An Exploration of AGN and Stellar Feedback Effects in the Intergalactic Medium via the Low Redshift Lyman-$\alpha$ Forest

We explore the role of galactic feedback on the low redshift Lyman-$\alpha$ (Ly$\alpha$)~forest ($z \lesssim 2$) statistics and its potential to alter the thermal state of the intergalactic medium. Using the Cosmology and Astrophysics with Machine Learning Simulations (CAMELS) suite, we explore variations of the AGN and stellar feedback models in the IllustrisTNG and Simba sub-grid models. We find that both AGN and stellar feedback in Simba play a role in setting the Ly$\alpha$ forest column density distribution function (CDD) and the Doppler width ($b$-value) distribution. The Simba AGN jet feedback mode is able to efficiently transport energy out to the diffuse IGM causing changes in the shape and normalization of the CDD and a broadening of the $b$-value distribution. We find that stellar feedback plays a prominent role in regulating supermassive black hole growth and feedback, highlighting the importance of constraining stellar and AGN feedback simultaneously. In IllustrisTNG, the AGN feedback variations explored in CAMELS do not affect the Ly$\alpha$ forest, but varying the stellar feedback model does produce subtle changes. Our results imply that the low-$z$ Ly$\alpha$ forest can be sensitive to changes in the ultraviolet background (UVB), stellar and black hole feedback, and that AGN jet feedback in particular can have a strong effect on the thermal state of the IGM.