NEW CONSTRAINTS ON COSMIC POLARIZATION ROTATION FROM B-MODE POLARIZATION IN THE COSMIC MICROWAVE BACKGROUND

The BICEP2 collaboration (Ade et al. 2014b) has recently detected the cosmic microwave background (CMB) B-mode (tensor mode) polarization and finds an excess power around l ∼ 80 over the gravitational lensing expectation with a significance of more than 5σ, which they interpret as due to inflationary gravitational waves with a tensor-to-scalar ratio r = 0.20$^{+0.07}_{-0.05}$. This result is in apparent contrast with the limit previously set by the Planck collaboration r < 0.11 at 95% CL (Ade et al. 2014a). Three processes can produce B-mode polarization: (1) gravitational lensing from E-mode polarization (Zaldarriaga & Seljak 1997), (2) local quadrupole anisotropies in the CMB within the last scattering region by large scale gravitational waves (Polnarev 1985), and (3) cosmic polarization rotation (CPR)³ due to pseudoscalar–photon interaction (Ni 1973; for a review, see Ni 2010). CPR is currently constrained to be less than about a couple of degrees by measurements of the linear polarization of radio galaxies and of the CMB (see di Serego Alighieri 2011 for a review). However, if CPR exists within the current upper limits, it would produce a non-negligible B-mode CMB polarization. The BICEP2 Collaboration (Ade et al. 2014b) has not considered this latter component in their model. To look for new constraints on the CPR and into the robustness of the BICEP2 fit, we include the CPR effect in the model. TE, TB, and EB correlations⁴ potentially give mean values of the CPR angle 〈α〉, while the contribution of CPR effects to the B-mode power potentially gives 〈α〉² plus the variations of the CPR angle squared 〈δα²〉. This method of constraining the CPR is new, is complementary to previous tests, and adds a new constraint for the sky area observed, although its application to current data is limited by the uniform angle derotation, which is applied to the measured CMB Q and U maps to compensate for insufficient calibrations of the polarization angle (Keating et al. 2013). We also include in our analysis of the B-mode power spectra the data available for l ⩾ 200, in particular, from SPTpol (Hanson et al. 2013) and POLARBEAR (Ade et al. 2014c), in addition those from BICEP2 for l ⩽ 335.

In Section 2, we review the pseudoscalar–photon interaction, its modifications to the Maxwell equations, and the associated electromagnetic propagation effect on polarization rotation. In Section 3 we present the results of our fits. In Section 4 we review and discuss various constraints on CPR, and in Section 5 we discuss a few issues and present an outlook for the future.

The Einstein equivalence principle (EEP) assumes local special relativity is a major cornerstone of general relativity and metric theories of gravity. Special relativity sprang from Maxwell–Lorentz electrodynamics. Maxwell equations in terms of field strength F_kl (${\boldsymbol E}$, ${\boldsymbol B}$) and excitation H^ij (${\boldsymbol D}$, ${\boldsymbol H}$) do not need metrics as a primitive concept. The field strength F_kl (${\boldsymbol E}$, ${\boldsymbol B}$) and excitation H^ij (${\boldsymbol D}$, ${\boldsymbol H}$) can all be independently defined operationally (e.g., Hehl & Obukhov 2003). To complete this set of equations, one needs the constitutive relation between the excitation and the field in both macroscopic electrodynamics and in the spacetime theory of gravity:

$$\begin{equation} H^{ij} = (1/2)\chi ^{ijkl} F_{kl}. \end{equation} \tag{ 1 }$$

When EEP is observed, this fundamental spacetime tensor density is induced by the metric g^ij of the form

$$\begin{equation} \chi ^{ijkl} = (1/2)(- g)^{1/2} (g^{ik} g^{jl} - g^{il} g^{kj}),\quad (g \equiv (\det g^{ij})^{ - 1}), \end{equation} \tag{ 2 }$$

and the Maxwell equations can be derived from a Lagrangian density. In the local inertial frame, the spacetime tensor has the special relativity form (1/2) (η^ik η^jl − η^il η^kj), with η^il the Minkowski metric. To study the empirical foundation of EEP, it is crucial to explore how the experiments/observations constrain the general spacetime tensor χ^ijkl to the general relativity/metric form.

Since both H^ij and F_kl are antisymmetric, χ^ijkl must be antisymmetric in i and j, and k and l. Hence the constitutive tensor density χ^ijkl has 36 independent components, and can be decomposed into the principal part (P), axion part (Ax), and Hehl–Obukhov–Rubilar skewon part (Sk; Hehl & Obukhov 2003). The skewon ^(Sk)χ^ijkl part is antisymmetric in the exchange of index pair ij and kl, satisfies a traceless condition, and has 15 independent components. The principal part and the axion (pseudoscalar) part constitute the parts that are symmetric in the exchange of the index pairs ij and kl. Together they have 21 independent components. The axion (pseudoscalar field) part is totally antisymmetric in all four indices and can be expressed as φ e^ijkl, with φ the pseudoscalar field and e^ijkl the completely antisymmetric Levi–Civita symbol. The principal part then has 20 degrees of freedom.

In constraining the general spacetime tensor from experiments and observations, we notice that EEP is already well tested and can only be violated weakly. Hence, one can start with Equation (2), adding a small general ^(Sk)χ^ijkl to look for a constraint on skewons first. From the dispersion relation it is shown that no dissipation/no amplification in the propagation implies that the additional skewon field must be of type II (Ni 2014). For a type I skewon field, the dissipation/amplification in the propagation is proportional to the frequency and the CMB spectrum would deviate from the Planck spectrum. From the high precision agreement of the CMB spectrum with the 2.755 K Planck spectrum (Fixsen 2009), the type I cosmic skewon field |^(SkI)χ^ijkl| is constrained to less than a few parts of 10⁻³⁵ (Ni 2014). A generic type II skewon field can be constructed from the antisymmetric part of an asymmetric metric and is allowed.

EEP implies that photons with the same initial position and direction follow the same world line and keep the same polarization state independent of energy (frequency) and polarization, i.e., no birefringence and no polarization rotation. This is observed to high precision for no birefringence and constrained for no polarization rotation. Since a type II skewon field in the weak-field limit and an axion (pseudoscalar) field do not contribute to the dispersion relation in the eikonal approximation (geometrical optics limit), the no-birefringence condition only limits the principal part of the spacetime tensor ^(P)χ^ijkl to

$$\begin{equation} ^{({\rm P})}\chi ^{ijkl} = 1/2(- h)^{1/2} [h^{ik} h^{jl} - h^{il} h^{kj} ]\psi, \end{equation} \tag{ 3 }$$

where the light cone metric h^ik can be expressed in terms of ^(P)χ^ijkl and the Minkowski metric with $h \equiv (\det h^{ij})^{ - 1}$ (Ni 1983a, 1984). In the skewonless case, the no-birefringence condition is

$$\begin{equation} \chi ^{ijkl} = 1/2(- h)^{1/2} [h^{ik} h^{jl} - h^{il} h^{kj} ]\psi + \varphi e^{ijkl}, \end{equation} \tag{ 4 }$$

(Ni 1983a, 1984). Equation (4) has also been proved more recently without weak-field approximation for a non-birefringent medium in the skewonless case (Favaro & Bergamin 2011; Lämmerzahl & Hehl 2004). Polarization measurements of electromagnetic waves from pulsars and cosmologically distant astrophysical sources yield stringent constraints agreeing with Equation (4) down to 2 × 10⁻³² fractionally (for a review, see Ni 2010).

The constraint of the light cone metric h^il to the matter metric g^il up to a scalar factor can be obtained from Hughes–Drever-type experiments and the constraint on the dilaton ψ to 1 (constant) can be obtained from Eötvös-type experiments to high precision (Ni 1983a, 1983b, 1984).

We note that in looking for the empirical foundation of EEP above, only the axion field and type II skewon field are not well constrained. In Section 4, observational constraints on the pseudoscalar–photon interaction (axion field) are reviewed. These give an upper limit of a few degrees for the mean value part of the difference of the pseudoscalar field at the last scattering surface and at the observation point. In this paper, we look into further constraints/evidence of pseudoscalar–photon interaction in CMB B-mode polarization, especially on the fluctuation/variation part.

The interaction Lagrangian density for the pseudoscalar–photon interaction is

$$\begin{eqnarray} L_{\rm I} ^{({\rm EM - Ax})} &=& - (1/(16\pi))\varphi \;e^{ijkl} F_{ij} \;F_{kl} \nonumber\\ &=& - (1/(4\pi))\;\varphi _{,i} e^{ijkl} A_j A_{k,l} (\rm mod \;{\rm div}), \end{eqnarray} \tag{ 5 }$$

where "mod div" means that the two Lagrangian densities are related by integration by parts in the action integral (Ni 1973, 1974, 1977). If we assume that the φ term is local charge conjugation, parity inversion, and time reversal (CPT) invariant, then φ should be a pseudoscalar (function) since e^ijkl is a pseudotensor density. Note that sometimes one inserts a constant parameter to this term; here we absorb this parameter into the definition of the pseudoscalar field φ. The Maxwell equations (Ni 1973, 1977) become

$$\begin{equation} F^{ik} _{;k} + (- g)^{ - 1/2} {\rm e}^{ikml} F_{km} \varphi _{,l} = 0, \end{equation} \tag{ 6 }$$

where the derivation ";" is with respect to the Christoffel connection. The Lorentz force law is the same as in metric theories of gravity or general relativity. Gauge invariance and charge conservation are guaranteed. The modified Maxwell equations are conformal invariants also.

In a local inertial (Lorentz) frame of the g-metric, Equation (6) is reduced to

$$\begin{equation} F^{ik} _{,k} + e^{ikml} F_{km} \varphi _{,l} = 0. \end{equation} \tag{ 7 }$$

Analyzing the wave into Fourier components, imposing the radiation gauge condition, and solving the dispersion eigenvalue problem, we obtain k = ω + (n^μφ_,μ + φ_,0) for a right circularly polarized wave and k = ω – (n^μφ_,μ + φ_,0) for a left circularly polarized wave (Ni 1973; see Ni 2010 for a review). Here n^μ is the unit 3-vector in the propagation direction. The group velocity is independent of polarization:

$$\begin{equation} v_g = \partial \omega /\partial k = 1. \end{equation} \tag{ 8 }$$

There is no birefringence. This property is well known (Ni 1973, 1984; Hehl & Obukhov 2003; Itin 2013). For the right circularly polarized electromagnetic wave, the propagation from a point P₁ (4-point) to another point P₂ adds a phase of α = φ(P₂) − φ(P₁) to the wave; for the 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.

When we integrate along the light (wave) trajectory in a global situation, the total polarization rotation (relative to no φ interaction) is again α = Δφ = φ(P₂) – φ(P₁) since φ is a scalar field where φ(P₁) and φ(P₂) 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).

In the CMB polarization observations, the variations and fluctuations due to pseudoscalar-modified propagation can be expressed as δφ(P₂) – δφ(P₁), where δφ(P₁) is the variation/fluctuation at the last scattering surface. δφ(P₂) at the present observation point (fixed) is zero. Therefore, the covariance of the fluctuation 〈[δφ(P₂) − δφ(P₁]²〉 is the covariance of δφ²(P₁) at the last scattering surface. Since our universe is isotropic and homogeneous at the last scattering surface to ∼10⁻⁵, this covariance is ∼(ξ × 10⁻⁵)²φ²(P₁), where the parameter ξ depends on various cosmological models (Ni 2008).

In the propagation, E-mode polarization will rotate into B-mode polarization with a sin²2α (≈4α² for small α) fraction of the power. For uniform rotation across the sky, the azimuthal eigenvalue l is invariant under polarization rotation and does not change. For small angles,

$$\begin{equation} \alpha = \varphi ({\rm P}_2) - \varphi ({\rm P}_1) = [\varphi ({\rm P}_2) - \varphi ({\rm P}_1)]_{{\rm mean}} + \delta \varphi ({\rm P}_1) = \left\langle \alpha \right\rangle + \delta \alpha, \end{equation} \tag{ 9 }$$

$$\begin{equation} \underline{\alpha}^2 \equiv \langle {\alpha ^2 } \rangle = ([\varphi ({\rm P}_2) - \varphi ({\rm P}_1)]_{{\rm mean}})^2 + \langle\delta \varphi ^2 ({\rm P}_1)\rangle = \langle \alpha \rangle ^2 + \langle\delta \alpha ^2\rangle, \end{equation} \tag{ 10 }$$

where $\underline{\alpha}$ ≡ 〈α²〉^1/2 is the rms-sum polarization rotation angle, [φ(P₂) − φ(P₁)]_mean = 〈α〉 and δα = δφ(P₁).

In translating the power distribution to the azimuthal eigenvalue variable l, we need to insert a factor ζ(l) ≈ l in front of δ_lφ²(2) to take care of the nonlinear conversion to l due to fluctuations. For a uniform rotation with angle α across the sky, the rotation of the (original) E-mode power $C_l^{EE}$ into the B-mode power $C_l^{{BB},\rm obs}$ and EB correlation power is given by (see, e.g., Keating et al. 2013):

$$\begin{equation} C_l ^{{BB},\rm obs} = C_l ^{BB} \cos ^2 (2\alpha) + C_l ^{EE} \sin ^2 (2\alpha),\quad \end{equation} \tag{ 11a }$$

$$\begin{equation} C_l ^{{EB},\rm obs} = \left(C_l ^{EE} - C_l ^{BB}\right)\;\sin \;(2\alpha)\;\cos \;(2\alpha). \end{equation} \tag{ 11b }$$

The rotation of the (original) B-mode power $C_l^{BB}$ into the E-mode power $C_l^{{EE},\rm obs}$ and EB correlation power is small and negligible in our analysis since the primordial B-mode is small compared with the E-mode power. For CPR fluctuation in a patch of sky, if we consider only the components deriving from the E-mode power $C_l^{EE}$ for l > 20, the rotated-part B-mode l-power spectrum $C_l^{{BB},\rm obs}$ and the rotated EB correlation power $C_l^{{EB},\rm obs}$ for a small CPR angle $\underline{\alpha}$ is accurately given by

$$\begin{equation} C_l ^{{BB},\rm obs} \approx C_l ^{EE} \;\sin ^2 (2\alpha) \approx 4\underline{\alpha}^2 \;C_l ^{EE},\quad\quad\ \end{equation} \tag{ 12a }$$

$$\begin{equation} C_l ^{{EB},\rm obs} \approx C_l ^{EE} \;\sin (2\alpha)\;\cos \;(2\alpha) \approx 2\langle\alpha\rangle\;C_l ^{EE}. \end{equation} \tag{ 12b }$$

The accuracy of Equation (12a) is shown in Figure 1, which compares the E-mode power spectrum from Figure 10 of Lewis & Challinor (2006) and the CPR B-mode power spectrum from Figure 5 of Zhao & Li (2014). They are almost identical up to a scale 4$\underline{\alpha}^2$ although their input parameters are slightly different.

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The present BICEP2 data group about 32 azimuthal eigenmodes into one band, with the lowest l contribution greater than 20; ζ(l) is virtually equal to one. We will set it to one in our analysis. For precise measurement of variations/fluctuations, direct processing of data without first evaluating the l components may be an alternative method.

Some CMB polarization projects apply a uniform angle derotation to the measured Q and U maps by minimizing the TB and EB power to compensate for insufficient calibrations of the polarization angle (Keating et al. 2013). This procedure will automatically eliminate the sum of any systematic error in the polarization angle calibration and of any uniform CPR, if it exists. If calibration errors of the polarization angle were small compared to the uniform CPR, in principle this minimization procedure could provide an estimate of the uniform CPR angle 〈α〉. However, since the systematic errors in the polarization angle are of the same order as current upper limits to uniform CPR, this procedure in fact is equivalent to assuming no uniform CPR and will preclude any information on it.

SPTpol first estimates the lensing potential from a Herschel-SPERE map of the cosmic infrared background and constructs a template for the lensing B-mode signal by combining the SPTpol-measured E-mode polarization with the estimated lensing potential. SPTpol then compares this constructed template to its directly measured B-modes to determine the lensing B-mode by correlation method. In this way the uniform (mean) CPR would not be included. The CPR fluctuations incurred at the lensing site would be included and correlated with the pseudoscalar fluctuations at the lensing site (z ∼ 2–5). Assuming that the linear perturbation scheme works, the uniform (mean) part of the pseudoscalar field at lensing is developed from the uniform (mean) part of the pseudoscalar field at the last scattering surface, and the pseudoscalar fluctuations at lensing are also developed from the fluctuations at the last scattering surface (z ∼ 1100). In most theoretical models, the pseudoscalar fluctuations are correlated to the density fluctuations at the last scattering surface. Therefore, the pseudoscalar field fluctuations at lensing would also be correlated with density fluctuations at lensing. The strength of this correlation depends on cosmological models with a pseudoscalar field. Effects from re-ionization are minor but could also be included. Therefore, in the fits with SPTpol data, we include a percentage correlation parameter κ (this parameter could also be greater than one in some models).

In this section, we model the available data for the BB power spectrum with the three effects mentioned in the Introduction. The theoretical spectrum of the inflationary gravitational waves and the lensing contribution to B-mode are extracted from the BICEP2 paper (Ade et al. 2014b). The power spectrum $C_l^{{BB},\rm obs}$, induced by any existing CPR angle (Equation (12a)), is obtained from the theoretical E-mode power spectrum $C_l^{EE}$ of Lewis & Challinor (2006) and is shown in Figure 1.

The data on the BB power spectrum are the nine points of BICEP2 in Figure 14 of Ade et al. (2014b) for the low multipole part (21 ⩽ l ⩽ 335), the four points of SPTpol in Figure 2 of Hanson et al. (2013) for 200 ⩽ l ⩽ 3000, and the four points of POLARBEAR (Ade et al. 2014c) for 500 ⩽ l ⩽ 2100, based on observations at 150 GHz on three regions of the sky for a total of 30 deg². Although the signal-to-noise ratio of the recently published results of POLARBEAR for the CMB B-mode polarization is considerably lower than that obtained by the SPTpol collaboration in the same range of l, the SPTpol data, depending on theoretical models, only include CPR partially in their lensing correlation measurement. Therefore, we have considered both data sets in our fits. As it turned out, the three experiments give similar constraints on the relevant CPR parameters.

Since a uniform derotation is implemented by both the BICEP2 and POLARBEAR experiments, by minimizing the EB and TB power to compensate for the relatively large errors in the calibration of the polarization angle, their data can only give constraints on the CPR fluctuations (variance), but not on the CPR mean angle. SPTpol have not applied such a derotation, although their systematic uncertainty on the polarization angle is about 1° at 150 GHz. However, they give the cross-correlation of their measured B-modes with the lensing B-modes inferred from cosmic infrared background fluctuations as measured by Herschel, rather than the BB autocorrelation. Therefore, also from the SPTpol data, it is possible to derive a constraint only on the CPR fluctuations, not on the mean angle.

Table 1 and Figures 2–7 show the results of our fits for various combinations of the available data and for different values of the correlation percentage κ of the pseudoscalar field fluctuations with the density fluctuations at lensing in the SPTpol case. Figure 2 shows the results of the fitting of the CPR fluctuation and of the scalar-to-tensor ratio for BICEP2. Figure 3 shows the results of the fitting to the POLARBEAR experiment. Figure 4 shows the results of the joint fitting to BICEP2 and POLARBEAR. Figure 5 shows the results of the fitting to the SPTpol experiment. Figure 6 shows the joint BICEP2–SPTpol fitting and Figure 7 the joint BICEP2–POLARBEAR–SPTpol fitting. Figure 6(b) shows how the χ² and the CPR angle fluctuations vary with the CPR–SPTpol correlation percentage κ, considering BICEP2 and SPTpol data: the minimum χ² is obtained at 70% (but it is a shallow minimum) and the largest CPR fluctuations are obtained at 23%. Figure 7(b) show the same for all three experiments: in this case the minimum χ² is obtained at 86% and the largest CPR variance is at 29%.

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Table 1. Results of Fitting the CPR Fluctuation δα² and/or the Scalar-to-tensor Ratio r to BICEP2 (Ade et al. 2014b), POLARBEAR (Ade et al. 2014c), and SPTpol (Hanson et al. 2013), and Their Joint Combinations

Experiment	Fitting Parameter		χmin2 (No. of Data Points −	1σ Upper Limit on CPR Fluctuation
	〈δα2〉	r	No. of Fitting	Amplitude 〈δα2〉1/2
	(mrad2)		Parameters)	(mrad)
BICEP2	339 ± 455	0.196 ± 0.033	10.424 (9 – 2)	28.2 (161)
POLARBEAR	89 ± 535	0.2a	3.73 (4 – 1)	25.0 (143)
POLARBEAR + BICEP2	265 ± 397	0.198 ± 0.033	14.31 (13 – 2)	25.7 (147)
SPTpol	κ−1(233 ± 148)	0.2a	2.61 (4 – 1)	κ−119.5 (κ−1112)
SPTpol (23%) + BICEP2	486 ± 411	0.190 ± 0.033	13.81 (13 – 2)	29.9 (172)
SPTpol (70%) + BICEP2	340 ± 270	0.196 ± 0.033	13.05 (13 – 2)	24.7 (142)
SPTpol (100%) + BICEP2	244 ± 202	0.199 ± 0.033	13.13 (13 – 2)	21.1 (121)
POLARBEAR + BICEP2 + SPTpol (29%)	392 ± 353	0.194 ± 0.033	19.27 (17 – 2)	27.3 (156)
POLARBEAR + BICEP2 + SPTpol (86%)	264 ± 218	0.198 ± 0.033	18.34 (17 – 2)	22.0 (126)

Note.^a r is set to 0.2 to conform to BICEP2 data; the effect of setting r to 0.2 or 0 to the CPR fluctuation fitting is small since the power of a non-vanishing r contribution to the total power is small for the multipoles measured in the POLARBEAR and SPTpol experiments.

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The CPR has not yet been detected. Upper limits have been obtained from radio galaxy polarization, both in the radio and in the optical/UV (di Serego Alighieri et al. 2010), and from CMB polarization anisotropies. These limits were reviewed by di Serego Alighieri (2011): all methods have reached an accuracy of about 1° and 3σ upper limits to any rotation of a few degrees. Since this review, only minor revisions of the limits from the CMB have appeared (see Table 2). Gluscevic et al. (2012) searched for direction-dependent CPRs with Wilkinson Microwave Anisotropy Probe (WMAP) 7 yr data and obtained an upper limit on the rms rotation angle 〈α²〉^1/2 < 95 for 0 < l < 512 and an upper limit at 68% CL of about 1° on the quadrupole of a scale-independent rotation-angle power spectrum.

Table 2. Recent Constraints on Uniform CPR Angle from CMB E-mode Polarization

Experiment	Frequencies	l-range	CPR Angle	Reference
WMAP9	41, 61,94 GHz	2–800	−036 ± 124 ± 15	Hinshaw et al. (2013)
BICEP1	100,150 GHz	20–335	−277 ± 086 ± 13	Kaufman et al. (2014)

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The rotation angle given by the revision of BICEP1 data by Kaufman et al. (2014) formally corresponds to a 1.78σ detection of CPR. However, taking into account the uncertainties involved in the calibration of the BICEP1 polarization angle, the problems in correcting for galactic Faraday rotation (Section VI.B of Kaufman et al. 2014), and the inconsistency of this result with the result of QUaD, for which Brown et al. (2009) give a CPR angle of 064 ± 05(stat.) ± 05(syst.), we do not consider this CPR "detection" as final. Recently, Gubitosi & Paci (2013) have reviewed the constraints on CPR, but they considered only those derived from CMB polarization data.

By modeling the SPTpol, POLARBEAR, and BICEP2 data on the B-mode polarization power spectrum, taking into account the inflationary gravitational waves, the gravitational lensing, and the component induced by CPR, as explained in the previous sections, we can derive a new constraint on CPR fluctuations, since the present data limit 〈δα²〉. In fact in our case, since we cannot predict which should be the correlation percentage between the lensing E-mode and B-mode CPR effects, the most conservative CPR constraint is set at the percentage that gives the maximum 〈δα²〉, i.e., 29% (Figure 7(b) and upper part of Figure 7(a)). In fact, we also cannot consider instead the correlation percentage giving the lowest χ², i.e., 86% (Figure 7(b) and lower part of Figure 7(a)) because all correlation percentages between 30% and 100% give χ² values within 10% of the best one, hence they all are statistically acceptable. Therefore, using the data from the three available experiments, we can set a limit: 〈δα²〉 ⩽ 392+353 = 745 mrad², i.e., 〈δα²〉^1/2 ⩽ 27.3 mrad or 156 at 68% C.L.

Our models give a value for r which is slightly lower than the value given by the BICEP2 collaboration (Ade et al. 2014b), but still in disagreement with the Planck six-parameter-fit limit (Ade et al. 2014a). However, the Planck limit is relaxed to r < 0.26 when running is allowed with dn_s/dlnk = −0.022 ± 0.010 (Ade et al. 2014b). Our fitted r value gives a 6σ detection, which agrees with the BICEP2 results and shows robustness to adding CPR effects.

Lee et al. (2014) recently modeled the power spectrum of the B-mode CMB polarization, including a component derived from the CPR (they call it "cosmological birefringence"), therefore decreasing the "primordial" component and bring it within the limit r < 0.11 set by Ade et al. (2014a). However, they considered only the BICEP2 data (Ade et al. 2014b) at low l, not the SPTpol nor POLARBEAR data (Hanson et al. 2013; Ade et al. 2014c) at high l, which together better constrain the CPR. Therefore, they allow for a much larger CPR component than we do. We believe that our approach is a considerable improvement.

In this paper, we investigated, both theoretically and experimentally, the possibility of detecting CPR or setting new constraints on it, using its coupling with the B-mode power spectra of the CMB. Three experiments have detected B-mode polarization in the CMB: SPTpol (Hanson et al. 2013) for 200 ⩽ l ⩽ 3000, BICEP2 (Ade et al. 2014b) for 21 ⩽ l ⩽ 335, and POLARBEAR (Ade et al. 2014c) for 500 ⩽ l ⩽ 2100.

The practical realization of our suggestion has a problem: as explained in Section 3, currently available data on B-mode CMB polarization only allow the fluctuations of the CPR angle to be constrained, not its mean value. Using all available data we set a constraint to the fluctuations of the CPR: 〈δα²〉^1/2 ⩽ 27.3 mrad (156) at 68% CL.

However, we believe that the method of investigating the CPR from data on the B-mode CMB polarization will become useful with future experiments, which will solve the problem of calibrating the polarization angle. For example, Naess et al. (2014) very recently reported a new measurement of the polarization of the Crab Nebula, an important calibration source, at 146 GHz with the Atacama Cosmology Telescope, giving an accuracy of 05 on the angle, an improvement over previous measurements. Concerning the prospects for improvements in the detection/constraints on CPR in the near future, the Planck satellite (http://www.rssd.esa.int/index.php?project=Planck) is expected to allow for a considerable step forward, since it will have a sensitivity to polarization rotation on the order of a tenth of a degree and systematic errors of the same order (Gruppuso et al. 2012), provided that calibration procedures of sufficient accuracy for the polarization orientation are implemented. In any case the Planck polarization results are foreseen only toward the end of this year. Also, the Keck Array, which is a set of five telescopes very similar to the BICEP2 one, is expected to bring considerable improvements to current CMB polarization measurements. Three of these telescopes are already operating and results are expected soon, particularly for B-mode polarization due to inflationary gravitational waves (Pryke & BICEP2 and Keck-Array Collaborations 2013).

As far as we are aware, there is no current plan for extending the optical/UV polarization measurement of distant radio galaxies to a level that would bring considerable improvements on the CPR constraints obtained so far from these objects. However, it would be desirable to perform detailed spatially resolved observations of the optical/UV polarization of a few radio galaxies, of the kind performed on 3C 265 (Wardle et al. 1997), and selected to be in the sky areas also observed by CMB polarization experiments on the ground. In fact these observations are likely to constrain the CPR angle with an accuracy of better than 1°, thereby providing a better calibration for the absolute polarization angle than available so far to these experiments.

If pseudoscalar–photon interactions exist, a natural cosmic variation of the pseudoscalar field at the decoupling era is 10⁻⁵ fractionally. The CPR fluctuation is then of the order of 10⁻⁵φ_{decoupling-era}. We will search for the possibility of its detection or for more constraints in future experiments.

We thank Brian Keating and the referee for useful comments and Matteo Galaverni for useful discussion. W.T.N. would also like to thank the National Science Council (grant Nos. NSC101–2112-M-007–007 and NSC102–2112-M-007–019) and the National Center for Theoretical Sciences (NCTS) for supporting this work in part.