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Constraints on pseudoscalar-photon interaction from CMB polarization observation

Effective pseudoscalar-photon interaction(s) would induce a rotation of linear polarization of electromagnetic wave propagating with cosmological distance in various cosmological models. Pseudoscalar-photon interaction is proportional to the gradient of the pseudoscalar field. From phenomenological point of view, this gradient could be neutrino number asymmetry, other density current, or a constant vector. In these situations, Lorentz invariance or CPT may effectively be violated. CMB polarization observations are superb tests of these models and have the potential to discover new fundamental physics. In this paper, we review the constraints on pseudoscalar-photon interaction from CMB polarization observation.


Introduction
In 1973, we studied the relationship of Galilio Equivalence Principe (WEP I) and Einstein Equivalence Principle in a framework (the χ-g framework) of electromagnetism and charged particles, we found the following theory with (gravitational) interaction Lagrangian density as an example which obeys WEP I, but not EEP [1,2,3]. The nonmetric part of this theory is where 'mod div' means that the two Lagrangian densities are related by partial integration in the action integral. The Maxwell equations [1,3] are 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 Maxwell equations are also conformally invariant.
The rightest term in equation (2) is reminiscent of Chern-Simons [4] term e αβγ A α F β γ. There are two differences: (i) Chern-Simons term is in 3 dimensional space; (ii) Chern-Simons term is a total divergence.
A term similar to the one in equation (2) (axion-gluon interaction) occurs in QCD in an effort to solve the strong CP problem (Peccei and Quinn [5], Weinberg [6], Wilczek [7]). Carroll, Field and Jackiw [8] proposed a modification of electrodynamics with an additional e ijkl V i A j F kl term with V i a constant vector. This term is a special case of the term e ijkl ϕF ij F kl (mod div) with ϕ ,i = − 1 2 V i . Various terms in the Lagrangians discussed in this section are listed in Table 1. Empirical tests of the pseudoscalar-photon interaction (2) from CMB polarization observation will be discussed in section 2. Section 3 will present an outlook.
vector coupling

Constraints from CMB polarization observation
Pseudoscalar-photon interaction induces polarization rotation in electromagnetic propagation. From (3), for the right circularly polarized electromagnetic wave, the propagation from a point P 1 to another point P 2 adds a phase of α = ϕ(P 2 ) − ϕ(P 1 ) to the wave; for left circularly polarized light, the added phase will be opposite in sign [1]. Linearly polarized electromagnetic wave is a superposition of circularly polarized waves. Its polarization vector will then rotate by an angle α. When the propagation distance is over a large part of our observed universe, we call this phenomenon cosmic polarization rotation [9,10].
Since the first successful polarization observation of the cosmological microwave background (CMB) in 2002 by DASI [11] (Degree Angular Scale Interferometer), there have been a number of observations [12][13][14][15][16] with better precision. These observations set up limits on the electromagnetic polarization rotation due to effective pseudoscalar-photon interaction.
In the CMB polarization observations, there are variations and fluctuations. The variations and fluctuations due to scalar-modified propagation can be expressed as δϕ(2) − δϕ(1), where 1 denotes a point at the last scattering surface in the decoupling epoch and 2 observation point. δϕ(2) is the variation/fluctuation at the last scattering surface. δϕ(1) at the present observation point is fixed. Therefore the covariance of fluctuation < [δϕ(2) − δϕ(1)] 2 > gives the covariance of δϕ 2 (2) at the last scattering surface. Since our Universe is isotropic to ∼ 10 −5 , this covariance is ∼ (ξ × 10 −5 ) 2 where the parameter ξ depends on various cosmological models [10,17].
In 2002, DASI microwave interferometer observed the polarization of the cosmic background [11]. 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 (2), 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 [12]. This gives a constraint of about 10 −1 for ∆ϕ [9,18].
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 in more precise observations.

Discussion and Outlook
Better accuracy in CMB polarization observation is expected from PLANCK mission launched on May 14, 2009. Dedicated CMB polarization observers like B-Pol mission, CMBpol mission and LiteBIRD mission would improve the sensitivity further. These development would probe the fundamental issues of effective pseudoscalar-photon interaction discussed in this paper more deeply in the future.