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.
