Can the Low Redshift Lyman Alpha Forest Constrain AGN Feedback Models?

We investigate the potential of low-redshift Lyman alpha (Ly$\alpha$) forest for constraining active galactic nuclei (AGN) feedback models by analyzing the Illustris and IllustrisTNG simulation at z=0.1. These simulations are ideal for studying the impact of AGN feedback on the intergalactic medium (IGM) as they share initial conditions with significant differences in the feedback prescriptions. Both simulations reveal that the IGM is significantly impacted by AGN feedback. Specifically, feedback is stronger in Illustris and results in reducing cool baryon fraction to 23% relative to 39% in IllustrisTNG. However, when comparing various statistics of Ly$\alpha$ forest such as 2D and marginalized distributions of Doppler widths and H I column density, line density, and flux power spectrum with real data, we find that most of these statistics are largely insensitive to the differences in feedback models. This lack of sensitivity arises because of the fundamental degeneracy between the fraction of cool baryons and the H I photoionization rate ($\Gamma_{\rm HI}$) as their product determines the optical depth of the Ly$\alpha$ forest. Since the $\Gamma_{\rm HI}$ cannot be precisely predicted from first principles, it needs to be treated as a nuisance parameter adjusted to match the observed Ly$\alpha$ line density. After adjusting $\Gamma_{\rm HI}$, the distinctions in the considered statistics essentially fade away. Only the Ly$\alpha$ flux power spectrum at small spatial scales exhibits potentially observable differences, although this may be specific to the relatively extreme feedback model employed in Illustris. Without independent constraints on either $\Gamma_{\rm HI}$ or cool baryon fraction, constraining AGN feedback with low-redshift Ly$\alpha$ forest will be very challenging.