**8. Cosmic polarization rotation**

For the electromagnetism in gravity with an effective pseudoscalar-photon interaction discussed in the last section, the electromagnetic wave propagation equation is governed by equation (99). In a local inertial (Lorentz) frame of the *g*-metric, it is reduced to

60 Trends in Electromagnetism – From Fundamentals to Applications

should be a pseudoscalar (function) since *εijkl* is a pseudotensor. The pseudoscalar(scalar) photon interaction part (or the nonmetric part) of the Lagrangian density of this theory is

 *L*(φγγ) = *L*(NM) = - (1/16π) *φ eijklFijFkl* = - (1/4π) *φ,i eijklAjAk,l* (mod div), (98) where 'mod div' means that the two Lagrangian densities are related by integration by parts in the action integral. This term gives pseudoscalar-photon-photon interaction in the quantum regime and can be denoted by *L*(φγγ). This term is also the *ξ*-term in the PPM Lagrangian density *LPPM* with the *φ* ≡ (1/4)*ξΦ* correspondence. The Maxwell equations (Ni

> ϕ*,l = -4*π*ji*

where the derivation ; is with respect to the Christoffel connection of the metric. The Lorentz force law is the same as in metric theories of gravity or general relativity. Gauge invariance and charge conservation are guaranteed. For discussions on the tests of charge conservation, and on the limits of differences in active and passive charges, please see Lämmerzahl et al.

The rightest term in equation (99) is reminiscent of Chern-Simons (1974) term *eαβγAαFβγ*. There are two differences: (i) Chern-Simons term is in 3 dimensional space; (ii) Chern-Simons term in the integral is a total divergence (Table 1). However, it is interesting to notice that the cosmological

*,* (99)

invarinat

Pseudoscalar-gluon coupling

(1973, 1974, 1977) Pseudoscalar-photon coupling

(1990) External constant vector coupling

*Fik;k + εikml Fkm*

(2005, 2007). The modified Maxwell equations (99) are also conformally invariant.

time may be defined through the Chern-Simons invariant (Smolin and Soo, 1995).

*eijkl φ Fij Fkl* <sup>4</sup>Ni

*eijkl Vi Aj Fkl* <sup>4</sup>Carroll-Field-Jackiw

Table 1. Various terms in the Lagrangian and their meaning.

special case of the term *eijkl φ Fij Fkl* (mod div) with *φ,i =* - ½*Vi*.

*eijkl φ FQCDij FQCDkl* 4

**8. Cosmic polarization rotation** 

Term Dimension Reference Meaning *<sup>e</sup>αβγ <sup>A</sup><sup>α</sup> <sup>F</sup>βγ* 3 Chern-Simons (1974) Intergrand for topological

> Peccei-Quinn (1977) Weinberg (1978) Wilczek (1978)

A term similar to the one in equation (98), axion-gluon interaction term, occurs in QCD in an effort to solve the strong CP problem (Peccei & Quinn, 1977; Weinberg, 1978; Wilczek, 1978). Carroll, Field and Jackiw (1990) proposed a modification of electrodynamics with an additional *eijkl Vi Aj Fkl* term with *Vi* a constant vector (See also Jackiw, 2007). This term is a

Various terms in the Lagrangians discussed in this subsection are listed in Table 1. Empirical

For the electromagnetism in gravity with an effective pseudoscalar-photon interaction discussed in the last section, the electromagnetic wave propagation equation is governed by

tests of the pseudoscalar-photon interaction (98) will be discussed in next section.

equation (99). In a local inertial (Lorentz) frame of the *g*-metric, it is reduced to

1973, 1977) from Eq. (97) become

$$F^{ik}{}\_{,k} + e^{ikml} \; F\_{km} \; \!\!\/ p\_{,l} = 0. \tag{100}$$

Analyzing the wave into Fourier components, imposing the radiation gauge condition, and solving the dispersion eigenvalue problem, we obtain *k* = *ω* + (*nμφ,*<sup>μ</sup> + *φ,0*) for right circularly polarized wave and *k* = *ω* – (*nμφ,*<sup>μ</sup> + *φ,0*) for left circularly polarized wave in the eikonal approximation (Ni 1973). Here *nμ* is the unit 3-vector in the propagation direction. The group velocity is

$$
v\_{\mathcal{S}} = \partial \alpha / \partial k = 1,\tag{101}$$

which is independent of polarization. There is no birefringence. For the right circularly polarized electromagnetic wave, the propagation from a point P1 = {x(1)*<sup>i</sup>* } = {x(1)0; x(1)μ} = {x(1)0, x(1)1, x(1)2, x(1)3} to another point P2 = {x(2)*<sup>i</sup>* } = {x(2)0; x(2)μ} = {x(2)0, x(2)1, x(2)2, x(2)3} adds a phase of *α* = *φ*(P2) - *φ*(P1) to the wave; for left circularly polarized light, the added phase will be opposite in sign (Ni 1973). Linearly polarized electromagnetic wave is a superposition of circularly polarized waves. Its polarization vector will then rotate by an angle *α*. Locally, the polarization rotation angle can be approximated by

$$\rho = \varrho(\mathbf{P}\_2) \cdot \varrho(\mathbf{P}\_1) = \Sigma\_{\mathbf{i}=0} \,^3 \left[ \varrho\_{,i} \times (\mathbf{x}\_{(2)} \,^i \cdot \mathbf{x}\_{(1)}) \right] = \Sigma\_{\mathbf{i}=0} \,^3 \left[ \varrho\_{,i} \Delta \mathbf{x} \right] = \varrho\_{,0} \Delta \mathbf{x}^0 + \left[ \Sigma\_{\mathbf{i}=1} \,^3 \varrho\_{,i} \Delta \mathbf{x}^0 \right]$$

$$= \left. \begin{pmatrix} \mathbf{i} \end{pmatrix} \right\rangle \Sigma\_{\mathbf{i}=0} \,^3 \left[ V\_{,i} \Delta \mathbf{x} \right] = \left. \begin{pmatrix} \mathbf{i} \end{pmatrix} \,^3 V\_{0} \Delta \mathbf{x}^0 - \begin{pmatrix} \mathbf{i} \end{pmatrix} \begin{bmatrix} \Sigma\_{\mathbf{i}=1} \,^3 V\_{,\mathbf{i}} \Delta \mathbf{x} \mathbf{i} \end{bmatrix} \tag{102}$$

The rotation angle in (102) consists of 2 parts -- *φ,0*Δx0 and [Σμ=13*φ,*μΔxμ]. For light in a local inertial frame, |Δxμ| = |Δx0|. In Fig. 4, space part of the rotation angle is shown. The amplitude of the space part depends on the direction of the propagation with the tip of magnitude on upper/lower sphere of diameter |Δxμ| × |*φ,μ*|. The time part is equal to Δx0 *φ,*0. (∇*φ* ≡ [*φ,μ*]) When we integrate along light (wave) trajectory in a global situation, the total polarization rotation (relative to no *φ*-interaction) is again Δ*φ = φ*<sup>2</sup> – *φ*1 for *φ* is a scalar field where *φ*1 and *φ*2 are the values of the scalar field at the beginning and end of the wave. When the propagation distance is over a large part of our observed universe, we call this phenomenon cosmic polarization rotation (Ni, 2008, 2009a, 2010).

In the CMB polarization observations, there are variations and fluctuations. The variations and fluctuations due to scalar-modified propagation can be expressed as δ*φ*(2) - δ*φ*(1), where 2 denotes a point at the last scattering surface in the decoupling epoch and 1 observation point. δ*φ*(2) is the variation/fluctuation at the last scattering surface. δ*φ*(1) at the present observation point is zero or fixed. Therefore the covariance of fluctuation <[δ*φ*(2) - δ*φ*(1)]2> gives the covariance of δ*φ*2(2) at the last scattering surface. Since our Universe is isotropic to ~ 10-5, this covariance is ~ (ς× 10-5)2 where the parameter ς depends on various cosmological models. (Ni, 2008, 2009a, 2010)

Now we must say something about nomenclature.

Birefringence, also called double refraction, refers to the two different directions of propagation that a given incident ray can take in a medium, depending on the direction of polarization. The index of refraction depends on the direction of polarization.

Dichroic materials have the property that their absorption constant varies with polarization. When polarized light goes through dichroic material, its polarization is rotated due to difference in absorption in two principal directions of the material for the two polarization components. This phenomenon or property of the medium is called dichroism.

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 63

Feng, Li, Xia, Chen & Zhang (2006) -105 ± 70 WMAP3 (Spergel *et al* 2007) &

Ni (2005a, b) ±100 WMAP1 (Bennett *et al* 2003)

Liu, Lee & Ng (2006) ±24 BOOMERANG (B03) (Montroy *et al* 2006) Kostelecky & Mews (2007) 209 ± 122 BOOMERANG (B03) (Montroy *et al* 2006) Cabella, Natoli & Silk (2007) -43 ± 52 WMAP3 (Spergel *et al* 2007) Xia, Li, Wang & Zhang (2008) -108 ± 67 WMAP3 (Spergel *et al* 2007) &

Komatsu *et al* (2009) -30 ± 37 WMAP5 (Komatsu *et al* 2009) Xia, Li, Zhao & Zhang (2008) -45 ± 33 WMAP5 (Komatsu *et al* 2009) &

Kostelecky & Mews (2008) 40 ± 94 WMAP5 (Komatsu *et al* 2009) Kahniashvili, Durrer & Maravin (2008) ± 44 WMAP5 (Komatsu *et al* 2009)

Table 2. Constraints on cosmic polarization rotation from CMB polarization observations.

Both magnetic field and potential new physics affect the propagation of CMB propagation and generate BB power spectra from EE spectra of CMB. The Faraday rotation due to magnetic field is wavelength dependent while the cosmic polarization rotation due to effective pseudoscalar-photon interaction is wavelength-independent. This property can be used to separate the two effects. With the tensor mode generated by these two effects measured and subtracted, the remaining tensor mode perturbations could be analyzed for signals due to primordial (inflationary) gravitational waves (GWs). In Ni (2009a,b), we have discussed the direct detectability of these primordial GWs using space GW

Observations of radio and optical/UV polarization of radio galaxies are also sensitive to measure/test the cosmic polarization rotation, and give comparable constraints of tens of mrad. These observations have the capability of determining the polarization rotation in

We have looked at the foundations of electromagnetism in this chapter. For doing this, we have used two approaches. The first one is to formulate a Parametrized Post-Maxwellian framework to include QED corrections and a pseudoscalar photon interaction. We discuss various vacuum birefringence experiments – ongoing and proposed -- to measure these parameters. The second approach is to look at electromagnetism in gravity and various experiments and observations to determine its empirical foundation. We found that the foundation is solid with the only exception of a potentially possible pseudoscalar-photon interaction. We discussed its experimental constraints and look forward to more future

various directions. For a recent work, see di Serego Alighieri et al. (2010).

Wu *et al* (2009) 9.6 ± 14.3 ± 8.7 QuaD (Pryke *et al* 2009) Brown *et al.*(2009) 11.2 ± 8.7 ± 8.7 QuaD (Brown *et al* 2009) Komatsu *et al.*(2011) -19 ± 22 ± 26 WMAP7 (Komatsu *et al* 2011)

[mrad] Source data

BOOMERANG (B03) (Montroy *et al* 2006)

BOOMERANG (B03) (Montroy *et al* 2006)

BOOMERANG (B03) (Montroy *et al* 2006)

Analysis Constraint

[See Ni (2010) for detailed references.]

detectors.

**9. Outlook** 

experiments.

Fig. 4. Space contribution to the local polarization rotation angle -- [Σμ=13*φ,* μΔxμ] = |∇*φ*| cos *θ* Δx0. The time contribution is *φ,0* Δx0. The total contribution is (|∇*φ*| cos *θ* + *φ,0*) Δx0. (Δx0 > 0).

In a medium with optical activity, the direction of a linearly polarized beam will rotate as it propagates through the medium. A medium subjected to magnetic field becomes optically active and the associated polarization rotation is called Faraday rotation.

Cosmic polarization rotation is neither dichroism nor birefringence. It is more like optical activity, with the rotation angle independent of wavelength. Conforming to the common usage in optics, one should not call it cosmic birefringence -- *a misnomer*.

Now we review and compile the constraints of various analyses from CMB polarization observations.

In 2002, DASI microwave interferometer observed the polarization of the cosmic background (Kovac et al., 2002). E-mode polarization is detected with 4.9 σ. The TE correlation of the temperature and E-mode polarization is detected at 95% confidence. This correlation is expected from the Raleigh scattering of radiation. However, with the (pseudo)scalar-photon interaction under discussion, the polarization anisotropy is shifted differently in different directions relative to the temperature anisotropy due to propagation; the correlation will then be downgraded. In 2003, from the first-year data (WMAP1), WMAP found that the polarization and temperature are correlated to more than 10 σ (Bennett *et al* 2003). This gives a constraint of about 10-1 for *Δφ* (Ni, 2005a, 2005b).

Further results and analyses of CMB polarization observations came out after 2006. In Table 2, we update our previous compilations (Ni 2008, 2010). Although these results look different at 1 σ level, they are all consistent with null detection and with one another at 2 σ level.


Table 2. Constraints on cosmic polarization rotation from CMB polarization observations. [See Ni (2010) for detailed references.]

Both magnetic field and potential new physics affect the propagation of CMB propagation and generate BB power spectra from EE spectra of CMB. The Faraday rotation due to magnetic field is wavelength dependent while the cosmic polarization rotation due to effective pseudoscalar-photon interaction is wavelength-independent. This property can be used to separate the two effects. With the tensor mode generated by these two effects measured and subtracted, the remaining tensor mode perturbations could be analyzed for signals due to primordial (inflationary) gravitational waves (GWs). In Ni (2009a,b), we have discussed the direct detectability of these primordial GWs using space GW detectors.

Observations of radio and optical/UV polarization of radio galaxies are also sensitive to measure/test the cosmic polarization rotation, and give comparable constraints of tens of mrad. These observations have the capability of determining the polarization rotation in various directions. For a recent work, see di Serego Alighieri et al. (2010).
