halo, and thereafter cools via H$_{2}$ rotational and vibrational line emission. The H$_{2}$ that enables the gas to cool is primarily formed via the gas-phase reactions $$\begin{aligned}
{\rm H} + {\rm e^{-}} & \rightarrow & {\rm H^{-}} + \gamma, \\
{\rm H^{-}} + {\rm H} & \rightarrow & {\rm H_{2}} + {\rm e^{-}},\end{aligned}$$ where the required free electrons are those that remain in the gas after cosmological recombination at $z \sim 1100$. The low abundance of these electrons, plus the fact that the gas starts to recombine further once it collapses into the dark matter halo, limits the amount of H$_{2}$ that can form in this manner; the typical fractional abundances of H$_{2}$ that result are a few times $10^{-3}$. The limited H$_{2}$ abundance, the low critical density of the H$_{2}$ molecule, and the large energy separation of its lowest accessible rotational levels combine to significantly limit the extent to which the gas can cool. The minimum temperature reached depends upon the dynamical history of the collapse, but is typically around $200 \: {\rm K}$. This minimum temperature is reached at a density of around $10^{4}
\: {\rm cm^{-3}}$, comparable to the critical density of H$_{2}$ (the density at which 