Prospects for ${\rm kSZ^2}$-${\rm 21cm^2}$ Cross-Correlations during Reionization

The 21\,cm line and the patchy kinetic Sunyaev-Zel'dovich (kSZ) effect are promising and complementary probes of the Epoch of Reionization (EoR). A challenge for cross-correlating these two signals is that foreground avoidance or removal algorithms applied to the 21\,cm data inevitably sacrifice Fourier modes with long wavelengths along the line-of-sight (i.e., low-$k_\parallel$ modes), yet \textit{only} these same modes contribute to the kSZ signal. Here we show that a suitable kSZ$^2$ $\times$ 21\,cm$^2$ cross-correlation statistic nevertheless remains non-vanishing, even after filtering out the corrupted low-$k_\parallel$ Fourier modes from the 21\,cm data. We simulate the kSZ$^2$ $\times$ 21\,cm$^2$ cross-correlation signal across reionization-era redshifts and find distinctive redshift evolution. This signal peaks early in the reionization history, when the volume-averaged fraction is around $0.1 \lesssim x_\mathrm{HII} \lesssim 0.2$, after which it changes sign and reaches a minimum near reionization's midpoint ($x_\mathrm{HII} \sim 0.5$), while the signal gradually vanishes as reionization completes. These trends appear generic across three simulated models which differ in their reionization histories. We forecast the detectability of the kSZ$^2$ $\times$ 21\,cm$^2$ cross-power spectrum for the HERA and SKA1-Low 21\,cm experiments in combination with current and next-generation CMB surveys including the Simons Observatory, CMB-S4, and CMB-HD. We find that a high-significance detection ($\mathrm{S/N} \gtrsim 5\sigma$) is possible with SKA1-Low and CMB-S4.