Searching for the shadows of giants II: the effect of local ionisation
  on the Lyman-$\alpha$ absorption signatures of protoclusters at redshift
  $z\sim2.4$
  Local variations in the intergalactic medium (IGM) neutral hydrogen fraction
will affect the Ly-$\alpha$ absorption signature of protoclusters identified in
tomographic surveys. Using the IllustrisTNG simulations, we investigate how the
AGN proximity effect and hot, collisionally ionised gas arising from
gravitational infall and black hole feedback changes the Ly-$\alpha$ absorption
associated with $M_{z=0}\simeq10^{14}\,M_\odot$ protoclusters at $z\simeq2.4$.
We find that protocluster galaxy overdensities exhibit a weak anti-correlation
with Ly-$\alpha$ transmission in IGM transmission maps, but local HI ionisation
enhancements due to hot $T>10^{6}\rm\,K$ gas or nearby AGN can disrupt this
relationship within individual protoclusters. On average, however, we find that
strong reductions in the IGM neutral fraction are limited to within $\lesssim
5h^{-1}\,\textrm{cMpc}$ of the dark matter haloes. Local ionisation
enhancements will therefore have a minimal impact on the completeness of
protocluster identification in tomographic surveys if smoothing Ly-$\alpha$
transmission maps over scales of $\sim4 h^{-1}\,\textrm{cMpc}$, as is typically
done in observations. However, if calibrating the relationship between the
matter density and Ly-$\alpha$ transmission in tomographic maps using simple
analytical models for the Ly-$\alpha$ forest opacity, the presence of hot gas
around haloes can still result in systematically lower estimates of $M_{z=0}$
for the most massive protoclusters.
