  We explore the use of mm-wave emission line ratios to trace molecular gas
density when observations integrate over a wide range of volume densities
within a single telescope beam. For observations targeting external galaxies,
this case is unavoidable. Using a framework similar to that of Krumholz and
Thompson (2007), we model emission for a set of common extragalactic lines from
lognormal and power law density distributions. We consider the median density
of gas producing emission and the ability to predict density variations from
observed line ratios. We emphasize line ratio variations, because these do not
require knowing the absolute abundance of our tracers. Patterns of line ratio
variations have the prospect to illuminate the high-end shape of the density
distribution, and to capture changes in the dense gas fraction and median
volume density. Our results with and without a high density power law tail
differ appreciably; we highlight better knowledge of the PDF shape as an
important area. We also show the implications of sub-beam density distributions
for isotopologue studies targeting dense gas tracers. Differential excitation
often implies a significant correction to the naive case. We provide tabulated
versions of many of our results, which can be used to interpret changes in
mm-wave line ratios in terms of changes in the underlying density
distributions.
