Colour-magnitude diagrams, probabilistic synthesis models and the upper mass limit of the initial mass function

We present the underlying relations between colour-magnitude diagrams (CMDs) and synthesis models through the use of stellar luminosity distribution func- tions. CMDs studies make a direct use of the stellar luminosity distribution function while, ingeneral, synthesis models only use its mean value, even though high-order moments can also be obtained. We show that the mean, high-order moments and in- tegrated luminosity distribution functions of stellar ensembles are related to the stellar luminosity distribution function, within the formalism of probabilistic synthesis mod- els. More details have been yet presented in Cervin ~ o & Luridiana (2006) and references therein. As a direct application of this formalism, we discuss two key issues. First, in- ferences on the upper mass limit of the initial mass function as a function of the total mass of clusters. Second, we apply extreme value theory to show that that the cluster mass obtained from normalising the IMF between mmax and mup does not provide the cluster mass in the case where only one star in this mass range is present, as assumed in the IGIMF theory. It provides instead the cluster mass with a 60% probability to have a star with mass larger than mmax, and we argue that in light of this result the basic formulation ofthe IGIMF theory must be revised.