a scale-invariant magnetic spectrum, to wit P B (k) independent on k. The attractive figure of this model resides in the fact that, as firstly pointed out in [80], all the existing constraints on cosmic magnetic fields do not strongly peak over a specific range of either small or large scales. Hence, a scaling-invariant magnetic field can satisfy, in a "natural way", all the current experimental bounds, included the one in Eq. (3). For the scaling-invariant case, the electric and magnetic spectra are, roughly speaking, On super-horizon scales (−kη 1), then, the dominant contribution to the electromagnetic energy-momentum tensor is provided by the electric part. Its maximum value is attained at the end of inflation, η = η end . Therefore, backreaction on inflation is negligible on scales λ = 1/k such that where we used the fact that −kη end ∼ 10 −22 (Mpc/λ)(10 16 GeV/M ).
To simplify the analysis, we have considered here the case of instantaneous reheating, to wit, we have assumed that after inflation the Universe entered directly in the radiation dominated era. Needless to say, the electromagnetic backreaction on inflation has to be negligible on all observable scales. This, in turns, means that λ max has to be greater than the present horizon scale, H −1 0 4000Mpc, where H 0 is the Hubble constant. Accordingly, the scale of inflation has to be below 10 9 GeV. Such a low scale seems to be incompatible with recent results on the detection of inflation-produced gravitational waves, which require a scale of inflation around 10 16 GeV [81]. However, even assuming a scale as low as M ∼ 10 9 GeV, the amplitude of the inflation-produced magnetic field would be today too small to directly explain cosmic magnetism. Indeed, the actual magnetic field for the scaling-invariant case is and it assumes the extremely low value B 0 ∼ 10 −26 G for M ∼ 10 9 GeV.

IV. LORENTZ-VIOLATING COUPLINGS
The arguments in section III clearly show that the generation of (scaling-invariant) magnetic fields during (de Sitter) inflation, able to directly explain cosmic magnetization, is problematic due to their strong backreaction effects. The validity of this sort of no-go theorem for inflationary magnetogenesis, however, is not general, but restricted to the specific model described by the Lagrangian density (5). This leaves open the possibility to explore different couplings between the inflaton and the photon that may eventually generate, in a self-consistent way, cosmic magnetic fields.
In the following, we investigate one such a possibility, by looking at a possible new interaction of the inflaton with the electromagnetic field, this time in the context of the Lorentz-violating extension of the standard model of particle physics (for other mechanisms of cosmic magnetic fields at inflation reposing on the violation of Lorentz symmetry see [82][83][84][85][86][87][88][89][90]).

IVa. Lagrangian
The photon sector of the minimal Lorentz-violating standard model extension (SME) is described by the action where L M is the Lorentz-and CPT-invariant Maxwell Lagrangian density, while contains all Lorentz-violating terms that involve the photon field. They can be separated into two parts, L CPT−even and L CPT−odd , with the former preserving and the latter violating CPT symmetry, respectively. In Minkowski spacetime, Lorentz violation is achieved by coupling the electromagnetic field to rank-n, constant spacetime tensors k a1a2...an , known as external or background tensors. The passage from Minkowski to a general curved spacetime is obtained via the vierbein e a µ , k µ1µ2...µn = e a1 µ1 e a2 µ2 ...e an µn k a1a2...an [91]. In this passage, however, the external tensors acquire a spacetime dependence and then cease to be constant. This is due to the fact that the vierbein e a µ (x) is, generally, a function of the spacetime position x.
In the photon sector, the most general renormalizable Lagrangian density contains only three Lorentz-violating terms, where (k F ) µναβ , (k AF ) µ , and (k A ) µ , are background tensors. Their components are arbitrary real spacetime functions and are known as coefficients for Lorentz violation. Although the presence of the external tensors may indicate an explicit breaking of Lorentz violation, the form of the Lagrangian terms (19)- (20) is completely general and independent of the origin of the Lorentz violation. Indeed, these terms would have the same form also in the case where Lorentz violation were spontaneous, deriving, for example, from the fact that the external tensor k µ1µ2...µn are vacuum expectation values of corresponding field operators K µ1µ2...µn , We now generalize the coupling in Eq. (5) by assuming that the inflaton is coupled to the photon field, via the generic function f (φ), to both the standard photon kinetic term L M , and the Lorentz-violating term L LV , For the sake of simplicity, we consider only the CPT-even terms in the Lagrangian density (18). The electromagnetic Lagrangian density then reads where is referred to as the Maxwell-Kostelecký Lagrangian density. The dimensionless rank-4 background tensor (k F ) µναβ is antisymmetric on the first two and last two indices, and it is symmetric for the interchange of the first and last pair of indices. These symmetries reduce the number of independent components of (k F ) µναβ to 21. It is useful to decompose (k F ) µναβ into irreducible multiplets [91], 21 = 1 a +1 s +9 s +10 s , where 1 a represents an antisymmetric singlet (pseudoscalar), 1 s a symmetric singlet (scalar), 9 s a symmetric traceless rank-2 tensor, and 10 s a rank-4 tensor possessing the same symmetries of (k F ) µναβ and such that any contraction is identically zero.
Let us restrict our analysis to the case where the background tensor in Eq. (24) is constructed from fundamental (not composite) tensors which appear just once in the definition of (k F ) µναβ . Excluding the cases where such fundamental tensors are a scalar and/or a pseudoscalar, in which case the resulting theory is Lorentz invariant, we are left with the cases of a fundamental rank-2 symmetric tensor and/or a fundamental rank-4 tensor. In this paper, and for the sake of simplicity, we consider just the case of a fundamental rank-2 tensor (k F ) µν , and leave the case of a rank-4 tensor to future investigations. In this specific case, the independent components of (k F ) µναβ reduce to 10, and are the given by which are, respectively, the trace and the traceless part of the tensor (k F ) µν . Accordingly, the electromagnetic Lorentz-violating Lagrangian density, which describe the coupling between the inflaton and the photon, can be written as where ξ i are real numerical factors. The background tensor (k F ) µναβ , when expressed as a function of the fundamental rank-2 tensor (k F ) µν , has the form

IVb. Equation of motion
As in section III, we restrict our analysis to the case of a spatially flat, Friedmann-Robertson-Walker universe. Since g µν = a 2 η µν , we can take for the vierbein e Let now assume that the background tensor (k F ) ab is homogeneous and isotropic, so that the number of its independent components reduces to 2. In this case, (k F ) ab can be generally written as (k F ) cb = diag(ρ K , p K , p K , p K ), where ρ K and p K are two scalar functions which depend only on the conformal time η. In curved spacetime, then, the background tensor assumes the form It is useful, for the following discussion, to introduce the electromagnetic Lagrangian L em through Taking into account Eq. (4) and the fact that √ −g = a 4 , we have with L em given by Eq. (25). Working in the Coulomb gauge, we have where we have defined the time-dependent functions Varying the action (30) with respect to A, we find the equation of motion for the vector potential, where we have defined n = √ εµ, and we assume that ε and µ are positive-defined quantities.

IVc. Analogy with continuous media
It is well known in the literature that there exists an analogy between the photon sector of the minimal SME and the electrodynamics of continuous (or macroscopic) media [92]. In our case, this analogy works as follows. We rewrite the electromagnetic Lagrangian density as where is the susceptibility tensor, and δ µν αβ the generalized Kronecker delta. Introducing the polarization-magnetization tensor M µν as the equation of motion is where is the displacement tensor. In the Coulomb gauge, Eq. (37) reduces to 0 = 0 for ν = 0, and to Eq. (33) for ν = i. Let us introduce the electric and magnetic fields, a 2 E i = −F 0i and a 2 B i = 1 2 ijk F jk , the displacement and magnetizing fields, a 2 D i = −D 0i and a 2 H i = 1 2 ijk D jk , and the polarization and magnetization fields, a 2 P i = M 0i and a 2 M i = 1 2 ijk M jk . Equation (38) can be then rewritten, in three-dimensional form, as D = E + P and , and X stands for E, B, D, H, P, or M. Equation (36) gives, instead, The equations connecting the displacement and magnetizing fields to the electric and magnetic fields are known, in the electrodynamic theory of continuous media, as "constitutive relations", and completely determine (together with the boundary conditions) the propagation properties of electromagnetic signals. In particular, Eqs. (39) and (40) describe a isotropic linear medium with electric permittivity ε and magnetic permeability µ. Accordingly, the evolution in vacuum of electromagnetic fields described by the Lorentz-violating electromagnetic Lagrangian density (34), is formally equivalent to the evolution of electromagnetic fields described by the standard Maxwell theory in a continuous medium with ε and µ given by Eqs. (31) and (32). Continuing with the analogy of continuous media, the quantity n defined below Eq. (33) can be interpreted as the refractive index of the medium. Finally, and for the sake of completeness, we observe that the equation of motion, in terms of the displacement and magnetizing fields, assume the form while the Bianchi identities are Inserting Eqs. (39) and (40) in the second equation of Eq. (41), we recover equation (33).

V. QUANTIZATION
Let us now quantize the electromagnetic field whose dynamics is described by Lagrangian (30).

Va. Wronskian condition
We expand the electromagnetic vector potential as where k is the comoving wavenumber, with k = |k|, and ε k,λ are the standard circular polarization vectors. 2 The annihilation and creation operators a k,λ and a † k,λ satisfy the usual commutation relations The vacuum state |0 is defined by a k,λ |0 = 0 for all k and λ, and it is normalized as 0|0 = 1.
The equation of motion for the two photon polarization states, A k,λ , is obtained by inserting Eq. (43) in Eq. (33), In order to have a consistent quantization of the electromagnetic field, the solutions of the above equation must satisfies a normalization condition, know as Wronskian condition, that can be obtained as follows. Let us introduce the electromagnetic conjugate momentum, π = (π 1 , π 2 , π 3 ), as usual as where in the last equality we used Eq. (30), and let us impose the canonical commutation relation where is the transverse delta function. Inserting Eqs. (43) and (47) in the left hand side of Eq. (48), we find that the latter equation is satisfied only if where we used Eqs. (44)- (45). Here, is the Wronskian of any two independent solutions, A k,λ (η) and A k,λ (η), of Eq. (46). Using the Abel's identity [93], the above Wronskian can be found explicitly, 2 The vectors ε k,λ satisfy the following properties: (i) k · ε k,λ = 0, where η i is an arbitrary time. Accordingly, Eq. (50) can be written as We take now η i as the initial time, namely when inflation begins, and we assume that in which case we have for all η. Equation (55) represents the wanted condition that must be satisfied by any solution A k,λ (η) of the equation of motion in order to have a consistent quantization of the electromagnetic field.

Vb. Bunch-Davies normalized solutions
Let us assume, for the sake of simplicity, that the external tensor (k F ) µ ν is constant during inflation 3 (so that ρ k and p K are constant as well), and that the coupling function f (φ) evolves in time following a simple power law, where f i = f (φ(η i )), and γ is a free index. In this case, the permittivity ε and the permeability µ evolve in time as η γ , while the refractive index n is a constant. The solution of Eq. (46) is then easily found, are integration constants. The latter can be fixed by the choice of the vacuum, that we take to be the Bunch-Davies vacuum [12,13]. It reduces to the standard Minkowski vacuum in the short wavelength limit, k → ∞. To find it, let us re-scale the electromagnetic field as Inserting Eq. (58) in Eq. (46), we get that the re-scaled ψ-modes satisfy the equation of motion where we have defined Let us observe that Eq. (59) is formally equal to the zeromode, one-dimensional Schrodinger equation with potential energy U k , η taking the place of the spatial coordinate, and k playing the role of a free constant parameter. If ψ k,λ and ψ (2) k,λ are any two solutions of Eq. (59), the following inner product is conserved, Moreover, using Eq. (55), we get that ψ-modes are normalized as For k → ∞, the potential energy is dominated by the first term in the right-hand-side of Eq. (60). Therefore, the positive-frequency solution of Eq. (59) in the short wavelength limit is ψ k,λ = c k e −iω k η , where ω k = k/n and c k is an integration constant. The latter is fixed the the normalization condition (62), where Z = µ/ε is, in the language of the electrodynamics of continuous media, the the wave impedance of the medium. Equation (57) must then reduce to Eq. (63) in the limit k → ∞. This happens only if c is the wanted Bunch-Davies vacuum normalized solution.

VI. BACKREACTION ON INFLATION
We now draw our attention to the electromagnetic backreaction on inflation in the model described by Lagrangian (25). We will find the conditions under which such a backreaction is completely negligible.
Let us first observe that Lagrangian density (34) can be conveniently re-written as with D µν given by Eq. (38). The electromagnetic energymomentum tensor is obtained by inserting Eq. (65) in Eq. (7). We find Here, is the standard electromagnetic energy-momentum tensor in a medium described by the displacement tensor (38) The rank-five tensor is antisymmetric on the first two and second two indices, symmetric in the last two indices, and it is symmetric for the interchange of the first and second pair of indices. When the susceptibility tensor has the form in Eq. (35), the electromagnetic energy-momentum tensor can be written, in its full form, as Due to symmetry, we have that the only (possible) nonnull components of the vacuum expectation value of the electromagnetic energy-momentum tensor, (T em ) µ ν , are the electromagnetic energy density, (T em ) 0 0 , and Here, (T em ) µ µ is the trace of the electromagnetic energy-momentum tensor, which is, in general, different from zero due to the coupling of the photon to the background tensor (k F ) µ ν . In particular, we have where we have defined When the susceptibility tensor has the form in Eq. (35), we have The vacuum expectation value of the squared magnetic and electric fields operator are easily found, where P B (k, η) and P E (k, η) are the so-called magnetic and electric power spectra, Defining also the electromagnetic energy density spectrum, ρ em (k, η), and the electromagnetic trace spectrum, we recast Eqs. (71) and (72) as where we have defined Let us now specialize our results to the case of de Sitter spacetime and for large-scale, super-horizon modes. Inserting the asymptotic expansion for −kη → 0 of the solution (64) in Eqs. (78) and (79), we get respectively. We are principally interested to the case of a scaling-invariant magnetic spectrum (the general case goes along the same lines as below), so that we take ν = −5/2 [corresponding to γ = 6 in Eq. (56)]. In this case, we have Looking at Eqs. (82)- (83) and Eqs. (90)- (91), and observing that Z = n/ε ∼ 1/f (φ), we conclude, following the discussion in section IIIb, that the electromagnetic backreaction on inflation is not generally negligible. This conclusion could be avoided if the coefficients τ 1 and τ 3 , which enter in the definition of the electric part of the electromagnetic energy-momentum tensor, are vanishing. This happens only if the background tensor (k F ) µ ν assumes a particular form that we are now going to determine. Assuming that τ 3 = 0, we straightforwardly get .
The above equations, when combined with the condition τ 1 = 0 and Eq. (31), give which determine the form of (k F ) µ ν in Eq. (27) as a function of ξ i and, accordingly, Imposing the reality condition n 2 > 0, we find where Taking into account Eqs. (92) and (93), we find where we have defined We observe that all the three of the above spectra are scaling-invariant, and that ρ em (k, η) and T em (k, η) are time-independent, while the time-dependence of P B (k, η) is all encoded in f (φ).
Finally, imposing that (T em ) µ ν (T inf ) µ ν , we get which are the wanted conditions that must be satisfied in order to have a negligible electromagnetic backreaction on inflationary dynamics.

VII. ACTUAL MAGNETIC FIELD
We have seen that inflation is able to produce superhorizon magnetic field fluctuations whose intensity is given by the magnetic power spectrum (97). For the following discussion, it is useful to define the magnetic field strength on the scale λ = 1/k as At the end of inflation, we have then the scale-invariant magnetic field where B end = B(λ, η end ) and f end = f (φ(η end )). Such a field will evolve from the end of inflation until today. In this section, we will find the actual value of the inflation-produced magnetic field as a function of the free parameters of the model, namely the constants ξ 1 and ξ 2 [the other two free parameters, ρ K and p K , are assumed to be fixed by Eq. (93)] and, consequently, find the regions in the parameter space (ξ 1 , ξ 2 ) where it satisfies both the constraint in Eq. (2) and those in Eqs. (102)- (103).

VIIa. Evolution after reheating
In order to find the present intensity of the magnetic field, we must evolve it from the end of inflation until the present time η 0 . As in section IIIb, and to simplify the analysis, we consider the case of instantaneous reheating. After the end of reheating (which corresponds in this case to the end of inflation and the beginning of radiation era), the dynamics of the inflation-produced electromagnetic field is governed by standard Lagrangian since f (φ(η)) = 1 after inflation. Here, we have assumed, for the sake of simplicity, that the background tensor field is vanishingly small for η > η end . This assures that the experimental constraints on the coefficients of the Lorentz-violation (k F ) µναβ are automatically fulfilled [94]. The post-inflationary external electric current j µ is vanishing on superhorizon scales due to causality [52], while inside the horizon it can be written as j µ = (0, −σ c E), whereσ c = aσ c is the comoving conductivity and σ c is the standard conductivity of the plasma. Accordingly, the equation of the motion for the comoving magnetic field a 2 B, also known as the magnetic flux F, isF − ∇ 2 F = 0 (107) for modes that live outside the horizon, and [14] F − ∇ 2 F = −σ cḞ (108) for subhorizon modes. Going in Fourier space, F k (η) = d 3 x e ikx F(x, η), and observing that |k 2 F k |/|F k | ∼ (−kη) 2 , we find that superhorizon magnetic modes (−kη 1) evolve according toF k = 0, so that they scale adiabatically, B ∝ a −2 . Modes inside the horizon (−kη 1), instead, evolve according to the socalled (comoving) autoinduction equation (see, e.g., [95]), where is the comoving dissipation length and RH indicates the time of reheating. Accordingly, modes with wavenumber k 2π/ d evolve adiabatically, while modes with k 2π/ d are dissipated. 4 Putting all together, we conclude that, during the expansion of the Universe after reheating, magnetic modes are washed out on scales below the dissipation length and diluted adiabatically on larger scales. However, the actual dissipation length is very small compared to the scale of interest for cosmic magnetic fields.
To see this, we firstly remember the the conductivity σ c depends, generally, on the temperature T [14]. In the radiation-dominated era, and for temperatures much greater than the electron mass m e , we have σ c (T ) ∼ T /e 2 [14], where e is the absolute value of the electric charge. After the epoch of e + e − annihilation (T anh ∼ m e ), the conductivity is given by σ c (T ) ∼ (T /e 2 ) T /m e [111], while in the matter-dominated era (T < T eq 3eV), and after electrons and ions recombine (T rec 0.3eV), it drops to the constant value σ c (T ) ∼ 10 −13 m e /e 2 8 × 10 8 s −1 [14]. Taking into account that η ∝ a and η ∝ a 1/2 in the radiationdominated and matter-dominated eras, respectively, and that a ∝ g Since I 1 /I 2 ∼ (T eq /m e ) 3/2 ∼ 10 −8 , I 2 /I 3 ∼ T rec /T eq ∼ 10 −1 , and I 3 /I 4 ∼ 10 −13 (m e /T rec ) 3/2 ∼ 10 −4 , the integral I 4 dominates over the other three in the expression for the actual dissipation length. Accordingly, we have where we used the fact that in the matted-dominated era a(t) (t/t 0 ) 2/3 and then η(t) 3t 0 (t/t 0 ) 1/3 , and t rec 8 × 10 12 s [112].

VIIb. Actual magnetic field strength
As anticipated, the actual dissipation length is negligibly small compared to the scale of interest for cosmic magnetic fields, which is of order of 1Mpc. We conclude that the inflation-produced magnetic field evolve adiabatically, from the time of reheating until today. Its actual intensity is then where B 0 = B(λ, η 0 ), g * S,0 = g * S (T 0 ) = 43/11 [113], g * S,RH = g * S (T RH ) [113], T 0 2.37 × 10 −4 eV [112] is the actual temperature, and T RH is the reheat temperature. Above the electroweak phase transition (when we assume inflation is taking place) the U (1) gauge field which is quantum mechanically excited is indeed the hypercharge field, not the electromagnetic one [15]. Below the electroweak phase transition, however, the hypercharge field is projected onto the electromagnetic field, and this gives the cosine of the Weinberg angle θ W . The reheat temperature can be related to the energy scale of inflation by observing that the energy density of radiation at the beginning of radiation era, ρ rad = (π 2 /30)g * ,RH T 4 RH , where g * ,RH is the effective number of degrees of freedom at the time of reheating and can be taken equal to g * S,RH [112], must be equal to the energy density at the end of inflation. We get T RH = [30/(π 2 g * ,RH )] 1/4 M . Taking g * S,RH = 427/4 [112], referring to the massless degrees of freedom of the standard model of particle physics above the electroweak scale, the actual, scale-invariant magnetic field is Let us now take f end ∼ 1 and M ∼ 10 16 GeV. Accordingly, we have The condition that the electromagnetic backreaction on inflation is negligible, expressed by Eqs. (102) and (103), becomes Let us now analyze, separately, the three cases in Eq. (95), namely ξ 1 = 0, ξ 2 = 0, and ξ 1 /ξ 2 ∈ X.
(iii) ξ 1 /ξ 2 ∈ X. -In Fig. 1, we plot the function n 5/2 , entering in the expression of B 0 in Eq. (114), at the varying of ξ 1 /ξ 2 . Remembering the discussion in section II [see, in particular, Eq. (2)], we have that, in order to explain cosmic magnetization, the quantity n 5/2 should be in the range [ 0.1, f ew]. This is realized for all values of ξ 1 /ξ 2 ∈ X, with the exclusion of those values very close to the boundary In fact, n diverges for ξ 1 /ξ 2 → −1/3 and ξ 1 /ξ 2 → 1 + 2/ √ 3, while it goes to zero for ξ 1 /ξ 2 → 1 − 2/ √ 3 and ξ 1 /ξ 2 → 1. When ξ 1 /ξ 2 is not so close to the above boundary values, the conditions (115) and (116), which assure that the electromagnetic field does not back-react on the inflationary dynamics, are fulfilled. This is clear from Fig. 1, where we show n 5 Υ 1 and n 5 Υ 2 at the varying of ξ 1 /ξ 2 . We conclude that, apart from some particular values of ξ 1 /ξ 2 , the inflation-produced magnetic field can be, also in this case, at the origin of cosmic magnetic fields. Let us now impose some physically "reasonable" conditions on the inflation-produced electromagnetic energymomentum tensor. Some of these conditions, as the positivity of the energy, are often assumed to be "necessary" in the literature. It is worth noticing, however, that there exist examples of physically "reasonable" matter that violates some or of all of them. For example, all the conditions that we are going to discuss are violated in particular setups of the Casimir effect [12], and even the inflaton violates the strong energy condition (discussed below) when it drives de Sitter inflation.
Weak energy condition. -Looking at the left panel of Fig. 1, we see that the quantity n 5 Υ 1 is negative for ξ 1 /ξ 2 < −1/3. This corresponds to have a negative electromagnetic energy density on large super-horizon scales during inflation [see Eq. (98)]. On these scales, we expect that the electromagnetic field behaves classically, so that one could wonder if having classical negative energies is reasonable physically. Let us then impose the condition of positivity of the energy. In a generalcovariant formulation, this condition is known as "weak energy condition" and it is, indeed, a condition on the energy-momentum tensor. In our specific case, the electromagnetic energy-momentum tensor can be written as This is the energy-momentum tensor of a (isotropic) perfect fluid of type I (according to the Hawking-Ellis classification [114]) with energy density ρ em and pressure density p em = − (T em ) i i (no sum on i). For perfect fluids, the weak energy condition states that [114] ρ em ≥ 0, ρ em + p em ≥ 0.
The light gray areas in Fig. 2 show the regions in the parameter space (ξ 1 , ξ 2 ) where 10 −13 G ≤ B 0 ≤ 5×10 −12 G and electromagnetic backreaction on inflation is completely negligible. The dark gray areas represent, instead, the shrunk regions where a specific supplementary condition on the electromagnetic energy-momentum has been imposed (from up to down and from left to right: strong energy condition, weak energy condition, dominant energy condition, and trace condition).

IX. CURVATURE PERTURBATIONS
Recently enough, it has been pointed out in the literature that the production of electromagnetic fields during inflation may significantly affect the primordial spectrum of both scalar and tensor curvature perturbations (see, e.g., [61,[116][117][118]). In order to have a self-consistent model of inflationary magnetogenesis then, we have to check that the curvature perturbations introduced by the inflation-produced electromagnetic field are compatible with CMB results.

IXa. Scalar curvature perturbation
In order to find how a primordial magnetic field can generate curvature perturbations, let us consider the curvature perturbation ζ(t, x) on the uniform energy density hypersurface [119] on which δρ(t, x) = 0, where δρ is the energy density perturbation, and t is the cosmic time. The curvature perturbation, as a function of the scale factor a(t, x), is [119] ζ(t, x) = ln a(t, x) − ln a(t), where a(t) is the global scale factor, namely the one introduced in the unperturbed metric (8). On super-Hubble scales, the curvature perturbation evolves according to [118] ζ (t, where a prime denote the differentiation with respect to the cosmic time. Here, ρ(t) and p(t) are the total energy and pressure densities, H(t) the Hubble parameter, and δp rel (t, x) is the so-called nonadiabatic pressure density perturbation defined by δp rel (t, x) = δp(t, x) − δρ(t, x)p /ρ , with δp being the pressure density perturbation. Assuming that the electromagnetic field is just a small perturbation with respect to the background (which is dominated by the inflaton field), we can write ρ = ρ inf and p = p inf . The evolution equation for the curvature perturbation introduced by the electromagnetic field, ζ em (t, x), is then where δp rel,em (t, x) = δp em (t, x) − δρ em (t, x)p inf /ρ inf is the nonadiabatic pressure perturbation due to the relative entropy perturbation between the electromagnetic and the inflaton fields. Here, δρ em and δp em are the electromagnetic energy and pressure density perturbations, respectively, and they are the same quantities defined in Eq. (117), to wit, δρ em = ρ em and δp em = p em . Assuming a quasi-de Sitter inflation characterized by the slow-roll parameter 1, p inf = (1 − 2 /3)ρ inf , and introducing the electromagnetic equation-of-state w em = ρ em /p em = δρ em /δp em , the solution of Eq. (124) reads where t i is the time when electromagnetic fluctuations begin to develop, ζ em (t i , x) = 0, and In obtaining Eq. (125), we assumed, as in Ref. [118], that H, , and ρ inf are constant during inflation, and that p inf ρ inf . Equation (126), instead, comes from Eqs. (98) and (99). In the case w em = 1/3, we recover the result of Ref. [118]. The curvature perturbation in Eq. (125) is the key quantity from which observable quantities can be constructed and then compared to CMB results.
In Ref. [118], the standard kinetically coupled scenario for magnetogenesis was studied. In this case, the electromagnetic energy density is dominated by the electric part, so that, working in Fourier space, the electromagnetic energy spectrum can be approximated by δρ em (k, η) = 1 2 f (φ)P E (k, η). The expression of the electric power spectrum (in the standard kinetically coupled scenario) can be obtained by using Eqs. (88)- (89), and taking Z = 1/f (φ) and n = 1. It is P E (k, η) = (ν 2 |c ν | 2 /π)(−kη) 3+2ν H 4 /f (φ). The scalinginvariant case corresponds to taking ν = −3/2, and it gives P E (k, η) = (9/2π 2 )H 4 /f (φ). Accordingly, we have where, here and in the following, a star indicates that the corresponding result is obtained in the standard kinetically coupled scenario for the case of a scaling-invariant electric power spectrum. In our case, instead, the electric part does not make any contribution to the electromagnetic energy which is then dominated by the magnetic part (see discussion in section VIb). The expression for δρ em (k, η) is given in Eqs. (98), which we rewrite here for the sake of convenience, Comparing Eqs. (127) and (128), and taking into account Eq. (125), we see that the curvature perturbation in our case can be obtained by multiplying the result of Ref. [118] (for the scaling-invariant case) by a constant factor ϑ, where ϑ(ξ 1 , ξ 2 ) = 3(1 + w em ) 2 and where, in the last equality of the above equation, we used Eqs. (100), (101), and (126).

IXb. Scalar modes: spectrum, bispectrum, and trispectrum
The observable quantities that can be constructed starting from the curvature perturbation ζ(t, x), are the corresponding n-points correlation functions in Fourier space. In particular, the actual sensitivity of CMB experiments allows us to put constraints on the 2-points correlator, the power spectrum of curvature perturbations, on the 3-point correlator, the bispectrum, and on the 4points correlator, the trispectrum, defined via respectively, where a scaling invariant power spectrum P ζ (k) is assumed, and all quantities are evaluated at the end of inflation η = η end . Here, ζ k is the Fouriertransformed curvature perturbation, ... indicates an ensemble average, k i = |k i |, and k ij = |k i + k j |. The observable quantities, other than P ζ (k), are the local-type non-linearity (dimensionless) parameters f local NL and τ NL , which parameterize the non-Gaussian features of the primordial spectrum of curvature perturbations.
Let us observe that the 3-points correlator can be generally written as ζ k1 ζ k2 ζ k3 = (2π) 3 δ(k 1 + k 2 + k 3 )f NL P ζ (k 1 , k 2 , k 3 ), where the bispectrum P ζ (k 1 , k 2 , k 3 ) measures the correlation among three perturbation modes [120]. The bispectrum can assume different forms which depend on the type of triangle formed by the three wavenumbers k 1 , k 2 and k 3 . Local-type non-linearities are characterized by a bispectrum that is maximal for "squeezed" triangles with k 1 k 2 k 3 (or permutations). Other types of configurations are possible, such as the equilateral and the orthogonal. They are, however, inessential for our discussion since a scalinginvariant electromagnetic field produces (under appropriate approximations [118]) only local-type shapes for the bispectrum and trispectrum.
Using the results of Ref. [118], appropriately re-scaled by using Eq. (129), we find for the electromagnetic part of the power spectrum of curvature perturbations, P em ζ , and for the electromagnetic part of the local-type non-linearity parameters, f em NL and τ em NL , P em respectively. Here, it has been assumed that the dominant component of the power spectrum of curvature perturbations is generated by the inflaton, where the power spectrum of curvature perturbations in slow-roll inflation is [118] Moreover, it has been assumed that ∆N > 1, where with N tot = − ln |k min η end | and N cmb = − ln |k cmb η end |.
Here, k min is the wavenumbers of the mode that crosses the horizon when magnetogenesis begins, and k cmb , referred to as the CMB scale, is the wavenumber of the largest scale CMB mode. We have assumed, in the previous sections, that magnetogenesis begins simultaneously with inflation, so that k min = −1/η i , and in turns N tot is the total number of e-fold of inflation. On the other hand, N cmb is the number of e-folds between the moment at which the CMB mode leaves the horizon and the end of inflation. Assuming for simplicity an instantaneous reheating, we have [112] N cmb 61 + ln λ cmb 4000Mpc + ln M 10 16 GeV , where λ cmb = 2π/k cmb . Taking M = 10 16 GeV and the CMB scale as large as the the present Hubble radius, λ cmb = H −1 0 4000Mpc, we get N cmb 61. We use the experimental results derived from Planck data [121][122][123], where P inf ζ (k) is evaluated at the pivot scale k 0 = 0.05Mpc −1 . Taking into account Eq. (141), we can conveniently re-write Eqs. (134), (135), and (136) as where the above three constraints come from the spectrum, bispectrum, and trispectrum of the curvature perturbation, respectively. Since ∆N is a quantity greater than unity, we obtain that the stringent constraint on |ϑ| comes either form the bispectrum if ∆N (25/2) 6 4 × 10 6 , or from the spectrum otherwise.
Let us analyze the three cases in Eq. (95), by assuming, for the sake of simplicity, that the stringent constraint on |ϑ| comes form the bispectrum.
(iii) ξ 1 /ξ 2 ∈ X. -If (N cmb /61)(∆N ) 1/3 is of order unity, we have n 5 |2Υ 1 − Υ 2 | = |ϑ| 2. In this case, looking at Fig. 1 and remembering the discussion at the end of section VIc, we conclude that, with the exclusion of those values very close to the boundary 3}, the inflation-produced electromagnetic field (whose magnetic component directly accounts for cosmic magnetic fields) generates curvature perturbations compatible with CMB data. The regions in the parameter space (ξ 1 , ξ 2 ) with acceptable curvature perturbations, instead, progressively shrink for increasing values of (N cmb /61)(∆N ) 1/3 , as it appears from Fig. 3. Here, in the light gray regions the strength of the actual, scale-invariant magnetic field is in the range 10 −13 G ≤ B 0 ≤ 5×10 −12 G with no constraints on curvature perturbations imposed, while in the shrunk dark gray regions the constraint (149) has been imposed, [the darkness increases as (N cmb /61)(∆N ) 1/3 increases].

IXc. Tensor modes
The inflation-produced electromagnetic field affects, other than the scalar part of metric perturbation, also its tensor part, namely it produces gravitational waves. However, we expect that if P em ζ P inf ζ , then the same it is true for the gravitational wave spectra, P em GW P inf GW . This is because, as discussed in [61], the tensor modes are only produced gravitationally, while the dominant source of the scalar modes is the direct coupling between the inflaton and the electromagnetic field. In particular, the latter is enhanced with respect to the gravitational one by a factor inversely proportional to the slow-roll parameter. Therefore, we expect that on physical grounds. Accordingly, due to the assumption (137), we can safely neglect the possible constraints on ξ 1 and ξ 2 coming from the production of gravitational waves in our model of inflationary magnetogenesis.

X. DISCUSSION
So far, we have not distinguished between explicit (not dynamical) and spontaneous (dynamical) Lorentz violation. In the model at hand (described by a background rank-2 tensor), the simplest realization of spontaneous Lorentz violation is realized when a dynamical tensor operator (K F ) µν is coupled to the electromagnetic field and it acquires a vacuum expectation value different from zero, (k F ) µν = 0|(K F ) µν |0 . In this case, the coupling between the rank-2 tensor field and the electromagnetic field would be described by Lagrangian (25) with (k F ) µν replaced by (K F ) µν .
Decomposing (K F ) µν in a classical part, (k F ) µν , and a quantum part, δ(k F ) µν , to wit, writing all the above analysis remains valid only if the quantum fluctuations δ(k F ) µν are dynamically negligible. Assuming homogeneity and isotropy, we can write δ(k F ) µ ν = diag(δρ K , −δp K , −δp K , −δp K ). All the arguments and results in section VI are then preserved if, roughly speaking, In this case, in fact, the contribution of the quantum fluctuations of (K F ) µν to the electromagnetic energymomentum tensor are negligible in the scaling-invariant case [see Eqs. (84)- (84) and Eqs. (90)-(91)]. Although the condition (152) seems to be very restrictive, it could be realized if, for example, δ(k F ) µ ν ∼ (−kη) α , with α 2. Finally, let us observe that the dynamics of the field (K F ) µν is described by a Lagrangian of the type L K = L K,kin + L K,int , where L K,kin is the kinetic term (whose involved expression, for the case of de Sitter spacetime, can be found in [124]), while the interaction term L K,int contains, other than self-coupling terms, all the couplings to other fields, included those with the photon and the inflaton. In a complete and self-consistent model, which is beyond the aim of this paper, one should also consider such a dynamics and consistently check that the field (K F ) µν does not appreciably back-react on the dynamics of inflation and it generates curvature perturbations in agreement with CMB results.

XI. CONCLUSIONS
Astrophysical observations definitely indicate the presence of microgauss magnetic fields in any type of galaxies, and they give compelling indications of existence of strong, large-scale magnetic fields in galaxy clusters and cosmic voids.
A plausible hypothesis about their nature is that they have a primordial origin. In particular, presentlyobserved magnetic fields could be nothing but primordial quantum electromagnetic fluctuations excited during an inflationary epoch of the universe which have survived until today.
However, since standard electrodynamics is conformally invariant, large-scale magnetic fields cannot be generated in a conformally invariant background spacetime, as a result of the Parker theorem. Accordingly, whatever is the mechanism responsible for generation of quantum electromagnetic fluctuations it must break conformal invariance of Maxwell theory.
In this paper, we have analyzed the generation of cosmic magnetic fields during (de-Sitter) inflation in a nonconformally-invariant, Lorentz-violating effective model of electrodynamics. We have considered a Lorentzviolating extension of the kinetically coupled scenario for magnetogenesis, where the latter is described by a Lagrangian of the form L em = f (φ)L M . Here, L M is the Maxwell Lagrangian and f (φ) is a generic coupling function between the photon and the inflaton φ. Lorentz violation is introduced in our model by considering the Lagrangian L em = f (φ)(L M + L LV ), where L LV incorporates all possible Lorentz-violating renormalizable operators.
We have restricted our analysis to the case where Lorentz symmetry breaking is implemented by the presence of a classical, homogeneous, and time-independent rank-2 symmetric background tensor. Working in the weak-coupling regime, we have shown that the creation of inflationary magnetic fields in this model proceeds similarly to the case of magnetogenesis in the standard kinetically coupled scenario. The key difference is that the new degrees of freedom represented by the components of the background tensor can be tuned in such a way to suppress the electric part of the inflation-produced electromagnetic energy-momentum tensor. This allows us to have, in de Sitter inflation with scale ∼ 10 16 GeV, a selfconsistent model of magnetogenesis where the inflationproduced electromagnetic field (i) does not appreciably back-react on inflation, (ii) it does not significantly affect the primordial spectrum of both scalar (Gaussian and non-Gaussian) and tensor curvature perturbations, and (iii) it evolves after inflation to give a strong, scalinginvariant magnetic field that directly accounts for galactic magnetic fields.