Particle initialization effects on Lyman-$\alpha$ forest statistics in cosmological SPH simulations

Confronting measurements of the Lyman-$\alpha$ forest with cosmological hydrodynamical simulations has produced stringent constraints on models of particle dark matter and the thermal and ionization state of the intergalactic medium. We investigate the robustness of such models of the Lyman-$\alpha$ forest, focussing on the effect of particle initial conditions on the Lyman-$\alpha$ forest statistics in cosmological SPH simulations. We study multiple particle initialization algorithms in simulations that are designed to be identical in other respects. In agreement with the literature, we find that the correct linear theory evolution is obtained when a glass-like configuration is used for initial unperturbed gas particle positions alongside a regular grid configuration for dark matter particles and the use of non-identical initial density perturbations for gas and dark matter. However, we report that this introduces a large scale-dependent distortion in the one-dimensional Lyman-$\alpha$ transmission power spectrum at small scales ($k > 0.05$ s/km). The effect is close to $50\%$ at $k\sim 0.1$ s/km, and persists at higher resolution. This can severely bias inferences in parameters such as the dark matter particle mass. By considering multiple initial conditions codes and their variations, we also study the impact of a variety of other assumptions and algorithmic choices, such as adaptive softening, background radiation density, particle staggering, and perturbation theory accuracy, on the matter power spectrum, the Lyman-$\alpha$ flux power spectrum, and the Lyman-$\alpha$ flux PDF. This work reveals possible pathways towards more accurate theoretical models of the Lyman-$\alpha$ forest to match the quality of upcoming measurements.