  We explore how the assumption of ionization equilibrium modulates the modeled
intergalactic medium (IGM) at the end of the hydrogen Epoch of Reionization
using the cosmological radiation hydrodynamic \textsc{Technicolor Dawn}
simulation. In neutral and partially-ionized regions where the metagalactic
ultraviolet background (UVB) is weak, the ionization timescale
$t_\mathrm{ion}\equiv \Gamma^{-1}$ exceeds the Hubble time. Assuming
photoionization equilibrium in such regions artificially boosts the ionization
rate, accelerating reionization. By contrast, the recombination time
$t_\mathrm{rec} < t_\mathrm{ion}$ in photoionized regions, with the result that
assuming photoionization equilibrium artificially increases the neutral
hydrogen fraction. Using snapshots between $8 \geq z \geq 5$, we compare the
predicted Lyman-$\alpha$ forest flux power spectrum with and without the
assumption of ionization equilibrium. Small scales ($k > 0.1$ rad s km$^{-1}$)
exhibit reduced power from $7 \leq z \leq 5.5$ in the ionization equilibrium
case while larger scales are unaffected. This occurs for the same reasons:
ionization equilibrium artificially suppresses the neutral fraction in
self-shielded gas and boosts ionizations in voids, suppressing small-scale
fluctuations in the ionization field. When the volume-averaged neutral fraction
drops below $10^{-4}$, the signature of non-equilibrium ionizations on the
Lyman-$\alpha$ forest (LAF) disappears. Comparing with recent observations
indicates that these non-equilibrium effects are not yet observable in the LAF
flux power spectrum.
