tokens: Ly α power spectrum. Although the impact of inhomogeneous reionization in the Ly α forest is slightly larger than in our previous work, if one takes into account the statistical error in the eBOSS measurement (plotted in Fig. 2 ) and the systematic error in modelling reionization, it is currently challenging to use the 1D Ly α power spectrum to extract information from the reionization epoch via the impact of inhomogeneous reionization in the Ly α forest. However, in future work we will investigate this possibility. On the other hand, the expected effect of patchy reionization in the 3D Ly α forest power spectrum are significantly larger. We find that the percentage deviation is in the order of tens of per cent at z obs = 4.0 (e.g. for our model R2), and in the order of a few per cent at z obs = 2.0 (e.g. for our model T3). Our results for the survival of these thermal relics from reionization to low redshift might seem unconventional . However, the reason we obtain such strong ripples in the Ly α forest power spectra is because we are able to carefully track the small-scale structure prior to reionization. In particular, mini-voids at reionization get later compressed to mean-density while they simultaneously get reionized to high entropy. This HEMD gas gets usually heated to ∼3 × 10 4 K in our simulations and remains above the temperature–density relation even at z 4. As a result the temperature–density relation of the IGM becomes bimodal with the usual low-entropy phase and this high-entropy phase. The ultimate fate of the HEMD phase is to merge with the low-entropy mode; however, it takes cosmological time-scales for the merging to complete (see figs 4 and 5 of Hirata 2018 ). We stress here that it is really both the patchy nature of reionization , and the HEMD gas, i.e. ∂ψ/∂ z in equation ( 3 ) together that result in the memory of reionization in the Ly α forest computed in this work. To illustrate the former point, we turn-off the inhomogeneous recombination prescription of Sobacchi Mesinger ( 2014 ) in 21cmFAST , and consider the resulting, faster, reionization. Our results for the 3D Ly α power spectrum are reduced by a factor of 2 at z = 2.0 and ∼1.8 at z = 4.0. This decrease of deviations in the Ly α power spectrum is expected since a faster reionization model will have a narrower range of redshifts for the peak of that contribute the most to the integration in equation ( 3 ). Regarding the importance of the HEMD gas, on the other hand, we artificially limit the impact of the HEMD gas by only considering our transparency results for higher redshifts, say z re = {8.0, 9.0, 10.0, 11.0, 12.0}. Note that patches of the sky that reionize earlier will have the high-entropy gas merge with the low-entropy gas sooner, thus recovering the temperature–density relation of the IGM faster. In this case we obtain a decrease in the 3D Ly α results with a factor of 2.2 at z = 2.0 and 1.8 at z = 4.0. In Fig. 3 , we plot the k -dependence of both the
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