Numerical Calibration of the HCN$-$Star Formation Correlation
  HCN(1$-$0) emission traces dense gas and correlates very strongly with star
formation rates (SFRs) on scales from small Milky Way clouds to whole galaxies.
The observed correlation offers strong constraints on the efficiency of star
formation in dense gas, but quantitative interpretation of this constraint
requires a mapping from HCN emission to gas mass and density. In this paper we
provide the required calibration by postprocessing high-resolution simulations
of dense, star-forming clouds to calculate their HCN emission ($L_{\rm HCN}$)
and to determine how that emission is related to the underlying gas density
distribution and star formation efficiency. We find that HCN emission traces
gas with a luminosity-weighted mean number density of $0.8-1.7 \times
10^4\,{\rm cm}^{-3}$ and that HCN luminosity is related to mass of dense gas of
$\gtrsim 10^4\,{\rm cm}^{-3}$ with a conversion factor of ${\alpha}_{\rm HCN}
\approx 14\,\rm M_{\odot}/(K\,km\,s^{-1}\,{pc}^2)$. We also measure a new
empirical relationship between the SFR per global mean freefall time
(${\epsilon}_{\rm ff}$) and the SFR$-$HCN relationship, ${\rm SFR}/L_{\rm HCN}
= 2.0 \times 10^{-7}\,({\epsilon}_{\rm ff}/0.01)^{1.1}\,\rm
M_{\odot}\,{yr}^{-1}/(K\,km\,s^{-1}\,{pc}^2)$. The observed SFR$-$HCN
correlation strongly constrains ${\epsilon}_{\rm ff} \approx 1\%$ with a factor
of $\sim 3$ systematic uncertainty. The scatter in ${\epsilon}_{\rm ff}$ from
cloud to cloud within the Milky Way is a factor of a few. We conclude that
$L_{\rm HCN}$ is an effective tracer of dense gas and that the IR$-$HCN
correlation is a significant diagnostic of the microphysics of star formation
in dense gas.
