The low redshift Lyman-$\alpha$ Forest as a constraint for models of AGN
  feedback
  We study the low redshift Lyman-$\alpha$ Forest in the Illustris and
IllustrisTNG (TNG) cosmological simulations to demonstrate their utility in
constraining aspects of sub-grid models of feedback from active galactic nuclei
(AGN). The two simulations share an identical Ultraviolet Background
prescription and similar cosmological parameters, but TNG features an entirely
reworked AGN feedback model. Therefore a comparison of these simulations is
useful to assess the effects of an altered AGN sub-grid model on the low
redshift Lyman-$\alpha$ Forest. We find significant differences in the IGM
temperature-density relation between the two simulations due to changes in the
gas heating rate due to AGN. We investigate Lyman-$\alpha$ Forest observables
such as the column density distribution function, flux PDF, and Doppler width
($b$-parameter) distribution. Due to the AGN radio mode model, the original
Illustris simulations have a factor of 2-3 fewer absorbers than TNG at column
densities $N_{\rm HI}< 10^{15.5}$ cm$^{-2}$. We show that TNG is in much better
agreement with the observed $z=0.1$ flux power spectrum than Illustris. The
differences in the amplitude and shape of the flux PDF and power spectrum
between Illustris and TNG cannot be attributed to simple changes in the
photoheating rate. We also compare the simulated Forest statistics to UV data
from the Cosmic Origins Spectrograph (COS) and find that neither simulation can
reproduce the slope of the absorber distribution. Both Illustris and TNG also
produce significantly smaller $b$-parameter distributions than observed in the
COS data, possibly due to unresolved or missing sources of turbulence.
