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.
The intergalactic medium (IGM) prior to the epoch of reionization consists mostly of neutral hydrogen gas. Ly-alpha photons produced by early stars resonantly scatter off hydrogen atoms, causing energy exchange between the radiation field and the gas
. This interaction results in moderate heating of the gas due to the recoil of the atoms upon scattering, which is of great interest for future studies of the pre-reionization IGM in the HI 21 cm line. We investigate the effect of this Ly-alpha heating in the IGM with linear density, temperature, and velocity perturbations. Perturbations smaller than the diffusion length of photons could be damped due to heat conduction by Ly-alpha photons. The scale at which damping occurs and the strength of this effect depend on various properties of the gas, the flux of Ly-alpha photons and the way in which photon frequencies are redistributed upon scattering. To find the relevant length scale and the extent to which Ly-alpha heating affects perturbations, we calculate the gas heating rates by numerically solving linearized Boltzmann equations in which scattering is treated by the Fokker-Planck approximation. We find that (1) perturbations add a small correction to the gas heating rate, and (2) the damping of temperature perturbations occurs at scales with comoving wavenumber k>10^4 Mpc^{-1}, which are much smaller than the Jeans scale and thus unlikely to substantially affect the observed 21 cm signal.
