Nonlinear electrodynamics and CMB polarization

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JCAP03(2011)033

ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Nonlinear electrodynamics and CMB

polarization

Herman J. Mosquera Cuestaa,b,c,d and G. Lambiasee,f

aDepartmento de Fısica Universidade Estadual Vale do Acarau,Avenida da Universidade 850, Campus da Betania, CEP 62.040-370, Sobral, Ceara, Brazil

bInstituto de Cosmologia, Relatividade e Astrofısica (ICRA-BR),Centro Brasileiro de Pesquisas Fısicas,Rua Dr. Xavier Sigaud 150, CEP 22290-180, Urca Rio de Janeiro, RJ, Brazil

cInternational Center for Relativistic Astrophysics Network (ICRANet),International Coordinating Center, Piazzalle della Repubblica 10, 065112, Pescara, Italy

dInternational Institute for Theoretical Physics and Mathematics Einstein-Galilei,via Bruno Buozzi 47, 59100 Prato, Italy

eDipartimento di Fisica “E.R. Caianiello”, Universita di Salerno,84081 Baronissi (SA), Italy

f INFN, Sezione di Napoli,Napoli, Italy

E-mail: herman@icra.it, lambiase@sa.infn.it

Received July 1, 2010Revised January 23, 2011Accepted February 8, 2011Published March 22, 2011

Abstract. Recently WMAP and BOOMERanG experiments have set stringent constraintson the polarization angle of photons propagating in an expanding universe: ∆α = (−2.4 ±1.9). The polarization of the Cosmic Microwave Background radiation (CMB) is reviewedin the context of nonlinear electrodynamics (NLED). We compute the polarization angle ofphotons propagating in a cosmological background with planar symmetry. For this purpose,we use the Pagels-Tomboulis (PT) Lagrangian density describing NLED, which has the formL ∼ (X/Λ4)δ−1 X, where X = 1

4FαβFαβ , and δ the parameter featuring the non-Maxwelliancharacter of the PT nonlinear description of the electromagnetic interaction. After lookingat the polarization components in the plane orthogonal to the (x)-direction of propagationof the CMB photons, the polarization angle is defined in terms of the eccentricity of theuniverse, a geometrical property whose evolution on cosmic time (from the last scatteringsurface to the present) is constrained by the strength of magnetic fields over extragalacticdistances.

Keywords: CMBR polarisation, CMBR theory, extragalactic magnetic fields, cosmic mag-netic fields theory

c© 2011 IOP Publishing Ltd and SISSA doi:10.1088/1475-7516/2011/03/033

JCAP03(2011)033

Contents

1 Introduction 1

2 Minimally coupling gravity to nonlinear electrodynamics 2

3 Cosmological setting: Space-time with planar symmetry −→ universe ec-

centricity −→ polarization angle 4

3.1 Space-time anisotropy and magnetic energy density evolution 4

3.2 Space-time eccentricity and polarization angle 5

3.3 Eccentricity evolution on cosmic time 6

3.4 Constraints on parameter Λ from extragalactic B strengths in an ellipsoidalUniverse 7

4 Light propagation in NLED and birefringence 8

5 Stokes parameters, rotated CMB spectra and constraints on parameter Λ 9

5.1 Estimative of Λ 10

6 Discussion and closing remarks 11

1 Introduction

Modifications to the standard (Maxwell) electrodynamics were proposed in the literature inorder to avoid infinite physical quantities from theoretical descriptions of electromagneticinteractions. Born and Infeld [1], for instance, proposed a model in which the infinite selfenergy of point particles (typical of Maxwell’s electrodynamics) are removed by introducingan upper limit on the electric field strength, and by considering the electron as an electricparticle with finite radius. Along this line, other models of nonlinear electrodynamics (NLED)Lagrangians were proposed by Plebanski, who also showed that Born-Infeld model satisfiesphysically acceptable requirements [2]. Consequences of nonlinear electrodynamics have beenstudied in many contexts, such a, for example, cosmological models [3], black holes andwormhole physics [4, 5], primordial magnetic fields in the Universe [8, 9, 11], gravitationalbaryogenesis [8], and astrophysics [12, 17].

In this paper we investigate the CMB polarization of photons described by nonlinearelectrodynamics. We compute the polarization angle of photons propagating in an expand-ing Universe, by considering in particular cosmological models with planar symmetry. Thepolarization angle does depend on the parameter characterizing the nonlinearity of electro-dynamics, which will be constrained by making use of the recent data from WMAP andBOOMERANG. This kind of investigations has received a lot of interest because they rep-resent a probe of models beyond the standard model, which may violate the fundamentalsymmetries such as CPT and Lorentz invariance [13, 14]. In what follows we will followthe main lines of the paper on “Cosmological CPT violation, baryo/leptogenesis and CMBpolarization” by Li-Xia-Li-Zhang [6].

– 1 –

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2 Minimally coupling gravity to nonlinear electrodynamics

The action of (nonlinear) electrodynamics coupled minimally to gravity is

S =1

2κ

∫

d4x√−gR +

1

4π

∫

d4x√−gL(X,Y ) , (2.1)

where κ = 8πG, L is the Lagrangian of nonlinear electrodynamics depending on the invariantX = 1

4FµνFµν = −2(E2 − B2) and Y = 14Fµν

∗Fµν , where Fµν ≡ ∇µAν − ∇νAµ, and∗Fµν = ǫµνρσFρσ is the dual bivector, and ǫαβγδ = 1

2√−g

εαβγδ , with εαβγδ the Levi-Civita

tensor (ε0123 = +1).The equations of motion are [9]

∇µ (−LXFµν − LY∗Fµν) = 0 , (2.2)

where LX = ∂L/∂X and LY = ∂L/∂Y ,

∇µFνλ + ∇νFλµ + ∇λFµν = 0 . (2.3)

After a swift grasp on this set of equations one realizes that is difficult to find solutionsin closed form of these equations. Therefore to study the effects of nonlinear electrodynamics,we confine ourselves to consider the abelian Pagels-Tomboulis theory [16], proposed as aneffective model of low energy QCD. The Lagrangian density of this theory involves only theinvariant X in the form

L(X) = −(

X2

Λ8

)

δ−12

X = −γXδ , (2.4)

where γ (or Λ) and δ are free parameters that, with appropriate choice, reproduce the wellknown Lagrangian already studied in the literature. γ has dimensions [energy]4(1−δ).

Following Kunze [9], the energy momentum tensor corresponding to the Lagrangiandensity L(X) is given by

Tµν =1

4π

[

LXgαβFµαFβν + gµνL]

(2.5)

and the decomposition of the electromagnetic tensor with respect to a fundamental observerwith 4-velocity uµ (uµuµ = −1)

Fµν = 2E[µuν] − ηµνςτuςBτ . (2.6)

The electric and magnetic fields are therefore given by Eµ = Fµνuν and Bµ =12ηµνκλuνF κλ (ηαβγδ =

√−g εαβγδ). The energy density turns out to be

ρ = Tµνuµuν = − 1

8π

L

X

[

(2δ − 1)EαEα + BαBα]

. (2.7)

The positivity of ρ (weak energy condition) imposes, in general, the constraint on δ. Forthe Lagrangian (2.4) one gets δ ≥ 1

2 . However, this condition can be relaxed because weshall consider cosmological scenarios in which the electric field is zero, and only the magneticfields survive (this is justified by the fact that during the radiation dominated era the plasmaeffects induce a rapid decay of the electric field, whereas magnetic field remains (see thepaper by Turner and Widrow in [23])).

– 2 –

JCAP03(2011)033

The equation of motion for the Pagels-Tomboulis theory follows from eq. (2.2) withY = 0

∇µFµν = −(δ − 1)∇µX

XFµν . (2.8)

In terms of the potential vector Aµ, and imposing the Lorentz gauge ∇µAµ = 0, eq. (2.8)becomes

∇µ∇µAν + RνµAµ = −(δ − 1)

∇µX

X(∇µAν −∇νAµ) , (2.9)

where the Ricci tensor Rνµ appears because the relation [∇µ,∇ν ]A

ν = −RµµAµ.

To proceed onward, we apply the geometrical optics approximation. This means thatthe scales of variation of the electromagnetic fields are smaller than the cosmological scaleswe consider next. In this approximation, the 4-vector Aµ(x) can be written as [18]

Aµ(x) = Re[

(aµ(x) + ǫbµ(x) + . . .)eiS(x)/ǫ]

(2.10)

with ǫ ≪ 1 so that the phase S/ǫ varies faster than the amplitude. By defining the wavevector kµ = ∇µS, which defines the direction of the photon propagation, one finds that thegauge condition implies kµaµ = 0 and kµbµ = 0. It turns out to be convenient to introducethe normalized polarization vector εµ so that the vector aµ can be written as

aµ(x) = A(x)εµ , εµεµ = 1 . (2.11)

As a consequence of (2.11), one also finds kµεµ = 0, i.e. the wave vector is orthogonal to thepolarization vector.

By making use of eq. (2.10), one obtains

∇µX

X=

2ikµ

ǫ[1 + Ω(ǫ)] , (2.12)

where

Ω ≡ǫik[αaβ]∇µk[αaβ] + O(ǫ2)

−(k[αaβ])2 + iǫ2k[αaβ](∇[αaβ] + ik[αaβ]) + O(ǫ2).

To leading order in ǫ, the term depending on the Ricci tensors can be neglected in (2.9).Inserting (2.10) into (2.9) and collecting all terms proportional to ǫ−2 and ǫ−1, one obtains

1

ǫ2: kµkµaσ = 2(δ − 1)kµk[µaσ] , (2.13)

1

ǫ: 2kµ∇µaσ + aσ∇µkµ = −2(δ − 1)kµ∇[µaσ] . (2.14)

Taking into account the gauge condition kµaµ = 0, the first equation implies

(2δ − 1)k2 = 0 → kµkµ = 0 , provided δ 6= 1

2, (2.15)

hence photons propagate along null geodesics. Multiplying eq. (2.14) by aσ, and using (2.11)one obtains

1

2∇µkµ = −δkµ∇µ ln A + (δ − 1)kµεσ∇σεµ ,

so that eq. (2.14) can be recast in the form

kµ∇µεσ =δ − 1

δΥσ (2.16)

whereΥσ ≡ kµ [∇σεµ − (ερ∇ρε

µ)εσ ] . (2.17)

– 3 –

JCAP03(2011)033

3 Cosmological setting: Space-time with planar symmetry −→ universe

eccentricity −→ polarization angle

3.1 Space-time anisotropy and magnetic energy density evolution

Let us consider cosmological models with planar symmetry, i.e., having a similar scale factoron the first two spatial coordinates. The most general line-element of a geometry with plane-symmetry is [19]

ds2 = dt2 − b2(dx2 + dy2) − c2dz2 , (3.1)

where b(t) and c(t) are the scale factors, which are normalized in order that b(t0) = 1 = c(t0)at the present time t0. As eq. (3.1) shows, the symmetry is on the (xy)-plane. The coherenttemperature and polarization patterns produced in homogeneous but anisotropic cosmologicalmodels (Bianchi type with a Friedman-Robertson-Walker limit has been studied in [15]).

The Christoffel symbols corresponding to the metric (3.1) are

Γ011 = Γ0

22 = bb , Γ033 = cc , (3.2)

Γ101 = Γ2

02 =b

b, Γ3

03 =c

c.

The dot stands for derivative with respect to the cosmic time t.

To make an estimate on the parameter δ, we have to investigate in more detail thegeometry with planar symmetry. As pointed out by Campanelli-Cea-Tedesco (CCT) in [20],the most general tensor consistent with the geometry (3.1) is

T µν = diag(ρ,−p‖,−p‖,−p⊥) = T µ

(I)ν + T µ(A)ν ,

in which T µ(I)ν = diag(ρ,−p,−p,−p) is the standard isotropic energy-momentum tensor de-

scribing matter, radiation, or cosmological constant, and T µ(A)ν = diag(ρA,−pA,−pA. − pA)

represents the anisotropic contribution which induces the planar symmetry, and can be givenby a uniform magnetic field, a cosmic string, a domain wall.1 In what follows, we shall con-sider a Universe matter dominated (p = 0) with planar symmetry generated by a uniformmagnetic field B(t).

Magnetic fields have been observed in galaxies, galaxy clusters, and extragalactic struc-tures [22], and it is assumed that they may have a primordial origin [9, 23]. Due to thehigh conductivity of the primordial plasma, the magnetic field evolves as B(t) ∼ b−2 be-ing frozen into the plasma [21, 22] (see below). Denoting with ρB the magnetic fielddensity, the energy-momentum tensor for a uniform magnetic field can be written asT µ

(B) ν = ρBdiag(1,−1,−1,−1).

According to (2.7), we find that the energy density of the magnetic field is given by

ρB =B2

8π

(

B2

2Λ4

)δ−1

. (3.3)

1Notice that the cases discussed in [20], i.e. the magnetic fields or the topological defects as responsible ofthe planar symmetry of the Universe, allow to explain the anomaly concerning the low quadruple amplitudeprobed by WMAP. The interesting analysis performed by CCT leads to the conclusion that an eccentricityat the decoupling of the order of e ∼ 10−2 is indeed able to explain the drastic reduction in the quadrupleanisotropy without affecting higher multiples of the angular power spectrum of the temperature anisotropies.

– 4 –

JCAP03(2011)033

The evolution law of the energy density ρB is given by [10]

ρB +4

3ΘρB + 16πσabΠ

ab = 0 , (3.4)

where Θ is the volume expansion (contraction) scalar, σab is the shear, and Πab the anisotropicpressure of the fluid. In a highly conducting medium we still have with good approximationB ∼ b−2 provided that anisotropies can be neglected (this means that we neglect radiativeeffect of the primordial fluid).

3.2 Space-time eccentricity and polarization angle

We shall assume that photons propagate along the (positive) x-direction, so that kµ =(k0, k1, 0, 0) [7]. Gauge invariance assures that the polarization vector of photons has onlytwo independent components, which are orthogonal to the direction of the photons motion.Therefore, we are only interested in how the components of the polarization vector (Iµ 2 andIµ3) change. It then follows that Υσ defined in (2.17) assumes the form

Υσ = −k0

[

δσ2 b

bε2 + δσ3 c

cε3 +

(

bb(ε2)2 + cc(εc)2)

εσ

]

(3.5)

The components of Υσ given by (3.5) vanish in the case of a Friedman-Robertson-Walkergeometry.

By defining the affine parameter λ which measures the distance along the line-element,kµ ≡ dxµ/dλ, one obtains that ε2 and ε3 satisfy the following geodesic equation (fromeq. (2.16))

dε2

dλ+

b

bk0ε2 = −δ − 1

δk0

[

b

b+ bb(ε2)2 + cc(ε3)2

]

ε2

dε3

dλ+

c

ck0ε3 = −δ − 1

δk0

[

c

c+ bb(ε2)2 + cc(ε3)2

]

ε3

These equations can be further simplified if one observes that k0 = dt/dλ

1

k0D ln(bε2) =

d ln(bε2)

dt= −δ − 1

δ

(

− b

b+

c

c

)

(cε3)2 , (3.6)

1

k0D ln(cε3) =

d ln(cε3)

dt= −δ − 1

δ

(

− c

c+

b

b

)

(bε2)2 . (3.7)

whereD ≡ kµ∇µ . (3.8)

Moreover, the difference of the Hubble expansion rate b/b and c/c can be written as

b

b− c

c=

1

2(1 − e2)

de2

dt(3.9)

where we have introduced the eccentricity

e(t) =

√

1 −(c

b

)2. (3.10)

– 5 –

JCAP03(2011)033

The polarization angle α is defined as α = arctan[(cε3)/(bε2)]. Its time evolution isgoverned by equation

Dα − δ − 1

2δk0

(

b

b− c

c

)

[(bε2)3 + (cε3)3] = 0 . (3.11)

However, eqs. (3.6) and (3.7) implies that both bε2 and cε3 evolves as Ai + (δ − 1)fi(t),i = 2, 3 , where fi(t) is a function of time and Ai are constant of integration. Therefore, toleading order (δ − 1) eq. (3.11) reads

Dα − δ − 1

δ

K

2k0

(

b

b− c

c

)

+ O((δ − 1)2) = 0 . (3.12)

where K = A2 + A3.To compute the rotation of the polarization angle, one needs to evaluate α at two distinct

instants. In the cosmological context that we are considering is assumed that the referencetime t corresponds to the moment in which photons are emitted from the last scatteringsurface, and the instant t0 corresponds to the present time. One, therefore, gets2

∆α = α(t) − α(t0) =δ − 1

4δKe2(z) , (3.13)

where we have used e(t0) = 0 because of the normalization condition b(t0) = c(t0) = 1 andlog(1 − e2) ∼ −e2.

Notice that for δ = 1 or e2 = 0 there is no rotation of the polarization angle, as expected.Moreover, in the case in which photons propagate along the direction z-direction, so thatka = (ω0, 0, 0, k), we find that the NLED have no effects as concerns to the rotation of thepolarization angle.

As arises from (3.13), ∆α vanishes in the limit δ = 1, so that no rotation of thepolarization angle occurs in the standard electrodynamics, even if the background is describedby a geometry with planar symmetry. Moreover, even if δ 6= 1, ∆α still vanishes for anisotropic and homogeneous cosmology described by the Friedman-Robertson-Walker elementline (b = c) ds2 = dt2 − b2(dx2 + dy2 + dz2), because in such a case the eccentricity vanishes(this agrees with the fact that for this background the components of Υσ, eq. (3.5), are zero).

3.3 Eccentricity evolution on cosmic time

The time evolution of the eccentricity is determined from the Einstein field equations

1

1 − e2

d(ee)

dt+ 3Hb(ee) +

(ee)2

(1 − e2)2= 2κρB , (3.14)

where Hb = b/b.

2Preliminary calculations [38] performed in terms of the electromagnetic field Fµν and of time evolution ofthe Stokes parameters I,Q, U, V (this approach is alternative to one presented in the section II of the paperwhere the analysis is performed in terms of the 4-potential Aµ) yield again the result (3.13). Calculationsshow that the total flux I is not the same along the three spatial directions, as expected owing to the differentexpansion of the Universe along the x, y and z directions. Moreover the time evolution of the Stokes parametersturns out to be a mixture of each others, which reduce to standard results as δ = 1. The polarization angleis defined as 2α = arctan(U/Q).

– 6 –

JCAP03(2011)033

It is extremely difficult to exactly solve this equation. We shall therefore assume thatthe e2-terms can be neglected. Since b(t) ∼ t2/3 during the matter-dominated era, eq. (3.14)implies

e2(z) = 18Fδ(z)Ω(0)B , (3.15)

where we used 1 + z = b(t0)/b(t), e(t0) = 0, and

Fδ ≡ 3

(9 − 8δ)(4δ − 3)− 2 − 3(1 + z)4δ−3

(9 − 8δ)(4δ − 3)+ 2(1 + z)

32 . (3.16)

Ω(0)B is the present energy density ratio

Ω(0)B =

ρB

ρcr=

B2(t0)

8πρcr

(

B2(t0)

2Λ4

)δ−1

≃ 10−11

(

B(t0)

10−9G

)2(B2(t0)

2Λ4

)δ−1

, (3.17)

with ρcr = 3H2b (t0)/κ = 8.1h210−47 GeV4 (h = 0.72 is the little-h constant), and B(t0) is the

present magnetic field amplitude.From eq. (3.13) then follows

∆α =δ − 1

4δK e2(zdec) . (3.18)

where e(zdec)2 the eccentricity (3.15) evaluated at the decoupling z = 1100.

3.4 Constraints on parameter Λ from extragalactic B strengths in an ellipsoidal

Universe

To make an estimate on the parameter δ, we need the order of amplitude of the presentmagnetic field strength B(t0). In this respect, observations indicate that there exist, incluster of galaxies, magnetic fields with field strength (10−7 − 10−6) G on 10 kpc - 1 Mpcscales, whereas in galaxies of all types and at cosmological distances, the order of magnitudeof the magnetic field strength is ∼ 10−6 G on (1-10) kpc scales. The present acceptedestimations is [22]3

B(t0) . 10−9 G . (3.20)

Moreover, for an ellipsoidal Universe the eccentricity satisfies the relation 0 ≤ e2 < 1. Thecondition e2 > 0 means Fδ > 0, with Fδ defined in (3.16). The function Fδ given by eq. (3.16)is represented in figure 1. Clearly the allowed region where Fδ is positive does depend onthe redshift z. On the other hand, the condition e2 < 1 poses constraints on the magneticfield strength. By requiring e2 < 10−1 (in order that our approximation to neglect e2-termsin (3.14) holds), from eqs. (3.15)–(3.17) it follows

B(t0) . 9 × 10−8G . (3.21)

3The bound (3.19) is consistent with the estimation on the present value of the magnetic field strengthobtained from Big Bang Nucleosynthesis (BBN). As before pointed out, the magnetic fields scales as B ∼ b−2

where the scale factor does depend on the temperature T and on the total number of effectively masslessdegree of freedom g∗S as b ∝ g

−1/3∗S T−1 [26]. The upper bound on the magnetic field at the epoch of the

BBN is given by [27] B(TBBN) . 1011G, where according to the standard cosmology TBBN = 109K≃ 0.1MeV.Referred to the present value of the magnetic field, the bound on B(TBBN) becomes [20, 24]

B(t0) =

„

g∗S(T0)

g∗S(TBBN)

«2/3 „

T0

TBBN

«2

B(TBBN) . 6 × 10−7G , (3.19)

where T0 = T (t0) ≃ 2.35 × 10−4eV and g∗S(TBBN) ≃ g∗S(T0) ≃ 3.91 [26].

– 7 –

JCAP03(2011)033

0.85 0.90 0.95 1.00∆

72 400

72 600

72 800

73 000

F∆

z=1100

1.05 1.10 1.15 1.20∆

-400 000

-200 000

200 000

400 000

600 000

800 000

1.´106F∆

z=1100

Figure 1. In this plot is represented Fδ vs δ for δ ≤ 1 (upper plot) and δ ≥ 1 (lower plot). Thecondition that the eccentricity is positive follows for Fδ > 0.

It must also be noted that such magnetic fields does not affect the expansion rate of theuniverse and the CMB fluctuations because the corresponding energy density is negligiblewith respect to the energy density of CMB.

4 Light propagation in NLED and birefringence

In this section we discuss the modification of the light velocity (birefringence effect) for themodel of nonlinear electrodynamics L(X,Y ). We shall follow the paper [32] (see also [2, 33]),in which is studied the propagation of wave in local nonlinear electrodynamics by making useof the Fresnel equation for the wave covectors kµ. The latter are related to phase velocity

v of the wave propagation by the relation ki =k0

vki, where ki are the components of the

unit 3-covector. Thus, in what follows we confine ourselves to the phase velocity. It isstraightforward to show that for the models under consideration (2.4) the group velocity isalways greater or equal to the phase velocity [32].

The main result in ref. [32] corresponds to the optic metric tensors

gµν1 = X gµν + (Y +

√Y −XZ)tµν , (4.1)

gµν2 = X gµν + (Y −

√Y −XZ)tµν , (4.2)

which describe the effect of birefringent light propagation in a generic model for nonlinearelectrodynamics. The quantities X , Y, and Z are related to the derivatives of the LagrangianL(X,Y ) with respect to the invariant X and Y , and tµν = FµαF ν

α.

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JCAP03(2011)033

For our model, expressed by eq. (2.4), the quantities X , Y, and Z are given by

X ≡ K21 =

γ2δ2

4X2(δ−1) , Y ≡ K1K2 =

γ2δ2

4(δ − 1)X2(δ−1)−1 , Z = 0 ,

where K1 = 4∂L

∂Xand K2 = 8

∂2L

∂X2, while the metrics (4.1) and (4.2) are

gµν1 = K1(K1g

µν + 2K2tµν) , gµν

2 = K21gµν .

As a consequence, birefringence is present in our model. This means that some photons prop-agate along the standard null rays of spacetime metric gµν , whereas other photons propagatealong rays null with respect to the optical metric K1g

µν + 2K2tµν .

The velocities of the light wave can be derived by using the light cone equations (effectivemetric)

gµν1 kµkν = 0 and gµν

2 kµkν = 0 .

It is worthwhile to report the general expression for the average value of the velocity scalar [32]

〈v2〉 = 1 +4

3

T 00(Y + Zt00)

X + 2Yt00 + Z(t00)2+

2

3S2 2Y2 −XZ + Z(t00)2 + 2YZt00

[X + 2Yt00 + Z(t00)2]2

where T 00 = −t00 + X = (E2γ + B2

γ)/2 (t00 = −E2γ), and S2 = δµνt0µt0ν , where S = E × B

is the energy flux density. The subscript γ is introduced for distinguishing the photon fieldfrom the magnetic background. The value of the mean velocity has been derived averagingover the directions of propagation and polarization. For our model, we get

〈v2〉 ≃ 1 + (δ − 1)R + (δ − 1)2S , (4.3)

R ≡ 4

3

T 00

4X + 2(δ − 1)t00, S =

4

3

S2

[4X + 2(δ − 1)t00]2

The high accuracy of optical experiments in laboratories requires tiny deviations fromstandard electrodynamics. This condition is satisfied provided |δ − 1| ≪ 1. Moreover, thereare two aspects related to (4.3):

• The average velocity does depend on (only) the parameter δ, so that γ or Λ in ourmodel can be fixed independently. This task is addressed in the next section.

• Because R is positive, one has to demand that δ − 1 < 0 in order that v2 < 1.

The above considerations hold for flat spacetime, and can be straightforwardly generalizedto the case of curved space time [32].

5 Stokes parameters, rotated CMB spectra and constraints on parameter

Λ

The propagation of photons can be described in terms of the Stokes parameters I, Q, U ,and V . The parameters Q and V can be decomposed in gradient-like (G) and a curl-like(C) components [30] (G and C are also indicated in literature as E and B), and characterizethe orthogonal modes of the linear polarization (they depend on the axes where the linear

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JCAP03(2011)033

polarization are defined, contrarily to the physical observable I and V which are independenton the choice of coordinate system).

The polarization G and C and the temperature (T ) are crucial because they allow tocompletely characterize the CMB on the sky. If the Universe is isotropic and homogeneousand the electrodynamics is the standard one, then the TC and GC cross-correlations powerspectrum vanish owing to the absence of the cosmological birefringence. In presence of thelatter, on the contrary, the polarization vector of each photons turns out to be rotated bythe angle ∆α, giving rise to TC and GC correlations.

Using the expression for the power spectra CXYl ∼

∫

dk[k2∆X(t0)∆Y (t0)], where X,Y =T,G,C and ∆X are the polarization perturbations whose time evolution is controlled by theBoltzman equation, one can derive the correlation for T , G and C in terms of ∆α [14]4

C ′TCl = CTC

l sin 2∆α , C ′TGl = CTG

l cos 2∆α , (5.1)

C ′GCl =

1

2

(

CGGl − CCC

l

)

sin 4∆α , (5.2)

C ′GGl = CGG

l cos2 2∆α + CCCl sin2 2∆α , (5.3)

C ′CCl = CCC

l cos2 2∆α + CGGl sin2 2∆α . (5.4)

The prime indicates the rotated quantities. Notice that the CMB temperature powerspectrum remains unchanged under the rotation.

Experimental constraints on ∆α have been put from the observation of CMB polariza-tion by WMAP and BOOMERanG [14, 25].

∆α = (−2.4 ± 1.9) = [−0.0027π,−0.0238π] . (5.5)

The combination of eqs. (5.5) and (3.18), and the laboratory constraints |δ− 1| ≪ 1 allow toestimate Λ.

5.1 Estimative of Λ

To estimate Λ we shall write

B = 10−9+b G b . 2 , (5.6)

Fδ = 2z3/2 z = 1100 ≫ 1 . (5.7)

The bound (5.5) can be therefore rewritten in the form

10−3

A . |δ − 1| .10−2

A , (5.8)

where

A ≡ 9K

14FδΩ

(0)B ≃ K 10−6+2b

[

0.24 × 10−56+2b

(

GeV

Λ

)4]δ−1

. (5.9)

The condition |δ − 1| ≪ 1 requires A ≫ 1. It turns out convenient to set

A = 10a , a > O(1) . (5.10)

4Notice that in ref. [29] the analysis did not include the rotation of the CMB spectra, and in ref. [30] theanalysis focused on only the TC and TG modes. Other approximated approaches to discuss the rotation anglecan be found in refs. [28, 31].

– 10 –

JCAP03(2011)033

22 026.2 22 026.2 22 026.3 22 026.3K

10

15

20

25

Log@LGeVD

b = 0

Lx = 19

a=4

∆-1 = - 10-7

2980.91 2980.92 2980.93 2980.94K

5

10

15

20

25

30

35

Log@LGeVD

b = 1

Lx = 19

a = 4

∆-1 = - 10-7

442 413 442 413 442 414 442 414K

-1000

-500

500

1000

Log@LGeVD

b = 0

Lx = 19

a = 7

∆-1 = - 10-10

59 874.1 59 874.1 59 874.1 59 874.1 59 874.2 59 874.2K

-1000

1000

2000

3000

Log@LGeVD

b = 1

Lx = 19

a = 7

∆-1 = - 10-10

Figure 2. Λ vs K for different values of the parameter δ − 1, a and b. The parameter a is related tothe range in which δ − 1 varies, i.e. −10−3−a . δ − 1 . −10−2−a, while b parameterizes the magneticfield strength B = 10−9+b G. The red-shift is z = 1100. Plot refers to Planck scale Λ = 10Λx GeV,with Λx=Pl = 19. Similar plots can be also obtained for GUT (ΛGUT = 16) and EW (ΛEW = 3)scales.

From eqs. (5.9) and (5.10) it then follows

Λ = 10−14+b/2

[

1

K10a−2b+6

]− 14(δ−1)

GeV , (5.11)

or equivalently

Log

[

Λ

GeV

]

=

(

−14 +b

2

)

+(−1)

4(δ − 1)[a − 2b + 6 − LogK] . (5.12)

The constant K can now be determined to fix the characteristic scale Λ. WritingΛ = 10Λx GeV, where Λx=P l = 19, ΛGUT = 16 and ΛEW = 3 for the Planck, GUT andelectroweak (EW) scales, respectively, eq. (5.12) yields

K = 10a−2b+6−ǫ , ǫ ≡ 4(δ − 1)Λx

14 − b/2≪ 1 . (5.13)

In figure 2 is plotted Log(Λ/GeV) vs K for fixed values of the parameters a, b and δ − 1.Similar plots can be derived for GUT and EW scales

6 Discussion and closing remarks

In conclusion, in this paper we have calculated, in the framework of the nonlinear electro-dynamics, the rotation of the polarization angle of photons propagating in a Universe withplanar symmetry. We have found that the rotation of the polarization angle does depend on

– 11 –

JCAP03(2011)033

the parameter δ, which characterizes the degree of nonlinearity of the electrodynamics. Thisparameter can be constrained by making use of recent data from WMAP and BOOMERang.Results show that the CMB polarization signature, if detected by future CMB observations,would be an important test in favor of models going beyond the standard model, includingthe nonlinear electrodynamics.

Some comments are in order. In our investigation we have assumed that the planar-symmetry is induced by a magnetic field. This is not the unique case. In fact, a planargeometry can also be induced by topological defects, such as cosmic string (cs) or domainwall (dw) [20]. In such a case, one has [20]

e2

∣

∣

∣

∣

∣

dw

=2

7Ω

(0)dw

[

3

(1 + z)2+ 4(1 + z)3/2 − 7

]

, (6.1)

and

e2

∣

∣

∣

∣

∣

cs

=4

5Ω(0)

cs

[

3

(1 + z)+ 2(1 + z)3/2 − 5

]

, (6.2)

where Ω(0)(dw,cs) are the present energy densities, in units of critical density, of the domain wall

and cosmic string. At the decoupling, one obtains

e2(zdec)

∣

∣

∣

∣

∣

dw

≃ 10−4 Ω(0)dw

5 × 10−7, (6.3)

and

e2(zdec)

∣

∣

∣

∣

∣

cs

≃ 10−4 Ω(0)cs

4 × 10−7. (6.4)

The analysis leading to determine the bounds on δ from CMB polarization goes along theline above traced.

Moreover, a complete analysis of the planar-geometry is required to fix the parameterδ. From a side, in our calculations in fact we have assumed that the Universe is matterdominated. A more precise calculation should require to use (to solve (3.14)) the relation

t =1

H0

∫ z

0

1 + z√

Ωm(1 + z)3 + ΩΛ

dz , (6.5)

where Ωm = 0.3, ΩΛ = 0.7 and H0 = 72 km sec−1 Mpc−1 (z = 1100). From the other side,a complete study of the eq. (3.14) is necessary in order to put stringent constraints on theparameter δ.

As closing remark, we would like to point out that the approach to analyze the CMBpolarization in the context of NLED that we have presented above can also be applied todiscuss the extreme-scale alignments of quasar polarization vectors [36], a cosmic phenomenonthat was discovered by Hutsemekers [34] in the late 1990’s, who presented paramount evidencefor very large-scale coherent orientations of quasar polarization vectors (see also Hutsemekersand Lamy [35]).5 As far as the authors of the present paper are awared of, the issue has

5Hutsemekers and Lamy, and collaborators, have presented, in a long series of papers (not all cited here)published over the period 1998 to 2008, a tantamount evidence that the alignment of quasar polarizationvectors is a factual cosmological enigma deserving to be properly addressed in the framework of the standardmodel of cosmology. The papers quoted here are intended to call to the attention of attentive readers theparamount evidence presenting this cosmic phenomenon.

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JCAP03(2011)033

remained as an open cosmological connundrum, with a few workers in the field having focusedtheir attention on to those intriguing observations. Nonetheless, we quote “en passant” thatin a recent paper [37] Hutsemekers et al. discussed the possibility of such phenomenon to beunderstood by invoking very light pseudoscalar particles mixing with photons. They claimedthat the observations of a sample of 355 quasars with significant optical polarization presentstrong evidence that quasar polarization vectors are not randomly oriented over the sky, asnaturally expected. Those authors suggest that the phenomenon can be understood in termsof a cosmological-size effect, where the dichroism and birefringence predicted by a mixingbetween photons and very light pseudoscalar particles within a background magnetic fieldcan qualitatively reproduce the observations. They also point out at a finding indicating thatcircular polarization measurements could help constrain their mechanism.

Since cosmic magnetic fields have a typical strength of ∼ 10−7 − 10−8 G, on average,for a characteristic distance scale of 10-30 Mpc, it is our view that such phenomenology canbe understood in the framework of a nonlinear description of photon propagation (NLED)over cosmic background magnetic fields and the use of a planar symmetry for the space-time.Specifically, phenomena involving light propagation as dichroism and birefringence can beinscribed on to the framework of Heisenberg-Euler NLED, which predicts the occurrence ofbirefringence on cosmological distance scales. We plan to present such analysis in a forth-coming communication [38].

Acknowledgments

The authors express their gratitude to the referee for relevant comments. The authors alsothank Prof. Xin-min Zhang, and Dr. Mingzhe Li for reading the manuscript and theirsuggestions. H.J.M.C. thanks ICRANet International Coordinating Center, Pescara, Italyfor hospitality during the early stages of this work. G.L. acknowledges the financial supportof MIUR through PRIN 2006 Prot. 1006023491−003, and of research funds provided by theUniversity of Salerno. H.J.M.C. is fellow of the Ceara State Foundation for the Developmentof Science and Technology (FUNCAP), Fortaleza, Ceara, Brazil.

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