Constraining the Primordial Magnetic Field from Cosmic Microwave   Background Anisotropies at Higher Multipoles
  The cosmological magnetic field is one of the important physical quantities which affect strongly the cosmic microwave background (CMB) power spectrum. Recent CMB observations have been extended to higher multipoles $l\gtrsim$1000, and they resultantly exhibit an excess power than the standard model prediction in cosmological theory which best fits the Wilkinson Microwave Anisotropy Probe (WMAP) data at lower multipoles $l\lesssim$900. We calculate the CMB temperature anisotropies generated by the power-law magnetic field at the last scattering surface (LSS) in order to remove the tension between theory and observation at higher multipoles and also place an upper limit on primordial magnetic field. In our present calculation we take account of the effect of ionization ratio exactly without approximation. This effect is very crucial to precisely estimate the effect of the magnetic field on CMB power spectrum. We consider both effects of the scalar and vector modes of magnetic field on the CMB anisotropies, where current data are known to be insensitive to the tensor mode which we ignore in the present study. In order to constrain the primordial magnetic field, we evaluate likelihood function of the WMAP data in a wide range of parameters of the magnetic field strength $|\mathbf{B}|_\lambda$ and the power-law spectral index $n$, along with six cosmological parameters in flat Universe models, using the technique of the Markov Chain Monte Carlo(MCMC) method. We find that the upper limit at $2\sigma$ C.L. turns out to be $|\mathbf{B}_\lambda|\lesssim 3.9 $nG at 1 Mpc for any $n_B$ values, which is obtained by comparing the calculated result including the Sunyaev-Zeldovich(SZ) effect with recent WMAP data of the CMB anisotropies. 
D. G. Yamazaki, K. Ichiki, and T. Kajino