Conventional wisdom was that thermal relics from the epoch of reionization (EOR) would vanish swiftly. Recently, however, it was shown that these relics can survive to lower redshifts ($z sim 2$) than previously thought, due to gas at mean density being heated to $T sim 3 times 10^4$ K by reionization, which is inhomogeneous, and shocks. Given the high sensitivities of upcoming Ly$alpha$ forest surveys, this effect will be a novel broadband systematic for cosmological application. From the astrophysical point of view, however, the imprint of inhomogeneous reionization can shed light on the EOR and cosmic dawn. We utilize a hybrid method -- which includes two different simulation codes capable of handling the huge dynamical range -- to show the impact of patchy reionization on the Ly$alpha$ forest and its dependence on different astrophysical scenarios. We found statistically significant deviations in the 1D Ly$alpha$ power spectrum at $k = 0.14$ cMpc$^{-1}$ that range from $sim 1%$ at $z = 2$ up to almost $sim 20%$ at $z = 4$. The deviations in the 3D Ly$alpha$ power spectrum, at the same wavenumber, are large and range from a few per cent at $z = 2$ up to $sim 50%$ at $z = 4$, although these deviations ignore the effect of He II reionization and AGN feedback at $z<4$. By exploiting different $k$-dependence of power spectrum among various astrophysical scenarios, the effect of patchy reionization on the Ly$alpha$ forest power spectrum can open a new window into cosmic reionization and possibly cosmic dawn.
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