Probing Feedback via IGM tomography and Ly$\alpha$ forest with Subaru
  PFS, TMT/ELT, and JWST
  In preparation for the IGM tomography study by Subaru Prime Focus
Spectrograph (PFS) survey and other large future telescopes such as
TMT/ELT/GMT, we present the results of our pilot study on Ly$\alpha$ forest and
IGM tomography statistics using the GADGET3-OSAKA cosmological smoothed
particle hydrodynamical simulation. Our simulation includes models for star
formation and supernova feedback, which enables more realistic
cross-correlation studies between galaxies, neutral hydrogen (HI) and metals in
circumgalactic and intergalactic medium. We create a light-cone data set at
$z=2-3$ from our simulations and generate mock Ly$\alpha$ forest data. As a
first step, in this paper, we focus on the distribution of HI and galaxies, and
present statistical results on 1D flux PDF, 1D power spectrum, flux contrast
vs. impact parameter, HI$-$galaxy cross-correlations. Our results show overall
agreement with current observational data, with some interesting discrepancies
on small scales that are due to either feedback effects or varying
observational conditions. Our simulation shows stronger HI absorption with
decreasing transverse distance from galaxies. We find that the massive galaxies
with $M_\star \ge 10^{10}\,M_\odot$ contribute strongly to the flux contrast
signal, and that the lower-mass galaxies with $M_\star \sim
10^8-10^{10}\,M_\odot$ tend to dilute the flux contrast signal from massive
galaxies. On large scales, the average flux contrast smoothly connects to the
IGM level, supporting the concordance $\Lambda$ cold dark matter model. We also
find an increase in the HI absorption toward the center of a protocluster.
