Lyman-$\alpha$ forest power spectrum and its cross-correlation with dark matter halos in different astrophysical models

The Ly$\alpha$ forest, a series of HI absorption lines in the quasar spectra, is a powerful tool for probing the large-scale structure of the intergalactic medium. Its three-dimensional (3D) correlation and cross-correlations with quasars allow precise measurements of the baryon acoustic oscillation feature and redshift space distortions at redshifts $z>2$. Understanding small-scale astrophysical phenomena, such as star formation and feedback, is crucial for full-shape analyses. In this study, we measure the 3D auto-power spectrum of the Ly$\alpha$ forest and its cross-power spectrum with halos using hydrodynamic simulations from the GADGET3-OSAKA code, which includes models for star formation and supernova feedback. Across five astrophysical models, we find significant deviations from the Fiducial model, with $5-10\,\%$ differences for wavenumbers $k>2\,h\mathrm{Mpc}^{-1}$ in the Ly$\alpha$ auto-power spectrum. The Ly$\alpha\,\times\,$halo cross-power spectra show even larger deviations, exceeding $10\,\%$ in some cases. Using the fitting models of Arinyo-i-Prats et al. (2015) and Givans et al. (2022), we jointly fit the Ly$\alpha$ auto- and Ly$\alpha$ $\times$ halo cross-power spectra, and assess the accuracy of the estimated $f\sigma_8$ parameter by comparing it with the ground truth from the simulations, while varying the maximum wavenumber $k_\mathrm{max}$ and minimum halo mass $M_h$. Our results demonstrate that the extended model of Givans et al. (2022) is highly effective in reproducing $f\sigma_8$ at $k_\mathrm{max}\leq3.0\,h\mathrm{Mpc}^{-1}$ for $M_h>10^{10.5} M_\odot$, and remains robust against astrophysical uncertainties.