On the CMB circular polarization. I. The Cotton-Mouton effect

# On the CMB circular polarization. I. The Cotton-Mouton effect

Damian Ejlli Department of Physics, Novosibirsk State University, Novosibirsk 630090 Russia
###### Abstract

Generation of cosmic microwave background (CMB) elliptic polarization due to the Cotton-Mouton (CM) effect in a cosmic magnetic field is studied. We concentrate on the generation of CMB circular polarization and on the rotation angle of the CMB polarization plane from the decoupling time until at present. For the first time, a rather detailed analysis of the CM effect for an arbitrary direction of the cosmic magnetic field with respect to photon direction of propagation and arbitrary magnetic field profile is done. Considering the CMB linearly polarized at the decoupling time, it is shown that the CM effect is one of the most substantial effects in generating circular polarization especially in the low part of the CMB spectrum. It is shown that in the frequency range Hz Hz, the degree of circular polarization of the CMB at present for perpendicular propagation with respect to the cosmic magnetic field is in the range or Stokes circular polarization parameter K K for realistic values of the cosmic magnetic field amplitude. On the other hand, for not perpendicular propagation with respect to the cosmic magnetic field we find or K K. Estimates on the rotation angle of the CMB polarization plane due to the CM effect and constraints on the cosmic magnetic field amplitude from current constraints on due to a combination of the CM and Faraday effects are found.

## 1 Introduction

In the last two decades, there have been many established observational facts about the nature and properties of the CMB and their possible implications in cosmology. Among these, it has already been established the fact that the CMB has a linear polarization with a degree of polarization at present of the order . This linear polarization is believed to have been generated at the decoupling time mostly due to the Thomson scattering of the CMB photons on electrons. In general, if the incident electromagnetic radiation has an isotropic intensity distribution, Thomson scattering does not generate a net linear polarization. In the specific case of the CMB the fact that linear polarization has been initially observed by DASI, WMAP and BOOMERANG collaborations [1] and then re-confirmed by other collaborations, implies that at the decoupling time the CMB intensity did not have an isotropic distribution, a fact which is widely confirmed from the observation of the CMB temperature anisotropy. Another important consequence of the Thomson scattering is that it does not generates circular polarization in the case when electrons are assumed to be unpolarized. Based on this fact, during these years it has been erroneously assumed, at least from the theoretical point of view, that the CMB does not have a circular polarization at all even though there have been initial studies that might support its existence [2] and also initial experimental efforts to detect it [3].

In the recent years there have been several other theoretical studies exploring the possibility of CMB circular polarization from standard and non-standard effects and also new experiments such as MIPOL [4] and SPIDER [5] aiming to detect it. The MIPOL [4] collaborations reported an upper limit on the degree of circular polarization at present of at the frequency 33 GHz and at angular scales between and . On the other hand, the SPIDER collaboration reported an upper limit on the CMB circular polarization power spectrum for multipole momenta at the CMB frequencies GHz and GHz. From the theoretical point of view, studies based on non-standard effects that generate circular polarization include; the interaction of the CMB with a vector field via a Chern-Simons term [6], non commutative geometry [7] and free photon-photon scattering due to the Euler-Heisenberg Lagrangian term [8]. On the other hand, some theoretical studies of standard effects include; the electron-positron scattering in magnetized plasma at the decoupling time [9], the propagation of the CMB photons in magnetic field of supernova remnants of the first stars [10], the scattering of the CMB photons with cosmic neutrino background [11] and also the alignment of the cosmological matter particles in the post-decoupling epoch which results in an anisotropic susceptibility matter tensor [12]. For a recent and not complete review of the CMB circular polarization see Ref. [13].

Apart from the circular polarization generation effects mentioned above, there is a class of effects called magneto-optic effects which generate CMB circular polarization as well. In Ref. [14] and Ref. [15], I studied the most important magneto-optic effects which can generate CMB circular polarization when the CMB interacts with large-scale cosmic magnetic fields. Among the effects which I studied one of them is a standard effect, namely the CM effect, and the other effects are non-standard and include the vacuum polarization in an external magnetic field due to one loop electron-positron, one loop millicharged fermion-antifermion and the photon-pseudoscalar mixing in a magnetic field. For all these effects to occur it is necessary the presence of a magnetic field which gives rise to birefringence effects due to the fact that each of the photon states acquires different indexes of refraction in the presence of the magnetized plasma.

While it is well known that it does exist a magnetic field in galaxies and galaxy clusters with an order of magnitude of few G, it is still not known if such a field is present also in the intergalactic space. The only information that we have about intergalactic magnetic fields are only in forms of upper and lower limits on the field magnitude at the present epoch. The upper limits on the magnetic field amplitude are found from observations of the CMB temperature anisotropy and from the rotation angle of the CMB polarization plane due to the Faraday effect. The temperature anisotropy upper limit is usually stronger than the Faraday effect limit, as reported by the Planck collaboration [16], where the limit from CMB temperature anisotropy is nG while the limit from the Faraday effect is nG. One important aspect of these limits is that they differ from each other roughly speaking by three orders of magnitude and most importantly these limits do not mutually exclude each other from the simple fact that they are model depended. For a general review on large-scale cosmic magnetic field see Ref. [17].

One key aspect which distinguishes the CMB linear polarization with the CMB circular polarization, is that the former being generated at the decoupling time due to the Thomson scattering does not depend on the CMB photon frequency because of the nature of Thomson scattering, while the latter in most cases strongly depends on the CMB frequency. Because of this frequency dependence of the circular polarization, there is in some sense a kind of uncertainty on how to use and interpret the current limits obtained by experiments such as MIPOL and SPIDER since their limits are usually derived by observing the CMB in a specific frequency and it is not known how much substantial could be the signal at other frequencies.

In order to study and detected CMB circular polarization, it is very important to first identify the circular polarization (possibly standard) effects that generate substantial CMB circular polarization and identify their frequency band where the signal is the strongest. So far, there has been a tendency in the literature to study the circular polarization in the high-frequency range, namely for frequencies above ten or few hundred GHz. This tendency has been partially influenced by the fact that most important CMB experiments such as WMAP and Planck operates at these frequencies and therefore their data in these frequencies might be useful in some way. In addition, there are some effects such as photon-photon scattering in a magnetic field [14] and the free photon-photon scattering [8], [12] which are linearly proportional to the CMB frequency and one might hope that the higher is the frequency the stronger is the circular polarization signal. Even though this is true, the signal for such effects is still very weak even at very high frequencies to be detected in the near future.

Based on the facts discussed above, it is rather logical to explore the CMB circular polarization at low frequencies and study the magnitude of the signal. In this work, I study such possibility and concentrate on the CM effect in a large-scale magnetic field. As we will see, the CM effect is proportional to the square of the magnetic field amplitude, , and inversely proportional to the third power of the CMB frequency, namely . It is especially the scaling law with the frequency of which makes the CM effect one of the most important effects in generating circular polarization of the CMB. I partially studied this effect in a previous work [14] where some estimates of the degree of circular polarization were made for a specific configuration of the magnetic field with the respect to the photon direction of propagation. In this work, I study the CM effect in details for an arbitrary configuration of the magnetic field direction and for arbitrary magnetic field amplitude profile. By generalizing the CM effect to an arbitrary direction of the magnetic field with respect to the observer’s direction, the system of differential equations for the Stokes parameters has additional terms with respect to the case studied in Ref. [14]. In addition, I also study in details the impact that the CM effect has on the rotation angle of the CMB polarization plane and its interaction with the Faraday effect.

This paper is organized in the following way: in Sec. 2, I discuss in a concise way the propagation of the electromagnetic radiation in a magnetized plasma and derive the elements of the photon polarization tensor in a cold magnetized plasma. In Sec. 3, I derive the system of differential equations for the Stokes parameters in an expanding universe. In Sec. 4, I find perturbative solutions of the equations of motion in various regimes. In Sec. 5, I calculate in details the generation of the CMB circular polarization due to the CM effect at present. In Sec. 6, I study the rotation angle of the CMB polarization plane due to the CM effect alone and also due to a combination of the CM and Faraday effects. In Sec. 7, I conclude. In this work I use the metric with signature and work with the rationalized Lorentz-Heaviside natural units () with . In addition in this work we use the values of the cosmological parameters found by the Planck collaboration [18] with with zero spatial curvature with .

## 2 Propagation of the electromagnetic waves in a magnetized plasma

In this section we give a detailed description of the propagation of electromagnetic waves in a cold magnetized plasma. This description is useful because it would allow us to understand in details how electromagnetic waves propagate in a cold magnetized plasma and which are the most common effects which give rise to birefringence effects in the medium. In this section we use the same notation as in Ref. [19] where basics of propagation of the electromagnetic waves in a cold magnetized plasma are presented in the appendix.

When electromagnetic waves (photons) propagate in a medium several effects manifest which include dispersion, absorption and scattering of the electromagnetic radiation. In connection with the dispersion phenomena, the effects of the medium on the incident electromagnetic wave are usually described in terms of the photon polarization tensor () with components in a given cartesian coordinate system where the medium is at rest. Consequently, in a medium the free Maxwell equations in momentum space, in absence of external currents, get modified to

 ω2(δij−n2ij)=Πij (1)

for a plane electromagnetic wave travelling into the medium. Here is the photon energy and we used the expression with being the photon momentum tensor and being the index of refraction tensor of photons in medium. We may see that the role of in (1) is to give to photons an ”effective mass” in the medium. In the case when the medium is isotropic, we have that is a diagonal tensor with diagonal entries corresponding to the photon indexes of refraction in medium where . In the case when photons propagate in vacuum, we have that and we get the on-shell photon relation where is the photon wave-vector and .

The explicit expression of the photon polarization tensor depends on the induced currents that enter a given problem. In this work we are interested in a cold magnetized plasma which is quite common situation in astrophysics and cosmology. We assume that the magnetized plasma is with almost no collisions, globally neutral and homogeneous. In addition, there is not an external electric field, namely and the presence of the external magnetic field locally breaks the isotropy of the plasma since it singles out a preferred direction in a given region of space where the plasma is located.

In the cold magnetized plasma approximation, consider now an incident electromagnetic wave propagating along the observer’s axis which points to the East, in a magnetized plasma with external magnetic field vector . Here is a unit vector in the direction of the external magnetic field and are, respectively, the polar and azimutal angles between the magnetic field and and axes. As shown in Ref. [19], the medium polarization vector satisfies the equation of motion

 ¨P=ω2plE−ωc˙P×^n, (2)

where is the electric field of the incident electromagnetic wave, is the plasma frequency, is the free electron number density and is the cyclotron frequency. In Eq. (2) the dot symbol () above denotes the derivative with respect to the time.

Assume that the fields evolve in time harmonically at a given point and then let us write

 P(x,t)=P(x,ω)e−iωt,E(x,t)=E(x,ω)e−iωt, (3)

where is the incident electromagnetic wave energy. By using the expressions in (3) into Eq. (2) and then solving for the components of , after we get the following solution in terms of the incident electric field components , in the case when and

 Pi(x,ω)=χij(ω)Ej(x,ω),(i,j=x,y,z), (4)

where are the components of the electric susceptibility tensor

 χxx=−ω2plω2−ω2c+ω2plω2ccos2(Θ)ω2(ω2−ω2c),χxy=ω2plω2csin(2Θ)cos(Φ)2ω2(ω2−ω2c)+iω2plωcsin(Θ)sin(Φ)ω(ω2−ω2c),χxz=ω2plω2csin(2Θ)sin(Φ)2ω2(ω2−ω2c)−iω2plωcsin(Θ)cos(Φ)ω(ω2−ω2c),χyx=χ∗xy,χyy=−ω2plω2−ω2c+ω2plω2csin2(Θ)cos2(Φ)ω2(ω2−ω2c),χyz=ω2plω2csin(2Φ)sin2(Θ)2ω2(ω2−ω2c)+iω2plωccos(Θ)ω(ω2−ω2c),χzx=χ∗xz,χzy=χ∗yz,χzz=−ω2plω2−ω2c+ω2plω2csin2(Θ)sin2(Φ)ω2(ω2−ω2c). (5)

The expressions for the components of in (5) are valid for an incident electromagnetic wave with an arbitrary direction of propagation with respect to . In addition, the components do not explicitly depend on but only implicitly through which enters in . Another fact is that the expressions for in (5) are valid for arbitrary external magnetic profile . After these general comments about (5), let us find the components of the photon polarization tensor in a cold magnetized plasma. In order to do that we have to relate the components of with . It is well known that the components of the index of refraction tensor are related to the relative permittivity tensor111In general, the index of refraction is given by where is the relative magnetic permeability tensor. However, in most cases of magnetized plasmas we have that to good accuracy. through the relation . On the other hand, the relative permittivity tensor is related to the electric susceptibility tensor , through the relation . By using these relations into the relation (1), we get

 Πij=−ω2χij. (6)

By using the expressions for in (5) into (6), we get

 Πxx=ω2ω2plω2−ω2c−ω2plω2ccos2(Θ)ω2−ω2c,Πxy=−ω2plω2csin(2Θ)cos(Φ)2(ω2−ω2c)−iω2plωcωsin(Θ)sin(Φ)ω2−ω2c,Πxz=−ω2plω2csin(2Θ)sin(Φ)2(ω2−ω2c)+iωω2plωcsin(Θ)cos(Φ)ω2−ω2c,Πyx=Π∗xy,Πyy=ω2ω2plω2−ω2c−ω2plω2csin2(Θ)cos2(Φ)ω2−ω2c,Πyz=−ω2plω2csin(2Φ)sin2(Θ)2(ω2−ω2c)−iωω2plωccos(Θ)ω2−ω2c,Πzx=Π∗xz,Πzy=Π∗yz,Πzz=ω2ω2plω2−ω2c−ω2plω2csin2(Θ)sin2(Φ)ω2−ω2c. (7)

The expression for in (7) quantify the dispersive effects induced by the medium on the incident electromagnetic wave. One thing which is very well known, is that in the presence of a medium, appears also a longitudinal component for the incident electromagnetic wave. So, the difference with respect to vacuum propagation, is that in the presence of a plasma one has also to deal with the induced longitudinal electric field. Indeed, as it is evident from the expression of , all components , which mean that also the longitudinal component of the electromagnetic wave manifest dispersive phenomena. However, one important fact about the longitudinal photon state is that it does not transport energy and it does not propagate in space. Moreover, the longitudinal component of the electromagnetic wave has no magnetic field associated to it but only an electric field. It essentially correspond to a density wave of electrons in the plasma in the presence of that does not propagate in space.

After these general comments on the longitudinal component of the electromagnetic wave, let us consider a harmonic electromagnetic wave with electric field vector with components which propagates along the direction in a given coordinate system where the medium is at rest. Here the (T) symbol indicates the transpose of a row vector. For an electromagnetic wave propagating in the direction, we have that all electric field components depend only on , namely . In the presence of a medium, the Maxwell equation for the electric field is given by

 1v2ph∂2Ei∂t2−∇2Ei=0 (8)

where is the phase velocity of the electromagnetic wave in the plasma and is its index of refraction. Here and are respectively the electric permittivity and magnetic permeability in the plasma. By assuming that the electromagnetic wave evolves harmonically in time at a given point in space and considering the propagation in an anisotropic plasma, we get from (8) and (6)

 ∂2zEi(z,ω)=[Πij(ω)−ω2δij]Ej(z,ω). (9)

One important aspect of the wave equation (9) is that for , namely for the component of the electric field, we have that . In this case the equation (9) for gives a constraint on which implies that it depends linearly on the transverse components of the electric field through the relation

 Ez=ΠzxEx+ΠzyEyω2−Πzz. (10)

Now by using the constraint (10) into the () components of the electric field in (9), we get the following equation for the transverse components of the electric field

 ∂2zEi(z,ω)=[~Πij−ω2δij]Ej(z,ω),for(i,j=x,y), (11)

where for is the effective photon polarization tensor of the transverse electromagnetic field in the cold magnetized plasma

 ~Πij=Πij+ΠizΠziω2−Πzz, (12)

where the components of are given in (7). The effective expression for the polarization tensor in (12) takes into account the mixing of the longitudinal electromagnetic wave in plasma with the usual transverse electromagnetic waves. From expressions (12) and (7), we find the following expressions for the components of

 ~Πxx=ω2ω2% plω2−ω2c⎡⎢ ⎢⎣1−ω2cω2cos2(Θ)+ω2plω2c(4ω2cos2(Φ)sin2(Θ)+ω2csin2(2Θ)sin2(Φ))4ω2(ω4−ω2ω2pl−ω2ω2c+ω2cω2plsin2(Θ)sin2(Φ))⎤⎥ ⎥⎦,~Πxy=−ω2plω2c2(ω2−ω2c)⎡⎢ ⎢⎣sin(2Θ)cos(Φ)+ω2pl(2ω2sin(2Θ)cos(Φ)−ω2csin2(Θ)sin(2Θ)sin(Φ)sin(2Φ))2(ω4−ω2ω2pl−ω2ω2c+ω2cω2plsin2(Θ)sin2(Φ))⎤⎥ ⎥⎦−iω2plωcωω2−ω2c⎡⎢ ⎢⎣sin(Θ)sin(Φ)+ω2plω2c(sin(Φ)cos(Θ)sin(2Θ)+sin3(Θ)sin(2Φ)cos(Φ))2(ω4−ω2ω2pl−ω2ω2c+ω2cω2plsin2(Θ)sin2(Φ))⎤⎥ ⎥⎦,~Πyx=~Π∗xy,~Πyy=ω2ω2plω2−ω2c⎡⎢ ⎢⎣1−ω2cω2sin2(Θ)cos2(Φ)+ω2plω2c(4ω2cos2(Θ)+ω2csin4(Θ)sin2(2Φ))4ω2(ω4−ω2ω2pl−ω2ω2c+ω2cω2plsin2(Θ)sin2(Φ))⎤⎥ ⎥⎦. (13)

The expressions given in (13) are the most general form of the elements of in a cold magnatized plasma. As already mentioned above they take into account the mixing of the longitudinal electric field with the usual transverse electric field. We may note that this contribution in (13) is inversely proportional to . The latter condition is satisfied as far as and

 ω≠√ω2pl+ω2c2⎡⎢⎣1±⎛⎝1−4ω2cω2plsin2(Φ)sin2(Θ)ω2pl+ω2c⎞⎠1/2⎤⎥⎦, (14)

where we must have in order to have real and positive roots of the quadratic equation. Another important question to ask is for what minimum frequencies we have propagating transverse electromagnetic waves? This can be seen by requiring that all spatial derivatives on the left hand side in Eq. (9) are zero, namely a non propagating electric field in space. In that case we would have where nontrivial solution exist only if det with . However, the solution of det in terms of would be quite complicated in the case when all components of . For this reason it would be convenient to rotate the coordinate system in such a way that and , namely is along the direction of propagation of the electromagnetic wave. Under a rotation of the coordinate system we have that in the new coordinate system is related to the old through and where is an orthogonal rotation matrix with unit determinant. In the rotated coordinate system the equation becomes . Consequently, the condition det is equivalent to det. Now by requiring that det and after doing some algebra we find that the lower bound on the frequencies for propagation are and .

## 3 Solutions of the equations of motion of the Stokes parameters

In the previous section we derived the most general form of the elements of the photon polarization tensor in a cold magnetized plasma for arbitrary direction of propagation of the electromagnetic waves with respect to the external magnetic field . In this section, we focus on our attention on deriving the equations of motion of the Stokes parameters in an expanding universe and provide perturbative solutions of the equations of motion. As in the previous section, let us consider an electromagnetic wave propagating along the direction in a cartesian reference system with wave vector in a cold magnetized plasma with arbitrary direction of the external magnetic field . The linearized equations of motion for the vector potential transverse components and in an unperturbed FRW metric for the CMB photons are given by [14]

 i∂tΨ(k,t)=[M(k,Be,Φ,Θ)−32H(t)I]Ψ(k,t), (15)

where and are respectively the transverse components of the vector potential of the CMB photons with respect to the and axes, is a two component field, is the Hubble parameter, is a identity matrix and is the mixing matrix which is given by

 M(k,Be,Φ,Θ)=[k−My−MCF−M∗CFk−Mx], (16)

where , and . The term takes into account the combination of the CM and Faraday effects in a magnetized plasma.

In order to describe the polarization of the light and more precisely in our case of the CMB photons, it is better to work with the Stokes parameters rather than the wave equation (15). The procedure in obtaining the equations of motion of the Stokes parameters has been presented in [14] and it consists on two steps; first write the equations of motion for the polarization density matrix based on the wave equation (15) and second, express the polarization density matrix in terms of the Stokes parameters in order to get the equations of motion of the latter quantities. The equations of motion of the polarization density matrix in an unperturbed FRW metric are given by [14]

 ∂ρ∂t=−i[M,ρ]−{D,ρ}, (17)

where is the damping matrix which takes into account the damping of the electromagnetic waves in an expanding universe due to the Hubble friction. In our case the field mixing matrix is Hermitian, namely since in our case we do not include any process which might change the number of photons due to decay or absorption in the medium222 If there is any process that may change the number of the CMB photons, its magnitude is a very small quantity at post decoupling epoch.. Now by using the connection between the Stokes parameters and the polarization density matrix elements as shown in Ref. [20], see also the appendix of Ref. [14], we get the following equations of motion of the effective Stokes parameters

 ˙I(k,^n,t) =−3H(t)I(k,^n,t), ˙Q(k,^n,t) =−2MF(k)U(k,^n,t)−2MC(k)V(k,^n,t)−3H(t)Q(k,^n,t), (18) ˙U(k,^n,t) =2MF(k)Q(k,^n,t)+ΔM(k)V(k,^n,t)−3H(t)U(k,^n,t), ˙V(k,^n,t) =2MC(k)Q(k,^n,t)−ΔM(k)U(k,^n,t)−3H(t)V(k,^n,t),

where we have defined with the dot sign above Stokes parameters indicating the time derivative with respect to the cosmological time . For simplicity, in (3) we have dropped the symbols and which do appear in the elements of .

The system of linear differential equations (3) can be written in a more compact form as where is the Stokes vector formed with the Stokes parameters and is the time dependent coefficient matrix which is given by

 A(k,t)=⎛⎜ ⎜ ⎜ ⎜⎝−3H0000−3H−2MF−2MC02MF−3HΔM02MC−ΔM−3H⎞⎟ ⎟ ⎟ ⎟⎠. (19)

In most cases is more convenient to express the quantities in as a function of the photon temperature rather than the cosmological time , so, in this case one needs to express the time derivative in an expanding universe as in the equations of motion of the Stokes vector, namely . At this stage is more convenient to write the matrix as the sum of , where the matrix is given by

 B(k,T)=1HT⎛⎜ ⎜ ⎜ ⎜⎝0000002MF(T)2MC(T)0−2MF(T)0−ΔM(T)0−2MC(T)ΔM(T)0⎞⎟ ⎟ ⎟ ⎟⎠, (20)

where in an expanding universe the wave-vector is a function of the temperature . We may note that with respect to the case when the direction of is in the plane as studied in Ref. [14], for arbitrary magnetic field direction, do appear the terms in the matrix . The appearance of these terms which makes possible the mixing of the parameter with and parameters, complicate the situation with respect to the case when .

## 4 Series solution of the polarization equations of motion

In the previous section, Sec. 3, we found the equations of motion of the Stokes parameters in an expanding universe for an arbitrary direction of the external magnetic field with respect to the electromagnetic wave direction of propagation. In this section we focus on our attention on perturbative solutions of the equations of motion in some limiting cases. Before aiming to find these solutions, it is very important to explicitly calculate each term which enters the matrix since it will be very useful in what follows. Let us recall the definitions of , and . Now by using the expressions of the photon polarization tensor given in (13) we get

 MC=ω2plω2c4ω(ω2−ω2c)⎡⎢ ⎢⎣sin(2Θ)cos(Φ)+ω2pl(2ω2sin(2Θ)cos(Φ)−ω2csin2(Θ)sin(2Θ)sin(Φ)sin(2Φ))2(ω4−ω2ω2pl−ω2ω2c+ω2cω2plsin2(Θ)sin2(Φ))⎤⎥ ⎥⎦,MF=ω2plωc2(ω2−ω2c)⎡⎢ ⎢⎣sin(Θ)sin(Φ)+ω2plω2c(sin(Φ)cos(Θ)sin(2Θ)+sin3(Θ)sin(2Φ)cos(Φ))2(ω4−ω2ω2pl−ω2ω2c+ω2cω2plsin2(Θ)sin2(Φ))⎤⎥ ⎥⎦,ΔM=ω2cω2pl2ω(ω2−ω2c)[sin2(Θ)cos2(Φ)−cos2(Θ)+ω2pl[4ω2(cos2(Φ)sin2(Θ)−cos2(Θ))+ω2c(sin2(2Θ)sin2(Φ)−sin4(Θ)sin2(2Φ))]4(ω4−ω2ω2pl−ω2ω2c+ω2cω2plsin2(Θ)sin2(Φ))⎤⎥ ⎥⎦. (21)

The expression for the elements of the matrix given in (21), which are the most general ones for arbitrary magnetic field direction and magnitude, can be further simplified by making some reasonable assumptions on the parameters. Since in this work we concentrate on the CMB frequency spectrum we have that and . In order to see this, let us calculate explicitly the numerical values of the parameters. The numerical value of the angular plasma frequency which enters the expressions in (21) can be written as (rad/s) or (Hz) for the frequency. On the other hand the numerical value of the cyclotron angular frequency is given by (rad/s) or (Hz). However, in the case of CMB photons propagating in an expanding universe, we can express the time in terms of the cosmological temperature as as we did in the previous section. Therefore, the conditions and , in an expanding universe, are respectively satisfied when

 (22)

where we expressed with being the frequency of the electromagnetic radiation at the present time at the temperature , being the universe expansion scale factor and is the magnetic field strength at the present time. Here we expressed the number density of free electrons as where is the total baryon number density at the present time and is the ionization function of the free electrons. The factor of takes into account the contribution of hydrogen atoms to the free electrons at the post decoupling time. By taking for example cm as given by the Planck collaboration [18], and expressing , we can write the conditions (22) as

 (ν0Hz)≫3.88X1/2e(T)(T/T0)1/2and(ν0Hz)≫2.8×106(B0G)(T/T0). (23)

Given the fact that the present day CMB photon frequencies are in the frequency part above Hz, the condition given in (23) is well satisfied for physically reasonable values of and . With these considerations in mind, we can simplify in (21) for and . Consequently, we can write the expressions in (21) as

 MC≃ω2plω2c4ω3⎡⎣sin(2Θ)cos(Φ)+ω2pl(2ω2sin(2Θ)cos(Φ)−ω2csin2(Θ)sin(2Θ)sin(Φ)sin(2Φ))2ω4⎤⎦,MF≃ω2plωc2ω2⎡⎣sin(Θ)sin(Φ)+ω2plω2c(sin(Φ)cos(Θ)sin(2Θ)+sin3(Θ)sin(2Φ)cos(Φ))2ω4⎤⎦,ΔM≃ω2cω2pl2ω3[sin2(Θ)cos2(Φ)−cos2(Θ)+ω2pl[4ω2(cos2(Φ)sin2(Θ)−cos2(Θ))+ω2c(sin2(2Θ)sin2(Φ)−sin4(Θ)sin2(2Φ))]4ω4⎤⎦. (24)

From the expressions (24) we may note that each expression within the square brackets is composed of a first term of trigonometric functions and a second term which is the product of trigonometric functions with terms or . However, since we are in the regime when and , we also have and . This fact tells us that in the case when the trigonometric functions in the first and second terms within the square brackets in (24) are different from zero, the second term is usually much smaller than the first term. In order to see this, let us consider the case when , namely when the magnetic field has components only along the . In this case and , where , so the contribution coming from the second term can be completely neglected. One can see that by making similar examples, the contribution of the second terms within the square brackets in (24), which arise due to the mixing of the longitudinal electromagnetic wave with the transverse waves, can be neglected with respect to the first terms. Consequently, in the regime studied in this work and , we have that

 MC≃ω2plω2c4ω3sin(2Θ)cos(Φ),MF≃ω2plωc2ω2sin(Θ)sin(Φ),ΔM≃ω2cω2pl2ω3[sin2(Θ)cos2(Φ)−cos2(Θ)]. (25)

### 4.1 Neumann series solutions

Here we present a Neumann series solutions of the equations of motion by making use of the perturbation theory. Let us concentrate on the full equation and omit from now on the dependence of the Stokes vector on and and matrix on . From the equation of motion of the Stokes vector, the term is a term which takes into account the damping of the fields in an expanding universe. In the case when there is not a magnetic field the solution of the equation is for . It is worth to stress since now that the effective scaling of the Stokes vector in an expanding universe is not