Molecular hydrogen in the cosmic recombination epoch

The advent of precise measurements of the cosmic microwave background (CMB) anisotropies has motivated correspondingly precise calculations of the cosmic recombination history. Cosmic recombination proceeds far out of equilibrium because of a "bottleneck" at the $n=2$ level of hydrogen: atoms can only reach the ground state via slow processes: two-photon decay or Lyman-$\alpha$ resonance escape. However, even a small primordial abundance of molecules could have a large effect on the interline opacity in the recombination epoch and lead to an additional route for hydrogen recombination. Therefore, this paper computes the abundance of the H$_2$ molecule during the cosmic recombination epoch. Hydrogen molecules in the ground electronic levels X$^1\Sigma^+_g$ can either form from the excited H$_2$ electronic levels B$^1\Sigma^+_u$ and C$^1\Pi_u$ or through the charged particles H$_2^+$, HeH$^+$ and H$^-$. We follow the transitions among all of these species, resolving the rotational and vibrational sub-levels. Since the energies of the X$^1\Sigma^+_g$--B$^1\Sigma^+_u$ (Lyman band) and X$^1\Sigma^+_g$-C$^1\Pi_u$ (Werner band) transitions are near the Lyman-$\alpha$ energy, the distortion of the CMB spectrum caused by escaped H Lyman-line photons accelerates both the formation and the destruction of H$_2$ due to this channel relative to the thermal rates. This causes the populations of H$_2$ molecules in X$^1\Sigma^+_g$ energy levels to deviate from their thermal equilibrium abundances. We find that the resulting H$_2$ abundance is $10^{-17}$ at $z=1200$ and $10^{-13}$ at $z=800$, which is too small to have any significant influence on the recombination history.