tokens: sample into three metallicity bins ([Fe/H] −1.5, −1.5 [Fe/H] −0.7, and [Fe/H] > −0.7) and plot the difference between CaT based and literature [Ca/H] as a function of half-mass relaxation time for each bin in Fig. 26 . For the low-metallicity bin, Kendall’s test provides weak evidence (τ = 0.26 and p = 0.07) for a correlation. Using pymc3 , we fit a significant slope () and find that a linear relationship is favoured over a constant value (ΔBIC = 9.2) for 24 GCs in the low-metallicity bin. The nuclear remnant NGC 5139 appears to be a significant outlier from the low-metallicity GC relationship. For both the 20 GCs in the intermediate-metallicity bin and the eight GCs in the high-metallicity bin, Kendall’s test provides no evidence (τ = 0.18, p = 0.27 and τ = 0.00, p = 1.0, respectively) for a correlation. The pymc3 fits finds an insignificant slope for the intermediate-metallicity subsample (), but a marginally significant slope for the metal-rich subsample (). Neither subsample shows strong evidence that a linear relationship is preferred over a constant value (ΔBIC = −5.7 and 2.3, respectively). A significant relationship between the CaT and the half-mass relaxation time only at low metallicity is consistent with the behaviour of the CaT with the slope of the stellar mass function. For both the mass function slope and the relaxation time relations, the relationship for all GCs in our sample is driven by the low-metallicity GCs. Figure 26. Differences between literature and our CaT-based [Ca/H] as a function of the half-mass radius relaxation time. From top to bottom, we show the difference for GCs with [Fe/H] −1.5, −1.5 [Fe/H] −0.7, and [Fe/H] −0.7. The black line in the top panel shows our fitted relationship between Δ[Ca/H] and the relaxation time for the lowest metallicity GCs. The colours, shapes, and size of the points are the same as in Fig. 18 . Since the slope of the mass function and the relaxation time are anticorrelated, the relaxation time increases from right to left to match the trend seen between the CaT and the mass function slope. As in Fig. 19 , the dashed line shows Δ[Ca/H] = 0, while the dotted lines show Δ[Ca/H] = ±0.1. The lowest metallicity GCs with longer relaxation times show stronger CaT strengths than GCs with shorter relaxation times and the same metallicity. However, intermediate- and high-metallicity GCs show no relationship between CaT strength and relaxation time at fixed metallicity. The observed qualitative behaviour of the CaT with the stellar mass function slope and metallicity is consistent with the predictions of stellar population synthesis models. The Vazdekis et al. ( 2003 , 2012 ) models predict that the slope of the relationship between the CaT strength and the slope of the stellar mass function as a strong function of metallicity, with a strong anticorrelation between the two at low metallicities, little or no mass function dependence at intermediate metallicities and a strong correlation at solar
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