  A generation of magnetic fields from cosmological density perturbations is
investigated. In the primordial plasma before cosmological recombination, all
of the materials except dark matter in the universe exist in the form of
photons, electrons, and protons (and a small number of light elements). Due to
the different scattering nature of photons off electrons and protons, electric
currents and electric fields are inevitably induced, and thus magnetic fields
are generated. We present a detailed formalism of the generation of
cosmological magnetic fields based on the framework of the well-established
cosmological perturbation theory following our previous works. We numerically
obtain the power spectrum of magnetic fields for a wide range of scales, from
$k\sim 10^{-5}$ Mpc$^{-1}$ up to $k\sim 2\times 10^{4}$ Mpc$^{-1}$ and provide
its analytic interpretation. Implications of these cosmologically generated
magnetic fields are discussed.
