Accounting for stochastic fluctuations when analysing the integrated light of star clusters I. First systematics

Context. Star clusters are studied widely both as benchmarks for stellar evolution models and in their own right. Cluster age distributions and mass distributions within galaxies are probes of star formation histories and of cluster formation and disruption processes. The vast majority of clusters in the Universe are small, and it is well known that the integrated fluxes and colours of all but the most massive ones have broad probability distributions, owing to small numbers of bright stars. Aims. This paper goes beyond describing predicted probability distributions to present results of the analysis of cluster energy distributions in an explicitly stochastic context. Methods. The method developed is Bayesian. It provides posterior probability distributions in the age-mass-extinction space, using multi-wavelength photometric observations and a large collection of Monte-Carlo simulations of clusters of finite stellar masses. The main priors are the assumed intrinsic distributions of current mass and current age for clusters in a galaxy. Both UBVI and UBVIK data sets are considered, and the study conducted in this paper is restricted to the solar metallicity. Results. We first use the collection of simulations to reassess and explain errors arising from the use of standard analysis methods, which are based on continuous population synthesis models: systematic errors on ages and random errors on masses are large, while systematic errors on masses tend to be smaller. The age-mass distributions obtained after analysis of a synthetic sample are very similar to those found for real galaxies in the literature. The Bayesian approach, on the other hand, is very successful in recovering the input ages and masses over ages ranging between 20 Myr and 1.5 Gyr, with only limited systematics that we explain. Conclusions. Considering stochasticity is important, more important for instance than the choice of adding or removing near-IR data in many cases. We found no immediately obvious reason to reject priors inspired by previous (standard) analyses of cluster populations in galaxies, i.e., cluster distributions that scale with mass as M-2 and are uniform on a logarithmic age scale.