  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.
