Assuming Ionization Equilibrium and the Impact on the Lyman-$\mathrm{\alpha}$ Forest Power Spectrum during the End of Reionization at $8 \geq z \geq 5

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.