Nonlinear Evolution of Cosmic Magnetic Fields and Cosmic Microwave Background Anisotropies

In this work we investigate the effects of primordial magnetic fields on cosmic microwave background anisotropies (CMB). Based on cosmological magneto-hydro dynamic (MHD) simulations [1] we calculate the CMB anisotropy spectra and polarization induced by fluid fluctuations (Alfvén modes) generated by primordial magnetic fields. The strongest effect on the CMB spectra comes from the transition epoch from a turbulent regime to a viscous regime. The balance between magnetic and kinetic energy until the onset of the viscous regime provides a one to one relation between the comoving coherence length L and the comoving magnetic field strength B, such as L ∼ 30 (B/10Gauss)pc. The resulting CMB temperature and polarization anisotropies for the initial power law index of the magnetic fields n > 3/2 are somewhat different from the ones previously obtained by using linear perturbation theory. In particular, differences can appear on intermediate scales l < 2000 and small scales l > 20000. On scales l < 2000 the CMB anisotropy and polarization spectra are flat in the case of our nonlinear calculations whereas the spectra have a blue index calculated with linear perturbation theory if we assume the velocity fields of baryons induced by the magnetic fields achieved Alfvén velocity due to the turbulent motions on large scales in the early universe. Our calculation gives a constraint on the magnetic field strength in the intermediate scale of CMB observations. Upper limits are set by WMAP and BOOMERANG results for comoving magnetic field strength of B < 28 nGauss with a comoving coherence length of L > 0.7Mpc for the most extreme case, or B < 30 nGauss and L > 0.8Mpc for the most conservative case. We may also expect higher signals on large scales of the polarization spectra compared to linear calculations. The signal may even exceed the B-mode polarization from gravitational lensing depending on the strength of the primordial magnetic fields. On very small scales, the diffusion damping scale of nonlinear calculations turns out to be much smaller than the one of linear calculations if the comoving magnetic field strength B > 16 nGauss. If the magnetic field strength is smaller, the diffusion scales become smaller too. Therefore we expect to have both, temperature and polarization anisotropies, even beyond l > 10000 regardless of the strength of the magnetic fields. The peak values of the temperature anisotropy and the B-mode polarization spectra are approximately 40μK and a few μK, respectively.