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

## 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 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 \(10^8\) Hz \(\le \nu _0\le 10^9\) 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 \( 10^{-13}\lesssim P_C(t_0)\lesssim 7.65\times 10^{-7}\) or Stokes circular polarization parameter \(2.7 \times 10^{-13}\) K \(\lesssim |V(t_0)|\lesssim 2 \times 10^{-6}\) K for values of the cosmic magnetic field amplitude at present in the range \(10^{-9}\) G \(\lesssim B\lesssim 8\times 10^{-8}\) G. On the other hand, for not perpendicular propagation with respect to the cosmic magnetic field we find \(10^{-15}\lesssim P_C(t_0)\lesssim 6\times 10^{-12}\) or \(2.72 \times 10^{-15}\) K \(\lesssim |V(t_0)| \lesssim 10^{-11}\) K, for the same values of the cosmic magnetic field amplitude and same frequency range. Estimates on the rotation angle of the CMB polarization plane \(\delta \psi _0\) due to the CM effect and constraints on the cosmic magnetic field amplitude from current constraints on \(\delta \psi _0\) due to a combination of the CM and Faraday effects are found.

## 1 Introduction

In the last 2 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 \(P_L(t_0)\simeq 10^{-6}\). 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, 2, 3] 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 [4, 5, 6] and also initial experimental efforts to detect it [7, 8].

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 [9] and SPIDER [10] aiming to detect it. The MIPOL [9] collaborations reported an upper limit on the degree of circular polarization at present of \(P_C(t_0)\lesssim 7\times 10^{-5}\) to \(5\times 10^{-4}\) at the frequency 33 GHz and at angular scales between \(8^\circ \) and \(24^\circ \). On the other hand, the SPIDER collaboration reported an upper limit on the CMB circular polarization power spectrum \(\ell (\ell +1) C_\ell ^{VV}/(2\pi )< 255 {(\upmu \text {K})}^2\) for multipole momenta \(33<\ell <307\) at the CMB frequencies \(\nu _0=95\) GHz and \(\nu _0=150\) 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 [11], non commutative geometry [12] and free photon-photon scattering due to the Euler–Heisenberg Lagrangian term [13]. On the other hand, some theoretical studies of standard effects include; the electron-positron scattering in magnetized plasma at the decoupling time [14], the propagation of the CMB photons in magnetic field of supernova remnants of the first stars [15], the scattering of the CMB photons with cosmic neutrino background [16] and also the alignment of the cosmological matter particles in the post-decoupling epoch which results in an anisotropic susceptibility matter tensor [17]. For a recent and not complete review of the CMB circular polarization see Ref. [18].

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 Refs. [19, 20], 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 \(\upmu \)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 [21], where the limit from CMB temperature anisotropy is \(B_{e0}\lesssim 3\) nG at a scale \(\lambda _B=1\) Mpc, while the limit from the Faraday effect is \(B_{e0}\lesssim 1380\) nG at \(\lambda _B=1\) Mpc. One important aspect of these limits is that they differ from each other roughly speaking by three orders of magnitude and most importantly the stronger limit on the magnetic field amplitude from the CMB temperature anisotropy does not exclude the weaker limit from the Faraday effect because these upper limits depend on how the magnetic field is modelled and other assumptions, see Ref. [21] for details. For simplicity, in this work we assume that the cosmic magnetic field amplitude changes in time *t* and it is an almost constant function of the position \(\varvec{x}\). For a general review on large-scale cosmic magnetic field see Refs. [22, 23, 24, 25, 26, 27].

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 frequency because of the nature of Thomson scattering which is frequency independent at lower energies, 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 detect the 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 where the CMB intensity is the highest and therefore their data at these frequencies might be useful in some way. In addition, there are some effects such as the photon-photon scattering in a cosmic magnetic field [19] and the free photon-photon scattering [13, 17] 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 cosmic magnetic field. As we will see, the CM effect is proportional to the square of the magnetic field amplitude, \(B^2\), and inversely proportional to the third power of the CMB frequency, namely \(\nu ^{-3}\) in the case of perpendicular propagation with respect to the cosmic magnetic field. It is especially the scaling law with the frequency of \(\nu ^{-3}\) which makes the CM effect the most important effect in generating CMB circular polarization at low frequencies. I partially studied this effect in a previous work [19] where some estimates of the degree of circular polarization were made for a specific configuration of the cosmic 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 cosmic magnetic field direction. By generalizing the CM effect to an arbitrary direction of the cosmic 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. [19]. 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 Sect. 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 the cold magnetized plasma approximation. In Sect. 3, I derive the system of differential equations for the Stokes parameters in an expanding universe. In Sect. 4, I find perturbative solutions of the equations of motion in various regimes. In Sect. 5, I calculate in details the generation of the CMB circular polarization due to the CM effect at present. In Sect. 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 Sect. 7, I conclude. In this work I use the metric with signature \(\eta _{\mu \nu }=\text {diag}[1, -1, -1, -1]\) and work with the rationalized Lorentz–Heaviside natural units (\(k_B=\hbar =c=\varepsilon _0=\mu _0=1\)) with \(e^2=4\pi \alpha \). In addition in this work I use the values of the cosmological parameters found by the Planck collaboration [28] with \(\Omega _\Lambda \simeq 0.68, \Omega _\text {M}\simeq 0.31, h_0\simeq 0.67\) with zero spatial curvature where \(\Omega _\kappa =0\).

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

In this section we give a compact description of propagation of the electromagnetic waves in the cold magnetized plasma approximation. This description is useful because it would allow us to understand 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. [29] where basics of propagation of the electromagnetic waves in a cold magnetized plasma are presented in the appendix.

The explicit expression of the photon polarization tensor \(\Pi _{ij}\) 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 \(\varvec{E}_e=0\) and the presence of the external magnetic field \(\varvec{B}_e\) 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.

*z*axis which points to the East, in a magnetized plasma with external magnetic field vector \(\varvec{B}_e = B_e \hat{\varvec{n}}\). Here \(\hat{\varvec{n}}=[\cos (\Theta ), \sin (\Theta )\cos (\Phi ), \sin (\Theta )\sin (\Phi )]\) is a unit vector in the direction of the external magnetic field \(\varvec{B}_e\) and \(\Theta , \Phi \) are, respectively, the polar and azimuthal angles between the magnetic field \(\varvec{B}_e\) and

*x*and

*y*axes. As shown in Ref. [29], the medium polarization vector \(\varvec{P}\) satisfies the equation of motion

^{1}\(\varepsilon _{ij}\) through the relation \(\Pi _{ij}=\omega ^2(\delta _{ij} - \varepsilon _{ij})\). On the other hand, the relative permittivity tensor \(\varepsilon _{ij}\) is related to the electric susceptibility tensor \(\chi _{ij}\), through the relation \(\varepsilon _{ij}=\chi _{ij} + \delta _{ij}\). By using these relations, we get

*z*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

*z*direction, we have that all electric field components depend only on

*z*, namely \(E_i(\varvec{x}, \omega )=E_i(z, \omega )\). In the presence of a medium, the Maxwell equation for the electric field \(\varvec{E}_i(\varvec{x}, t)\) is given by

*n*is its index of refraction. Here \(\epsilon \) and \(\mu \) 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

*z*in space and considering the propagation in an anisotropic plasma, we get from (6) and (8)

*z*component of the electric field, we have that \(\partial _z^2 E_z(z, \omega )=0\). In this case the Eq. (9) for \(i=z\) gives a constraint on \(E_z\) which implies that it depends linearly on the transverse components of the electric field through the relation

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

*z*direction in a cartesian reference system with wave vector \(\varvec{k}=(0, 0, k)\) in a cold magnetized plasma with arbitrary direction of the external magnetic field \(\varvec{B}_e\). The linearized equations of motion for the vector potential transverse components \(A_x\) and \(A_y\) in an unperturbed FRW metric for the CMB photons are given by [19]

*x*and

*y*axes, \(\Psi (k, t)=[A_x(k, t), A_y(k, t)]^\text {T}\) is a two component field,

*H*(

*t*) is the Hubble parameter, \(\varvec{I}\) is a \(2\times 2\) identity matrix and

*M*is the mixing matrix which is given by

*M*is Hermitian, namely \(M=M^\dagger \) since in our case we do not include any process which might change the number of photons due to decay or absorption in the medium.

^{2}Now by using the connection between the Stokes parameters and the polarization density matrix elements as shown in Refs. [30, 31, 32, 33], see also the appendix of Ref. [19], we get the following equations of motion of the effective Stokes parameters

*t*. For simplicity, in (18) we have dropped the symbols \(B_e, \Phi \) and \(\Theta \) which do appear in the elements of

*M*.

*A*(

*k*,

*t*) is the time dependent coefficient matrix which is given by

*A*as a function of the photon temperature

*T*rather than the cosmological time

*t*, so, in this case one needs to express the time derivative in an expanding universe as \(\partial _t= - H T\partial _T\) in the equations of motion of the Stokes vector, namely \(S^\prime (k, \hat{\varvec{n}}, T)= {\tilde{A}}(k, T) S(k, \hat{\varvec{n}}, T)\). At this stage is more convenient to write the matrix \({\tilde{A}}(k, T)\) as the sum of \({\tilde{A}}(k, T)=B(k, T) + (3/T) \varvec{I}_{4\times 4}\), where the matrix

*B*(

*k*,

*T*) is given by

*T*. We may note that with respect to the case when the direction of \(\varvec{B}_e\) is in the

*xz*plane as studied in Ref. [19], for arbitrary magnetic field direction, do appear the terms \(2 M_\text {C}\) in the matrix

*B*. The appearance of these terms which makes possible the mixing of the

*Q*parameter with

*U*and

*V*parameters, complicate the situation with respect to the case when \(M_\text {C}=0\).

## 4 Series solution of the polarization equations of motion

*B*(

*k*,

*T*) since it will be very useful in what follows. Let us recall the definitions of \(M_\text {F} \equiv -\text {Im}\{{{\tilde{\Pi }}}_{xy}\}/(2\omega )\), \(M_\text {C} \equiv -\text {Re}\{{{\tilde{\Pi }}}_{xy}\}/(2\omega )\) and \(\Delta M \equiv M_x-M_y=({{\tilde{\Pi }}}_{yy}-{{\tilde{\Pi }}}_{xx})/(2\omega )\). Now by using the expressions of the photon polarization tensor given in (13) we get

*B*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 \(\omega \gg \omega _\text {pl}\) and \(\omega \gg \omega _c\). 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 \(\omega _\text {pl}=5.64\times 10^4 \sqrt{n_e/\text {cm}^3}\) (rad/s) or \(\nu _\text {pl}=\omega _\text {pl}/(2\pi )=8976.33 \sqrt{n_e/\text {cm}^3}\) (Hz) for the frequency. On the other hand the numerical value of the cyclotron angular frequency is given by \(\omega _c=1.76\times 10^7( B_{e0}/\text {G})\) (rad/s) or \(\nu _c=2.8\times 10^6( B_{e0}/\text {G})\) (Hz). However, in the case of CMB photons propagating in an expanding universe, we can express the time

*t*in terms of the cosmological temperature

*T*as \(t=t(T)\) as we did in the previous section. Therefore, the conditions \(\omega \gg \omega _\text {pl}\) and \(\omega \gg \omega _c\), in an expanding universe, are respectively satisfied when

*a*(

*t*) being the universe expansion scale factor and \( B_{e0}= B_e(t_0)=B_e(T_0)\) is the magnetic field strength at the present time. Here we expressed the number density of free electrons as \(n_e(t)=n_e(T)\simeq 0.76\, n_B(T_0) X_e(T)(T/T_0)^3\) where \(n_B(T_0)\) is the total baryon number density at the present time and \(X_e(T)\) is the ionization function of the free electrons. The factor of 0.76 takes into account the contribution of hydrogen atoms to the free electrons at the post decoupling time. By taking for example \(n_B(T_0)\simeq 2.47\times 10^{-7}\) cm\(^{-3}\) as given by the Planck collaboration [28], and expressing \(a(t_0)/a(t)=T/T_0\), we can write the conditions (22) as

*x*. In this case \(M_\text {C}=M_\text {F}=0\) and \(\Delta M= [\omega _\text {pl}^2\omega _c^2/(2\omega ^3)][1+(\omega _\text {pl}/\omega )^2]\), where \((\omega _\text {pl}/\omega )^2\ll 1\), 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 (21), 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 \(\omega \gg \omega _\text {pl}\) and \(\omega \gg \omega _c\), we have that

### 4.1 Neumann series solutions

*k*and matrix

*B*on

*k*. Starting from now, in what follows in this work, we use the expressions found in (24) for the elements of the matrix

*B*in (20). From the equation of motion of the Stokes vector, the term 3 /

*T*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 \(S^\prime (T)= \left[ B(T) + (3/T) \varvec{I}_{4\times 4}\right] S(T)\) is \(S(T)=\exp [3 \varvec{I}_{4\times 4}\int _{T_i}^T dT^\prime /T^\prime ] S(T_i)= (T/T_i)^3 \varvec{I}_{4\times 4} S(T_i)\) for \(B(T)=0\). It is worth to stress since now that the effective scaling of the Stokes vector in an expanding universe is not \((T/T_i)^3\) but \((T/T_i)^2\) as discussed in details in Ref. [19]. In the case when the magnetic field is present, namely when \(B(T) \ne 0\), it is convenient to define \( S(T)\equiv (T/T_i)^3 \varvec{I}_{4\times 4} {\tilde{S}}(T)\). In this case, the equations of motion for \(S^\prime (T)= \left[ B(T) + (3/T) \varvec{I}_{4\times 4}\right] S(T)\) in components become

*H*(

*T*)

*T*, where the Hubble parameter in the case of zero spatial curvature is given by

*T*. On the other hand, the conditions \(|{\mathcal {M}}_\text {C}(T)|<1\) and \(|\Delta {\mathcal {M}}(T)|<1\) are respectively satisfied by the stronger conditions

*T*.

*T*are shown. In the temperature interval 57.22 K\(\le T \le 2970\) K the curve of the ionization function \(X_e(T)\) is obtained by solving the differential equation for \(X_e(T)\) as given in Refs. [34, 35], where the lower limit \(T = 57.22\) K corresponds to the start of reionization epoch at redshift \(z_\text {ion}\sim 20\) and the upper limit corresponds to the CMB decoupling temperature \(T_i=2970\) K for redshift \(1+z\simeq 1090\). The complete re-ionization is reached approximately at \(z_\text {ion}\simeq 7\). The evolution of \(X_e(T)\) in the temperature interval 21.8 K \(\le T \le \) 57.22 K has been obtained by a smooth interpolation of the curve \(X_e(T)\) in the interval 57.22 K \(\le T \le \) 2970 K with \(X_e(T) = 1\) in the interval 2.725 K \(\le T \le \) 21.8 K. By using the numerical solutions found for \(X_e(T)\) as described above and plotted in Fig. 1a, we get the following values for \(\left| \int _{T_0}^{T_i} dT^\prime X_e(T^\prime ) T^{\prime 3/2} \right| \simeq 4.45\times 10^6\) (K\(^{5/2}\)) and \(\left| \int _{T_0}^{T_i} dT^\prime X_e(T^\prime ) T^{\prime 1/2} \right| \simeq 1790.3\) (K\(^{3/2}\)). With these values of the integrals, the stronger conditions (30) and (31) are respectively satisfied when

*B*(

*T*) in (20), we may note that \(B_{1j}=B_{j1}=B_{jj}=0\) with the rest of the elements different from zero. Let us define for commodity

### 4.2 Power series solution for dominant Faraday effect

*B*(

*T*) and require that \(|{\mathcal {M}}_\text {C}(T_0)|<1\) and \(|\Delta {\mathcal {M}}(T_0)|<1\). In addition, the conditions \(|2M_\text {F}(T)| \gg |2M_\text {C}(T)|, |\Delta M(T)|\) explicitly require that \(|\sin (\Theta )\sin (\Phi )|\ne 0\). The conditions \(|2M_\text {F}(T)| \gg |2M_\text {C}(T)|\) and \(|2M_\text {F}(T)| \gg |\Delta M(T)|\), for fixed values of the angles \(\Theta \) and \(\Phi \), are respectively satisfied for any temperature in the interval \(T_0\le T\le T_i\), only when \(T=T_i\)

^{3}of \(B_1(T)\) with the initial condition \({\tilde{M}}^{(0)}(T_i)=\varvec{I}_{4\times 4}\). After doing several calculations we get

*n*order matrix equation in (38) and let us define \(n-1=m\). Next, let us note that the non zero elements of the matrix \(B_1(T)\) are \(B_{1, (23)}(T)=G_\text {F}(T)\) and \(B_{1, (32)}(T)= -G_\text {F}(T)\). On the other hand the non zero elements of the matrix \(\epsilon B_2(T)\) are \(\epsilon B_{2, (24)}(T)=G_\text {C}(T)\), \(\epsilon B_{2, (42)}(T)=-G_\text {C}(T)\), \(\epsilon B_{2, (34)}(T)=-\Delta G(T)\) and \(\epsilon B_{2, (43)}(T)=\Delta G(T)\). After these consideration, we obtain the following results for the components of the

*n*order matrix equation in (38)

*i*) and after sum it with the second equation. Then we get

*m*are known. Since we already know the elements of \({\tilde{M}}_{ij}\) at the order \(m=0\) as given in (39), we can recursively calculate those at the order \(m+1\). Let us define for simplicity

### 4.3 Another perturbative solution

*T*, is equivalent to \(\sin (\Theta )\sin (\Phi )\ne 0\). This fact tells us that the condition on the Faraday effect term \({\mathcal {M}}_\text {F}(T)\ne 0\) is not necessary in order to find the solution (50) and that the condition \({\mathcal {M}}_\text {F}(T)\ne 0\) comes out only in the regular perturbation theory. On the other hand, in this section in order to use the Neumann series expansion we required that \(\left| \int _{T_i}^T dT^\prime {\bar{M}}_{ij}(T^\prime ) \right| <1\) or equivalently that \(| L_4^{(0)}(T)|<1\) and \(| K_4^{(0)}(T)|<1\). For example we may note

## 5 Degree of circular polarization

In the previous section we found perturbative solutions of the equations of motion of the Stokes parameters in two different regimes by using perturbation theory. In this section, we focus on our attention on generation of circular polarization, where in specific, we study the expected degree of circular polarization at present time and the expected rotation angle of the CMB polarization plane. We separate our analysis by first studying the solutions found in Sect. 4.1 and second study those found in Sect. 4.2. In what follows, we consider the evolution of the CMB polarization and rotation angle of the polarization plane starting from the decoupling epoch at the temperature \(T=T_i\) until at the present time at the temperature \(T=T_0\). Moreover, we consider the CMB at the decoupling epoch partially polarized where it acquires only a linear polarization due to the Thomson scattering off the CMB photons on electrons with no initial circular polarization, namely \(Q_i\ne 0, U_i\ne 0\) and \(V_i=0\) as studied in Ref. [19].

### 5.1 Case when \(|{\mathcal {M}}_\text {F}(T_0|< 1, |{\mathcal {M}}_\text {C}(T_0)|< 1,|\Delta {\mathcal {M}}(T_0)|< 1\).

*r*being a parameter which can have either sign and which value is not a priori known. In addition, we have chosen those values of \(B_{e0}\) and \(\nu _0\) that satisfy the constraints (32). Usually if the stronger constraint on the Faraday effect term is satisfied, namely the first constraint on the left hand side in (32), the remaining two stronger constraints which arise from \(|{\mathcal {M}}_\text {C}(T_0)|<1\) and \(|\Delta {\mathcal {M}}(T_0)|<1\) are also satisfied. We may observe from Figs. 2 and 3 that \(P_C^\text {rms}(T_0)\) is usually a very small quantity where it gets bigger values for smaller values of \(\nu _0\) and bigger values of \(B_{e0}\). The main reason why \(P_C^\text {rms}(T_0)\) is small is because we are working under the constraints (32) where there is not too much choice on the values of \(\nu _0\) and \(B_{e0}\) which would give much large values of \(P_C^\text {rms}(T_0)\). The main reason for this situation is because of the constraint \(0<|{\mathcal {M}}_\text {F}(T_0)|<1\) gives very tight constraints on \(\nu _0\) and \(B_{e0}\).

### 5.2 Case when \(|{\mathcal {M}}_\text {F}(T_0)|= 0\) and \(|{\mathcal {M}}_\text {C}(T_0)|< 1,|\Delta {\mathcal {M}}(T_0)|< 1\)

In the case when \(|{\mathcal {M}}_\text {F}(T_0)|= 0\) and \(|{\mathcal {M}}_\text {C}(T_0)|< 1,|\Delta {\mathcal {M}}(T_0)|< 1\) the constraints on \(\nu _0\) and \(B_{e0}\) are much less stringent than in the previous section. In fact, for finite values of \(\nu _0\) and \(B_{e0}\) which interest us, the only possibility for the condition \(|{\mathcal {M}}_\text {F}(T_0)|= 0\) to hold is only when \(|\sin (\Theta )\sin (\Phi )|= 0\) which occurs either when \(\Theta = n \pi \) or \(\Phi = n \pi \) with \(n\ge 0\). In both cases the direction of the magnetic field is perpendicular to the direction of photon propagation where \({\mathcal {M}}_\text {F}(T_0)= 0\) and also \({\mathcal {M}}_\text {C}(T_0)= 0\). Consequently, the constraints in (32) reduce to only the constraint \(\left( \text {Hz}/\nu _0\right) ^3 \left( B_{e0}/\text {G} \right) ^2 < 8.35 \times 10^{-39}\) which correspond to the stronger constraint on \(|\Delta {\mathcal {M}}(T_0)|<1\), namely the region within the black line in Fig. 1b.

*r*| are shown. We may observe in Fig. 4 that for higher values of \(B_{e0}\) and lower values of \(\nu _0\), the acquired degree of circular polarization of the CMB is quite substantial and be comparable with that of the linear polarization for some values of the parameters. For example, as we can see from Fig. 4, for \(B_{e0}=8 \times 10^{-8}\) G, we get \(P_C(T_0)\simeq 7.65 \times 10^{-7}\) for \(\nu _0=10^8\) Hz and \(P_C(T_0)\simeq 7.65 \times 10^{-10}\) for \(\nu _0=10^9\) Hz. It is worth to point out that the expression (58) can also be obtained by using the perturbative approach used in the previous section.

### 5.3 Case when \(|{\mathcal {M}}_\text {F}(T_0)|\ge 1\) and \(|{\mathcal {M}}_\text {C}(T_0)|< 1,|\Delta {\mathcal {M}}(T_0)|< 1\)

*s*and

*z*are complex numbers and \(\Gamma (s, z)=\int _{z}^{\infty } dx\, x^{s-1} e^{-x}\) is the incomplete Euler Gamma function. By using (62) into expression (61) and by summing all together we get

One thing which is worth to mention is that (66) is valid as far as the condition (65) is satisfied and when \(\sin (\Theta )\sin (\Phi ) \ne 0\). The latter condition still appears for the simple fact that we had to divide by \({\mathcal {A}}\) in the integration procedure. Another important fact is that \(P_C(T_0)\) in (66) can assume zero values as far as \(\sin (\Theta )\sin (\Phi ) \ne 0\) and \(r \sin (2\Theta ) \cos (\Phi ) + 2 \left[ -\sin ^2(\Theta )\cos ^2(\Phi ) +\cos ^2(\Theta )\right] =0\). In Fig. 6 plots of the degree of circular polarization [given by expression (66)] as a function of the CMB frequency \(\nu _0\) for various values of the parameters are shown. The values of the parameters have been chosen in such a way that expression (65) is satisfied for \(T=T_i\). As we may observe from Fig. 6 the presence of the Faraday effect significantly reduces the degree of circular polarization by many orders of magnitude with respect to the case of absent Faraday effect. In Fig. 7a we show the plot of \(P_C(T_0)\) as a function of the angle \(\Theta \) as given in (66) for some given values of the angle \(\Phi \). We can see that in the case when \(\Theta \rightarrow 0\) the degree of circular polarization significantly increases by many orders of magnitude. In this case we recover the results of the previous section where the magnetic filed has been assumed to be purely transverse, namely \({\mathcal {M}}_\text {F}(T_0)=0\). The cusp-like behaviour that appear in the plots in Fig. 7a correspond to those values of \(\Theta \) where \(P_C(T_0)= 0\) due to the trigonometric function \(r \sin (2\Theta ) \cos (\Phi ) + 2 \left[ -\sin ^2(\Theta )\cos ^2(\Phi ) +\cos ^2(\Theta )\right] =0\). However, these points are not real cusps but simply arise due to the fact that we have to take the absolute value of \({\tilde{V}}(T_0)\) in order to calculate \(P_C(T_0)\). In Fig. 7b plot of the degree of circular polarization \(P_C(T_0)\) obtained by using expression (59) for \(X_e(T)\simeq {\bar{X}}_e=0.023\) are shown. These plots essentially correspond to the case when we include all terms which do appear in the integrals (63)–(64) and to their similar integral functions. The oscillating nature of the plots arises due to the fact that for higher values of \(\nu _0\) and \(B_{e0}\) do contribute to the integrals the terms proportional to \(\sin (x_i)\) and \(\cos (x_i)\) which are fast oscillating functions of the parameters.

## 6 Rotation angle of the polarization plane

*T*the rotation angle of the polarization plane is given by

*T*and it is the quantity which interests us. Since for the frequency range of interest in this work, the magnitude of the effects which we study are in general small, namely \(|{\mathcal {M}}_\text {F}(T)|<1, |\Delta {\mathcal {M}}(T)|<1, |{\mathcal {M}}_\text {C}(T)|<1\), and because experimentally \(\delta \psi (T_0)\) is constrained to a small quantity (in radians), we expect that the rotation angle of the CMB polarization plane from decoupling epoch until at present to be a small quantity \(|\delta \psi (T)|\ll 1\). In this case by using the trigonometric identity we can write

*r*. In Fig. 8 plots of the present epoch CMB rotation angle of the polarization plane \(\delta \psi _0=\delta \psi (T_0)\) given by expression (74) are shown for various values of the parameters. We may note that substantial rotation of the polarization plane occurs only at low frequencies and for higher values of \(B_{e0}\). On the other hand for higher values of the frequency and lower values of the magnetic field amplitude \(\delta \psi _0\) is extremely small.

^{4}\(|{\mathcal {M}}_\text {F}(T)|\ge 1\) and as far as the stronger conditions for \(|\Delta {\mathcal {M}}(T)|, |{\mathcal {M}}_\text {C}(T)| <1\) are satisfied, the rotation angle of the CMB polarization plane is given by \(\delta \psi (T) \simeq {\mathcal {M}}_\text {F}(T)/2\) where the contribution of the CM effect is sub-leading. However, if we are interested to take simply the average value of \(\delta \psi (T) \simeq {\mathcal {M}}_\text {F}(T)/2\) over \(\Theta \) and \(\Psi \), the Faraday effect gives on average zero contribution. So, in the case of simple average value, we also need to keep the second order terms of the CM effect in (76), which gives a small contribution to \(\delta \psi \) but not zero. On the other hand, if we still insist on average values over the angles \(\Theta \) and \(\Phi \), the Faraday effect dominates when we consider the root mean square of \(\delta \psi (T)\) as far as \(|{\mathcal {M}}_\text {F}(T)| \gg |\Delta {\mathcal {M}}(T)|, |{\mathcal {M}}_\text {C}(T)| \). In this case we explicitly have

^{5}on \(|\delta \psi _0|\), represented by the black points, are within the grey region between the magnetic field values \(10^{-8}\) G \(\le B_{e0} \le 8\times 10^{-8}\) G. The only exception is the constraint found by WMPA9 where the magnetic field amplitude corresponding to \(|\delta \psi _0|=0.36^\circ \) is by Eq. (77) \(B_{e0}= 7.47\times 10^{-10}\) G. In Fig. 9b plots of the root mean square \(\langle \delta \psi _0 \rangle \) as a function of the CMB frequency are shown. Similarly to the Fig. 9a, the black points represent the constraints on \(|\delta \psi _0|\). For example, the QUaD constraint on \(|\delta \psi _0|=0.83^\circ \) is consistent with \(B_{e0}=1.38\times 10^{-8}\) G, while the BICEP 1 constraint on \(|\delta \psi _0|=2.77^\circ \) is consistent with \(B_{e0}=3.4 \times 10^{-8}\) G.

## 7 Conclusions

In this work, we have studied the generation of the CMB circular polarization and rotation angle of the CMB polarization plane due to the CM effect in a large-scale cosmic magnetic field. We worked with the Stokes parameters and derived a system of differential equations for their evolution in an expanding universe. In the equations governing the evolution of the Stokes vector, we included all standard magneto-optic effects which manifest in a magnetized plasma which are the CM and Faraday effects. Then we looked to solutions of the equations of motion of the Stokes parameters in different regimes by using several perturbative approaches such as the regular perturbation theory and the Neumann series expansion. The equations of motion that we found in (18) are a generalization to the equations of motion found in Ref. [19] in the case of an arbitrary direction of \(\varvec{B}_{e}\) with respect to the photon direction of propagation and for arbitrary magnetic field profile. For an arbitrary direction of \(\varvec{B}_e\), the equations of motion (18) include two additional terms proportional to \(M_\text {C}(T)\), which, would be absent in the particular case when the magnetic field \(\varvec{B}_e\) is in the same plane with the wave-vector \(\varvec{k}\). These two terms proportional to \(M_\text {C}(T)\) make possible the mixing of the *Q*(*T*) and *V*(*T*) parameters with each other.

The magnitude of the degree of circular polarization for the CM effect depends on several parameters where the most important ones are the CMB frequency \(\nu _0\) and the magnetic field amplitude \(B_{e}(\varvec{x}, t_0)\). In addition, other parameters which play also an important role are the angles \(\Theta \) and \(\Phi \). Consequently, depending on the values of these parameters, in this work, we divided our analysis of the CM effect in three major regimes. In the regime where \(|{\mathcal {M}}_\text {C, F}(T_0)|<1\) and \(|\Delta {\mathcal {M}}(T_0)|<1\), the degree of circular polarization assumes the lowest values as shown in Figs. 2 and 3, where at best its value reaches \(P_C(T_0)\simeq 10^{-17}\). The reason for such low values of \(P_C(T_0)\) stands from the fact that the condition \(|{\mathcal {M}}_\text {F}(T_0)|<1\), drastically restricts the values of the parameters \(\nu _0\) to very high frequencies and the values of \(B_{e0}\) to very low ones.

In the case when the Faraday effect is completely absent, which happens when the direction of \(\varvec{B}_{e}\) is perpendicular to the direction of propagation of the CMB photons, we essentially have that \({\mathcal {M}}_\text {F}(T)=0\) and the generation of circular polarization is maximal. The absence of the Faraday effect for such specific configuration results in an enhancement of the generation of the CMB circular polarization. For such case, we have found in Sect. 5.2 that the degree of circular polarization can reach values close to the CMB degree of linear polarization in the CMB low-frequency part of the spectrum. The maximum values of the degree of circular polarization are reached in the case when we concentrate at the frequency \(\nu \simeq 10^{8}\) Hz, where depending on the magnetic field amplitude, the degree of circular polarization is in the range \(1.19 \times 10^{-10}\lesssim P_C(T_0)\lesssim 7.65\times 10^{-7}\) for magnetic field values \(10^{-10}\) G \(\le B_{e0}\le 8\times 10^{-8}\) G. These results are plotted in Figs. 4 and 5 for different values of the parameters.

In the case when the Faraday effect is present and in particular when \(|{\mathcal {M}}_\text {F}(T)|\ge 1\), the generation of the CMB circular polarization is strongly suppressed with respect to the case of absent Faraday effect where \({\mathcal {M}}_\text {F}(T)=0\). However, the generation of the CMB circular polarization in the case when \(|{\mathcal {M}}_\text {F}(T)|\ge 1\) is usually much efficient than that in the case when \(0<|{\mathcal {M}}_\text {F}(T)|< 1\) which we studied in Sect. 5.1. Even in the case \(|{\mathcal {M}}_\text {F}(T)|\ge 1\) the degree of circular polarization depends on \(B_{e0}\) and \(\nu _0\), where in some specific range of these parameters, the degree of circular polarization scales with the frequency as \(P_C(T_0) \propto \nu _0^{-1}\) and with the magnetic field amplitude as \(P_C(T_0) \propto B_{e0}\), see for example the expression (66). As shown in Fig. 6b, the degree of circular polarization can reach values in the range \( 10^{-14}\lesssim P_C(T_0)\lesssim 6\times 10^{-12}\) for magnetic field values \(10^{-9}\) G \(\le B_{e0}\le 8\times 10^{-8}\) G at the frequency \(\nu _0\simeq 10^{8}\) Hz and \(|Q_i|=10^{-6}\) and \(|r|=1\). At the frequency \(\nu _0\simeq 10^9\) Hz, the values of \(P_C(T_0)\) decrease exactly by an order of magnitude since \(P_C(T_0) \propto \nu _0^{-1}\) in the frequency range considered. On the other hand, \(P_C(T_0) \propto |r|\), so, higher values of |*r*| give higher values of \(P_C(T_0)\) and vice-versa for smaller values of |*r*|.

Apart from generating circular polarization, the CM effect also generates linear polarization and this fact is evident in all expressions of the Stokes parameters that we found in Sect. 4. In connection with linear polarization, in this work, we have studied the rotation angle of the CMB polarization plane due to the CM effect in the case when the Faraday effect is absent and in combination with the Faraday effect when it is present. In the case when it is present only the CM effect, the rotation angle is \(\delta \psi (T_0) \propto \nu _0^{-6} B_{e0}^4\) and consequently, significant rotation of the polarization plane occurs in the low-frequency part of the CMB spectrum and for higher values of the magnetic field amplitude. We have found in Sect. 6 that at \(\nu _0\simeq 10^8\) Hz, the rotation angle in units of degrees is in the range \(10^{-3}\le \delta \psi (T_0)\le 1\) for magnetic field amplitude in the range \(10^{-9}\) G \(\le B_{e0}\le 8\times 10^{-8}\) G, \(|r|=1\) and \(|Q_i|=10^{-6}\), see Fig. 8. For higher frequencies, \(|\delta \psi (T_0)|\) acquires extremely smaller values which are uninteresting for any practical purpose.

In the case when the rotation angle \(\delta \psi (T_0)\) is due to a combination of the CM and Faraday effects the situation slightly changes with respect to the case of absent Faraday effect. If we are interested in taking the average value of \(\delta \psi (T_0)\), the Faraday effect gives null contribution while the CM effect gives in average almost the same contribution as it does in the case of absent Faraday effect. If we take the root mean square of \(\delta \psi \), the Faraday effect usually dominates over the CM effect in the case when it is present unless the magnetic field is almost transverse with respect to the direction of propagation. One important aspect is that in case we take the root mean square of \(\delta \psi (T_0)\), the Faraday effect generates significant rotation of the polarization plane depending on the CMB frequency and magnetic field amplitude. As shown in Fig. 9, the Faraday effect generates substantial rotation of the polarization plane especially in the low-frequency part especially for \(\nu _0\lesssim 10^{10}\) Hz. In the high-frequency part of the spectrum, namely for frequencies above 10 GHz, the rotation angle is still large depending on the magnetic field amplitude. An interesting fact is that most of the constraints on \(\delta \psi (T_0)\) experimentally found correspond to magnetic field amplitudes in the range \(10^{-8}\) G \(\lesssim B_{e0}\lesssim 8\times 10^{-8}\) G.

If we have to consider current limits on \(\delta \psi (T_0)\) as a potential indicator of the existence of the large-scale cosmic magnetic field and consequently a non zero rotation angle of the polarization, these limits would allow us to make some predictions on the signal of the circular polarization due to the CM effect. Indeed, if we consider the hypothesis that the rotation angle is due to the Faraday effect only (root mean square value) and that most experimental constraints on \(\delta \psi (T_0)\) would suggest a magnetic field with amplitude approximately \(10^{-8}\) G \(\lesssim B_{e0}\lesssim 8\times 10^{-8}\) G, we would have that the signal of circular polarization for these values of \(B_{e0}\) would be quite substantial. For these values of the magnetic field, in the case when the field is perpendicular to the photon direction of propagation, we would have a circular polarization signal at present in the range \(3\times 10^{-8}\) K \(\lesssim |V(T_0)|\lesssim 2 \times 10^{-6}\) K at \(\nu _0\simeq 10^{8}\) Hz and a signal of \(3\times 10^{-11}\) K \(\lesssim |V(T_0)|\lesssim 2 \times 10^{-9}\) K at \(\nu _0\simeq 10^9\) Hz, see Fig. 4. In finding these values we used \(|V(T_0)|=P_C(T_0)I(T_0)\) with \(I(T_0)=T_0\) due to most CMB physics conventions. In the case when the magnetic field is not perpendicular, the signal of the circular polarization is reduced by many orders of magnitude and is in the range \(2.7\times 10^{-14}\) K \(\lesssim |V(T_0)|\lesssim 1.6\times 10^{-11}\) K at \(\nu _0=10^8\) Hz and depending on the angles \(\Theta \) and \(\Phi \), see Fig. 6.

Based on the arguments presented so far, it seems quite plausible that the CM effect is probably the most substantial effect in generating CMB circular polarization. However, the strongest signal of the circular polarization is located in the CMB frequency range \(10^{8}\) Hz \(\lesssim \nu _0\lesssim 10^9\) Hz. In the high-frequency range the signal of circular polarization due to the CM effect is much smaller than in the low-frequency part of the spectrum, but still, the signal is not negligible and can be comparable with the vacuum polarization circular polarization signal in a cosmic magnetic field and to that due to free photon-photon scattering. If we assume that there is not any major difficulty in arranging an experiment aiming to detect the circular polarization in the low-frequency part of the CMB spectrum, then it is quite logical to concentrate our attention in this frequency part of the spectrum where the signal is the strongest and more likely to be detected in a relatively short time.

## Footnotes

- 1.
Here we are assuming that in most cases of magnetized plasmas we have that the magnetic permeability tensor \(\mu _{ij}\simeq 1\).

- 2.
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.

- 3.
Here we are assuming that the reader is familiar on how to find the matrix exponential.

- 4.
In the case when \(0<|{\mathcal {M}}_\text {F}(T)|<1\), the situation is slightly more complicated and depending on the values of the parameters either the Faraday effect term or CM effect term, gives the biggest contribution to \(\delta \psi (T)\). In this case one can use directly the results obtained in expressions (69), (72)–(73) in order to calculate \(\delta \psi \).

- 5.
Here we consider for simplicity only the value of \(|\delta \psi _0|\) without the error and assume that the magnetic field configuration is statistically the same in every direction in the sky.

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