Does the HCN/CO ratio trace the star-forming fraction of gas? II. Variations in CO and HCN Emissivity

We model emissivities of the HCN and CO $J=1-0$ transitions using measured properties of clouds found in normal star forming galaxies and more extreme systems. These models are compared with observations of HCN and CO $J=1-0$ transitions. We combine these model emissivities with predictions of gravoturbulent models of star formation, explore the impact of excitation and optical depth on CO and HCN emission, and assess if observed HCN/CO ratios track the fraction of gravitationally-bound dense gas, $f_\mathrm{grav}$, in molecular clouds. Our modeled HCN/CO ratios and emissivities are consistent with measurements from observations. CO emission shows a range of optical depths across different environments, from optically thick in normal galaxies to moderately optically thin in extreme systems. HCN is only moderately optically thick, with significant subthermal excitation in both normal and extreme galaxies. We find an anticorrelation between HCN/CO and $f_\mathrm{grav}$ as predicted by gravoturbulent models of star formation. Instead this ratio tracks gas at moderate densities ($n>10^{3.5}\ \mathrm{cm}^{-3}$), which is below the standard dense gas threshold of $n>10^{4.5}\ \mathrm{cm}^{-3}$. Variations in CO emissivity depend strongly on optical depth, due to variations in the dynamics of the cloud gas. HCN emissivity depends more strongly on excitation, and thus does not directly track variations in CO emissivity. We conclude that a single line ratio, such as HCN/CO, will not consistently track the fraction of gravitationally-bound, star-forming gas if the critical density for star formation varies in molecular clouds. This work highlights important uncertainties that need to be considered when observationally applying an HCN conversion factor in order to estimate the dense (i.e. $n>10^{4.5}\ \mathrm{cm}^{-3}$) gas content in nearby galaxies.