  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 an ensemble
average energy density of a 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
effects 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 effects on the MPS. The CMB polarizations are affected 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.
