Reconstructing galactic feedback history via the Lyman-$\alpha$ forest with Habitable Worlds Observatory

Recent studies have focused on the low-$z$ Ly$\alpha$ forest as a potential constraint on galactic feedback, as different AGN and stellar feedback models in hydrodynamic simulations produce varying intergalactic medium (IGM) statistics. However, existing low-$z$ Ly$\alpha$ forest data provide insufficient observational constraints for simulations due to their low precision and the lack of observations from $z \sim 0.4$ to $1.8$. The Habitable Worlds Observatory, equipped with an ultraviolet (FUV / NUV) spectrograph, could provide transformative data for this science by increasing both absorbers in the redshift range where current data are lacking and increasing the precision of the Ly$\alpha$ forest observational data at $z \leq 0.4$. This spectrograph should cover the wavelengths $1215-3402$ \AA~and have a spectral resolution of $R =$~40,000. Distinguishing between certain active-galactic nuclei feedback models requires high precision ($\sim 5-10$\%) measurements of the Ly$\alpha$ forest 1D transmitted flux power spectrum for $z\leq 1.8$. This requires $\sim 830$ QSO spectra with S/N~$\geq5$ - $25$ depending on redshift. This data would also enable a comprehensive study on the thermal state of the IGM across time, and address the tension between the observed and simulated Ly$\alpha$ forest $b$-value distribution.