Co-evolution of the Brightest Cluster Galaxies and their Host Clusters in IllustrisTNG

We use the IllustrisTNG simulations to explore the dynamic scaling relation between massive clusters and their central galaxies (BCGs). The Illustris TNG300-1 simulation we use includes 280 massive clusters with $M_{200} > 10^{14}$ M$_{\odot}$ enabling a robust statistical analysis. We derive the line-of-sight velocity dispersion of the stellar particles of the BCGs ($\sigma_{*, BCG}$), analogous to the observed BCG stellar velocity dispersion. We also compute the subhalo velocity dispersion to measure the cluster velocity dispersion ($\sigma_{cl}$). Both $\sigma_{*, BCG}$ and $\sigma_{cl}$ are proportional to the cluster halo mass, but the slopes differ slightly. Thus like the observed relation, $\sigma_{*, BCG} / \sigma_{cl}$ declines as a function of $\sigma_{cl}$, but the scatter is large. We explore the redshift evolution of $\sigma_{*, BCG} - \sigma_{cl}$ scaling relation for $z \lesssim 1$ in a way that can be compared directly with observations. The scaling relation has a similar slope at high redshift, but the scatter increases because of the large scatter in $\sigma_{*, BCG}$. The simulations imply that high redshift BCGs are dynamically more complex than their low redshift counterparts.