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.