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.
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