How Dense is Your Gas? On the recoverability of LVG model parameters
  We explore the recoverability of gas physical conditions with the Large
Velocity Gradient (LVG) model, using the public code RADEX and the molecules
HCN and CO. Examining a wide parameter range with a series of models of
increasing complexity we use both grid and Monte Carlo Markov Chain (MCMC)
methods to recover the input conditions, and quantify the inherent and noise
induced uncertainties in the model results. We find that even with the benefit
of generous assumptions the LVG models struggle to recover any parameter better
than to within half a dex, although we find no evidence of systemic offsets.
Examining isotopologue lines we demonstrate that it is always preferable to
model the isotopologue abundance ratio as a free parameter, due to large biases
introduced in all other parameters when an incorrect ratio is assumed. Finally,
we explore the effects of the background radiation temperature on CO and HCN
line ratios, with an emphasis on the effect of the CMB at $z>4$, and show that
while the effect on the line ratios is minor, the effect on the SLED peak is
significant and that the CO$(1-0)$ line luminosity to H$_2$ mass conversion
factor ($\alpha_{\rm CO}$) needs to be altered to account for the loss of
contrast against the hotter CMB as redshift increases.
