  We provide a model for how Kennicutt-Schmidt (KS) laws, which describe the
correlation between star formation rate and gas surface or volume density,
depend on the molecular line chosen to trace the gas. We show that, for lines
that can be excited at low temperatures, the KS law depends on how the line
critical density compares to the median density in a galaxy's star-forming
molecular clouds. High critical density lines trace regions with similar
physical properties across galaxy types, and this produces a linear correlation
between line luminosity and star formation rate. Low critical density lines
probe regions whose properties vary across galaxies, leading to a star
formation rate that varies superlinearly with line luminosity. We show that a
simple model in which molecular clouds are treated as isothermal and homogenous
can quantitatively reproduce the observed correlations between galactic
luminosities in far infrared and in the CO(1->0) and HCN(1->0) lines, and
naturally explains why these correlations have different slopes. We predict
that IR-line luminosity correlations should change slope for galaxies in which
the median density is close to the line critical density. This prediction may
be tested by observations of lines such as HCO^+(1->0) with intermediate
critical densities, or by HCN(1->0) observations of intensely star-forming high
redshift galaxies with very high densities. Recent observations by Gao et al.
hint at just such a change in slope. We argue that deviations from linearity in
the HCN(1->0)-IR correlation at high luminosity are consistent with the
assumption of a constant star formation efficiency.
