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.
