The latest measurements of CMB electron scattering optical depth reported by Planck significantly reduces the allowed space of HI reionization models, pointing toward a later ending and/or less extended phase transition than previously believed. Reionization impulsively heats the intergalactic medium (IGM) to $sim10^4$ K, and owing to long cooling and dynamical times in the diffuse gas, comparable to the Hubble time, memory of reionization heating is retained. Therefore, a late ending reionization has significant implications for the structure of the $zsim5-6$ Lyman-$alpha$ (ly$alpha$) forest. Using state-of-the-art hydrodynamical simulations that allow us to vary the timing of reionization and its associated heat injection, we argue that extant thermal signatures from reionization can be detected via the Ly$alpha$ forest power spectrum at $5< z<6$. This arises because the small-scale cutoff in the power depends not only the the IGM temperature at these epochs, but is also particularly sensitive to the pressure smoothing scale set by the IGM full thermal history. Comparing our different reionization models with existing measurements of the Ly$alpha$ forest flux power spectrum at $z=5.0-5.4$, we find that models satisfying Plancks $tau_e$, constraint favor a moderate amount of heat injection consistent with galaxies driving reionization, but disfavoring quasar-driven scenarios. We explore the impact of different reionization histories and heating models on the shape of the power spectrum, and find that they can produce similar effects, but argue that this degeneracy can be broken with high enough quality data. We study the feasibility of measuring the flux power spectrum at $zsimeq 6$ using mock quasar spectra and conclude that a sample of $sim10$ high-resolution spectra with an attainable signal-to-noise ratio will allow distinguishing between different reionization scenarios.
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