  The redshifted 21-cm background is expected to be a powerful probe of the
early Universe, carrying both cosmological and astrophysical information from a
wide range of redshifts. In particular, the power spectrum of fluctuations in
the 21-cm brightness temperature is anisotropic due to the line-of-sight
velocity gradient, which in principle allows for a simple extraction of this
information in the limit of linear fluctuations. However, recent numerical
studies suggest that the 21-cm signal is actually rather complex, and its
analysis likely depends on detailed model fitting. We present the first
realistic simulation of the anisotropic 21-cm power spectrum over a wide period
of early cosmic history. We show that on observable scales, the anisotropy is
large and thus measurable at most redshifts, and its form tracks the evolution
of 21-cm fluctuations as they are produced early on by Lyman-a radiation from
stars, then switch to X-ray radiation from early heating sources, and finally
to ionizing radiation from stars. In particular, we predict a redshift window
during cosmic heating (at z ~ 15), when the anisotropy is small, during which
the shape of the 21-cm power spectrum on large scales is determined directly by
the average radial distribution of the flux from X-ray sources. This makes
possible a model-independent reconstruction of the X-ray spectrum of the
earliest sources of cosmic heating.
