The long-lasting effect of X-ray preheating in the post-reionization intergalactic medium

X-ray photons can penetrate deep into the intergalactic medium (IGM), leading to preheating of the IGM prior to cosmic reionization. X-ray preheating wipes out some of the small-scale structures that would otherwise be present prior to the passage of an ionization front. Accurate modeling of the small-scale structure is vital to the post-reionization IGM since the small-scale structure is ultimately the dominant source of long-lasting relics from hydrogen reionization. However, the precise impact of X-ray preheating in the fossils from hydrogen reionization is highly uncertain. In this work, we explore and establish for the first time, the long-lasting impact of X-ray preheating in the post-reionization IGM via hydrodynamic simulations with high-mass resolution. We find that the addition of X-ray preheating astrophysics leads to an overall lesser impact of the effect of inhomogeneous reionization in the Lyman-$\alpha$ forest -- depending on specific X-ray prescription -- at low redshifts ($z \sim 2$) with respect to a model with no X-ray preheating. However, at high redshifts ($z \sim 4$), our results indicate a strengthening of the relics of reionization in the Lyman-$\alpha$ forest because the IGM becomes more transparent compared to the scenario with no preheating. Thus, the absence of X-ray preheating in Lyman-$\alpha$ modeling can lead to a biased inference of cosmological parameters. Nevertheless, optimistically, the inclusion of X-ray preheating emerges as a promising novel avenue to probe the astrophysics of cosmic dawn.