Redshift Evolution of the Fundamental Plane Relation in the IllustrisTNG
  Simulation
  We investigate the fundamental plane (FP) evolution of early-type galaxies in
the IllustrisTNG-100 simulation (TNG100) from redshift $z=0$ to $z=2$. We find
that a tight plane relation already exists as early as $z=2$. Its scatter stays
as low as $\sim 0.08$ dex across this redshift range. Both slope parameters $b$
and $c$ (where $R \propto \sigma^b I^c$ with $R$, $\sigma$, and $I$ being the
typical size, velocity dispersion, and surface brightness) of the plane evolve
mildly since $z=2$, roughly consistent with observations. The FP residual $\rm
Res$ ($\equiv\,a\,+\,b\log \sigma\,+\,c\log I\,-\,\log R$, where $a$ is the
zero point of the FP) is found to strongly correlate with stellar age,
indicating that stellar age can be used as a crucial fourth parameter of the
FP. However, we find that $4c+b+2=\delta$, where $\delta \sim 0.8$ for FPs in
TNG, rather than zero as is typically inferred from observations. This implies
that a tight power-law relation between the dynamical mass-to-light ratio
$M_{\rm dyn}/L$ and the dynamical mass $M_{\rm dyn}$ (where $M_{\rm dyn}\equiv
5\sigma^2R/G$, with $G$ being the gravitational constant) is not present in the
TNG100 simulation. Recovering such a relation requires proper mixing between
dark matter and baryons, as well as star formation occurring with correct
efficiencies at the right mass scales. This represents a powerful constraint on
the numerical models, which has to be satisfied in future hydrodynamical
simulations.
