tokens: of reionization, then decreases during the late stage. When the IGM global neutral fraction drops below , on the small scales a positive cross-correlation appears. At this stage, most of the IGM has been ionized, while the neutral hydrogen in haloes starts to dominate the cross-power spectrum, producing the positive correlation seen in the small-scale power spectrum. As the remaining neutral islands in the IGM shrink and decrease in number, the halo H i come to dominate the fluctuations on larger and larger scales, as a result this positive-to-negative transition point on the cross-power spectrum moves to larger scales. Figure 4. The cross-power spectrum between the neutral fraction and the dark matter density. The solid and the dashed lines represent positive and negative values, respectively. The cross-power spectrum between the 21cm brightness temperature and the DM density are plotted in Fig. 5 . We use the two panels to show the early and late stages of reionization, respectively. We find that at the very beginning of reionization, the 21cm-DM cross-power spectrum is positive on all scales, implying that the neutral hydrogen density fluctuations still follow the DM density fluctuations, although some rare density peaks have been ionized. As more and more IGM around density peaks becomes ionized, the 21cm-DM cross-power spectrum decreases, and transits to negative starting from the largest scales. Then, the large-scale 21cm-DM cross-power becomes more and more negative. The amplitude reaches its maximum at around , and decreases again. Different from the f H i -DM cross-power, the 21cm-DM cross-power spectrum is always positive at small scales, indicating that when weighted with the local density, the small-scale 21 cm signal is always dominated by the H i in haloes. At the very late stage of reionization, the positive-to-negative transition scale increases rapidly. Figure 5. The 21 cm brightness temperature and the dark matter density cross-power spectrum. Top panel: early EoR; Bottom panel: late EoR. The solid and dashed lines represent the positive and negative values, respectively. In this plot, the 21cm-DM cross-power spectrum at z = 9.90 shows some oscillations, which are likely statistical fluctuations. To check this interpretation, we estimate the errors on cross-power spectra using the jackknife resampling method. In Fig. 5 , we plot the errors for the z = 9.90 and z = 9.00 curves. To avoid clutter, errors of other curves are not shown there. We see that indeed at z = 9.90 the error bars are larger than the oscillation feature on the power spectrum curve, so it is likely that these are just statistical fluctuations. We also check that at z = 9.00 the errors are comparable with the z = 9.90. However, at z = 9.00 the cross-power spectrum is much larger than z = 9.90. In logarithmic axes apparently its errors are less obvious than z = 9.90. We then turn to the individual contribution from the IGM and haloes to the 21cm power
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