CMB polarization induced by stochastic magnetic fields

The complete calculation of the CMB polarization observables (i.e. Eand B-modes) is reported within the conventional ΛCDM paradigm supplemented by a stochastic magnetic field. Intriguing perspectives for present and forthcoming CMB polarization experiments are outlined. Large-scale magnetism recently became an intriguing triple point where cosmology, astronomy and high-energy astrophysics meet for complementary purposes [1]. Still we have no clues on its origin. The gravitational instability together with the subsequent galactic rotation could amplify a magnetic field spanning a collapsing region of the order of the Mpc in comoving units. The magnetic field regularized over such a scale L, i.e. BL, is not empirically observable at the epoch of the gravitational collapse of the protogalaxy. But the Universe is a good conductor: the magnetic flux and helicity are approximately conserved implying that large-scale magnetic fields could have been already present at the time when photons last-scattered electrons and ions, i.e., according to the WMAP 5-year data [2], at a redshift zdec ≃ 1090. Intriguing effects related to tangled magnetic fields have been discussed with semianalytical methods (see, in particular, [3]). More recently the impact of large-scale magnetic fields on scalar modes of the geometry have been addressed [4] and a dedicated numerical approach has been devised [5]. The complete calculation of the polarization angular power spectra (i.e., specifically, the EE, TE and BB angular power spectra) is here reported, for the first time, when the conventional ΛCDM paradigm is complemented by a stochastic magnetic field (i.e., according to the terminology of [5], mΛCDM scenario). In short the main theoretical impasse is the following. The large-scale description of temperature anisotropies demands a coarse grained (one-fluid) approach for the electronion system: this is the so called baryon fluid which is treated (with no exceptions) as a single fluid in popular Boltzmann solvers such as COSMICS [6] and CMBFAST [7]. On the other hand the dispersive propagation of electromagnetic disturbances demands to treat separately electrons and ions, at least at high frequencies. It is appropriate to start from the Vlasov-Landau equation written in the form: ∂f± ∂τ + v ∂f± ∂xi ± e(E + vjBkǫ k ) ∂f± ∂qi + 1 2 h′ijq i∂f± ∂qj = Ccoll. (1) where ~v = ~q/ √ m2a2 + q2 is the comoving three velocity, ~q is the comoving three-momentum and τ is the conformal time arising, in the line element, as ds = a2(τ){dτ 2 − [δij − hij(~x, τ)]dx idxj}. The prime denotes a derivation with respect to τ . The rescaled electromagnetic fields are denoted as ~ E = a2~ E and as ~ B = a ~ B. By choosing the plus (minus) sign in Eq. (1), the evolution equation for the one-body distribution function f±(~x, ~q, τ) of the ions (electrons) can be obtained. In the electron-ion system Ccoll is provided by Coulomb scattering. In the limit e → 0, Eq. (1) describes the evolution of neutral species. If Ccoll = 0, Eq. (1) leads, below the MeV, to the well known evolution equation for the reduced phase space The velocity-configuration space naturally arises since ions and electrons are all non-relativistic. Consequently, the comoving three-momentum is given by ~q = ma~v for each of the two charged species. The quasi-equilibrium distribution for electrons and protons is Maxwellian and the strength of Coulomb scattering guarantees Te ≃ Ti ≃ T . For relativistic (neutral) species q = nq. The equilibrium distribution for neutrinos and photons will be, respectively, Fermi-Dirac and Bose-EInstein.