The lya forest at high redshifts is a powerful probe of reionization. Modeling and observing this imprint comes with significant technical challenges: inhomogeneous reionization must be taken into account while simultaneously being able to resolve the web-like small-scale structure prior to reionization. In this work we quantify the impact of inhomogeneous reionization on the lya forest at lower redshifts ($2 < z < 4$), where upcoming surveys such as DESI will enable precision measurements of the flux power spectrum. We use both small box simulations capable of handling the small-scale structure of the lya forest and semi-numerical large box simulations capable of representing the effects of inhomogeneous reionization. We find that inhomogeneous reionization could produce a measurable effect on the lya forest power spectrum. The deviation in the 3D power spectrum at $z_{rm obs} = 4$ and $k = 0.14 rm{Mpc}^{-1}$ ranges from $19 - 36%$, with a larger effect for later reionization. The corrections decrease to $2.0 - 4.1%$ by $z_{rm obs} = 2$. The impact on the 1D power spectrum is smaller, and ranges from $3.3 - 6.5%$ at $z_{rm obs}=4$ to $0.35 - 0.75%$ at $z_{rm obs}=2$, values which are comparable to the statistical uncertainties in current and upcoming surveys. Furthermore, we study how can this systematic be constrained with the help of the quadrupole of the 21 cm power spectrum.
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