Physical Properties of 5,000 Cool LMC Supergiants with Gaia XP Spectra: A Detailed Portrait of the Upper HR Diagram Hints at Missing Supernova Progenitors

Characterizing the physical properties of cool supergiants allows us to probe the final stages of a massive star's evolution before it undergoes core collapse. Despite their importance, the fundamental properties for these stars -- $T_{\rm eff}$ and $\log L/L_\odot$ -- are only known for a limited number of objects. The third data release of the Gaia mission contains precise photometry and low-resolution spectroscopy of hundreds of cool supergiants in the LMC with well-constrained properties. Using these data, we train a simple and easily-interpretable machine learning model to regress effective temperatures and luminosities with high accuracy and precision comparable to the training data. We then apply our model to 5000 cool supergiants, many of which have no previously-published $T_{\rm eff}$ or $L$ estimates. The resulting Hertzprung-Russell diagram is well-populated, allowing us to study the distribution of cool supergiants in great detail. Examining the luminosity functions of our sample, we find a notable flattening in the luminosity function of yellow supergiants above $\log L/L_\odot=5$, and a corresponding steepening of the red supergiant luminosity function. We place this finding in context with previous results, and present its implications for the infamous red supergiant problem.