  Magnetic fields appear to be present in all galaxies and galaxy clusters.
Recent measurements indicate that a weak magnetic field may be present even in
the smooth low density intergalactic medium. One explanation for these
observations is that a seed magnetic field was generated by some unknown
mechanism early in the life of the Universe, and was later amplified by various
dynamos in nonlinear objects like galaxies and clusters. We show that a
primordial magnetic field is expected to be generated in the early Universe on
purely linear scales through vorticity induced by scale-dependent temperature
fluctuations or equivalently, a spatially varying speed of sound of the gas.
Residual free electrons left over after recombination tap into this vorticity
to generate magnetic field via the Biermann battery process. Although the
battery operates even in the absence of any relative velocity between dark
matter and gas at the time of recombination, the presence of such a relative
velocity modifies the predicted spatial power spectrum of the magnetic field.
At redshifts of order a few tens, we estimate a root mean square field strength
of order 1e-25 -1e-24 G on comoving scales~10 kpc. This field, which is
generated purely from linear perturbations, is expected to be amplified
significantly after reionization, and to be further boosted by dynamo processes
during nonlinear structure formation.
