The Lyman-$\alpha$ forest in optically-thin hydrodynamical simulations
  We study the statistics of the Lyman-$\alpha$ forest in a flat LCDM cosmology
with the N-body + Eulerian hydrodynamics code Nyx. We produce a suite of
simulations, covering the observationally relevant redshift range $2 \leq z
\leq 4$. We find that a grid resolution of 20 kpc/h is required to produce one
percent convergence of Lyman-$\alpha$ flux statistics, up to k = 10 h/Mpc. In
addition to establishing resolution requirements, we study the effects of
missing modes in these simulations, and find that box sizes of L > 40 Mpc/h are
needed to suppress numerical errors to a sub-percent level. Our optically-thin
simulations with the ionizing background prescription of Haardt & Madau (2012)
reproduce an IGM equation of state with $T_0 \approx 10^4 K$ and $\gamma
\approx 1.55$ at z=2, with a mean transmitted flux close to the observed
values. When using the ionizing background prescription of Faucher-Giguere et
al. (2009), the mean flux is 10-15 per cent below observed values at z=2, and a
factor of 2 too small at z = 4. We show the effects of the common practice of
rescaling optical depths to the observed mean flux and how it affects
convergence rates. We also investigate the common practice of `splicing'
results from a number of different simulations to estimate the 1D flux power
spectrum and show it is accurate at the 10 percent level. Finally, we find that
collisional heating of the gas from dark matter particles is negligible in
modern cosmological simulations.
