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

Context: When selecting a classifier to use for a supernova Ia (SN Ia) cosmological analysis, it is common to make decisions based on metrics of classification performance, i.e. contamination within the photometrically classified SN Ia sample, rather than a measure of cosmological constraining power. If the former is an appropriate proxy for the latter, this practice would save those designing an analysis pipeline from the computational expense of a full cosmology forecast. Aims: This study tests the assumption that classification metrics are an appropriate proxy for cosmology metrics. Methods: 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. Results: 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. Conclusions: 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.