  Of the many probes of reionization, the 21 cm line and the cosmic microwave
background (CMB) are among the most effective. We examine how the
cross-correlation of the 21 cm brightness and the CMB Doppler fluctuations on
large angular scales can be used to study this epoch. We employ a new model of
the growth of large scale fluctuations of the ionized fraction as reionization
proceeds. We take into account the peculiar velocity field of baryons and show
that its effect on the cross correlation can be interpreted as a mixing of
Fourier modes. We find that the cross-correlation signal is strongly peaked
toward the end of reionization and that the sign of the correlation should be
positive because of the inhomogeneity inherent to reionization. The signal
peaks at degree scales (l~100) and comes almost entirely from large physical
scales (k~0.01 Mpc). Since many of the foregrounds and noise that plague low
frequency radio observations will not correlate with CMB measurements, the
cross correlation might appear to provide a robust diagnostic of the
cosmological origin of the 21 cm radiation around the epoch of reionization.
Unfortunately, we show that these signals are actually only weakly correlated
and that cosmic variance dominates the error budget of any attempted detection.
We conclude that the detection of a cross-correlation peak at degree-size
angular scales is unlikely even with ideal experiments.
