  The baryon-acoustic oscillation (BAO) feature in the Lyman-$\alpha$ forest is
one of the key probes of the cosmic expansion rate at redshifts z~2.5, well
before dark energy is believed to have become dynamically significant. A key
advantage of the BAO as a standard ruler is that it is a sharp feature and
hence is more robust against broadband systematic effects than other
cosmological probes. However, if the Lyman-$\alpha$ forest transmission is
sensitive to the initial streaming velocity of the baryons relative to the dark
matter, then the BAO peak position can be shifted. Here we investigate this
sensitivity using a suite of hydrodynamic simulations of small regions of the
intergalactic medium with a range of box sizes and physics assumptions; each
simulation starts from initial conditions at the kinematic decoupling era
(z~1059), undergoes a discrete change from neutral gas to ionized gas thermal
evolution at reionization (z~8), and is finally processed into a Lyman-$\alpha$
forest transmitted flux cube. Streaming velocities suppress small-scale
structure, leading to less violent relaxation after reionization. The changes
in the gas distribution and temperature-density relation at low redshift are
more subtle, due to the convergent temperature evolution in the ionized phase.
The change in the BAO scale is estimated to be of the order of 0.12% at z=2.5;
some of the major uncertainties and avenues for future improvement are
discussed. The predicted streaming velocity shift would be a subdominant but
not negligible effect (of order $0.26\sigma$) for the upcoming DESI
Lyman-$\alpha$ forest survey, and exceeds the cosmic variance floor. It is
hoped that this study will motivate additional theoretical work on the
magnitude of the BAO shift, both in the Lyman-$\alpha$ forest and in other
tracers of large-scale structure.
