Quantifying the Impact of Incompleteness on Identifying and Interpreting Galaxy Protocluster Populations with the TNG-Cluster Simulation

We use the TNG-Cluster simulation to investigate how stellar mass and star formation rate (SFR) incompleteness affect the identification of density peaks within galaxy protoclusters at different redshifts. Our analysis focuses on a sample of $352$ protoclusters, defined as the progenitor populations of galaxies that reside within the virialized region of $z=0$ clusters with $M_{\rm{200}}^{z=0}\sim10^{14.3-15.5}~{\rm M}_{\odot}$. For comparison, we define our "baseline" protocluster population as galaxies with ${M}_{\star}> 10^{8.5}~{\rm M}_{\odot}$ at any redshift. We find that ${M}_{\star}$-limited (${M}_{\star} > 10^{9.5}~{\rm M}_{\odot}$) and SFR-limited ($\rm{SFR} > 10~{\rm M}_{\odot} \mathrm{yr}^{-1}$) subpopulations recover the baseline highest galaxy density peak in roughly $\sim60\%$ of cases within an accuracy of $1.0$ pMpc (corresponding to an angular scale of $\sim 2-2.5$ arcmin) at $z > 2$. This recovery fraction drops to $\sim40-50\%$ when restricting to galaxies with ${M}_{\star} > 10^{10.0}~{\rm M}_{\odot}$. We find that the baseline highest galaxy density peaks typically coincide with the highest dark matter and stellar mass density peaks, with separations less than $0.5$ pMpc in $\sim60-75\%$ of cases at $z>2$. This agreement drops to $\sim45-50\%$ when restricting to galaxies with ${M}_{\star} > 10^{10.0}~{\rm M}_{\odot}$. These results indicate that identifying the densest regions of protoclusters -- i.e., the core -- is highly sensitive to stellar mass and SFR completeness limits. Nevertheless, at $z>2$ we find that the baseline highest galaxy density peaks are generally sites of enhanced star formation and accelerated mass growth relative to the remainder of the protocluster, consistent with some observational studies.