A MODEL OF UNIVERSE ANISOTROPIZATION Leonardo Campanelli1,2∗ 1

Dipartimento di Fisica, Universit` a di Bari, I-70126 Bari, Italy and 2 INFN - Sezione di Bari, I-70126 Bari, Italy (Dated: September, 2009)

arXiv:0907.3703v2 [astro-ph.CO] 17 Sep 2009

Abstract The presence of a nonconformally invariant term in the photon sector of the Lorentz-violating extension of the standard model of particle physics, the “Kosteleck´ y term” LK ∝ (kF )αβµν F αβ F µν , enables a superadiabatic amplification of magnetic vacuum fluctuations during de Sitter inflation. For a particular form of the external tensor kF that parametrizes Lorentz violation, the generated field possesses a planar symmetry at large cosmological scales and can have today an intensity of order of nanogauss for a wide range of values of parameters defining inflation. This peculiar magnetic field could account for the presently observed galactic magnetic fields and induces a small anisotropization of the Universe at cosmological scales. The resulting Bianchi I model could explain the presumedly low quadrupole power in the cosmic microwave background radiation. PACS numbers: 98.62.En, 11.30.Cp, 98.80.-k, 98.70.Vc

I. INTRODUCTION The standard model of particle physics together with the Einstein theory of gravity encompasses, basically, all known fundamental physics. However, due to the classical character of general relativity, these theories are believed to be incomplete, and a search for a more fundamental theory which overcomes the stumbling block of quantizing gravity is a major goal of present-day theoretical physics. A promising candidate is string theory, which deals with gravity in a quantum and self-consistent way. Although string theory is well away from being an experimentally tested theory, some of its peculiar, low-energy manifestations, if ever detected in astrophysical and/or ground-based experiments, could be a strong signal of its correctness. Indeed, as pointed out by Kosteleck´ y [1], a possible detectable signature of string theory at low energies is violation of Lorentz symmetry [2]. If on the one hand, terrestrial and astrophysical experiments have not yet either confirmed or ruled out the existence of effects of Lorentz violation (LV), on the other hand astrophysical observations have definitely confirmed the presence of large-scale correlated, microgauss magnetic fields in any type of galaxies and clusters of galaxies. (For reviews on cosmic magnetic fields see Ref. [3].) This peculiar property of possessing, roughly speaking, the same intensity and correlation scale everywhere in the present Universe, has been interpreted as a strong hint that cosmic magnetic fields are, indeed, relic of the early Universe. Obviously, a damning evidence for the primordial origin of cosmic magnetic fields would be the detection of their effects on the Cosmic Microwave Background (CMB) radiation. Until now, however, exhaustive analyses have only given stringent constraints on their properties, without finding any evidence of their imprints on the CMB [4]. The use of the CMB radiation as a probe of the physics

of the early Universe has proved to be very fruitful in the last years. In particular, the high resolution data of temperature fluctuations of CMB angular power spectrum, provided by the Wilkinson Microwave Anisotropy Probe (WMAP) [5, 6] have, almost definitively, consecrated the so-called “Λ-dominated cold dark matter” (ΛCDM) as the standard cosmological model of the Universe. Nevertheless, the 1-, 3-, and 5-year WMAP data display at large angular scales some “anomalous” features, the most important ones being the low quadrupole moment and the presence of a preferred direction in the Universe. It is extremely important, however, to stress that, due to the importance of residual Galactic foreground emission, those anomalies in the CMB anisotropy are still subject to an intense debate [7]. If there is really a problem with the quadrupole moment, then its lowness, indicating a suppression of power at cosmological scales, may signal a nontrivial topology of the large-scale geometry of the Universe [8]. Indeed, several possibilities have been proposed in the recent literature to understand that suppression [9, 10] (for other large-scale anomalies in the angular distribution of CMB see, e.g., Ref. [11]). Recently enough [12, 13], it has been shown that a particular case of the simplest anisotropic cosmological model, i.e. the Bianchi I model, could account for the smallness of the quadrupole, without affecting higher multipoles of the angular power spectrum of the temperature anisotropy. Such a proposal of an “ellipsoidal universe” has been considered also in Ref. [14, 15]. Also, the WMAP data display a particular feature which has been deeply investigated in the last years: a statistically significant alignment and planarity of the quadrupole and octupole modes. This seems to indicate the existence of a preferred direction in the Universe, which has been named “axis of evil” (AE) [16]. Needless to say, there is no space in the isotropic, standard

2 cosmological model for such a type of features. In this paper, we investigate the possibility that effects of Lorentz violation at inflation could be responsible for the creation of large-scale magnetic fields possessing planar symmetry. As we will see, these fields can have the right intensity and correlation length to explain the existence of galactic and extragalactic magnetic fields. Moreover, because of their peculiar symmetry at cosmological scales, they could induce a modification of the (isotropic) Robertson-Walker metric, in such a way that the resulting cosmological model is well described by the ellipsoidal universe model. This, in turn, could naturally account for some peculiar and not-yet-explained anomalous features of CMB radiation discussed above. The plan of the paper is as follows. In Sec. II we discuss the generation at inflation of a plane-symmetric cosmic magnetic field in the framework of the Lorentzviolation extension of the standard model of particle physics. Section III deals with the analysis of CMB anisotropies including the asymmetric contribution due to the presence of the planar field. Finally, we draw our conclusions in Sec. IV. Some technical details are presented in the appendixes. II. LORENTZ-VIOLATING ELECTROMAGNETISM AND PLANAR COSMIC MAGNETIC FIELDS The large correlation scale of cosmic magnetic fields, ranging from ∼ 10kpc for magnetic fields in galaxies to ∼ 1Mpc for those in clusters, and the fact that they are found to have approximately the same intensity of a few microgauss seems to indicate a common and primordial origin, probably to ascribe to some unknown mechanism acting during an inflationary epoch of the Universe. If one takes into account that the collapse of primordial large-scale structures enhances the intensity of any preexisting magnetic field of about a factor 103 [3], a primeval field with comoving intensity of order of nanogauss and correlated on megaparsec scales could explain the “magnetization of the Universe”. During inflation all fields are quantum mechanically excited. Because the wavelength λ associated to a given fluctuation grows faster than the horizon, there will be a time, say t1 , when this mode crosses outside the horizon itself. After that, this fluctuation cannot collapse back into the vacuum being not causally self-correlated, and then “survives” as a classical real object [17]. The electromagnetic energy density at the time of crossing is then fixed by the Gibbons-Hawking temperature TGH [17]: 4 E ∼ TGH ∼ H 4,

(1)

where H is the Hubble parameter (in this paper we consider, for the sake of simplicity, just the case of de Sit-

ter inflation). Taking into account the expression for the electromagnetic energy in standard Maxwell electromagnetism, one arrives to the result that the spectrum of magnetic fluctuations at the time of horizon crossing is given by B1 ∼ H 2 ∼ M 4 /m2Pl [17, 18, 19], where in the last equality we used the Friedmann equation H 2 = (8π/3)M 4 /m2Pl . Here, M 4 is the total energy density during inflation (which is constant during de Sitter inflation) and mPl ∼ 1019 GeV is the Planck mass. Because of conformal invariance of Maxwell electromagnetism one finds, however, that the present magnitude of the inflation-produced field at the scale, say 10kpc, is vanishingly small, B0 ∼ 10−52 G [19]. (This is true only if the background metric is spatially-flat [20], which is the case discussed in this paper.) Since the pioneer work of Turner and Widrow [19], a plethora of mechanisms has been proposed for generating cosmic magnetic fields in the early Universe, all of which repose on the breaking of conformal invariance of standard electrodynamics (see references in Ref. [3] and, for recent papers, Ref. [21, 22]). In particular, Kosteleck´ y, Potting and Samuel [23] first pointed out that the breaking of conformal invariance is a natural consequence of LV. Indeed, they argued that the appearance of an effective photon mass, owing to spontaneous breaking of Lorentz invariance, could enable the generation of large-scale magnetic fields within inflationary scenarios. The idea that Lorentz symmetry breaking could result in the generation of cosmic magnetic fields has been pursued since then by others authors [24, 25, 26, 27, 28]. The aim of this paper is to show that within a particular Lorentz-violating model of particle physics, it is possible to generate magnetic fields of cosmological type possessing a peculiar spatial geometry. Then, in the next section, we will analyze their impact on the isotropy of the Universe and, in particular, on the cosmic microwave background radiation. The model we are going to study is the so-called standard model extension (SME) [29], which is an effective field theory including all admissible Lorentz-violating terms in the Glashow-Weinberg-Salam gauge theory. In curved spacetimes, the SME action for the photon field, here referred to as the Maxwell-Kosteleck´ y (MK) action, reads [30] Z SMK = d4 x e − 41 Fµν F µν − 41 (kF )αβµν F αβ F µν , (2)

where Fµν = ∂µ Aν − ∂ν Aµ is the electromagnetic field strength tensor and e the determinant of the vierbein. The presence of the external tensor (kF )αβµν breaks (particle) Lorentz invariance [30] and parametrizes then Lorentz violation. The external tensor (kF )αβµν is fixed in a given system of coordinates. Going in different systems of coordinates will, generally, induces a change of the form of (kF )αβµν . In the following, we assume that the form of (kF )αβµν refers to a system of coordinates at

3 rest with respect to the cosmic microwave background, the so-called “CMB frame.” It is worth noting that taking (kF )αβµν as a fixed tensor corresponds to have an explicit violation of Lorentz symmetry. This could introduce in the theory an instability associated to nonpositivity of the energy. Indeed, working in a flat isotropic universe described by a Robertson-Walker metric ds2 = a2 (dη 2 − dx2 ), where a(η) is the expansion parameter and η the conformal time, and introducing the electric and magnetic fields as F0i = −a2 Ei and Fij = ǫijk a2 Bk (Latin indices run from 1 to 3, while Greek ones from 0 to 3), the electromagnetic energy density turns out to be [31] 1 2 (E + B2 ) + (kF )i00j a−4 Ei Ej 2 1 + (kF )ijkl a−4 ǫijm ǫkln Bm Bn . 4

E=

(3)

The positivity of the above quadratic form depends on the particular form assumed by the fixed tensor (kF )αβµν , which is frame-dependent. This apparent paradox (frame-dependent positivity of the energy) is overcome when considering models in which LV is spontaneously broken. In this case, however, action (2) loses its character of generality, since the tensor (kF )αβµν is now regarded as a vacuum expectation value of some tensor field with its own dynamics. For this reason, and following Ref. [31], we will take (kF )αβµν to be a fixed tensor but, at the same time, we will impose positivity of the energy in the CMB frame. Needless to say, if in another system of coordinates the energy is not positive defined, this means that the effective theory with explicit Lorentz symmetry breaking becomes meaningless in that frame and one needs to consider the full theory with spontaneous Lorentz symmetry breaking in order to get physically acceptable results. The equations of motion follow from action (2) [31]: ∂η (a2 Ei ) − ǫijk ∂j (a2 Bk ) + ∂η [2(kF )i00j a−2 Ej − (kF )0ijk ǫjkl a−2 Bl ] + ∂j [2(kF )ijk0 a

−2

Ek − (kF )ijkl ǫklm a

−2

Bm ] = 0 (4)

and ∂i (a2 Ei ) + ∂i [2(kF )i00j a−2 Ej − (kF )0ijk ǫjkl a−2 Bl ] = 0. The Bianchi identities are ∂η (a2 B) + ∇ × (a2 E) = 0 and ∇ · B = 0. We are interested in the generation and evolution of superhorizon magnetic fields, that is to electromagnetic modes whose physical wavelength is much greater than the Hubble radius H −1 , λphys ≫ H −1 , where λphys = aλ and λ is the comoving wavelength. Since aη ∼ H −1 , introducing the comoving wavenumber k = 2π/λ, the above condition reads |kη| ≪ 1. Observing that the first Bianchi identity gives on large scales B ∼ kηE, where B and E stand for the average magnitude of the magnetic and electric field intensities, and assuming that all (non-null) components of (kF )αβµν have approximately

the same magnitude, we can neglect in Eq. (4), on large scales, the terms proportional to the magnetic field. Also, the next-to-last term is negligible with respect to the first one. Therefore, at large scales, Eq. (4) reduces to ∂η (a2 Ei ) + ∂η [2(kF )i00j a−2 Ej ] = 0.

(5)

Assuming that ||(kF )i00j || ≫ a4 we then have (kF )i00j a−2 Ej = ci ,

(6)

where ci are constants of integration. It is plausible to assume that the tensor (kF )αβµν is a nonincreasing function of time. Indeed, we can take the simple form (kF )i00j ∝ a−p , with p a non-negative real number. In this case, it is clear from Eq. (6), the first Bianchi identity, and the fact that η ∝ a−1 during de Sitter inflation, that the average intensity of the magnetic field grows “superadiabatically,” B ∝ a1+p . [If the elements of (kF )i00j are either all zero or negligibly small in the CMB frame, the inflation-produced field is, in general, small-scale correlated and then not astrophysically interesting.] In Ref. [31], the case was analyzed where (kF )i00j is a constant isotropic tensor, (kF )i00j ∝ δij , where δij is the Kronecker delta. In this paper, instead, we study the case in which (kF )i00j is “maximally” anisotropic, in the sense that only one component, say (kF )3003 , is different from zero. In this case, only the Ez component of the electric field is amplified and this in turn means that only the Bx and By components of the magnetic field grow superadiabatically during inflation. Accordingly, the amplified magnetic field will possess a planar symmetry, whose plane of symmetry is the xy-plane. Moreover, for the sake of simplicity, we will concentrate only on a single case, that corresponding to p = 2, so that we will assume that (kF )i00j ≡ a−2 δi3 δj3 kF ,

(7)

where kF is a constant which gives the magnitude of Lorentz violation effect at the actual time, a = a0 = 1. (It is worth noting that the condition ||(kF )i00j || ≫ a4 translates into kF ≫ a6 .) As we will see, the choice of Eq. (7) will correspond to the case of inflation-produced, scale-invariant, cosmic magnetic fields. 1 In this case,

1

Up to today there is no experimental evidence that the tensor (kF )αβµν is time-dependent or even different from zero (see below). Accordingly, it is worth stressing that our assumption that (kF )αβµν decreases in time should only be considered as a working hypothesis. On the other hand, the choice p = 2 follows by the observation that the actual cosmic magnetic fields possess, approximately, the same intensity on different scales (that of galaxies and that of clusters of galaxies), and by interpreting this occurrence as a hint that their spectrum is indeed scale-invariant (at least on cosmological scales).

4

a3 H 2 . B1 ∼ √1 kF

(8)

It is worth noting that, in order to have positivity of the energy, we are forced to assume kF > 0. In order to find the actual value of the inflationproduced magnetic field, we have to follow its evolution during reheating, radiation, and matter eras. Since the resulting analysis is very close to that performed in Ref. [31], we quote just the final result (full details are given in Appendix A): B0 ∼ nG

M 106 GeV

6

TRH MeV

7(n−1)/2

kF 10−37

n/2 ,

(9)

where TRH is the reheat temperature (see Appendix A), and n takes the values ±1 according to T∗ . TRH or T∗ & TRH . The temperature T∗ , defined in Eq. (48), is the temperature below which electric fields are washed out by dissipative effects of the primordial plasma (so that magnetic fields evolve adiabatically for T . T∗ ). The condition T∗ & TRH means, indeed, that the magnetic field evolves adiabatically from the end of reheating until today and is equivalent to having TRH /MeV . (10−37/kF )1/7 . Upper bounds on ||(kF )αβµν || come from the analysis of CMB polarization and polarized light of radiogalaxies and gamma-ray bursts: they are, respectively, 10−30 [32], 10−32 [33], and 10−37 [34]. (Although the former bound is less stringent than the latter ones, it covers the whole portion of coefficient space for Lorentz violation. The point-source nature of radiogalaxies and gamma-ray bursts, instead, allows us to put constraints only on limited portions of coefficient space [32].) Now, it is clear from Eq. (9) and Fig. 1 that, for a wide range of values of parameters defining inflation (i.e., M and TRH ) and Lorentz symmetry violation (i.e., kF ), the scaling-invariant, present-day magnetic field can be as strong as B0 ∼ nG, and then could naturally explain the presence of galactic and extragalactic magnetic fields.

0

Log10HMGeVL

the electric and magnetic fields scale as Ez ∝ a4 and Bx ∝ By ∝ a3 , respectively, while Ex , Ey , and Bz scale adiabatically. Before proceeding further, we estimate the spectrum of magnetic vacuum fluctuations generated during de Sitter inflation in Maxwell-Kosteleck´ y electromagnetism. If, as before, we assume that the only nonzero component of (kF )αβµν is just given by (kF )3003 = kF a−2 , the electromagnetic energy density turns out to be E = 2 2 1 −6 2 Ez . Therefore, at the time of crossing, 2 (E +B )+kF a where |kη| ∼ 1, and since kF ≫ a6 , we get E ∼ kF a1−6 B12 , where a1 = a(t1 ) and we used the first Bianchi identity. Here, B1 stands for the average intensity of the magnetic field on the plane xy at the time of crossing. Remembering that E ∼ H 4 , we obtain the spectrum of magnetic fluctuations when crossing the horizon,

1

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

2 kF kF kF kF kF

0

1

3

4

5

6

7

8 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

= 10-77 = 10-57 = 10-37 = 10-32 = 10-30

2

3

4

5

6

7

8

Log10HTRHGeVL FIG. 1: The plane-symmetric, scale-invariant, inflationproduced magnetic field has an actual intensity of order of nanogauss if the values of parameter defining inflation, i.e. the energy scale of inflation M and the reheat temperature TRH , stay on the curves. kF ∼ ||(kF )αβµν || estimates the magnitude of the (constant) external tensor which parametrizes Lorentz violation [see Eqs. (2) and (7)].

III. ELLIPSOIDAL UNIVERSE AND THE CMB QUADRUPOLE The cosmic magnetic field produced according to the mechanism discussed in the previous section possesses a planar symmetry. The energy-momentum tensor for a plane-symmetric magnetic field of the form B = (Bx , By , 0) is 1 2 0 0 0 2B 1 2 2 0 Bx By 0 2 (Bx − By ) . (TB)µν = 1 2 2 0 Bx By (B − B ) 0 y x 2 0 0 0 − 21 B2 (10) Taking an ensemble average, and assuming statistical isotropy in the plane of symmetry, gives hBx By i ≃ 0 and hBx2 i ≃ hBy2 i, so that Eq. (10) reduces to (TB )µν = ρB diag(1, 0, 0, −1), where ρB = hB2 i/2 is the average magnetic energy density. We want now to analyze the large-scale evolution of the matter-dominated universe filled with a plane-symmetric anisotropic component represented by the planar cosmic magnetic field. The total energy-momentum tensor is then given by T µν = diag(ρB + ρm , 0, 0, −ρB ),

(11)

where ρm is the matter energy density. We then consider a cosmological model with planar symmetry, whose most general plane-symmetric line element, compatible with Eq. (11), is [35] ds2 = dt2 − a2 (t)(dx2 + dy 2 ) − b2 (t) dz 2 ,

(12)

5 where a and b are the scale factors. The metric (12) corresponds to considering the xy-plane as a symmetry plane. Taking into account Eqs. (11) and (12), Einstein’s equations read [36] 2 a˙ b˙ a˙ +2 = 8πG(ρB + ρm ), (13) a ab a ¨ ¨b a˙ b˙ + + = 0, (14) a b ab 2 a ¨ a˙ 2 + = −8πGρB , (15) a a where a dot indicates the derivative with respect to the cosmic time. Since the conductivity of the primordial plasma is very high, we are safe in neglecting interaction effects between magnetic field and matter [3]. In this case, the magnetic component of the energy-momentum tensor is conserved, Dµ (TB )µν = 0, so that we have ! b˙ a˙ ρB = 0. (16) + ρ˙ B + 2 a b Let us introduce the so-called “eccentricity” r a 2 . e≡ 1− b

(18)

Here H = a/a ˙ is the usual Hubble expansion parameter for the isotropic universe. In the matter-dominated era, it results in a(t) ∝ t2/3 , so that H = 2/(3t). Moreover, to the zero-order in the eccentricity, from Eq. (16) it follows that the magnetic energy density scales in time as ρB ∝ a−4 . The solution of Eq. (18) is 2 3 (0) 2 (19) e = 4ΩB 1 − + 3/2 , a a (0)

(0)

(0)

where ΩB = ρB (t0 )/ρcr , ρcr = 3H02 /8πG is the actual critical energy density, and H0 = 100 h km sec−1 Mpc−1 is the current Hubble parameter with h ≃ 0.72 the littleh constant [6]. At the decoupling, t = tdec , we have (0) 3/2 e2dec ≃ 8 ΩB zdec , where edec = e(tdec ) and zdec ≃ 1091 is the red-shift at decoupling [6]. Accordingly, we get edec ≃ 4 × 10−3

B0 . nG

∞ l ∆T (θ, φ) X X = alm Ylm (θ, φ), hT i

(20)

(21)

l=2 m=−l

and introducing the power spectrum s 1 l(l + 1) X ∆Tl = |alm |2 , hT i 2π 2l + 1 m

(22)

the quadrupole anisotropy is defined by the multipole ℓ=2 Q ≡

(17)

The normalization of the scale factors is such that a(t0 ) = b(t0 ) = 1 at the present time t0 . In this paper, we restrict our analysis to the case of small eccentricities (that is, we consider the metric anisotropies as perturbations over the isotropic Friedmann-Robertson-Walker background). In this limit, from Eqs. (13)-(15), we get the following evolution equation for the eccentricity: d(ee) ˙ + 3H(ee) ˙ = 8πGρB . dt

If, for instance, we assume for the present cosmological magnetic field strength the estimate B0 ≃ nG, which is compatible with the constraints derived in Ref. [37], with the presence of galactic magnetic fields [3], and with the results obtained in Sec. II, we get an eccentricity at decoupling of about edec ≃ 0.4 × 10−2 . The presence of a nanogauss, plane-symmetric magnetic field at the surface of last scattering affects the temperature anisotropies in such a way, as we are going to see, to solve the quadrupole problem. Expanding the temperature anisotropy in terms of spherical harmonics [38]

∆T2 , hT i

(23)

where hT i ≃ 2.73K is the actual (average) temperature of the CMB radiation. The quadrupole problem resides in the fact that the observed quadrupole anisotropy, according to 3-year WMAP data (see Table I) and 5-year WMAP data (see Table II), is in the range 2

(24)

2

(25)

(∆T2 )obs ≃ (210 ÷ 276) µK2 and (∆T2 )obs ≃ (213 ÷ 302) µK2 ,

respectively, while the expected quadrupole anisotropy according the ΛCDM standard model is 2

(∆T2 )I ≃ 1252 µK2 ,

(26)

if we take into account the 3-year WMAP data, or (∆T2 )2I ≃ 1207 µK2 ,

(27)

according to 5-year WMAP data. If we admit that the large-scale spatial geometry of our Universe is plane-symmetric with a small eccentricity, then we have that the observed CMB anisotropy map is a linear superposition of two contributions [12, 39]: ∆T = ∆TA + ∆TI , where ∆TA represents the temperature fluctuations due to the anisotropic spacetime background, while ∆TI is the standard isotropic fluctuation caused by the inflation-produced gravitational potential

6 TABLE I: The cleaned maps SILC400, WILC3YR, and TCM3YR. Note that the values of a2m in this table correspond to the values of a2m given in Refs. [40, 41, 42] divided by hT i ≃ 2.73K. Moreover, the values of Re[a2m ] and Im[a2m ] are in units of 10−6 . Map

m 0 SILC400 1 2 0 WILC3YR 1 2 0 TCM3YR 1 2

Re[a2m ] 2.75 −0.56 −6.79 4.21 −0.02 −5.28 1.22 0.10 −5.45

Im[a2m ] (∆T2 )2 /µK2 Q/10−6 0.00 1.77 275.8 6.1 −6.60 0.00 1.78 248.8 5.8 −6.89 0.00 1.79 209.5 5.3 −6.32

TABLE II: Quadrupole power obtained from the cleaned maps “Hinshaw et al. cut sky”, WILC5YR, and HILCM5YR. Data from Ref. [43].

Map Hinshaw et al. cut sky WILC5YR HILCM5YR

(∆T2 )2 /µK2 213.4 242.7 301.7

Q/10−6 5.4 5.7 6.4

at the last scattering surface. As a consequence, we may write I alm = aA lm + alm .

(28)

We want now to analyze the distortion of the CMB radiation in a universe with planar symmetry described by the metric (12) in the small eccentricity approximation. The null geodesic equation gives that a photon emitted at the last scattering surface having energy Edec reaches the observer with an energy equal to E0 (b n) = hE0 i(1 + e2decn23 /2), where hE0 i ≡ Edec /(1 + zdec), and n b = (n1 , n2 , n3 ) are the direction cosines of the null geodesic in the symmetric (Robertson-Walker) metric. It is worth mentioning that the above result applies to the special case where the normal to the plane of symmetry is directed along the z-axis. The general case where this normal is directed along an arbitrary direction in a coordinate system (xG , yG , zG ) in which the xG yG -plane is the galactic plane, has been analyzed in Ref. [12]. Closely following [12], we perform a rotation R = Rx (ϑ) Rz (ϕ+π/2) of the coordinate system (x, y, z), where Rz (ϕ + π/2) and Rx (ϑ) are rotations of angles ϕ + π/2 and ϑ about the z- and x-axis, respectively. In the new coordinate system the z-axis is directed along the

direction defined by the polar angles (ϑ, ϕ). Therefore, the temperature anisotropy in this new reference system is E0 (nA ) − hE0 i 1 ∆TA ≡ = e2decn2A , hT i hE0 i 2 where nA ≡ (R n b)3 is given by

nA (θ, φ) = cos θ cos ϑ − sin θ sin ϑ cos(φ − ϕ).

(29)

(30)

Equations (29) and (30), then, give the general expression for the temperature anisotropy induced by a planar metric whose normal to the plane of symmetry points in the direction (ϑ, ϕ) in the galactic coordinate system. From Eqs. (29) and (30), it follows that only the quadrupole terms (ℓ = 2) are different from zero: √ π √ [1 + 3 cos(2ϑ)] e2dec , aA = 20 6 5 r π −iϕ A ∗ e sin(2ϑ) e2dec , (31) aA = −(a ) = − 21 2,−1 30 r π −2iϕ 2 2 A ∗ aA = (a ) = e sin ϑ edec . 22 2,−2 30 Since the temperature anisotropy is a real function, we have al,−m = (−1)m (al,m )∗ . Observing m A ∗ that aA = (−1) (a ) [see Eq. (31)], we get l,−m l,m aIl,−m = (−1)m (aIl,m )∗ . Moreover, because the standard inflation-produced temperature fluctuations are statistically isotropic, we will make the reasonable assumption that the aI2m coefficients are equals up to a phase factor. Therefore, we can write [12] r π iφ1 e QI , aI20 = 3 r π iφ2 aI21 = −(aI2,−1 )∗ = (32) e QI , 3 r π iφ3 I I ∗ a22 = (a2,−2 ) = e QI , 3 where 0 ≤ φ1 ≤ 2π, 0 ≤ φ2 ≤ 2π, and 0 ≤ φ3 ≤ 2π are unknown phases, and QI ≃ 13.0 × 10−6

(33)

QI ≃ 12.7 × 10−6

(34)

or

according to the 3-year WMAP data or 5-year WMAP data, respectively. We may fix the unknown direction (ϑ, ϕ) and the eccentricity by solving Eq. (28), which is (for ℓ = 2) a system of 5 equations containing 5 unknown parameters: edec, ϑ, ϕ, φ2 , and φ3 . Note that it is always possible to choose a20 real, so that φ1 = 0.

7 TABLE III: Numerical solutions of Eq. (28) obtained by using the map SILC400. The values of the angles ϑ, ϕ, φ2 , and φ3 are in degrees.

80

edec /10−2 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76

ϑ 67.9 67.8 67.8 67.7 67.9 67.9 67.7 67.7

ϕ 5.8 174.4 354.3 185.5 50.1 130.0 310.3 229.6

φ2 31.4 33.9 121.2 121.1 121.1 138.9 131.9 131.9

φ3 18.7 69.4 69.8 19.5 157.9 111.9 112.1 157.6

B0 /nG 1.88 1.89 1.89 1.89 1.88 1.88 1.89 1.90

TABLE IV: As in Table III, but for the map WILC3YR. edec /10−2 0.74 0.74 0.74 0.74 0.74 0.74 0.75 0.75

ϑ 66.4 66.3 66.3 66.3 66.5 66.5 66.2 66.2

ϕ 4.5 175.5 355.6 184.4 57.3 122.7 303.3 236.7

φ2 36.7 36.8 110.4 110.4 140.0 140.0 130.8 130.8

φ3 30.8 72.3 72.2 31.1 160.5 112.0 112.5 160.0

B0 /nG 1.84 1.84 1.84 1.84 1.83 1.83 1.85 1.85

TABLE V: As in Table III, but for the map TCM3YR. edec /10−2 0.78 0.78 0.78 0.78 0.78 0.78 0.78 0.78

ϑ 69.2 69.2 69.2 69.2 69.3 69.3 69.1 69.1

ϕ 0.5 179.4 359.5 180.6 49.0 131.1 311.6 228.4

φ2 89.3 89.0 96.6 96.9 140.5 140.4 130.6 130.6

φ3 46.2 52.5 52.3 49.5 154.5 116.4 116.6 154.3

B0 /nG 1.94 1.94 1.94 1.94 1.93 1.93 1.94 1.94

b

60

è SILC 400 ò WILC3YR TCM3YR ç Axis of Evil

40

ò 20è

ò è

ò è

èòòè

òè ç

òè

èò

ç 0 0

50

100

150

200

250

300

350

l FIG. 2: Numerical solutions of Eq. (28) obtained by using the three maps in Table I. Note that b = 90◦ −ϑ, and l = ϕ, where (b, l) are the galactic coordinates. The two open circles at (b, l) ≃ (20◦ , 130◦ ) and (b, l) ≃ (10◦ , 110◦ ) define the direction of the axis of evil determined in Ref. [44] for the 5-year WMAP temperature sky maps in V-band and W-band, respectively.

maps, are given in Tables III, IV, and V, respectively. In Appendix B, we will show that the system (28) (for ℓ = 2) admits 8 independent solutions. Moreover we observe that, for each independent solution (edec, ϑ, ϕ, φ2 , φ3 ) shown in the tables, there exists another one given by (edec , π − ϑ, ϕ ± π, φ2 , φ3 ), where we take the plus sign if ϕ < π and the minus sign if ϕ > π. Looking at Table I and Eq. (33), we see that the values of coefficients |a20 | and |a21 | are much smaller than QI . Moreover, comparing Eq. (33) with Table I, and Eq. (34) with Table II, we deduce, roughly speaking, that the value of QI is about twice the value of Q. Assuming QI ≫ |a20 |, |a21 |, and QI ≃ 2Q, we will show in Appendix B that approximate solutions for edec, ϑ, and ϕ are √ √ 15(5 + 3 73) 2 edec ≃ QI , (35) 24 " # √ √ 6(3 73 − 5) π 1 , (36) ϑ ≃ − arcsin 2 2 79 ϕ ≃ 0,

5π 7π π 3π , , π, , , 2π. 4 4 4 4

(37)

From Eqs. (35) and (33)-(34) we get To solve Eq. (28) for ℓ = 2, we need the observed values of the a2m ’s. We use the cleaned CMB temperature fluctuation map of the 3-year WMAP data obtained by using an improved internal linear combination method as galactic foreground subtraction technique. In particular, we adopt the three maps SILC400 [40], WILC3YR [41], and TCM3YR [42]. For completeness, we report in Table I the values of a2m corresponding to these maps. Numerical solutions of Eq. (28), referring to the three

edec ≃ 0.8 × 10−2 ,

(38)

which inserted in Eq. (20) gives B0 ≃ 2 nG.

(39)

As one can check, the approximate solutions Eqs. (35), (36), and (37) are quite close to the numerical values. In Fig. 2, we plot the numerical solutions ϑ versus ϕ which, defining a symmetry axis, singles out a preferred

8 direction in the CMB map. We adopt the so-called galactic coordinates system characterized by the galactic latitude, b, and galactic longitude, l. In our notation, the angle b corresponds to b = 90◦ − ϑ while l = ϕ. Equation (36) gives b ≃ 20◦ ,

(40)

while Eq. (37) can be written as l ≃ 0◦ , 45◦ , 135◦ , 180◦, 225◦ , 315◦, 360◦ .

(41)

From the above equations and Fig. 2, we see that the galactic latitude of the symmetry axis is remarkably independent on the adopted CMB temperature fluctuation map, while galactic longitude is poorly constrained. It is interesting to observe that, according to Ref. [44], the axis of evil points toward (b, l)AE ≃ (20◦ , 130◦ )

(42)

if one takes into account the 5-year WMAP data in the V-band, and toward (b, l)AE ≃ (10◦ , 110◦ )

(43)

if one uses the W-band 5-year WMAP data, so that it seems to be very close to the direction (b, l) ≃ (20◦ , 135◦ )

(44)

defined by the symmetry axis in our model. It is important to stress, however, that the apparent alignment of the above two axes could be just an accidental fact. Indeed, the axis of evil is determined by a statistical correlation (alignment and planarity) between the quadrupole and octupole modes, while the presence of a planar cosmic magnetic field affects only the quadrupole moment in the CMB radiation (this is true to the lowest order in the eccentricity parameter considered in this paper; to higher orders, modifications to the standard value of octupole intensity can occur, but they are vanishingly small). In any case, it is worth noting that there are already independent indications of a symmetry axis in the large-scale geometry of the Universe as, for example, those coming from the analysis of spiral galaxies in the Sloan Digital Sky Survey [45]. IV. CONCLUSIONS Our knowledge of the Universe has greatly improved after the discovery of the cosmic microwave background radiation. The observation and the analysis of the temperature fluctuations in this relic radiation, especially in the last decade, has confirmed at a surprising level of accuracy the canonical theoretical model describing

the evolution of the Universe, the standard cosmological model. This is the celebrated hot big-bang model, “equipped” with inflation, dark energy, and cold dark matter. Although these last three ingredients have not yet been framed in a definitely theoretical model of particle physics, from an operative point of view, this cosmological landscape appears satisfactory. However, the ubiquitous existence of large-scale correlated magnetic fields and some presumed anomaly discovered in the spectrum of the CMB radiation, together with the statement of fact that the evolution of the Universe (at least) before the Planck era cannot be understood without a self-consistent theory of quantum gravity, justify the study of both alternative cosmologies (such as anisotropic models of the Universe) and models of particle physics beyond the standard model. Indeed, in this paper, we analyzed the effects on standard cosmology of including in the photon sector of the standard model a Lorentz-violating term, the “Kosteleck´ y term” LK = − 14 (kF )αβµν F αβ F µν . Remarkably, we found that its presence is responsible for a superadiabatic amplification of magnetic vacuum fluctuations during de Sitter inflation. Moreover, the amplified magnetic field possesses a planar symmetry at large cosmological scales if the external tensor (kF )αβµν , which parametrizes Lorentz violation, assumes a particular (asymmetric) form. This peculiar magnetic field could account for the presence galactic magnetic fields and induces a small anisotropization of the Universe at cosmological scales (which can be described by a Bianchi I model). This, in turn, could explain the low quadrupole anomaly in the CMB radiation. Finally, we would like to stress that anisotropic cosmological models, such as that induced by the planar cosmic magnetic field discussed in this paper, are expected to induce a certain amount of polarization in the CMB radiation [46]. Although it is beyond the aim of this paper to discuss this kind of problems, we notice that it has been shown [47] that the 3-year WMAP data on large-scale polarization could be in agreement with an anisotropic model of a universe of Bianchi I type. An appropriate analysis of CMB polarization, resulting from the cosmological model presented in this paper, is in progress.

APPENDIX A In this appendix, we study the evolution during reheating, radiation and matter eras, of the magnetic field produced, according to the mechanism discussed in Sec. II, during de Sitter inflation. After inflation, the Universe enters into the so-called reheating phase, during which the energy of the inflaton is converted into ordinary matter. The reheating phase ends at the temperature TRH which is less than M and constrained as TRH . 108 GeV [48]. Moreover,

9 CMB analysis requires M . 10−2 mPl [19], otherwise it would be too much of a gravitational waves relic abundance, and also one must impose that TRH & 1GeV, so that the predictions of big-bang nucleosynthesis (BBN) are not spoiled [19, 49]. It is worth noting that the condition kF ≫ a6 is certainly fulfilled during inflation and reheating if kF ≫ a6RH , where aRH = a(TRH ). Since aRH ∼ T0 /TRH , where T0 ∼ 10−13 GeV is the actual temperature [50], we have 10−126 . a6RH . 10−78 for 1GeV . TRH . 108 GeV. Taking into account that the most stringent bound on ||(kF )αβµν || is 10−37 (as discussed in Sec. II) we get, for a wide range of allowed values of kF , that kF ≫ a6 during inflation and reheating. This, in turn means, according to Eqs. (6), (7) and the first Bianchi identity, that a superadiabatic amplification of magnetic vacuum fluctuations during these eras takes place. During reheating, in particular, taking into account that η ∝ a1/2 , we have B ∝ a9/2 . After reheating, the Universe enters the radiationdominated era. In this era, as well as in the subsequent matter era, the effects of the conducting primordial plasma are important when studying the evolution of a magnetic field. They are taken into account by adding to the electromagnetic Lagrangian the source term j µAµ [19]. Here, the external current j µ , expressed in terms of the electric field, has the form j µ = (0, σc E), where σc is the conductivity. Plasma effects introduce, in the right-hand side of Eq. (4), the extra term −aσc (a2 Ei ). In this case, it easy to see that modes well outside the horizon (assuming that kF ≫ a6 ) evolve as Z 1 7 4 (45) dη a σc . E ∝ a exp − 2kF R Approximating dη a7 σc with η a7 σc and using aη ∼ H −1 , we get a6 σc E ∝ a4 exp − . (46) 2kF H 2

In radiation era H ∼ T /mPl [50] and, for temperature much greater than the electron mass, the conductivity is approximately given by σc ∼ T /α [19], where α is the fine structure constant and T the temperature. Then we get " # 7 T∗ 4 E ∝ a exp − , (47) T where T∗ ∼ GeV

10−57 kF

1/7

.

(48)

This means that for T & T∗ we have E ∝ a4 (which in turn gives B ∝ a5 since η ∝ a in radiation era), while for

T . T∗ the electric field is dissipated, so the magnetic field evolves adiabatically, B ∝ a−2 . [We have assumed that kF ≫ a6 from the end of reheating until T∗ . As is easy to verify taking into account Eq. (48), this assumption is certainly satisfied in our case.] Finally, evolving along the lines discussed above the inflation-produced magnetic field from the time of horizon crossing until today, and taking into account Eq. (8), we easily recover Eq. (9). APPENDIX B In this appendix, we solve the system of Eqs. (28) for ℓ = 2. The numerical values of the a2m ’s are listed in I Table I, while the aA 2m ’s and a2m ’s are given by Eqs. (31) and (32), respectively. Since it is always possible to choose a20 real, we take φ1 = 0. Moreover, the temperature anisotropy is real so that we have a2,−m = (−1)m (a2m )∗ , aA 2,−m = ∗ m I ∗ I (−1)m (aA 2m ) , and a2,−m = (−1) (a2m ) . Therefore, the system of Eqs. (28) reduces to r √ π π 2 √ a20 = [1 + 3 cos(2ϑ)] edec + QI , (49) 3 6 5 r r π π cosϕ sin(2ϑ) e2dec + cosφ2 QI , Re[a21 ] = − 30 3 (50) r r π π Im[a21 ] = sinϕ sin(2ϑ) e2dec + sinφ2 QI , 30 3 (51) r r π π cosϕ sin2 ϑ e2dec + cosφ3 QI , Re[a22 ] = 30 3 (52) r r π π Im[a22 ] = − sinϕ sin2 ϑ e2dec + sinφ3 QI , 30 3 (53) where QI is given by Eq. (33). We see that these equations form a system of 5 transcendental equations containing 5 unknown parameters: edec , ϑ, ϕ, φ2 , and φ3 . Solving Eq. (49) with respect to ϑ we get two independent solutions:

where

e π − ϑ}, e ϑ = {ϑ,

(54)

! √ √ √ 1 5 15 a20 − 5 5πQI − 3π e2dec e √ . (55) ϑ = arccos 2 3 3π e2dec

Squaring Eqs. (52) and (53), adding side by side, and then solving with respect to ϕ, we obtain 8 independent solutions: ϕ = {ϕ e± , π − ϕ e± , 2π − ϕ e± , π + ϕ e± },

(56)

10 where

and

ϕ e± =

!

p 1 αγ ± β α2 + β 2 − γ 2 , arccos 2 α2 + β 2 r

2π α=− Re[a22 ] sin2 ϑ e2dec , 15 r 2π Im[a22 ] sin2 ϑ e2dec , β=− 15 π π sin4 ϑ e4dec + |a22 |2 − Q2I . γ= 30 3

(57)

(58) (59) (60)

By dividing side by side Eqs. (51) and (50), and solving with respect to φ2 , we get √ √ 30 Im[a21 ] − π sinϕ sin(2ϑ) e2dec √ tanφ2 = . √ 30 Re[a21 ] + π sinϕ sin(2ϑ) e2dec

(61)

Inserting the above equation in Eq. (66) gives Eq. (35), while its solution is indeed Eq. (36). Assuming that the main contribution to Q comes from the a22 , and that Re[a22 ] ≃ Im[a22 ], we can write r 5π Q. (68) Re[a22 ] ≃ Im[a22 ] ≃ − 12 Consequently, inserting Eqs. (66)-(68) in Eq. (57), we get ! r 1 1 2 ϕ e± ≃ arccos q ± (69) −q , 2 2

where

The same procedure applied to Eqs. (53) and (51) results in √ √ 30 Im[a22 ] + π sin(2ϕ) sin2 ϑ e2dec tanφ3 = √ . (62) √ 30 Re[a22 ] − π sin(2ϕ) sin2 ϑ e2dec Finally, by squaring Eqs. (50) and (51), and adding side by side, we get e4dec − 2c e2dec − d = 0,

Inserting the above expression in Eqs. (54)-(55), assuming that QI ≫ |a20 |, and after same algebraic manipulation, we get √ √ 6(3 73 − 5) | sin(2ϑ)| ≃ . (67) 79

(63)

where we have defined r 30 (Re[a21 ] cos ϕ − Im[a21 ] sin ϕ) csc(2ϑ), c(ϕ, ϑ) = π (64) 3 d(ϕ, ϑ) = 10 Q2I − |a21 |2 csc2(2ϑ). (65) π We observe that the couple (ϑ, ϕ) can assume 16 different values, according to Eqs. (54) and (56). Inserting these values in Eqs. (63)-(65) we arrive at 16 different equations for edec. It is straightforward to verify that only 8 of these are “independent,” in the sense that, given a solution (edec , ϑ, ϕ) of one of the independent equations, then (edec , π−ϑ, ϕ±π) is a solutions of one of the “dependent” ones (we must take the plus sign if ϕ < π and the minus sign if ϕ > π). The 8 independent equations can be solved numerically and their solutions are presented in Table III, IV, and V. Finally, we derive the approximate solutions (35)-(36). To this end,√ we may formally solve Eq. (63) to get e2dec = c ± c2 + d. If QI ≫ |a21 | then d ≫ c2 , and we obtain √ e2dec ≃ 10 QI | csc(2ϑ)|. (66)

√ √ √ Q 30( 73 − 5) QI q= (1 + 5 73) + 120 5760 Q QI QI Q ≃ 0.15 . (70) + 0.40 Q QI Observing that the value of the quadrupole according to the ΛCDM standard model is, approximately, twice the value of the observed one, QI ≃ 2Q [compare Eq. (33) with Table I, and Eq. (34) with Table II], we roughly get q ≃ 1/2 and, accordingly, ϕ e+ ≃ 0 and ϕ e− ≃ π/4. Inserting these values in Eq. (56), we recover Eq. (37).

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Dipartimento di Fisica, Universit` a di Bari, I-70126 Bari, Italy and 2 INFN - Sezione di Bari, I-70126 Bari, Italy (Dated: September, 2009)

arXiv:0907.3703v2 [astro-ph.CO] 17 Sep 2009

Abstract The presence of a nonconformally invariant term in the photon sector of the Lorentz-violating extension of the standard model of particle physics, the “Kosteleck´ y term” LK ∝ (kF )αβµν F αβ F µν , enables a superadiabatic amplification of magnetic vacuum fluctuations during de Sitter inflation. For a particular form of the external tensor kF that parametrizes Lorentz violation, the generated field possesses a planar symmetry at large cosmological scales and can have today an intensity of order of nanogauss for a wide range of values of parameters defining inflation. This peculiar magnetic field could account for the presently observed galactic magnetic fields and induces a small anisotropization of the Universe at cosmological scales. The resulting Bianchi I model could explain the presumedly low quadrupole power in the cosmic microwave background radiation. PACS numbers: 98.62.En, 11.30.Cp, 98.80.-k, 98.70.Vc

I. INTRODUCTION The standard model of particle physics together with the Einstein theory of gravity encompasses, basically, all known fundamental physics. However, due to the classical character of general relativity, these theories are believed to be incomplete, and a search for a more fundamental theory which overcomes the stumbling block of quantizing gravity is a major goal of present-day theoretical physics. A promising candidate is string theory, which deals with gravity in a quantum and self-consistent way. Although string theory is well away from being an experimentally tested theory, some of its peculiar, low-energy manifestations, if ever detected in astrophysical and/or ground-based experiments, could be a strong signal of its correctness. Indeed, as pointed out by Kosteleck´ y [1], a possible detectable signature of string theory at low energies is violation of Lorentz symmetry [2]. If on the one hand, terrestrial and astrophysical experiments have not yet either confirmed or ruled out the existence of effects of Lorentz violation (LV), on the other hand astrophysical observations have definitely confirmed the presence of large-scale correlated, microgauss magnetic fields in any type of galaxies and clusters of galaxies. (For reviews on cosmic magnetic fields see Ref. [3].) This peculiar property of possessing, roughly speaking, the same intensity and correlation scale everywhere in the present Universe, has been interpreted as a strong hint that cosmic magnetic fields are, indeed, relic of the early Universe. Obviously, a damning evidence for the primordial origin of cosmic magnetic fields would be the detection of their effects on the Cosmic Microwave Background (CMB) radiation. Until now, however, exhaustive analyses have only given stringent constraints on their properties, without finding any evidence of their imprints on the CMB [4]. The use of the CMB radiation as a probe of the physics

of the early Universe has proved to be very fruitful in the last years. In particular, the high resolution data of temperature fluctuations of CMB angular power spectrum, provided by the Wilkinson Microwave Anisotropy Probe (WMAP) [5, 6] have, almost definitively, consecrated the so-called “Λ-dominated cold dark matter” (ΛCDM) as the standard cosmological model of the Universe. Nevertheless, the 1-, 3-, and 5-year WMAP data display at large angular scales some “anomalous” features, the most important ones being the low quadrupole moment and the presence of a preferred direction in the Universe. It is extremely important, however, to stress that, due to the importance of residual Galactic foreground emission, those anomalies in the CMB anisotropy are still subject to an intense debate [7]. If there is really a problem with the quadrupole moment, then its lowness, indicating a suppression of power at cosmological scales, may signal a nontrivial topology of the large-scale geometry of the Universe [8]. Indeed, several possibilities have been proposed in the recent literature to understand that suppression [9, 10] (for other large-scale anomalies in the angular distribution of CMB see, e.g., Ref. [11]). Recently enough [12, 13], it has been shown that a particular case of the simplest anisotropic cosmological model, i.e. the Bianchi I model, could account for the smallness of the quadrupole, without affecting higher multipoles of the angular power spectrum of the temperature anisotropy. Such a proposal of an “ellipsoidal universe” has been considered also in Ref. [14, 15]. Also, the WMAP data display a particular feature which has been deeply investigated in the last years: a statistically significant alignment and planarity of the quadrupole and octupole modes. This seems to indicate the existence of a preferred direction in the Universe, which has been named “axis of evil” (AE) [16]. Needless to say, there is no space in the isotropic, standard

2 cosmological model for such a type of features. In this paper, we investigate the possibility that effects of Lorentz violation at inflation could be responsible for the creation of large-scale magnetic fields possessing planar symmetry. As we will see, these fields can have the right intensity and correlation length to explain the existence of galactic and extragalactic magnetic fields. Moreover, because of their peculiar symmetry at cosmological scales, they could induce a modification of the (isotropic) Robertson-Walker metric, in such a way that the resulting cosmological model is well described by the ellipsoidal universe model. This, in turn, could naturally account for some peculiar and not-yet-explained anomalous features of CMB radiation discussed above. The plan of the paper is as follows. In Sec. II we discuss the generation at inflation of a plane-symmetric cosmic magnetic field in the framework of the Lorentzviolation extension of the standard model of particle physics. Section III deals with the analysis of CMB anisotropies including the asymmetric contribution due to the presence of the planar field. Finally, we draw our conclusions in Sec. IV. Some technical details are presented in the appendixes. II. LORENTZ-VIOLATING ELECTROMAGNETISM AND PLANAR COSMIC MAGNETIC FIELDS The large correlation scale of cosmic magnetic fields, ranging from ∼ 10kpc for magnetic fields in galaxies to ∼ 1Mpc for those in clusters, and the fact that they are found to have approximately the same intensity of a few microgauss seems to indicate a common and primordial origin, probably to ascribe to some unknown mechanism acting during an inflationary epoch of the Universe. If one takes into account that the collapse of primordial large-scale structures enhances the intensity of any preexisting magnetic field of about a factor 103 [3], a primeval field with comoving intensity of order of nanogauss and correlated on megaparsec scales could explain the “magnetization of the Universe”. During inflation all fields are quantum mechanically excited. Because the wavelength λ associated to a given fluctuation grows faster than the horizon, there will be a time, say t1 , when this mode crosses outside the horizon itself. After that, this fluctuation cannot collapse back into the vacuum being not causally self-correlated, and then “survives” as a classical real object [17]. The electromagnetic energy density at the time of crossing is then fixed by the Gibbons-Hawking temperature TGH [17]: 4 E ∼ TGH ∼ H 4,

(1)

where H is the Hubble parameter (in this paper we consider, for the sake of simplicity, just the case of de Sit-

ter inflation). Taking into account the expression for the electromagnetic energy in standard Maxwell electromagnetism, one arrives to the result that the spectrum of magnetic fluctuations at the time of horizon crossing is given by B1 ∼ H 2 ∼ M 4 /m2Pl [17, 18, 19], where in the last equality we used the Friedmann equation H 2 = (8π/3)M 4 /m2Pl . Here, M 4 is the total energy density during inflation (which is constant during de Sitter inflation) and mPl ∼ 1019 GeV is the Planck mass. Because of conformal invariance of Maxwell electromagnetism one finds, however, that the present magnitude of the inflation-produced field at the scale, say 10kpc, is vanishingly small, B0 ∼ 10−52 G [19]. (This is true only if the background metric is spatially-flat [20], which is the case discussed in this paper.) Since the pioneer work of Turner and Widrow [19], a plethora of mechanisms has been proposed for generating cosmic magnetic fields in the early Universe, all of which repose on the breaking of conformal invariance of standard electrodynamics (see references in Ref. [3] and, for recent papers, Ref. [21, 22]). In particular, Kosteleck´ y, Potting and Samuel [23] first pointed out that the breaking of conformal invariance is a natural consequence of LV. Indeed, they argued that the appearance of an effective photon mass, owing to spontaneous breaking of Lorentz invariance, could enable the generation of large-scale magnetic fields within inflationary scenarios. The idea that Lorentz symmetry breaking could result in the generation of cosmic magnetic fields has been pursued since then by others authors [24, 25, 26, 27, 28]. The aim of this paper is to show that within a particular Lorentz-violating model of particle physics, it is possible to generate magnetic fields of cosmological type possessing a peculiar spatial geometry. Then, in the next section, we will analyze their impact on the isotropy of the Universe and, in particular, on the cosmic microwave background radiation. The model we are going to study is the so-called standard model extension (SME) [29], which is an effective field theory including all admissible Lorentz-violating terms in the Glashow-Weinberg-Salam gauge theory. In curved spacetimes, the SME action for the photon field, here referred to as the Maxwell-Kosteleck´ y (MK) action, reads [30] Z SMK = d4 x e − 41 Fµν F µν − 41 (kF )αβµν F αβ F µν , (2)

where Fµν = ∂µ Aν − ∂ν Aµ is the electromagnetic field strength tensor and e the determinant of the vierbein. The presence of the external tensor (kF )αβµν breaks (particle) Lorentz invariance [30] and parametrizes then Lorentz violation. The external tensor (kF )αβµν is fixed in a given system of coordinates. Going in different systems of coordinates will, generally, induces a change of the form of (kF )αβµν . In the following, we assume that the form of (kF )αβµν refers to a system of coordinates at

3 rest with respect to the cosmic microwave background, the so-called “CMB frame.” It is worth noting that taking (kF )αβµν as a fixed tensor corresponds to have an explicit violation of Lorentz symmetry. This could introduce in the theory an instability associated to nonpositivity of the energy. Indeed, working in a flat isotropic universe described by a Robertson-Walker metric ds2 = a2 (dη 2 − dx2 ), where a(η) is the expansion parameter and η the conformal time, and introducing the electric and magnetic fields as F0i = −a2 Ei and Fij = ǫijk a2 Bk (Latin indices run from 1 to 3, while Greek ones from 0 to 3), the electromagnetic energy density turns out to be [31] 1 2 (E + B2 ) + (kF )i00j a−4 Ei Ej 2 1 + (kF )ijkl a−4 ǫijm ǫkln Bm Bn . 4

E=

(3)

The positivity of the above quadratic form depends on the particular form assumed by the fixed tensor (kF )αβµν , which is frame-dependent. This apparent paradox (frame-dependent positivity of the energy) is overcome when considering models in which LV is spontaneously broken. In this case, however, action (2) loses its character of generality, since the tensor (kF )αβµν is now regarded as a vacuum expectation value of some tensor field with its own dynamics. For this reason, and following Ref. [31], we will take (kF )αβµν to be a fixed tensor but, at the same time, we will impose positivity of the energy in the CMB frame. Needless to say, if in another system of coordinates the energy is not positive defined, this means that the effective theory with explicit Lorentz symmetry breaking becomes meaningless in that frame and one needs to consider the full theory with spontaneous Lorentz symmetry breaking in order to get physically acceptable results. The equations of motion follow from action (2) [31]: ∂η (a2 Ei ) − ǫijk ∂j (a2 Bk ) + ∂η [2(kF )i00j a−2 Ej − (kF )0ijk ǫjkl a−2 Bl ] + ∂j [2(kF )ijk0 a

−2

Ek − (kF )ijkl ǫklm a

−2

Bm ] = 0 (4)

and ∂i (a2 Ei ) + ∂i [2(kF )i00j a−2 Ej − (kF )0ijk ǫjkl a−2 Bl ] = 0. The Bianchi identities are ∂η (a2 B) + ∇ × (a2 E) = 0 and ∇ · B = 0. We are interested in the generation and evolution of superhorizon magnetic fields, that is to electromagnetic modes whose physical wavelength is much greater than the Hubble radius H −1 , λphys ≫ H −1 , where λphys = aλ and λ is the comoving wavelength. Since aη ∼ H −1 , introducing the comoving wavenumber k = 2π/λ, the above condition reads |kη| ≪ 1. Observing that the first Bianchi identity gives on large scales B ∼ kηE, where B and E stand for the average magnitude of the magnetic and electric field intensities, and assuming that all (non-null) components of (kF )αβµν have approximately

the same magnitude, we can neglect in Eq. (4), on large scales, the terms proportional to the magnetic field. Also, the next-to-last term is negligible with respect to the first one. Therefore, at large scales, Eq. (4) reduces to ∂η (a2 Ei ) + ∂η [2(kF )i00j a−2 Ej ] = 0.

(5)

Assuming that ||(kF )i00j || ≫ a4 we then have (kF )i00j a−2 Ej = ci ,

(6)

where ci are constants of integration. It is plausible to assume that the tensor (kF )αβµν is a nonincreasing function of time. Indeed, we can take the simple form (kF )i00j ∝ a−p , with p a non-negative real number. In this case, it is clear from Eq. (6), the first Bianchi identity, and the fact that η ∝ a−1 during de Sitter inflation, that the average intensity of the magnetic field grows “superadiabatically,” B ∝ a1+p . [If the elements of (kF )i00j are either all zero or negligibly small in the CMB frame, the inflation-produced field is, in general, small-scale correlated and then not astrophysically interesting.] In Ref. [31], the case was analyzed where (kF )i00j is a constant isotropic tensor, (kF )i00j ∝ δij , where δij is the Kronecker delta. In this paper, instead, we study the case in which (kF )i00j is “maximally” anisotropic, in the sense that only one component, say (kF )3003 , is different from zero. In this case, only the Ez component of the electric field is amplified and this in turn means that only the Bx and By components of the magnetic field grow superadiabatically during inflation. Accordingly, the amplified magnetic field will possess a planar symmetry, whose plane of symmetry is the xy-plane. Moreover, for the sake of simplicity, we will concentrate only on a single case, that corresponding to p = 2, so that we will assume that (kF )i00j ≡ a−2 δi3 δj3 kF ,

(7)

where kF is a constant which gives the magnitude of Lorentz violation effect at the actual time, a = a0 = 1. (It is worth noting that the condition ||(kF )i00j || ≫ a4 translates into kF ≫ a6 .) As we will see, the choice of Eq. (7) will correspond to the case of inflation-produced, scale-invariant, cosmic magnetic fields. 1 In this case,

1

Up to today there is no experimental evidence that the tensor (kF )αβµν is time-dependent or even different from zero (see below). Accordingly, it is worth stressing that our assumption that (kF )αβµν decreases in time should only be considered as a working hypothesis. On the other hand, the choice p = 2 follows by the observation that the actual cosmic magnetic fields possess, approximately, the same intensity on different scales (that of galaxies and that of clusters of galaxies), and by interpreting this occurrence as a hint that their spectrum is indeed scale-invariant (at least on cosmological scales).

4

a3 H 2 . B1 ∼ √1 kF

(8)

It is worth noting that, in order to have positivity of the energy, we are forced to assume kF > 0. In order to find the actual value of the inflationproduced magnetic field, we have to follow its evolution during reheating, radiation, and matter eras. Since the resulting analysis is very close to that performed in Ref. [31], we quote just the final result (full details are given in Appendix A): B0 ∼ nG

M 106 GeV

6

TRH MeV

7(n−1)/2

kF 10−37

n/2 ,

(9)

where TRH is the reheat temperature (see Appendix A), and n takes the values ±1 according to T∗ . TRH or T∗ & TRH . The temperature T∗ , defined in Eq. (48), is the temperature below which electric fields are washed out by dissipative effects of the primordial plasma (so that magnetic fields evolve adiabatically for T . T∗ ). The condition T∗ & TRH means, indeed, that the magnetic field evolves adiabatically from the end of reheating until today and is equivalent to having TRH /MeV . (10−37/kF )1/7 . Upper bounds on ||(kF )αβµν || come from the analysis of CMB polarization and polarized light of radiogalaxies and gamma-ray bursts: they are, respectively, 10−30 [32], 10−32 [33], and 10−37 [34]. (Although the former bound is less stringent than the latter ones, it covers the whole portion of coefficient space for Lorentz violation. The point-source nature of radiogalaxies and gamma-ray bursts, instead, allows us to put constraints only on limited portions of coefficient space [32].) Now, it is clear from Eq. (9) and Fig. 1 that, for a wide range of values of parameters defining inflation (i.e., M and TRH ) and Lorentz symmetry violation (i.e., kF ), the scaling-invariant, present-day magnetic field can be as strong as B0 ∼ nG, and then could naturally explain the presence of galactic and extragalactic magnetic fields.

0

Log10HMGeVL

the electric and magnetic fields scale as Ez ∝ a4 and Bx ∝ By ∝ a3 , respectively, while Ex , Ey , and Bz scale adiabatically. Before proceeding further, we estimate the spectrum of magnetic vacuum fluctuations generated during de Sitter inflation in Maxwell-Kosteleck´ y electromagnetism. If, as before, we assume that the only nonzero component of (kF )αβµν is just given by (kF )3003 = kF a−2 , the electromagnetic energy density turns out to be E = 2 2 1 −6 2 Ez . Therefore, at the time of crossing, 2 (E +B )+kF a where |kη| ∼ 1, and since kF ≫ a6 , we get E ∼ kF a1−6 B12 , where a1 = a(t1 ) and we used the first Bianchi identity. Here, B1 stands for the average intensity of the magnetic field on the plane xy at the time of crossing. Remembering that E ∼ H 4 , we obtain the spectrum of magnetic fluctuations when crossing the horizon,

1

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

2 kF kF kF kF kF

0

1

3

4

5

6

7

8 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

= 10-77 = 10-57 = 10-37 = 10-32 = 10-30

2

3

4

5

6

7

8

Log10HTRHGeVL FIG. 1: The plane-symmetric, scale-invariant, inflationproduced magnetic field has an actual intensity of order of nanogauss if the values of parameter defining inflation, i.e. the energy scale of inflation M and the reheat temperature TRH , stay on the curves. kF ∼ ||(kF )αβµν || estimates the magnitude of the (constant) external tensor which parametrizes Lorentz violation [see Eqs. (2) and (7)].

III. ELLIPSOIDAL UNIVERSE AND THE CMB QUADRUPOLE The cosmic magnetic field produced according to the mechanism discussed in the previous section possesses a planar symmetry. The energy-momentum tensor for a plane-symmetric magnetic field of the form B = (Bx , By , 0) is 1 2 0 0 0 2B 1 2 2 0 Bx By 0 2 (Bx − By ) . (TB)µν = 1 2 2 0 Bx By (B − B ) 0 y x 2 0 0 0 − 21 B2 (10) Taking an ensemble average, and assuming statistical isotropy in the plane of symmetry, gives hBx By i ≃ 0 and hBx2 i ≃ hBy2 i, so that Eq. (10) reduces to (TB )µν = ρB diag(1, 0, 0, −1), where ρB = hB2 i/2 is the average magnetic energy density. We want now to analyze the large-scale evolution of the matter-dominated universe filled with a plane-symmetric anisotropic component represented by the planar cosmic magnetic field. The total energy-momentum tensor is then given by T µν = diag(ρB + ρm , 0, 0, −ρB ),

(11)

where ρm is the matter energy density. We then consider a cosmological model with planar symmetry, whose most general plane-symmetric line element, compatible with Eq. (11), is [35] ds2 = dt2 − a2 (t)(dx2 + dy 2 ) − b2 (t) dz 2 ,

(12)

5 where a and b are the scale factors. The metric (12) corresponds to considering the xy-plane as a symmetry plane. Taking into account Eqs. (11) and (12), Einstein’s equations read [36] 2 a˙ b˙ a˙ +2 = 8πG(ρB + ρm ), (13) a ab a ¨ ¨b a˙ b˙ + + = 0, (14) a b ab 2 a ¨ a˙ 2 + = −8πGρB , (15) a a where a dot indicates the derivative with respect to the cosmic time. Since the conductivity of the primordial plasma is very high, we are safe in neglecting interaction effects between magnetic field and matter [3]. In this case, the magnetic component of the energy-momentum tensor is conserved, Dµ (TB )µν = 0, so that we have ! b˙ a˙ ρB = 0. (16) + ρ˙ B + 2 a b Let us introduce the so-called “eccentricity” r a 2 . e≡ 1− b

(18)

Here H = a/a ˙ is the usual Hubble expansion parameter for the isotropic universe. In the matter-dominated era, it results in a(t) ∝ t2/3 , so that H = 2/(3t). Moreover, to the zero-order in the eccentricity, from Eq. (16) it follows that the magnetic energy density scales in time as ρB ∝ a−4 . The solution of Eq. (18) is 2 3 (0) 2 (19) e = 4ΩB 1 − + 3/2 , a a (0)

(0)

(0)

where ΩB = ρB (t0 )/ρcr , ρcr = 3H02 /8πG is the actual critical energy density, and H0 = 100 h km sec−1 Mpc−1 is the current Hubble parameter with h ≃ 0.72 the littleh constant [6]. At the decoupling, t = tdec , we have (0) 3/2 e2dec ≃ 8 ΩB zdec , where edec = e(tdec ) and zdec ≃ 1091 is the red-shift at decoupling [6]. Accordingly, we get edec ≃ 4 × 10−3

B0 . nG

∞ l ∆T (θ, φ) X X = alm Ylm (θ, φ), hT i

(20)

(21)

l=2 m=−l

and introducing the power spectrum s 1 l(l + 1) X ∆Tl = |alm |2 , hT i 2π 2l + 1 m

(22)

the quadrupole anisotropy is defined by the multipole ℓ=2 Q ≡

(17)

The normalization of the scale factors is such that a(t0 ) = b(t0 ) = 1 at the present time t0 . In this paper, we restrict our analysis to the case of small eccentricities (that is, we consider the metric anisotropies as perturbations over the isotropic Friedmann-Robertson-Walker background). In this limit, from Eqs. (13)-(15), we get the following evolution equation for the eccentricity: d(ee) ˙ + 3H(ee) ˙ = 8πGρB . dt

If, for instance, we assume for the present cosmological magnetic field strength the estimate B0 ≃ nG, which is compatible with the constraints derived in Ref. [37], with the presence of galactic magnetic fields [3], and with the results obtained in Sec. II, we get an eccentricity at decoupling of about edec ≃ 0.4 × 10−2 . The presence of a nanogauss, plane-symmetric magnetic field at the surface of last scattering affects the temperature anisotropies in such a way, as we are going to see, to solve the quadrupole problem. Expanding the temperature anisotropy in terms of spherical harmonics [38]

∆T2 , hT i

(23)

where hT i ≃ 2.73K is the actual (average) temperature of the CMB radiation. The quadrupole problem resides in the fact that the observed quadrupole anisotropy, according to 3-year WMAP data (see Table I) and 5-year WMAP data (see Table II), is in the range 2

(24)

2

(25)

(∆T2 )obs ≃ (210 ÷ 276) µK2 and (∆T2 )obs ≃ (213 ÷ 302) µK2 ,

respectively, while the expected quadrupole anisotropy according the ΛCDM standard model is 2

(∆T2 )I ≃ 1252 µK2 ,

(26)

if we take into account the 3-year WMAP data, or (∆T2 )2I ≃ 1207 µK2 ,

(27)

according to 5-year WMAP data. If we admit that the large-scale spatial geometry of our Universe is plane-symmetric with a small eccentricity, then we have that the observed CMB anisotropy map is a linear superposition of two contributions [12, 39]: ∆T = ∆TA + ∆TI , where ∆TA represents the temperature fluctuations due to the anisotropic spacetime background, while ∆TI is the standard isotropic fluctuation caused by the inflation-produced gravitational potential

6 TABLE I: The cleaned maps SILC400, WILC3YR, and TCM3YR. Note that the values of a2m in this table correspond to the values of a2m given in Refs. [40, 41, 42] divided by hT i ≃ 2.73K. Moreover, the values of Re[a2m ] and Im[a2m ] are in units of 10−6 . Map

m 0 SILC400 1 2 0 WILC3YR 1 2 0 TCM3YR 1 2

Re[a2m ] 2.75 −0.56 −6.79 4.21 −0.02 −5.28 1.22 0.10 −5.45

Im[a2m ] (∆T2 )2 /µK2 Q/10−6 0.00 1.77 275.8 6.1 −6.60 0.00 1.78 248.8 5.8 −6.89 0.00 1.79 209.5 5.3 −6.32

TABLE II: Quadrupole power obtained from the cleaned maps “Hinshaw et al. cut sky”, WILC5YR, and HILCM5YR. Data from Ref. [43].

Map Hinshaw et al. cut sky WILC5YR HILCM5YR

(∆T2 )2 /µK2 213.4 242.7 301.7

Q/10−6 5.4 5.7 6.4

at the last scattering surface. As a consequence, we may write I alm = aA lm + alm .

(28)

We want now to analyze the distortion of the CMB radiation in a universe with planar symmetry described by the metric (12) in the small eccentricity approximation. The null geodesic equation gives that a photon emitted at the last scattering surface having energy Edec reaches the observer with an energy equal to E0 (b n) = hE0 i(1 + e2decn23 /2), where hE0 i ≡ Edec /(1 + zdec), and n b = (n1 , n2 , n3 ) are the direction cosines of the null geodesic in the symmetric (Robertson-Walker) metric. It is worth mentioning that the above result applies to the special case where the normal to the plane of symmetry is directed along the z-axis. The general case where this normal is directed along an arbitrary direction in a coordinate system (xG , yG , zG ) in which the xG yG -plane is the galactic plane, has been analyzed in Ref. [12]. Closely following [12], we perform a rotation R = Rx (ϑ) Rz (ϕ+π/2) of the coordinate system (x, y, z), where Rz (ϕ + π/2) and Rx (ϑ) are rotations of angles ϕ + π/2 and ϑ about the z- and x-axis, respectively. In the new coordinate system the z-axis is directed along the

direction defined by the polar angles (ϑ, ϕ). Therefore, the temperature anisotropy in this new reference system is E0 (nA ) − hE0 i 1 ∆TA ≡ = e2decn2A , hT i hE0 i 2 where nA ≡ (R n b)3 is given by

nA (θ, φ) = cos θ cos ϑ − sin θ sin ϑ cos(φ − ϕ).

(29)

(30)

Equations (29) and (30), then, give the general expression for the temperature anisotropy induced by a planar metric whose normal to the plane of symmetry points in the direction (ϑ, ϕ) in the galactic coordinate system. From Eqs. (29) and (30), it follows that only the quadrupole terms (ℓ = 2) are different from zero: √ π √ [1 + 3 cos(2ϑ)] e2dec , aA = 20 6 5 r π −iϕ A ∗ e sin(2ϑ) e2dec , (31) aA = −(a ) = − 21 2,−1 30 r π −2iϕ 2 2 A ∗ aA = (a ) = e sin ϑ edec . 22 2,−2 30 Since the temperature anisotropy is a real function, we have al,−m = (−1)m (al,m )∗ . Observing m A ∗ that aA = (−1) (a ) [see Eq. (31)], we get l,−m l,m aIl,−m = (−1)m (aIl,m )∗ . Moreover, because the standard inflation-produced temperature fluctuations are statistically isotropic, we will make the reasonable assumption that the aI2m coefficients are equals up to a phase factor. Therefore, we can write [12] r π iφ1 e QI , aI20 = 3 r π iφ2 aI21 = −(aI2,−1 )∗ = (32) e QI , 3 r π iφ3 I I ∗ a22 = (a2,−2 ) = e QI , 3 where 0 ≤ φ1 ≤ 2π, 0 ≤ φ2 ≤ 2π, and 0 ≤ φ3 ≤ 2π are unknown phases, and QI ≃ 13.0 × 10−6

(33)

QI ≃ 12.7 × 10−6

(34)

or

according to the 3-year WMAP data or 5-year WMAP data, respectively. We may fix the unknown direction (ϑ, ϕ) and the eccentricity by solving Eq. (28), which is (for ℓ = 2) a system of 5 equations containing 5 unknown parameters: edec, ϑ, ϕ, φ2 , and φ3 . Note that it is always possible to choose a20 real, so that φ1 = 0.

7 TABLE III: Numerical solutions of Eq. (28) obtained by using the map SILC400. The values of the angles ϑ, ϕ, φ2 , and φ3 are in degrees.

80

edec /10−2 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76

ϑ 67.9 67.8 67.8 67.7 67.9 67.9 67.7 67.7

ϕ 5.8 174.4 354.3 185.5 50.1 130.0 310.3 229.6

φ2 31.4 33.9 121.2 121.1 121.1 138.9 131.9 131.9

φ3 18.7 69.4 69.8 19.5 157.9 111.9 112.1 157.6

B0 /nG 1.88 1.89 1.89 1.89 1.88 1.88 1.89 1.90

TABLE IV: As in Table III, but for the map WILC3YR. edec /10−2 0.74 0.74 0.74 0.74 0.74 0.74 0.75 0.75

ϑ 66.4 66.3 66.3 66.3 66.5 66.5 66.2 66.2

ϕ 4.5 175.5 355.6 184.4 57.3 122.7 303.3 236.7

φ2 36.7 36.8 110.4 110.4 140.0 140.0 130.8 130.8

φ3 30.8 72.3 72.2 31.1 160.5 112.0 112.5 160.0

B0 /nG 1.84 1.84 1.84 1.84 1.83 1.83 1.85 1.85

TABLE V: As in Table III, but for the map TCM3YR. edec /10−2 0.78 0.78 0.78 0.78 0.78 0.78 0.78 0.78

ϑ 69.2 69.2 69.2 69.2 69.3 69.3 69.1 69.1

ϕ 0.5 179.4 359.5 180.6 49.0 131.1 311.6 228.4

φ2 89.3 89.0 96.6 96.9 140.5 140.4 130.6 130.6

φ3 46.2 52.5 52.3 49.5 154.5 116.4 116.6 154.3

B0 /nG 1.94 1.94 1.94 1.94 1.93 1.93 1.94 1.94

b

60

è SILC 400 ò WILC3YR TCM3YR ç Axis of Evil

40

ò 20è

ò è

ò è

èòòè

òè ç

òè

èò

ç 0 0

50

100

150

200

250

300

350

l FIG. 2: Numerical solutions of Eq. (28) obtained by using the three maps in Table I. Note that b = 90◦ −ϑ, and l = ϕ, where (b, l) are the galactic coordinates. The two open circles at (b, l) ≃ (20◦ , 130◦ ) and (b, l) ≃ (10◦ , 110◦ ) define the direction of the axis of evil determined in Ref. [44] for the 5-year WMAP temperature sky maps in V-band and W-band, respectively.

maps, are given in Tables III, IV, and V, respectively. In Appendix B, we will show that the system (28) (for ℓ = 2) admits 8 independent solutions. Moreover we observe that, for each independent solution (edec, ϑ, ϕ, φ2 , φ3 ) shown in the tables, there exists another one given by (edec , π − ϑ, ϕ ± π, φ2 , φ3 ), where we take the plus sign if ϕ < π and the minus sign if ϕ > π. Looking at Table I and Eq. (33), we see that the values of coefficients |a20 | and |a21 | are much smaller than QI . Moreover, comparing Eq. (33) with Table I, and Eq. (34) with Table II, we deduce, roughly speaking, that the value of QI is about twice the value of Q. Assuming QI ≫ |a20 |, |a21 |, and QI ≃ 2Q, we will show in Appendix B that approximate solutions for edec, ϑ, and ϕ are √ √ 15(5 + 3 73) 2 edec ≃ QI , (35) 24 " # √ √ 6(3 73 − 5) π 1 , (36) ϑ ≃ − arcsin 2 2 79 ϕ ≃ 0,

5π 7π π 3π , , π, , , 2π. 4 4 4 4

(37)

From Eqs. (35) and (33)-(34) we get To solve Eq. (28) for ℓ = 2, we need the observed values of the a2m ’s. We use the cleaned CMB temperature fluctuation map of the 3-year WMAP data obtained by using an improved internal linear combination method as galactic foreground subtraction technique. In particular, we adopt the three maps SILC400 [40], WILC3YR [41], and TCM3YR [42]. For completeness, we report in Table I the values of a2m corresponding to these maps. Numerical solutions of Eq. (28), referring to the three

edec ≃ 0.8 × 10−2 ,

(38)

which inserted in Eq. (20) gives B0 ≃ 2 nG.

(39)

As one can check, the approximate solutions Eqs. (35), (36), and (37) are quite close to the numerical values. In Fig. 2, we plot the numerical solutions ϑ versus ϕ which, defining a symmetry axis, singles out a preferred

8 direction in the CMB map. We adopt the so-called galactic coordinates system characterized by the galactic latitude, b, and galactic longitude, l. In our notation, the angle b corresponds to b = 90◦ − ϑ while l = ϕ. Equation (36) gives b ≃ 20◦ ,

(40)

while Eq. (37) can be written as l ≃ 0◦ , 45◦ , 135◦ , 180◦, 225◦ , 315◦, 360◦ .

(41)

From the above equations and Fig. 2, we see that the galactic latitude of the symmetry axis is remarkably independent on the adopted CMB temperature fluctuation map, while galactic longitude is poorly constrained. It is interesting to observe that, according to Ref. [44], the axis of evil points toward (b, l)AE ≃ (20◦ , 130◦ )

(42)

if one takes into account the 5-year WMAP data in the V-band, and toward (b, l)AE ≃ (10◦ , 110◦ )

(43)

if one uses the W-band 5-year WMAP data, so that it seems to be very close to the direction (b, l) ≃ (20◦ , 135◦ )

(44)

defined by the symmetry axis in our model. It is important to stress, however, that the apparent alignment of the above two axes could be just an accidental fact. Indeed, the axis of evil is determined by a statistical correlation (alignment and planarity) between the quadrupole and octupole modes, while the presence of a planar cosmic magnetic field affects only the quadrupole moment in the CMB radiation (this is true to the lowest order in the eccentricity parameter considered in this paper; to higher orders, modifications to the standard value of octupole intensity can occur, but they are vanishingly small). In any case, it is worth noting that there are already independent indications of a symmetry axis in the large-scale geometry of the Universe as, for example, those coming from the analysis of spiral galaxies in the Sloan Digital Sky Survey [45]. IV. CONCLUSIONS Our knowledge of the Universe has greatly improved after the discovery of the cosmic microwave background radiation. The observation and the analysis of the temperature fluctuations in this relic radiation, especially in the last decade, has confirmed at a surprising level of accuracy the canonical theoretical model describing

the evolution of the Universe, the standard cosmological model. This is the celebrated hot big-bang model, “equipped” with inflation, dark energy, and cold dark matter. Although these last three ingredients have not yet been framed in a definitely theoretical model of particle physics, from an operative point of view, this cosmological landscape appears satisfactory. However, the ubiquitous existence of large-scale correlated magnetic fields and some presumed anomaly discovered in the spectrum of the CMB radiation, together with the statement of fact that the evolution of the Universe (at least) before the Planck era cannot be understood without a self-consistent theory of quantum gravity, justify the study of both alternative cosmologies (such as anisotropic models of the Universe) and models of particle physics beyond the standard model. Indeed, in this paper, we analyzed the effects on standard cosmology of including in the photon sector of the standard model a Lorentz-violating term, the “Kosteleck´ y term” LK = − 14 (kF )αβµν F αβ F µν . Remarkably, we found that its presence is responsible for a superadiabatic amplification of magnetic vacuum fluctuations during de Sitter inflation. Moreover, the amplified magnetic field possesses a planar symmetry at large cosmological scales if the external tensor (kF )αβµν , which parametrizes Lorentz violation, assumes a particular (asymmetric) form. This peculiar magnetic field could account for the presence galactic magnetic fields and induces a small anisotropization of the Universe at cosmological scales (which can be described by a Bianchi I model). This, in turn, could explain the low quadrupole anomaly in the CMB radiation. Finally, we would like to stress that anisotropic cosmological models, such as that induced by the planar cosmic magnetic field discussed in this paper, are expected to induce a certain amount of polarization in the CMB radiation [46]. Although it is beyond the aim of this paper to discuss this kind of problems, we notice that it has been shown [47] that the 3-year WMAP data on large-scale polarization could be in agreement with an anisotropic model of a universe of Bianchi I type. An appropriate analysis of CMB polarization, resulting from the cosmological model presented in this paper, is in progress.

APPENDIX A In this appendix, we study the evolution during reheating, radiation and matter eras, of the magnetic field produced, according to the mechanism discussed in Sec. II, during de Sitter inflation. After inflation, the Universe enters into the so-called reheating phase, during which the energy of the inflaton is converted into ordinary matter. The reheating phase ends at the temperature TRH which is less than M and constrained as TRH . 108 GeV [48]. Moreover,

9 CMB analysis requires M . 10−2 mPl [19], otherwise it would be too much of a gravitational waves relic abundance, and also one must impose that TRH & 1GeV, so that the predictions of big-bang nucleosynthesis (BBN) are not spoiled [19, 49]. It is worth noting that the condition kF ≫ a6 is certainly fulfilled during inflation and reheating if kF ≫ a6RH , where aRH = a(TRH ). Since aRH ∼ T0 /TRH , where T0 ∼ 10−13 GeV is the actual temperature [50], we have 10−126 . a6RH . 10−78 for 1GeV . TRH . 108 GeV. Taking into account that the most stringent bound on ||(kF )αβµν || is 10−37 (as discussed in Sec. II) we get, for a wide range of allowed values of kF , that kF ≫ a6 during inflation and reheating. This, in turn means, according to Eqs. (6), (7) and the first Bianchi identity, that a superadiabatic amplification of magnetic vacuum fluctuations during these eras takes place. During reheating, in particular, taking into account that η ∝ a1/2 , we have B ∝ a9/2 . After reheating, the Universe enters the radiationdominated era. In this era, as well as in the subsequent matter era, the effects of the conducting primordial plasma are important when studying the evolution of a magnetic field. They are taken into account by adding to the electromagnetic Lagrangian the source term j µAµ [19]. Here, the external current j µ , expressed in terms of the electric field, has the form j µ = (0, σc E), where σc is the conductivity. Plasma effects introduce, in the right-hand side of Eq. (4), the extra term −aσc (a2 Ei ). In this case, it easy to see that modes well outside the horizon (assuming that kF ≫ a6 ) evolve as Z 1 7 4 (45) dη a σc . E ∝ a exp − 2kF R Approximating dη a7 σc with η a7 σc and using aη ∼ H −1 , we get a6 σc E ∝ a4 exp − . (46) 2kF H 2

In radiation era H ∼ T /mPl [50] and, for temperature much greater than the electron mass, the conductivity is approximately given by σc ∼ T /α [19], where α is the fine structure constant and T the temperature. Then we get " # 7 T∗ 4 E ∝ a exp − , (47) T where T∗ ∼ GeV

10−57 kF

1/7

.

(48)

This means that for T & T∗ we have E ∝ a4 (which in turn gives B ∝ a5 since η ∝ a in radiation era), while for

T . T∗ the electric field is dissipated, so the magnetic field evolves adiabatically, B ∝ a−2 . [We have assumed that kF ≫ a6 from the end of reheating until T∗ . As is easy to verify taking into account Eq. (48), this assumption is certainly satisfied in our case.] Finally, evolving along the lines discussed above the inflation-produced magnetic field from the time of horizon crossing until today, and taking into account Eq. (8), we easily recover Eq. (9). APPENDIX B In this appendix, we solve the system of Eqs. (28) for ℓ = 2. The numerical values of the a2m ’s are listed in I Table I, while the aA 2m ’s and a2m ’s are given by Eqs. (31) and (32), respectively. Since it is always possible to choose a20 real, we take φ1 = 0. Moreover, the temperature anisotropy is real so that we have a2,−m = (−1)m (a2m )∗ , aA 2,−m = ∗ m I ∗ I (−1)m (aA 2m ) , and a2,−m = (−1) (a2m ) . Therefore, the system of Eqs. (28) reduces to r √ π π 2 √ a20 = [1 + 3 cos(2ϑ)] edec + QI , (49) 3 6 5 r r π π cosϕ sin(2ϑ) e2dec + cosφ2 QI , Re[a21 ] = − 30 3 (50) r r π π Im[a21 ] = sinϕ sin(2ϑ) e2dec + sinφ2 QI , 30 3 (51) r r π π cosϕ sin2 ϑ e2dec + cosφ3 QI , Re[a22 ] = 30 3 (52) r r π π Im[a22 ] = − sinϕ sin2 ϑ e2dec + sinφ3 QI , 30 3 (53) where QI is given by Eq. (33). We see that these equations form a system of 5 transcendental equations containing 5 unknown parameters: edec , ϑ, ϕ, φ2 , and φ3 . Solving Eq. (49) with respect to ϑ we get two independent solutions:

where

e π − ϑ}, e ϑ = {ϑ,

(54)

! √ √ √ 1 5 15 a20 − 5 5πQI − 3π e2dec e √ . (55) ϑ = arccos 2 3 3π e2dec

Squaring Eqs. (52) and (53), adding side by side, and then solving with respect to ϕ, we obtain 8 independent solutions: ϕ = {ϕ e± , π − ϕ e± , 2π − ϕ e± , π + ϕ e± },

(56)

10 where

and

ϕ e± =

!

p 1 αγ ± β α2 + β 2 − γ 2 , arccos 2 α2 + β 2 r

2π α=− Re[a22 ] sin2 ϑ e2dec , 15 r 2π Im[a22 ] sin2 ϑ e2dec , β=− 15 π π sin4 ϑ e4dec + |a22 |2 − Q2I . γ= 30 3

(57)

(58) (59) (60)

By dividing side by side Eqs. (51) and (50), and solving with respect to φ2 , we get √ √ 30 Im[a21 ] − π sinϕ sin(2ϑ) e2dec √ tanφ2 = . √ 30 Re[a21 ] + π sinϕ sin(2ϑ) e2dec

(61)

Inserting the above equation in Eq. (66) gives Eq. (35), while its solution is indeed Eq. (36). Assuming that the main contribution to Q comes from the a22 , and that Re[a22 ] ≃ Im[a22 ], we can write r 5π Q. (68) Re[a22 ] ≃ Im[a22 ] ≃ − 12 Consequently, inserting Eqs. (66)-(68) in Eq. (57), we get ! r 1 1 2 ϕ e± ≃ arccos q ± (69) −q , 2 2

where

The same procedure applied to Eqs. (53) and (51) results in √ √ 30 Im[a22 ] + π sin(2ϕ) sin2 ϑ e2dec tanφ3 = √ . (62) √ 30 Re[a22 ] − π sin(2ϕ) sin2 ϑ e2dec Finally, by squaring Eqs. (50) and (51), and adding side by side, we get e4dec − 2c e2dec − d = 0,

Inserting the above expression in Eqs. (54)-(55), assuming that QI ≫ |a20 |, and after same algebraic manipulation, we get √ √ 6(3 73 − 5) | sin(2ϑ)| ≃ . (67) 79

(63)

where we have defined r 30 (Re[a21 ] cos ϕ − Im[a21 ] sin ϕ) csc(2ϑ), c(ϕ, ϑ) = π (64) 3 d(ϕ, ϑ) = 10 Q2I − |a21 |2 csc2(2ϑ). (65) π We observe that the couple (ϑ, ϕ) can assume 16 different values, according to Eqs. (54) and (56). Inserting these values in Eqs. (63)-(65) we arrive at 16 different equations for edec. It is straightforward to verify that only 8 of these are “independent,” in the sense that, given a solution (edec , ϑ, ϕ) of one of the independent equations, then (edec , π−ϑ, ϕ±π) is a solutions of one of the “dependent” ones (we must take the plus sign if ϕ < π and the minus sign if ϕ > π). The 8 independent equations can be solved numerically and their solutions are presented in Table III, IV, and V. Finally, we derive the approximate solutions (35)-(36). To this end,√ we may formally solve Eq. (63) to get e2dec = c ± c2 + d. If QI ≫ |a21 | then d ≫ c2 , and we obtain √ e2dec ≃ 10 QI | csc(2ϑ)|. (66)

√ √ √ Q 30( 73 − 5) QI q= (1 + 5 73) + 120 5760 Q QI QI Q ≃ 0.15 . (70) + 0.40 Q QI Observing that the value of the quadrupole according to the ΛCDM standard model is, approximately, twice the value of the observed one, QI ≃ 2Q [compare Eq. (33) with Table I, and Eq. (34) with Table II], we roughly get q ≃ 1/2 and, accordingly, ϕ e+ ≃ 0 and ϕ e− ≃ π/4. Inserting these values in Eq. (56), we recover Eq. (37).

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