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 .