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2. J ournal of Cosmology and Astroparticle Physics
An IOP and SISSA journal
Nonlinear electrodynamics and CMB
polarization
Herman J. Mosquera Cuestaa,b,c,d and G. Lambiasee,f
JCAP03(2011)033
a Departmento de F´ısica Universidade Estadual Vale do Acara´,
u
Avenida da Universidade 850, Campus da Betˆnia, CEP 62.040-370, Sobral, Cear´, Brazil
a a
b Instituto de Cosmologia, Relatividade e Astrof´ ısica (ICRA-BR),
Centro Brasileiro de Pesquisas F´ ısicas,
Rua Dr. Xavier Sigaud 150, CEP 22290-180, Urca Rio de Janeiro, RJ, Brazil
c International Center for Relativistic Astrophysics Network (ICRANet),
International Coordinating Center, Piazzalle della Repubblica 10, 065112, Pescara, Italy
d International Institute for Theoretical Physics and Mathematics Einstein-Galilei,
via Bruno Buozzi 47, 59100 Prato, Italy
e Dipartimento di Fisica “E.R. Caianiello”, Universit` di Salerno,
a
84081 Baronissi (SA), Italy
f INFN, Sezione di Napoli,
Napoli, Italy
E-mail: herman@icra.it, lambiase@sa.infn.it
Received July 1, 2010
Revised January 23, 2011
Accepted February 8, 2011
Published March 22, 2011
Abstract. Recently WMAP and BOOMERanG experiments have set stringent constraints
on 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 reviewed
in the context of nonlinear electrodynamics (NLED). We compute the polarization angle of
photons propagating in a cosmological background with planar symmetry. For this purpose,
we use the Pagels-Tomboulis (PT) Lagrangian density describing NLED, which has the form
1
L ∼ (X/Λ4 )δ−1 X, where X = 4 Fαβ F αβ , and δ the parameter featuring the non-Maxwellian
character of the PT nonlinear description of the electromagnetic interaction. After looking
at the polarization components in the plane orthogonal to the (x)-direction of propagation
of the CMB photons, the polarization angle is defined in terms of the eccentricity of the
universe, a geometrical property whose evolution on cosmic time (from the last scattering
surface to the present) is constrained by the strength of magnetic fields over extragalactic
distances.
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

3. 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
JCAP03(2011)033
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 ellipsoidal
Universe 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 in
order to avoid infinite physical quantities from theoretical descriptions of electromagnetic
interactions. Born and Infeld [1], for instance, proposed a model in which the infinite self
energy of point particles (typical of Maxwell’s electrodynamics) are removed by introducing
an upper limit on the electric field strength, and by considering the electron as an electric
particle with finite radius. Along this line, other models of nonlinear electrodynamics (NLED)
Lagrangians were proposed by Plebanski, who also showed that Born-Infeld model satisfies
physically acceptable requirements [2]. Consequences of nonlinear electrodynamics have been
studied in many contexts, such a, for example, cosmological models [3], black holes and
wormhole physics [4, 5], primordial magnetic fields in the Universe [8, 9, 11], gravitational
baryogenesis [8], and astrophysics [12, 17].
In this paper we investigate the CMB polarization of photons described by nonlinear
electrodynamics. We compute the polarization angle of photons propagating in an expand-
ing Universe, by considering in particular cosmological models with planar symmetry. The
polarization 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 and
BOOMERANG. 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 fundamental
symmetries such as CPT and Lorentz invariance [13, 14]. In what follows we will follow
the main lines of the paper on “Cosmological CPT violation, baryo/leptogenesis and CMB
polarization” by Li-Xia-Li-Zhang [6].
–1–

4. 2 Minimally coupling gravity to nonlinear electrodynamics
The action of (nonlinear) electrodynamics coupled minimally to gravity is
1 √ 1 √
S= d4 x −gR + d4 x −gL(X, Y ) , (2.1)
2κ 4π
where κ = 8πG, L is the Lagrangian of nonlinear electrodynamics depending on the invariant
X = 1 Fµν F µν = −2(E2 − B2 ) and Y = 4 Fµν ∗ F µν , where F µν ≡ µ Aν − ν Aµ , and
4
1
∗ F µν = µνρσ F αβγδ = √ 1
ρσ is the dual bivector, and 2 −g
εαβγδ , with εαβγδ the Levi-Civita
tensor (ε0123 = +1).
JCAP03(2011)033
The equations of motion are [9]
µν
µ (−LX F − LY ∗ F µν ) = 0 , (2.2)
where LX = ∂L/∂X and LY = ∂L/∂Y ,
µ Fνλ + ν Fλµ + λ Fµν = 0. (2.3)
After a swift grasp on this set of equations one realizes that is difficult to find solutions
in 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 an
effective model of low energy QCD. The Lagrangian density of this theory involves only the
invariant X in the form δ−1
X2 2
L(X) = − X = −γX δ , (2.4)
Λ8
where γ (or Λ) and δ are free parameters that, with appropriate choice, reproduce the well
known Lagrangian already studied in the literature. γ has dimensions [energy]4(1−δ) .
Following Kunze [9], the energy momentum tensor corresponding to the Lagrangian
density L(X) is given by
1
Tµν = LX gαβ Fµα Fβν + gµν L (2.5)
4π
and the decomposition of the electromagnetic tensor with respect to a fundamental observer
with 4-velocity uµ (uµ uµ = −1)
ˆ ˆ
Fµν = 2E[µ uν] − ηµνςτ uς B τ . (2.6)
ˆ ˆ
The electric and magnetic fields are therefore given by Eµ = Fµν uν and Bµ =
1 ν F κλ (η √
2 ηµνκλ u αβγδ = −g εαβγδ ). The energy density turns out to be
1 L ˆ ˆ ˆ ˆ
ρ = Tµν uµ uν = − (2δ − 1)Eα E α + Bα B α . (2.7)
8π X
The positivity of ρ (weak energy condition) imposes, in general, the constraint on δ. For
the Lagrangian (2.4) one gets δ ≥ 1 . However, this condition can be relaxed because we
2
shall consider cosmological scenarios in which the electric field is zero, and only the magnetic
fields survive (this is justified by the fact that during the radiation dominated era the plasma
effects induce a rapid decay of the electric field, whereas magnetic field remains (see the
paper by Turner and Widrow in [23])).
–2–

5. The equation of motion for the Pagels-Tomboulis theory follows from eq. (2.2) with
Y =0
µν µX
µF = −(δ − 1) F µν . (2.8)
X
In terms of the potential vector Aµ , and imposing the Lorentz gauge µ Aµ = 0, eq. (2.8)
becomes
µ ν µX
µ A + Rνµ Aµ = −(δ − 1) ( µ Aν − ν Aµ ) , (2.9)
X
where the Ricci tensor Rνµ appears because the relation [ µ , ν ]Aν = −Rµ Aµ .
µ
To proceed onward, we apply the geometrical optics approximation. This means that
the scales of variation of the electromagnetic fields are smaller than the cosmological scales
JCAP03(2011)033
we consider next. In this approximation, the 4-vector Aµ (x) can be written as [18]
Aµ (x) = Re (aµ (x) + bµ (x) + . . .)eiS(x)/ (2.10)
with 1 so that the phase S/ varies faster than the amplitude. By defining the wave
vector kµ = µ S, which defines the direction of the photon propagation, one finds that the
gauge condition implies kµ aµ = 0 and kµ bµ = 0. It turns out to be convenient to introduce
the normalized polarization vector εµ so that the vector aµ can be written as
aµ (x) = A(x)εµ , εµ εµ = 1 . (2.11)
As a consequence of (2.11), one also finds kµ εµ = 0, i.e. the wave vector is orthogonal to the
polarization vector.
By making use of eq. (2.10), one obtains
µX 2ikµ
= [1 + Ω( )] , (2.12)
X
where
ik[α aβ] [α aβ] + O( 2 )
µk
Ω≡ .
−(k[α aβ] )2 + i 2k[α aβ] ( [α aβ] + ik[α aβ] ) + O( 2 )
To leading order in , 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 −2 and −1 , one obtains
1
2
: kµ kµ aσ = 2(δ − 1)kµ k[µ aσ] , (2.13)
1 µ σ
: 2kµ a + aσ µk
µ
= −2(δ − 1)kµ [µ σ]
a . (2.14)
Taking into account the gauge condition kµ aµ = 0, the first equation implies
1
(2δ − 1)k2 = 0 → kµ kµ = 0 , ,provided δ = (2.15)
2
hence photons propagate along null geodesics. Multiplying eq. (2.14) by aσ , and using (2.11)
one obtains
1 µ µ σ µ
µ k = −δk µ ln A + (δ − 1)kµ ε σε ,
2
so that eq. (2.14) can be recast in the form
δ−1 σ
kµ µε
σ
= Υ (2.16)
δ
where
Υσ ≡ kµ [ σ µ
ε − (ερ µ σ
ρ ε )ε ] . (2.17)
–3–

6. 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 factor
on the first two spatial coordinates. The most general line-element of a geometry with plane-
symmetry is [19]
ds2 = dt2 − b2 (dx2 + dy 2 ) − c2 dz 2 , (3.1)
where b(t) and c(t) are the scale factors, which are normalized in order that b(t0 ) = 1 = c(t0 )
JCAP03(2011)033
at the present time t0 . As eq. (3.1) shows, the symmetry is on the (xy)-plane. The coherent
temperature and polarization patterns produced in homogeneous but anisotropic cosmological
models (Bianchi type with a Friedman-Robertson-Walker limit has been studied in [15]).
The Christoffel symbols corresponding to the metric (3.1) are
˙
Γ0 = Γ0 = bb , Γ0 = cc ,
˙ (3.2)
11 22 33
˙
b c
˙
Γ1 = Γ2 = , Γ3 = .
01 02 03
b 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 the
geometry with planar symmetry. As pointed out by Campanelli-Cea-Tedesco (CCT) in [20],
the most general tensor consistent with the geometry (3.1) is
µ µ
T µ = diag(ρ, −p , −p , −p⊥ ) = T(I)ν + T(A)ν ,
ν
µ
in which T(I)ν = diag(ρ, −p, −p, −p) is the standard isotropic energy-momentum tensor de-
µ
scribing matter, radiation, or cosmological constant, and T(A)ν = diag(ρA , −pA , −pA . − pA )
represents the anisotropic contribution which induces the planar symmetry, and can be given
by 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 uniform
magnetic 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 the
high 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 field
density, the energy-momentum tensor for a uniform magnetic field can be written as
µ
T(B) ν = ρB diag(1, −1, −1, −1).
According to (2.7), we find that the energy density of the magnetic field is given by
δ−1
B2 B2
ρB = . (3.3)
8π 2Λ4
1
Notice that the cases discussed in [20], i.e. the magnetic fields or the topological defects as responsible of
the planar symmetry of the Universe, allow to explain the anomaly concerning the low quadruple amplitude
probed by WMAP. The interesting analysis performed by CCT leads to the conclusion that an eccentricity
at the decoupling of the order of e ∼ 10−2 is indeed able to explain the drastic reduction in the quadruple
anisotropy without affecting higher multiples of the angular power spectrum of the temperature anisotropies.
–4–

7. The evolution law of the energy density ρB is given by [10]
4
ρB + ΘρB + 16πσab Πab = 0 ,
˙ (3.4)
3
where Θ is the volume expansion (contraction) scalar, σab is the shear, and Πab the anisotropic
pressure of the fluid. In a highly conducting medium we still have with good approximation
B ∼ b−2 provided that anisotropies can be neglected (this means that we neglect radiative
effect of the primordial fluid).
3.2 Space-time eccentricity and polarization angle
JCAP03(2011)033
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 only
two independent components, which are orthogonal to the direction of the photons motion.
ˆ
Therefore, we are only interested in how the components of the polarization vector (Iµ 2 and
ˆ
Iµ3) change. It then follows that Υ σ defined in (2.17) assumes the form
˙
b c
˙ ˙
Υσ = −k0 δσ2 ε2 + δσ3 ε3 + bb(ε2 )2 + cc(εc )2 εσ
˙ (3.5)
b c
The components of Υσ given by (3.5) vanish in the case of a Friedman-Robertson-Walker
geometry.
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 (from
eq. (2.16))
dε2 ˙
b ˙
δ−1 0 b
+ k0 ε2 = − k ˙
+ bb(ε2 )2 + cc(ε3 )2 ε2
˙
dλ b δ b
dε3 c˙ δ−1 0 c
˙ ˙
+ k0 ε3 = − k + bb(ε2 )2 + cc(ε3 )2 ε3
˙
dλ c δ c
These equations can be further simplified if one observes that k0 = dt/dλ
1 d ln(bε2 ) δ−1 ˙
b c˙
0
D ln(bε2 ) = =− − + (cε3 )2 , (3.6)
k dt δ b c
1 d ln(cε3 ) δ−1 c b
˙ ˙
0
D ln(cε3 ) = =− − + (bε2 )2 . (3.7)
k dt δ c b
where
D ≡ kµ µ. (3.8)
˙
Moreover, the difference of the Hubble expansion rate b/b and c/c can be written as
˙
˙
b c˙ 1 de2
− = 2 ) dt
(3.9)
b c 2(1 − e
where we have introduced the eccentricity
c 2
e(t) = 1− . (3.10)
b
–5–

8. The polarization angle α is defined as α = arctan[(cε3 )/(bε2 )]. Its time evolution is
governed by equation
δ−1 0 ˙
b c ˙
Dα − k − [(bε2 )3 + (cε3 )3 ] = 0 . (3.11)
2δ b c
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, to
leading order (δ − 1) eq. (3.11) reads
˙
JCAP03(2011)033
δ−1 K 0 b c ˙
Dα − k − + O((δ − 1)2 ) = 0 . (3.12)
δ 2 b c
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 reference
time t corresponds to the moment in which photons are emitted from the last scattering
surface, and the instant t0 corresponds to the present time. One, therefore, gets2
δ−1
∆α = α(t) − α(t0 ) = Ke2 (z) , (3.13)
4δ
where we have used e(t0 ) = 0 because of the normalization condition b(t0 ) = c(t0 ) = 1 and
log(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 that
¯
ka = (ω0 , 0, 0, k), we find that the NLED have no effects as concerns to the rotation of the
polarization angle.
As arises from (3.13), ∆α vanishes in the limit δ = 1, so that no rotation of the
polarization angle occurs in the standard electrodynamics, even if the background is described
by a geometry with planar symmetry. Moreover, even if δ = 1, ∆α still vanishes for an
isotropic and homogeneous cosmology described by the Friedman-Robertson-Walker element
line (b = c) ds2 = dt2 − b2 (dx2 + dy 2 + dz 2 ), 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 d(ee)
˙ (ee)2
˙
2 dt
+ 3Hb (ee) +
˙ = 2κρB , (3.14)
1−e (1 − e2 )2
˙
where Hb = b/b.
2
Preliminary calculations [38] performed in terms of the electromagnetic field Fµν and of time evolution of
the Stokes parameters I, Q, U, V (this approach is alternative to one presented in the section II of the paper
where the analysis is performed in terms of the 4-potential Aµ ) yield again the result (3.13). Calculations
show that the total flux I is not the same along the three spatial directions, as expected owing to the different
expansion of the Universe along the x, y and z directions. Moreover the time evolution of the Stokes parameters
turns out to be a mixture of each others, which reduce to standard results as δ = 1. The polarization angle
is defined as 2α = arctan(U/Q).
–6–

9. It is extremely difficult to exactly solve this equation. We shall therefore assume that
the e2 -terms
can be neglected. Since b(t) ∼ t2/3 during the matter-dominated era, eq. (3.14)
implies
(0)
e2 (z) = 18Fδ (z)ΩB , (3.15)
where we used 1 + z = b(t0 )/b(t), e(t0 ) = 0, and
3 3(1 + z)4δ−3 3
Fδ ≡ −2− + 2(1 + z) 2 . (3.16)
(9 − 8δ)(4δ − 3) (9 − 8δ)(4δ − 3)
(0)
ΩB is the present energy density ratio
JCAP03(2011)033
δ−1 2 δ−1
(0) ρB B 2 (t0 ) B 2 (t0 ) −11 B(t0 ) B 2 (t0 )
ΩB = = 10 , (3.17)
ρcr 8πρcr 2Λ4 10−9 G 2Λ4
with ρcr = 3Hb (t0 )/κ = 8.1h2 10−47 GeV4 (h = 0.72 is the little-h constant), and B(t0 ) is the
2
present magnetic field amplitude.
From eq. (3.13) then follows
δ−1
∆α = K e2 (zdec ) . (3.18)
4δ
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 present
magnetic field strength B(t0 ). In this respect, observations indicate that there exist, in
cluster of galaxies, magnetic fields with field strength (10−7 − 10−6 ) G on 10 kpc - 1 Mpc
scales, whereas in galaxies of all types and at cosmological distances, the order of magnitude
of the magnetic field strength is ∼ 10−6 G on (1-10) kpc scales. The present accepted
estimations is [22]3
B(t0 ) 10−9 G . (3.20)
Moreover, for an ellipsoidal Universe the eccentricity satisfies the relation 0 ≤ e2 < 1. The
condition 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 on
the redshift z. On the other hand, the condition e2 < 1 poses constraints on the magnetic
field strength. By requiring e2 < 10−1 (in order that our approximation to neglect e2 -terms
in (3.14) holds), from eqs. (3.15)–(3.17) it follows
B(t0 ) 9 × 10−8 G . (3.21)
3
The bound (3.19) is consistent with the estimation on the present value of the magnetic field strength
obtained 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 massless
−1/3
degree of freedom g∗S as b ∝ g∗S T −1 [26]. The upper bound on the magnetic field at the epoch of the
BBN is given by [27] B(TBBN ) 1011 G, where according to the standard cosmology TBBN = 109 K 0.1MeV.
Referred to the present value of the magnetic field, the bound on B(TBBN ) becomes [20, 24]
„ «2/3 „ «2
g∗S (T0 ) T0
B(t0 ) = B(TBBN ) 6 × 10−7 G , (3.19)
g∗S (TBBN ) TBBN
where T0 = T (t0 ) 2.35 × 10−4 eV and g∗S (TBBN ) g∗S (T0 ) 3.91 [26].
–7–

10. F∆
73 000
72 800 z 1100
72 600
72 400
∆
0.85 0.90 0.95 1.00
JCAP03(2011)033
F∆
1. 106
800 000
600 000 z 1100
400 000
200 000
∆
1.05 1.10 1.15 1.20
200 000
400 000
Figure 1. In this plot is represented Fδ vs δ for δ ≤ 1 (upper plot) and δ ≥ 1 (lower plot). The
condition 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 the
universe and the CMB fluctuations because the corresponding energy density is negligible
with 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 the
model 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 use
of the Fresnel equation for the wave covectors kµ . The latter are related to phase velocity
k0 ˆ ˆ
v of the wave propagation by the relation ki = ki , where ki are the components of the
v
unit 3-covector. Thus, in what follows we confine ourselves to the phase velocity. It is
straightforward to show that for the models under consideration (2.4) the group velocity is
always greater or equal to the phase velocity [32].
The main result in ref. [32] corresponds to the optic metric tensors
µν √
g1 = X gµν + (Y + Y − X Z)tµν , (4.1)
µν µν
√ µν
g2 = X g + (Y − Y − X Z)t , (4.2)
which describe the effect of birefringent light propagation in a generic model for nonlinear
electrodynamics. The quantities X , Y, and Z are related to the derivatives of the Lagrangian
L(X, Y ) with respect to the invariant X and Y , and tµν = F µα F ν .
α
–8–

11. For our model, expressed by eq. (2.4), the quantities X , Y, and Z are given by
2 γ 2 δ2 2(δ−1) γ 2 δ2
X ≡ K1 = X , Y ≡ K1 K2 = (δ − 1)X 2(δ−1)−1 , Z = 0,
4 4
∂L ∂2L
where K1 = 4 and K2 = 8 , while the metrics (4.1) and (4.2) are
∂X ∂X 2
µν µν
g1 = K1 (K1 gµν + 2K2 tµν ) , g2 = K1 gµν .
2
As a consequence, birefringence is present in our model. This means that some photons prop-
JCAP03(2011)033
agate along the standard null rays of spacetime metric gµν , whereas other photons propagate
along rays null with respect to the optical metric K1 gµν + 2K2 tµν .
The velocities of the light wave can be derived by using the light cone equations (effective
metric)
µν µν
g1 kµ kν = 0 and g2 kµ kν = 0 .
It is worthwhile to report the general expression for the average value of the velocity scalar [32]
4 T 00 (Y + Zt00 ) 2 2Y 2 − X Z + Z(t00 )2 + 2YZt00
v2 = 1 + + S2
3 X + 2Yt00 + Z(t00 )2 3 [X + 2Yt00 + Z(t00 )2 ]2
where T 00 = −t00 + X = (E2 + B2 )/2 (t00 = −E2 ), and S2 = δµν t0µ t0ν , where S = E × B
γ γ γ
is the energy flux density. The subscript γ is introduced for distinguishing the photon field
from the magnetic background. The value of the mean velocity has been derived averaging
over the directions of propagation and polarization. For our model, we get
v2 1 + (δ − 1)R + (δ − 1)2 S , (4.3)
4 T 00 4 S2
R≡ , S=
3 4X + 2(δ − 1)t00 3 [4X + 2(δ − 1)t00 ]2
The high accuracy of optical experiments in laboratories requires tiny deviations from
standard electrodynamics. This condition is satisfied provided |δ − 1| 1. Moreover, there
are two aspects related to (4.3):
• The average velocity does depend on (only) the parameter δ, so that γ or Λ in our
model 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 v 2 < 1.
The above considerations hold for flat spacetime, and can be straightforwardly generalized
to 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 characterize
the orthogonal modes of the linear polarization (they depend on the axes where the linear
–9–

12. polarization are defined, contrarily to the physical observable I and V which are independent
on the choice of coordinate system).
The polarization G and C and the temperature (T ) are crucial because they allow to
completely characterize the CMB on the sky. If the Universe is isotropic and homogeneous
and the electrodynamics is the standard one, then the T C and GC cross-correlations power
spectrum vanish owing to the absence of the cosmological birefringence. In presence of the
latter, on the contrary, the polarization vector of each photons turns out to be rotated by
the angle ∆α, giving rise to T C and GC correlations.
Using the expression for the power spectra ClXY ∼ dk[k2 ∆X (t0 )∆Y (t0 )], where X, Y =
T, G, C and ∆X are the polarization perturbations whose time evolution is controlled by the
Boltzman equation, one can derive the correlation for T , G and C in terms of ∆α [14]4
JCAP03(2011)033
Cl T C = ClT C sin 2∆α , Cl T G = ClT G cos 2∆α , (5.1)
1
Cl GC = C GG − ClCC sin 4∆α , (5.2)
2 l
Cl GG = ClGG cos2 2∆α + ClCC sin2 2∆α , (5.3)
CC
Cl = ClCC 2
cos 2∆α + ClGG sin2 2∆α . (5.4)
The prime indicates the rotated quantities. Notice that the CMB temperature power
spectrum 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 to
estimate Λ.
5.1 Estimative of Λ
To estimate Λ we shall write
B = 10−9+b G b 2, (5.6)
3/2
Fδ = 2z z = 1100 1. (5.7)
The bound (5.5) can be therefore rewritten in the form
10−3 10−2
|δ − 1| , (5.8)
A A
where
4 δ−1
9K (0) −6+2b −56+2b GeV
A≡ Fδ ΩB K 10 0.24 × 10 . (5.9)
14 Λ
The condition |δ − 1| 1 requires A 1. It turns out convenient to set
A = 10a , a > O(1) . (5.10)
4
Notice that in ref. [29] the analysis did not include the rotation of the CMB spectra, and in ref. [30] the
analysis focused on only the TC and TG modes. Other approximated approaches to discuss the rotation angle
can be found in refs. [28, 31].
– 10 –

13. Log GeV Log GeV
∆ 1 10 7 35 7
∆ 1 10
25 b 0
30 b 1
a 4
25 a 4
20
x 19 20
x 19
15 15
10
10
5
K K
22 026.2 22 026.2 22 026.3 22 026.3 2980.91 2980.92 2980.93 2980.94
Log GeV Log GeV
1000 ∆ 1 10 10 ∆ 1 10 10
JCAP03(2011)033
b 0 3000 b 1
500 a 7 a 7
2000
x 19
K
442 413 442 413 442 414 442 414 1000
x 19
500 K
59 874.1 59 874.1 59 874.1 59 874.1 59 874.2 59 874.2
1000 1000
Figure 2. Λ vs K for different values of the parameter δ − 1, a and b. The parameter a is related to
the range in which δ − 1 varies, i.e. −10−3−a δ − 1 −10−2−a , while b parameterizes the magnetic
field strength B = 10−9+b G. The red-shift is z = 1100. Plot refers to Planck scale Λ = 10Λx GeV,
with Λx=P l = 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
1
− 4(δ−1)
−14+b/2 1 a−2b+6
Λ = 10 10 GeV , (5.11)
K
or equivalently
Λ b (−1)
Log = −14 + + [a − 2b + 6 − LogK] . (5.12)
GeV 2 4(δ − 1)
The constant K can now be determined to fix the characteristic scale Λ. Writing
Λ = 10Λx GeV, where Λx=P l = 19, ΛGUT = 16 and ΛEW = 3 for the Planck, GUT and
electroweak (EW) scales, respectively, eq. (5.12) yields
4(δ − 1)Λx
K = 10a−2b+6− , ≡ 1. (5.13)
14 − b/2
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 with
planar symmetry. We have found that the rotation of the polarization angle does depend on
– 11 –

14. the parameter δ, which characterizes the degree of nonlinearity of the electrodynamics. This
parameter 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, including
the 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 planar
geometry can also be induced by topological defects, such as cosmic string (cs) or domain
wall (dw) [20]. In such a case, one has [20]
2 (0) 3
JCAP03(2011)033
e2 = Ωdw + 4(1 + z)3/2 − 7 , (6.1)
7 (1 + z)2
dw
and
4 3
e2 = Ω(0)
cs + 2(1 + z)3/2 − 5 , (6.2)
5 (1 + z)
cs
(0)
where are the present energy densities, in units of critical density, of the domain wall
Ω(dw,cs)
and cosmic string. At the decoupling, one obtains
(0)
Ωdw
e2 (zdec ) 10−4 , (6.3)
5 × 10−7
dw
and
(0)
Ωcs
e2 (zdec ) 10−4 . (6.4)
4 × 10−7
cs
The analysis leading to determine the bounds on δ from CMB polarization goes along the
line above traced.
Moreover, a complete analysis of the planar-geometry is required to fix the parameter
δ. From a side, in our calculations in fact we have assumed that the Universe is matter
dominated. A more precise calculation should require to use (to solve (3.14)) the relation
z
1 1+z
t= dz , (6.5)
H0 0 Ωm (1 + z)3 + ΩΛ
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 the
parameter δ.
As closing remark, we would like to point out that the approach to analyze the CMB
polarization in the context of NLED that we have presented above can also be applied to
discuss the extreme-scale alignments of quasar polarization vectors [36], a cosmic phenomenon
that was discovered by Hutsemekers [34] in the late 1990’s, who presented paramount evidence
for very large-scale coherent orientations of quasar polarization vectors (see also Hutsemekers
and Lamy [35]).5 As far as the authors of the present paper are awared of, the issue has
5
Hutsemekers 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 polarization
vectors is a factual cosmological enigma deserving to be properly addressed in the framework of the standard
model of cosmology. The papers quoted here are intended to call to the attention of attentive readers the
paramount evidence presenting this cosmic phenomenon.
– 12 –

15. remained as an open cosmological connundrum, with a few workers in the field having focused
their attention on to those intriguing observations. Nonetheless, we quote “en passant” that
in a recent paper [37] Hutsemekers et al. discussed the possibility of such phenomenon to be
understood by invoking very light pseudoscalar particles mixing with photons. They claimed
that the observations of a sample of 355 quasars with significant optical polarization present
strong evidence that quasar polarization vectors are not randomly oriented over the sky, as
naturally expected. Those authors suggest that the phenomenon can be understood in terms
of a cosmological-size effect, where the dichroism and birefringence predicted by a mixing
between photons and very light pseudoscalar particles within a background magnetic field
can qualitatively reproduce the observations. They also point out at a finding indicating that
JCAP03(2011)033
circular 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 can
be 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 be
inscribed on to the framework of Heisenberg-Euler NLED, which predicts the occurrence of
birefringence 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 also
thank Prof. Xin-min Zhang, and Dr. Mingzhe Li for reading the manuscript and their
suggestions. H.J.M.C. thanks ICRANet International Coordinating Center, Pescara, Italy
for hospitality during the early stages of this work. G.L. acknowledges the financial support
of MIUR through PRIN 2006 Prot. 1006023491− 003, and of research funds provided by the
University of Salerno. H.J.M.C. is fellow of the Ceara State Foundation for the Development
of Science and Technology (FUNCAP), Fortaleza, Ceara, Brazil.
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