Modeling the Cosmological Lyman-$\alpha$ Forest at the Field Level

The distribution of absorption lines in the spectra of distant quasars, called the Lyman-$\alpha$ (Ly-$\alpha$) forest, is a unique probe of cosmology and the intergalactic medium at high redshifts and small scales. The statistical power of ongoing redshift surveys demands precise theoretical tools to model the Ly-$\alpha$ forest. We address this challenge by developing an analytic, perturbative forward model to predict the Ly-$\alpha$ forest at the field level for a given set of cosmological initial conditions. Our model shows a remarkable performance when compared with the Sherwood hydrodynamic simulations: it reproduces the flux distribution, the Ly-$\alpha$ - dark matter halo cross-correlations, and the count-in-cell statistics at the percent level down to scales of a few Mpc. Our work provides crucial tools that bridge analytic modeling on large scales with simulations on small-scales, enabling field-level inference from Ly-$\alpha$ forest data and simulation-based priors for cosmological analyses. This is especially timely for realizing the full scientific potential of the Ly-$\alpha$ forest measurements by the Dark Energy Spectroscopic Instrument.