Cosmic polarimetry in magnetoactive plasmas

Polarimetry of the Cosmic Microwave Background (CMB) represents one of the possible diagnostics aimed at testing large-scale magnetism at the epoch of the photon decoupling. The propagation of electromagnetic disturbances in a magnetized plasma leads naturally to a B-mode polarization whose angular power spectrum is hereby computed both analytically and numerically. Combined analyses of all the publicly available data on the B-mode polarization are presented, for the first time, in the light of the magnetized ΛCDM scenario. Novel constraints on pre-equality magnetism are also derived in view of the current and expected sensitivities to the B-mode polarization. Electronic address: [email protected] Electronic address: [email protected] 1 Formulation of the problem When (linearly) polarized electromagnetic radiation propagates in a diamagnetic material, the polarization plane of the wave changes by an angle ∆Φ(λ,B‖) ∆Φ(λ,B‖) = V(λ, T...) B‖ L, (1.1) where V denotes the Verdet constant, B‖ characterizes the magnetic field intensity along the direction of propagation of the electromagnetic radiation of wavelength λ; L is the distance travelled by the wave within the medium. The Verdet constant V not only depends upon the wavelength but also on the specifics of the given material such as, for instance, the temperature and the charge concentration. In electromagnetic plasmas, the kinetic energy of the charge carriers dominates against the potential energy, and the Verdet constant can be computed in simple terms: ∆Φ = RM(ñe, B‖)λ , RM = e 2πme ∫ L 0 ñe B‖ ds, (1.2) where me is the electron mass, ñe is the charge concentration of the electrons and the integral allows for the spatial variation of the magnetic field intensity as well as of the charge concentration. Equation (1.2) holds under two physical assumptions (see, e.g. [1, 2]): • the plasma is globally neutral, i.e. the charge concentrations of the electrons and of the ions are balanced, i.e. ñe = ñi = ñ0 (where ñ0 denotes generically the common value of ñe and ñi); • the plasma is cold, i.e. the kinetic temperature of the electrons and of the ions is always much smaller than the corresponding masses. Since the plasma is cold (i.e. non-relativistic) and globally neutral the values of the plasma and Larmor frequencies for the electrons are always smaller than the corresponding quantities but computed for the ions: this is why, in Eq. (1.2) only the electron mass and the electron concentration appears. In the case of a tokamak Eq. (1.2) is often used to determine the charge concentration ñe. Linearly polarized electromagnetic radiation can be sent through the tokamak and the ∆Φ(λ) can be directly measured, for instance, as a function of the frequency of the incident radiation. The latter experimental information is sufficient to determine, at least approximately, the electron concentration ñe since the geometry and intensity of the magnetic field are known from the design of the instrument. Given the profile of the electron concentration it is relevant, in diverse circumstances, to assess the magnetic field intensity. This is, in a nutshell, one of the objectives of cosmic polarimetry: the polarization properties of the cosmic microwave background radiation (CMB in what follows) are exploited as a diagnostic for the existence of large-scale magnetic fields. The polarization observable of the radiation field are customarily parametrized in terms of the Stokes parameters [3] whose evolution can be written in a schematic form which is closely analog to the set of equations often employed to describe the propagation of polarized radiation through stellar atmospheres [4, 5, 6]: dI dτ + ǫ′I = ǫSI , (1.3) dQ dτ + ǫ′Q = ǫSQ + 2ǫκF U, (1.4) dU dτ + ǫ′U = −2ǫκF Q, (1.5) where