Does the HCN/CO ratio trace the star-forming fraction of gas? I. A comparison with analytical models of star formation

We use archival ALMA observations of the HCN and CO $J=1-0$ transitions, in addition to the radio continuum at 93 GHz, to assess the relationship between dense gas, star formation, and gas dynamics in ten, nearby (U)LIRGs and late-type galaxy centers. We frame our results in the context of turbulent and gravoturbulent models of star formation to assess if the HCN/CO ratio tracks the gravitationally-bound, star-forming gas in molecular clouds ($f_\mathrm{grav}$) at sub-kpc scales in nearby galaxies. We confirm that the HCN/CO ratio is a tracer of gas above $n_\mathrm{SF}\approx10^{4.5}$ cm$^{-3}$, but the sub-kpc variations in HCN/CO do not universally track $f_\mathrm{grav}$. We find strong evidence for the use of varying star formation density threshold models, which are able to reproduce trends observed in $t_\mathrm{dep}$ and $\epsilon_\mathrm{ff}$ that fixed threshold models cannot. Composite lognormal and powerlaw models outperform pure lognormal models in reproducing the observed trends, even when using a fixed powerlaw slope. The ability of the composite models to better reproduce star formation properties of the gas provides additional indirect evidence that the star formation efficiency per free-fall time is proportional to the fraction of gravitationally-bound gas.