Effect of Separate Initial Conditions on the Lyman-$\alpha$ Forest in
  Simulations
  Using a set of high resolution simulations, we quantify the effect of species
specific initial transfer functions on probes of the IGM via the Lyman-$\alpha$
forest. We focus on redshifts $2-6$, after H~{\sc i} reionization. We explore
the effect of these initial conditions on measures of the thermal state of the
low density IGM: the curvature, Doppler width cutoff, and Doppler width
distribution. We also examine the matter and flux power spectrum, and potential
consequences for constraints on warm dark matter models. We find that the
curvature statistic is at most affected at the $\approx2\%$ level at $z=6$. The
Doppler width cutoff parameters are affected by $\approx5\%$ for the intercept,
and $\approx8\%$ for the fit slope, though this is subdominant to sample
variation. The Doppler width distribution shows a $\approx30\%$ effect at
$z=3$, however the distribution is not fully converged with simulation box size
and resolution. The flux power spectrum is at most affected by $\approx5\%$ at
high redshift and small scales. We discuss numerical convergence with
simulation parameters.
