The ratio of CO to total gas mass in high-redshift galaxies

Walter et al. (20012) have recently identified the J=6-5, 5-4, and 2-1 CO rotational emission lines, and [C_{II}] fine-structure emission line from the star-forming interstellar medium in the high-redshift submillimeter source HDF 850.1, at z = 5.183. We employ large velocity gradient (LVG) modeling to analyze the spectra of this source assuming the [C_{II}] and CO emissions originate from (i) separate unvirialized regions, (ii) separate virialized regions, (iii) uniformly mixed unvirialized region, and (iv) uniformly mixed virialized regions. We present the best fit set of parameters, including for each case the ratio $\alpha$ between the total hydrogen/helium gas mass and the CO(1-0) line luminosity. We also present computations of the ratio of H_{2} mass to [C_{II}] line-luminosity for optically thin conditions, for a range of gas temperatures and densities, for direct conversion of [C_{II}] line-luminosities to "dark-H_{2}" masses. For HDF 850.1 we find that a model in which the CO and C^{+} are uniformly mixed in gas that is shielded from UV radiation, requires a cosmic-ray or X-ray ionization rate of $\zeta \approx$ 10^{-13} s^{-1}, plausibly consistent with the large star-formation rate ($\sim$ 10^{3} M$_{\odot}$ yr^{-1}) observed in this source. Enforcing the cosmological constraint posed by the abundance of dark matter halos in the standard $\Lambda$CDM cosmology and taking into account other possible contributions to the total gas mass, we find that three of these four models are less likely at the 2$\sigma$ level. We conclude that modeling HDF 850.1's ISM as a collection of unvirialized molecular clouds with distinct CO and C^{+} layers, for which $\alpha$ = 0.6 M$_{\odot}$ (K km s^{-1} pc^{2})^{-1} for the CO to H_{2} mass-to-luminosity ratio, (similar to the standard ULIRG value), is most consistent with the $\Lambda$CDM cosmology.