  The relatively large Thomson optical depth, tau_e, inferred recently from the
WMAP observations suggests that the Universe was reionized in a more complex
manner than previously believed. However, the value of tau_e provides only an
integral constraint on the history of reionization and, by itself, cannot be
used to determine the nature of the sources responsible for this transition.
Here, we show that the evolution of the ionization state of the intergalactic
medium at high redshifts can be measured statistically using fluctuations in 21
centimeter radiation from neutral hydrogen. By analogy with the mathematical
description of anisotropies in the cosmic microwave background, we develop a
formalism to quantify the variations in 21 cm emission as a function of both
frequency and angular scale. Prior to and following reionization, fluctuations
in the 21 cm signal are mediated by density perturbations in the distribution
of matter. Between these epochs, pockets of gas surrounding luminous objects
become ionized, producing large HII regions. These ``bubbles'' of ionized
material imprint features into the 21 cm power spectrum that make it possible
to distinguish them from fluctuations produced by the density perturbations.
The variation of the power spectrum with frequency can be used to infer the
evolution of this process. As has been emphasized previously by others, the
absolute 21 cm signal from neutral gas at high redshifts is undetectable owing
to contamination by foreground sources. However, we argue that this source of
noise can be suppressed by comparing maps closely spaced in frequency, i.e.
redshift, so that 21 cm fluctuations from the IGM can be measured against a
much brighter, but smoothly varying (in frequency) background.
