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.
Download