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

When selecting a lc classifier for use as part of a photometric supernova Ia (SN Ia) cosmological analysis, it is common to make decisions based on metrics of classification performance, such as the 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 eliminate the computational expense of a full cosmology forecast in the analysis pipeline design process This study tests the assumption that lc classification metrics are an appropriate proxy for cosmology metrics. We emulated photometric SN Ia cosmology lc samples with controlled contamination rates of individual contaminant classes and evaluated each of them under a set of classification metrics. We then derived cosmological parameter constraints from all samples under two common analysis approaches and quantified 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 shown to be insensitive to the latter. Based on these findings, we discourage any exclusive reliance on classification-based metrics for analysis design decisions, which (counterintuitively) include but are not limited to the classifier choice. Instead, we recommend optimising science analysis pipeline design choices using a metric of the information gained about the physical parameters of interest.