Metal absorption lines provide a powerful way to probe the thermal, ionization, and enrichment history of cosmic gas, but their interpretation requires modelling ion populations beyond simple equilibrium assumptions. In this talk, I will discuss my work on fast and physically motivated models for tracking the non-equilibrium ionization state of the intergalactic medium. I will first present a metals-inclusive zero-dimensional non-equilibrium ionization framework that follows the coupled thermal and ionization evolution of H, He, and carbon in a redshift-dependent ultraviolet background. I will describe its validation against full three-dimensional non-equilibrium hydrodynamical simulations and its application to interpreting the cosmic abundance of C iv, ionization corrections, and IGM metallicities. I will then discuss future extensions of this framework, including local galactic radiation fields, patchy reionization histories, and a broader metal-ion network involving C iv, Si iv, O vi, N v etc. I will also outline how large suites of non-equilibrium ionization histories can be used to train neural-network emulators for predicting time-dependent ion fractions, and how post-processing cosmological simulations can bridge the gap between simplified zero-dimensional models and fully coupled non-equilibrium simulations.

I will also discuss my work on modelling the Lyα forest for cosmological and astrophysical applications. The Lyα forest is one of the most sensitive probes of the matter distribution, but precision analyses of upcoming survey data require large numbers of accurate mock spectra over cosmological volumes. I will begin by describing an end-to-end MCMC framework based on the lognormal approximation of the baryonic density field, which enables fast generation of Lyα spectra and parameter inference without interpolation over simulation grids. I will summarize its comparison with the hydrodynamical simulations, its extension across multiple redshifts, and its application to covariance estimation for the Lyα.flux power spectrum. Finally, I will present my current work on machine-learning models that map semi-analytical density fields, such as Zel’dovich and 2LPT approximations, as well as N-body dark matter fields, directly to Lyα transmitted flux. These networks are trained on relatively small hydrodynamical simulations and applied to much larger volumes, with the goal of producing Gpc-scale mock Lyα skewers for both covariance estimation and parameter inference.