Evolution of the cosmic matter density field with a primordial magnetic field

A cosmological magnetic field affects the time evolution of the cosmic matter density field. The squared Alfven velocity of the cosmic fluid is proportional to an ensemble average energy density of a primordial magnetic field (PMF), and it prevents the matter density field from collapsing in the horizon scale. The matter-radiation equality time also is delayed by the presence of the ensemble average energy density of the PMF. The ensemble average energy density of the PMF also affects the matter power spectrum (MPS) through the Meszaros effect and the potential decay. Since the ensemble average energy density of the PMF is not a first order perturbation but a zero order source in the linear perturbation equations for the cosmology, to correctly understand the overall effect of the PMF on the MPS, we should significantly revise previous approaches to research for the MPS with the PMF by considering both the effects of the zero and first order sources from the PMF in the linear perturbation theory. We apply the effects of the zero order sources from the PMF to theoretical computations of the MPS for the first time. We also analyze the overall PMF effect on the MPS. The CMB polarization is affected by the weak lensing. The weak lensing is determined by the MPS. Therefore, we have to consider the zero order sources of the PMF to gain a correct understanding not only of the MPS but also the CMB polarization.