Constraining the unexplored period between the dark ages and reionization with observations of the global 21 cm signal

Observations of the frequency dependence of the global brightness temperature of the redshifted 21 cm line of neutral hydrogen may be possible with single dipole experiments. In this paper, we develop a Fisher matrix formalism for calculating the sensitivity of such instruments to the 21 cm signal from reionization and the dark ages. We show that rapid reionization histories with duration $\ensuremath{\Delta}z\ensuremath{\lesssim}2$ can be constrained, provided that local foregrounds can be well modeled by low order polynomials. It is then shown that observations in the range $\ensuremath{\nu}=50--100\text{ }\text{ }\mathrm{MHz}$ can feasibly constrain the $\mathrm{Ly}\ensuremath{\alpha}$ and x-ray emissivity of the first stars forming at $z\ensuremath{\sim}15--25$, provided that systematic temperature residuals can be controlled to less than 1 mK. Finally, we demonstrate the difficulty of detecting the 21 cm signal from the dark ages before star formation.