ForestFlow: predicting the Lyman-$\alpha$ forest clustering from linear to nonlinear scales

On large scales, the Lyman-$\alpha$ forest provides insights into the expansion history of the Universe, while on small scales, it imposes strict constraints on the growth history, the nature of dark matter, and the sum of neutrino masses. This work introduces ForestFlow, a novel framework that bridges the gap between large- and small-scale analyses, which have traditionally relied on distinct modeling approaches. Using conditional normalizing flows, ForestFlow predicts the two Lyman-$\alpha$ linear biases ($b_\delta$ and $b_\eta$) and six parameters describing small-scale deviations of the three-dimensional flux power spectrum ($P_\mathrm{3D}$) from linear theory as a function of cosmology and intergalactic medium physics. These are then combined with a Boltzmann solver to make consistent predictions, from arbitrarily large scales down to the nonlinear regime, for $P_\mathrm{3D}$ and any other statistics derived from it. Trained on a suite of 30 fixed-and-paired cosmological hydrodynamical simulations spanning redshifts from $z=2$ to 4.5, ForestFlow achieves 3 and 1.5\% precision in describing $P_\mathrm{3D}$ and the one-dimensional flux power spectrum ($P_\mathrm{1D}$) from linear scales to $k=5\,\mathrm{Mpc}^{-1}$ and $k_\parallel=4\,\mathrm{Mpc}^{-1}$, respectively. Thanks to its conditional parameterization, ForestFlow shows similar performance for ionization histories and two $\Lambda$CDM model extensions $\unicode{x2013}$ massive neutrinos and curvature $\unicode{x2013}$ even though none of these are included in the training set. This framework will enable full-scale cosmological analyses of Lyman-$\alpha$ forest measurements from the DESI survey.