Co-evolution of the Brightest Cluster Galaxies and their Host Clusters
  in Illustris-TNG
  We use the Illustris-TNG 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.
