  We derive a physical model for the observed relations between star formation
rate (SFR) and molecular line (CO and HCN) emission in galaxies, and show how
these observed relations are reflective of the underlying star formation law.
We do this by combining 3D non-LTE radiative transfer calculations with
hydrodynamic simulations of isolated disk galaxies and galaxy mergers. We
demonstrate that the observed SFR-molecular line relations are driven by the
relationship between molecular line emission and gas density, and anchored by
the index of the underlying Schmidt law controlling the SFR in the galaxy.
Lines with low critical densities (e.g. CO J=1-0) are typically thermalized and
trace the gas density faithfully. In these cases, the SFR will be related to
line luminosity with an index similar to the Schmidt law index. Lines with high
critical densities greater than the mean density of most of the emitting clouds
in a galaxy (e.g. CO J=3-2, HCN J=1-0) will have only a small amount of
thermalized gas, and consequently a superlinear relationship between molecular
line luminosity and mean gas density. This results in a SFR-line luminosity
index less than the Schmidt index for high critical density tracers. One
observational consequence of this is a significant redistribution of light from
the small pockets of dense, thermalized gas to diffuse gas along the line of
sight, and prodigious emission from subthermally excited gas. At the highest
star formation rates, the SFR-Lmol slope tends to the Schmidt index, regardless
of the molecular transition. The fundamental relation is the Kennicutt-Schmidt
law, rather than the relation between SFR and molecular line luminosity. We use
these results to make imminently testable predictions for the SFR-molecular
line relations of unobserved transitions.
