Luminosity functions and IMF variations from large samples of HII regions and molecular clouds

Large high-quality samples of HII regions and their parent Giant Molecular Clouds (GMC) are now available for local galaxies. It is therefore possible to investigate links between the CO and H$\alpha$ luminosity functions and whether massive stars form in GMCs of all masses. The CO luminosity functions (LF), representing the distribution of GMC masses, are consistently steeper than the H$\alpha$ luminosity functions. The CO LF invariably steepens in the outer disk where fewer massive GMCs are present beyond the median cloud galactocentric distance. The H$\alpha$ LF also steepens in the outer disk for most of the galaxies examined. Using Salpeter, Kroupa, and Chabrier Initial Mass Functions (IMF) along with stellar mass-luminosity-radius relations, we compute numerically the bolometric luminosity and H$\alpha$ emission from young star clusters. The cluster masses are linked to the GMC mass by assuming that the cluster mass is a constant fraction (3\%) of the parent cloud mass. In particular, results for a fully stochastic IMF are compared to suggestions that very massive stars only form in massive clusters or clouds. Within the limits of the observations -- no small molecular clouds or low-luminosity HII regions can be detected at the typical $\sim 10$~Mpc distance of the sample galaxies -- we find no evidence for a maximum stellar mass which varies with cloud or cluster mass.