Evolution of Cosmological Perturbations in the Presence of Primordial
  Magnetic Fields
  Possible existence of the primordial magnetic fields has affected the
structure formation of the universe. In this paper it is shown that the initial
conditions for density perturbations with magnetic fields derived in previous
works are inconsistent with Einstein equations. We find that this inconsistency
arises due to the unwanted cancellation of contributions from the magnetic
fields and primordial radiations. A complete set of equations and consistent
initial conditions in the long wavelength limit are given with an explicit
derivation in the covariant approach with CDM frame, by newly taking into
account a non-relativistic matter contribution in the radiation dominated era.
By solving these equations numerically, we derive the angular spectrum of
cosmic microwave background anisotropies and the matter power spectrum with
magnetic fields. We find that the amplitude of the angular power spectrum of
CMB anisotropies can alter at most a order of magnitude at $l \lesssim 4000$
compared with the previous results in the literature.
