Detecting primordial gravitational waves with circular polarization of the redshifted 21 cm line: I. Formalism

We propose a new method to measure the tensor-to-scalar ratio $r$ using the circular polarization of the 21 cm radiation from the pre-reionization epoch. Our method relies on the splitting of the $F = 1$ hyperfine level of neutral hydrogen due to the quadrupole moment of the CMB. We show that unlike the Zeeman effect, where $M_{F}=\pm 1$ have opposite energy shifts, the CMB quadrupole shifts $M_{F}=\pm 1$ together relative to $M_{F}= 0$. This splitting leads to a small circular polarization of the emitted 21 cm radiation. In this paper (Paper I in a series on this effect), we present calculations on the microphysics behind this effect, accounting for all processes that affect the hyperfine transition. We conclude with an analytic formula for the circular polarization from the Dark Ages as a function of pre-reionization parameters and the value of the remote quadrupole of the CMB. We also calculate the splitting of the $F = 1$ hyperfine level due to other anisotropic radiation sources and show that they are not dominant. In a companion paper (Paper II) we make forecasts for measuring the tensor-to-scalar ratio $r$ using future radio arrays.