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, in general, 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 ens embles are related to the stellar luminosity distribution function, within the formalism of probabilistic synthesis mod- els. More details have been yet presented in Cervino & Luridiana (2006) and references therein. As a direct application of this formalism, we discu ss 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. 1. From stars to stellar ensembles and the mass-luminosity relation Our basic knowledge of the Universe stems from the light received from observed sources. In a first-order approximation (neglecting intera ctions with the interstellar medium and non-stellar components), we can consider two types of sources: individual stars and stellar ensembles. Since in this case the emission of an stellar ensemble is just the sum of its individual components, we can refer to this emission as integrated light, coming from an unresolved system such as a distant cluster, or a pixel/slit/IFU in an image of a galaxy. The problem of inferring physical properties from the observed data can thus be reduced to the analysis of the observed light of individual stars in terms of theoretical stellar models, or, in the case of integrated li ght, to decompose the integrated light into its (stellar) components. Obviously, the interpretation and physical inferences tha t can be obtained from the integrated light depends on our physical knowledge of the individual sources that