$Spitzer$-selected $z > 1.3$ protocluster candidates in the LSST Deep Drilling Fields

We have identified 189 candidate $z > 1.3$ protoclusters and clusters in the LSST Deep Drilling Fields. This sample will enable the measurement of the metal enrichment and star formation history of clusters during their early assembly period through the direct measurement of the rate of supernovae identified through the LSST. The protocluster sample was selected from galaxy overdensities in a $Spitzer$/IRAC colour-selected sample using criteria that were optimised for protocluster purity using a realistic lightcone. Our tests reveal that $60-80\%$ of the identified candidates are likely to be genuine protoclusters or clusters, which is corroborated by a $\sim4\sigma$ stacked X-ray signal from these structures. We provide photometric redshift estimates for 47 candidates which exhibit strong peaks in the photo-$z$ distribution of their candidate members. However, the lack of a photo-$z$ peak does not mean a candidate is not genuine, since we find a stacked X-ray signal of similar significance from both the candidates that exhibit photo-$z$ peaks and those that do not. Tests on the lightcone reveal that our pursuit of a pure sample of protoclusters results in that sample being highly incomplete ($\sim4\%$) and heavily biased towards larger, richer, more massive, and more centrally concentrated protoclusters than the total protocluster population. Most ($\sim75\%$) of the selected protoclusters are likely to have a maximum collapsed halo mass of between $10^{13}-10^{14}$ M$_{\odot}$, with only $\sim25\%$ likely to be collapsed clusters above $10^{14}$ M$_{\odot}$. However, the aforementioned bias ensures our sample is $\sim50\%$ complete for structures that have already collapsed into clusters more massive than $10^{14}$ M$_{\odot}$.