  The early stages of the Epoch of Reionization, probed by the 21 cm line, are
sensitive to the detailed properties and formation histories of the first
galaxies. We use 21cmFAST and a simple, self-consistent galaxy model to examine
the redshift evolution of the large-scale cross-power spectrum between the 21
cm field and line-emitting galaxies. A key transition in redshift occurs when
the 21 cm field shifts from being positively correlated with the galaxy
distribution to being negatively correlated. Importantly, this transition
redshift is insensitive to the properties of the galaxy tracers but depends
sensitively on the thermal and ionization histories traced through the 21 cm
field. Specifically, we show that the transition occurs when both ionization
fluctuations dominate over 21 cm spin temperature fluctuations and when the
average spin temperature exceeds the temperature of the cosmic microwave
background. We illustrate this with three different 21 cm models which have
largely the same neutral fraction evolution but different heating histories. We
find that the transition redshift has a scale dependence, and that this can
help disentangle the relative importance of heating and ionization
fluctuations. The best prospects for constraining the transition redshift occur
in scenarios with late X-ray heating, where the transition occurs at redshifts
as low as $z \sim 6-8$. In our models, this requires high-redshift galaxy
surveys with sensitivities of $\sim 10^{-18}~\rm erg/s/cm^2$ for optical lines
and $\sim 10^{-19}~\rm erg/s/cm^2$ for FIR lines. Future measurements of the
transition redshift can help discriminate between 21 cm models and will benefit
from reduced systematics.
