New probe of magnetic fields in the prereionization epoch. I. Formalism
  We propose a method of measuring extremely weak magnetic fields in the
intergalactic medium prior to and during the epoch of cosmic reionization. The
method utilizes the Larmor precession of spin-polarized neutral hydrogen in the
triplet state of the hyperfine transition. This precession leads to a
systematic change in the brightness temperature fluctuations of the 21-cm line
from the high-redshift universe, and thus the statistics of these fluctuations
encode information about the magnetic field the atoms are immersed in. The
method is most suited to probing fields that are coherent on large scales; in
this paper, we consider a homogenous magnetic field over the scale of the 21-cm
fluctuations. Due to the long lifetime of the triplet state of the 21-cm
transition, this technique is naturally sensitive to extremely weak field
strengths, of order $10^{-19}$ G at a reference redshift of $\sim 20$ (or
$10^{-21}$ G if scaled to the present day). Therefore, this might open up the
possibility of probing primordial magnetic fields just prior to reionization.
If the magnetic fields are much stronger, it is still possible to use this
method to infer their direction, and place a lower limit on their strength. In
this paper (Paper I in a series on this effect), we perform detailed
calculations of the microphysics behind this effect, and take into account all
the processes that affect the hyperfine transition, including radiative decays,
collisions, and optical pumping by Lyman-$\alpha$ photons. We conclude with an
analytic formula for the brightness temperature of linear-regime fluctuations
in the presence of a magnetic field, and discuss its limiting behavior for weak
and strong fields.
