tokens: of residuals on a cluster by cluster basis when compared to the grid of Kroupa IMF clusters. This increase in scatter is especially noticeable in mass, and originates from the M 3 degeneracy described above. As the mass of the cluster and its IMF slope are highly dependent on each other, the wide scatter in mass is further reflected in the inaccurate values inferred for 3 . Indeed, given the flat and broad posterior PDFs for the IMF slope, the medians of 3 have no physical meaning and simply reflect the median of the prior on 3 [in this case, a flat distribution between 3 (3.0, 1.5)]. Figure 12. Comparisons of the medians of the inferred posterior PDFs for all physical parameters to the true values adopted when simulating the mock clusters. Results for individual clusters are shown with crosses, while the median residuals and the corresponding 10th and 90th percentiles are shown with solid points. We plot analysis including broad-band photometry alone with (red, circles) and without (purple, diamonds) a variable IMF. We also show results of the variable IMF case when including H (blue, squares). Points are shifted along the x -axis for visualization purposes, with the leftmost point situated at the true x -axis value. Given this result, it is clear that obtaining refined constraints on the upper-end slope of the IMF requires a different approach. It is readily apparent that the ability to apply a prior to the cluster mass that is independent of the broad-band photometry may prove very useful in analysis such as this. Independent constraints on the masses of clusters that may then be used as priors may come, for instance, from dynamical measurements (e.g. Ho Filippenko 1996 ; Gerhard 2000 ). The era of 30m telescopes will make such measurements more feasible. To illustrate this point in more quantitative terms, we present in Fig. 13 a study of the accuracy with which 3 can be recovered as a function of the width of a prior on the mass. Results of three choices of IMF are presented, when leaving the mass unconstrained, and when applying a top-hat prior with width 0.5, 0.2 and 0.05 dex about the target cluster mass, for both broad-band filters only and broad-band filters combined with H. This analysis shows that, as expected, the medians of the PDFs on 3 converge to the true value as the precision with which the mass is constrained increases. Due to the difference between logarithmic intervals in mass and the linear scale in the IMF upper-end exponent, there is a weak scaling of 3 with the prior width, with the underlying values being recovered to within 3 0.2 when the mass is constrained to better than 10percent. However, more realistic constraints on the mass (to within a factor of 2), for instance via dynamical measurements, appear to be useful to rule out extreme variations in the IMF upper-end slope through multiwavelength broad-band photometry, given the assumptions of our model. The inclusion of H results in a wider scatter towards zero for
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