Magnetized birefringence and CMB polarization

The polarization plane of the cosmic microwave background radiation can be rotated either in a magnetized plasma or in the presence of a quintessential background with pseudoscalar coupling to electromagnetism. A unified treatment of these two phenomena is presented for cold and warm electron-ion plasmas at the pre-recombination epoch. The electron temperature is only relevant to the relativistic correction of the cold plasma results. The spectrum of plasma excitations is obtained from a generalized Appleton–Hartree equation, describing simultaneously the high-frequency propagation of electromagnetic waves in a magnetized plasma with a dynamical quintessence field. It is shown that these two effects are comparable for the plausible range of parameters allowed by present constraints. It is then argued that the generalized expressions derived in the present study may be relevant for direct searches of a possible rotation of the cosmic microwave background polarization. If linearly polarized radiation passes through a cold (or warm) plasma containing a magnetic field, the polarization plane of the wave can be rotated since the two circular polarizations (forming the linearly polarized beam) are travelling at different speeds. This effect has been studied in a variety of different (but related) frameworks even in relativistic QED plasmas and in the presence of extra-dimensions (see, for instance, [1]). The cosmic microwave background (CMB) has a weak degree of linear polarization, which is the direct result of Thompson scattering. If CMB is linearly polarized, then its polarization plane can be rotated provided a sufficiently strong magnetic field is present around the time of decoupling [2]. Typical values of magnetic fields, compatible with other astrophysical constraints, are of the order of B0 < ∼ 10 G at the decoupling time (see, for instance, [3]). An apparently unrelated possibility is that the quintessence field σ has pseudoscalar couplings to electromagnetism [4]. Couplings to ordinary matter, even if suppressed by the Planck scale, may lead to observable long-range forces and time dependence of the constants of nature. An approximate global symmetry may suppress these couplings even further as argued in [5] (second reference). This possibility may also imply a rotation of the plane of polarized emission, and radio-astronomical implications of this type of cosmic birefringence were investigated through various steps [5]. Typical values of the mass of σ are of the order of 10 eV in such a way that, between redshifts 0 and 3, σ can start dominating the background with energy density mσ 0 ∼ Λ, where Λ ≃ 10 eV and σ0 ≃ MP ∼ 10 GeV. In this paper a unified discussion of Faraday rotation and cosmic birefringence will be presented for the specific case of the pre-recombination plasma. The possible interference of these two effects will be scrutinized in both the cold and warm plasma approximations. A related aspect of the present analysis will be to study the range of validity of the Faraday rotation estimates. Let us start by discussing the typical scales involved in the problem. Right before decoupling, the temperature of the plasma is of the order of 0.3 eV and the typical free electron density can be estimated as n e ≃ xe(Ωbh0)(1 + z) × 10 cm, (1) where xe is the ionization fraction, h0 is the indetermination associated with the Hubble constant, Ωb is the fraction of critical density in baryons, and z is the redshift. For typical values of the parameters the electron density is of the order of ne ∼ 10 cm. Right before decoupling, the plasma is globally neutral and the ion density equals the electron density, i.e. ni ≃ ne = n0, denoting with n0 the common electron–ion number density. The global neutrality of the plasma occurs for typical length scales L ≫ λD where λD = √ Tei 8πen0 ≃ 10 ( n0 103 cm−3 )−1/2( Tei 0.3 eV )1/2 cm. (2) is the Debye screening length whose value is much smaller, for the same choice of physical parameters, than the electron and photon mean free paths, i.e. le ≃ 5.7 × 10 cm and lγ ≃ 1.22× 10 cm.