Left Panel: Distribution of densities (logΔ) from the hydro simulation (blue) and the semi-numerical setup (dashed orange). Middle Panel: Distribution of optical depths averaged at 4 h⁻¹cMpc scale (τ_i) for hydro simulation (blue), calibrated empirical relation (green), and calibrated semi-numerical setup (dashed orange). Right Panel: Distributions of effective optical depth skewers (τ_eff) for the previous three cases.