Towards physics responsible for large-scale Lyman-$\alpha$ forest bias parameters

Using a series of carefully constructed numerical experiments based on hydrodynamic cosmological SPH simulations, we attempt to build an intuition for the relevant physics behind the large scale density ($b_\delta$) and velocity gradient ($b_\eta$) biases of the Lyman-$\alpha$ forest. Starting with the fluctuating Gunn-Peterson approximation applied to the smoothed total density field in real-space, and progressing through redshift-space with no thermal broadening, redshift-space with thermal broadening and hydrodynamicaly simulated baryon fields, we investigate how approximations found in the literature fare. We find that Seljak's 2012 analytical formulae for these bias parameters work surprisingly well in the limit of no thermal broadening and linear redshift-space distortions. We also show that his $b_\eta$ formula is exact in the limit of no thermal broadening. Since introduction of thermal broadening significantly affects its value, we speculate that a combination of large-scale measurements of $b_\eta$ and the small scale flux PDF might be a sensitive probe of the thermal state of the IGM. We find that large-scale biases derived from the smoothed total matter field are within 10-20\% to those based on hydrodynamical quantities, in line with other measurements in the literature.