Detecting primordial gravitational waves with circular polarization of
  the redshifted 21 cm line: II. Forecasts
  In the first paper of this series, we showed that the CMB quadrupole at high
redshifts results in a small circular polarization of the emitted 21 cm
radiation. In this paper we forecast the sensitivity of future radio
experiments to measure the CMB quadrupole during the era of first cosmic light
($z\sim 20$). The tomographic measurement of 21 cm circular polarization allows
us to construct a 3D remote quadrupole field. Measuring the $B$-mode component
of this remote quadrupole field can be used to put bounds on the
tensor-to-scalar ratio $r$. We make Fisher forecasts for a future Fast Fourier
Transform Telescope (FFTT), consisting of an array of dipole antennas in a
compact grid configuration, as a function of array size and observation time.
We find that a FFTT with a side length of 100 km can achieve $\sigma(r)\sim
4\times 10^{-3}$ after ten years of observation and with a sky coverage
$f_{\mathrm{sky}}\sim 0.7$. The forecasts are dependent on the evolution of the
Lyman-$\alpha$ flux in the pre-reionization era, that remains observationally
unconstrained. Finally, we calculate the typical order of magnitudes for
circular polarization foregrounds and comment on their mitigation strategies.
We conclude that detection of primordial gravitational waves with 21 cm
observations is in principle possible, so long as the primordial magnetic field
amplitude is small, but would require a very futuristic experiment with
corresponding advances in calibration and foreground suppression techniques.
