Detecting preheating in proto-clusters with Lyman-$\alpha$ Forest
  Tomography
  Studies of low redshift galaxy clusters suggest the intra-cluster medium
(ICM) has experienced non-gravitational heating during the formation phase of
the clusters. Using simple phenomenological heating prescriptions, we simulate
the effect of this preheating of the nascent ICM in galaxy proto-clusters and
examine its effect on Lyman-$\alpha$ (Ly$\alpha$) forest tomographic maps. We
analyse a series of cosmological zoom-in simulations of proto-clusters within
the framework of the Ly$\alpha$ transmission-dark matter (DM) density
distribution. We find that the more energy is injected into the proto-ICM at
$z$ = 3, the more the distribution at high DM density tilts towards higher
Ly$\alpha$ transmission. This effect has been confirmed in both low-resolution
simulations adopting a preheating scheme based on entropy floors, as well as in
higher-resolution simulations with another scheme based on energy floors. The
evolution of the slope of this distribution is shown to vary with redshift. The
methodology developed here can be applied to current and upcoming Ly$\alpha$
forest tomographic survey data to help constrain feedback models in galaxy
proto-clusters.
