NEATH V: the relationship between line emission from dense gas tracers and the star formation rate

The Gao-Solomon relationship between the luminosity of the HCN $J=1-0$ line and the star formation rate (SFR) is observed to remain close to linear over scales ranging from individual star-forming clumps to entire galaxies. This is widely interpreted as the HCN line tracing the reservoir of dense gas directly associated with star formation. However, resolved observations of nearby molecular clouds have demonstrated that the threshold density above which star formation occurs is significantly higher than that of the gas traced by HCN emission. We perform radiative transfer modelling of molecular line emission from simulated clouds, based on magnetohydrodynamic simulations with realistic gas and dust thermodynamics and a time-dependent treatment of the molecular abundances. We find no correlation between HCN emission and the SFR in the simulations: the HCN line remains almost constant in brightness over several orders of magnitude in SFR. The N$_2$H$^+$ $J=1-0$ line correlates positively with the SFR, but weakly, and with a substantial dependence on environmental conditions. The strongest correlation between line emission and physical cloud properties is between the N$_2$H$^+$/HCN ratio and the dense gas fraction, which is close to linear. We argue that the observed HCN-SFR correlation on extragalactic scales is a result of each measurement integrating over many individual molecular clouds, which, on average, possess the same mass fraction of dense, star-forming gas. The HCN line does not directly trace this reservoir for star formation.