Linking Dense Gas from the Milky Way to External Galaxies

In a survey of 65 galaxies, Gao & Solomon (2004) found a tight linear relation between the infrared luminosity ($L_{\rm{IR}}$, a proxy for the star formation rate) and the HCN(1-0) luminosity ($L_{\rm{HCN}}$). Wu et al. (2005, 2010) found that this relation extends from these galaxies to the much less luminous Galactic molecular high-mass star-forming clumps ($\sim$1 pc scales), and posited that there exists a characteristic ratio $L_{\rm{IR}}$ /$L_{\rm{HCN}}$ for high-mass star-forming clumps. The Gao-Solomon relation for galaxies could then be explained as a summation of large numbers of high-mass star-forming clumps, resulting in the same $L_{\rm{IR}}$ /$L_{\rm{HCN}}$ ratio for galaxies. We test this explanation and other possible origins of the Gao-Solomon relation using high-density tracers (including HCN(1-0), N$_2$H$^+$(1-0), HCO$^+$(1-0), HNC(1-0), HC$_3$N(10-9), and C$_2$H(1-0)) for $\sim$300 Galactic clumps from the Millimetre Astronomy Legacy Team 90 GHz (MALT90) survey. The MALT90 data show that the Gao-Solomon relation in galaxies cannot be satisfactorily explained by the blending of large numbers of high-mass clumps in the telescope beam. Not only do the clumps have a large scatter in the $L_{\rm{IR}}$/$L_{\rm{HCN}}$ ratio, but also far too many high-mass clumps are required to account for the Galactic IR and HCN luminosities. We suggest that the scatter in the $L_{\rm{IR}}$/$L_{\rm{HCN}}$ ratio converges to the scatter of the Gao-Solomon relation at some size-scale $\gtrsim$1 kpc. We suggest that the Gao-Solomon relation could instead result from of a universal large-scale star formation efficiency, initial mass function, core mass function, and clump mass function.