Molecular hydrogen in the cosmic recombination epoch

The advent of precise measurements of the 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 ${\mathrm{H}}_2$ molecule during the cosmic recombination epoch. Hydrogen molecules in the ground electronic levels ${\mathrm{X}}^1{\Sigma }_g^{+}$ can either form from the excited ${\mathrm{H}}_2$ electronic levels ${\mathrm{B}}^1{\Sigma }_u^{+}$ and ${\mathrm{C}}^1{\Pi }_u$ or through the charged particles ${\mathrm{H}}_2^{+}$, ${\mathrm{HeH}}^{+}$, and ${\mathrm{H}}^{-}$. We follow the transitions among all of these species, resolving the rotational and vibrational sublevels. Since the energies of the ${\mathrm{X}}^1{\Sigma }_g^{+}-{\mathrm{B}}^1{\Sigma }_u^{+}$ (Lyman band) and ${\mathrm{X}}^1{\Sigma }_g^{+}-{\mathrm{C}}^1{\Pi }_u$ (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 ${\mathrm{H}}_2$ due to this channel relative to the thermal rates. This causes the populations of ${\mathrm{H}}_2$ molecules in ${\mathrm{X}}^1{\Sigma }_g^{+}$ energy levels to deviate from their thermal equilibrium abundances. We find that the resulting ${\mathrm{H}}_2$ 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.

Figure 1

The $P$-branch contribution to the spontaneous dissociation of ${\mathrm{H}}_2({\mathrm{B}}^1{\Sigma }_u^{+})$ from the $J=14$, $\nu =9$ level, as a function of the center-of-mass frame kinetic energy of the final H atoms. Note the quasibound resonance peak at $E_k=0.0009\mathrm{hartree}$, which contains $\sim 4\%$ of the integrated rate.

Figure 2

The radiation spectrum at $z=1142$, showing both the CMB blackbody spectrum (straight line) and the full calculation including H spectral distortions.

Figure 3

The network of routes for the formation and destruction of an ${\mathrm{H}}_2$ molecule that we consider in this paper.

Figure 4

The abundance of ${\mathrm{H}}_2$ molecules, $x[{\mathrm{H}}_2]$, as a function of redshift. Short dashed blue and dotted magenta lines show the cases where we do and do not include quadrupole transitions between X levels, respectively, and only the Lyman and Werner band reactions can create and destroy ${\mathrm{H}}_2$. The long dashed red curve shows the case when all the relevant reactions are included. The green solid curve shows the thermal abundances of Eq. (7).

Figure 5

The ratios of the abundances of the ${\mathrm{H}}_2$ ${\mathrm{X}}^1{\Sigma }_g^{+}$ levels with $J=0$ to their thermal values at $z=1142$, plotted as a function of the vibrational quantum number $\nu$. Blue stars and magenta squares show the cases with and without quadrupole transitions between the X electronic levels, respectively. This plot included only the Lyman and Werner band reactions as sources and sinks for ${\mathrm{H}}_2$.

Figure 6

The abundance of each ${\mathrm{H}}_2$ level plotted versus the level energy at $z=1142$. In the left panel we show the abundances of X levels and in the right panel the B, ${\mathrm{C}}^{+}$, and ${\mathrm{C}}^{-}$ levels. The solid line shows the thermal equilibrium abundances.

Figure 7

The minimum eigenvalue of the rate matrix $T$ (solid blue line) as compared to the Hubble rate (dashed magenta line).

Figure 8

Here we show the abundances of the intermediate species, ${\mathrm{H}}^{-}$, ${\mathrm{H}}_2^{+}$, and ${\mathrm{HeH}}^{+}$ in the left panel and ${\mathrm{H}}_2(\mathrm{B},{\mathrm{C}}^{\pm })$ in the right panel, as a function of redshift during the recombination epoch.

Figure 9

Total Sobolev optical depth of all of the Lyman and Werner lines as a function of redshift.

Figure 10

The net rate of emission (i.e. emission minus absorption) of photons at $E>E(\mathrm{Ly}\alpha )$ in the ${\mathrm{H}}_2$ Lyman and Werner bands, measured in photons per H nucleus per Hubble time.