Are classification metrics good proxies for SN Ia cosmological constraining power?

We emulate photometric SN Ia cosmology samples with controlled contamination rates of individual contaminant classes and evaluate each of them under a set of classification metrics. We then derive cosmological parameter constraints from all samples under two common analysis approaches and quantify the impact of contamination by each contaminant class on the resulting cosmological parameter estimates. We observe that cosmology metrics are sensitive to both the contamination rate and the class of the contaminating population, whereas the classification metrics are insensitive to the latter. We therefore discourage exclusive reliance on classification-based metrics for cosmological analysis design decisions, e.g. classifier choice, and instead recommend optimizing using a metric of cosmological parameter constraining power.