Probing Early Phases of the Epoch of Reionization with the ${\mathrm{k}\mathrm{S}\mathrm{Z}}^2\times 21{\mathrm{c}\mathrm{m}}^2$ Cross-Correlation

Observations of the Epoch of Reionization (EoR) provide key insights into the nature of the first luminous sources and the thermal state of the intergalactic medium (IGM). Cross-correlations between the kinetic Sunyaev–Zel’dovich (kSZ) effect and neutral hydrogen (HI) 21-cm emission offer a promising probe of this era, sensitive to both the ionization and thermal histories of the early Universe. We investigate the cross-correlation of the squared kSZ field with the squared 21-cm intensity-mapping field (${\mathrm{k}\mathrm{S}\mathrm{Z}}^2\times 21{\mathrm{c}\mathrm{m}}^2$) over $z\simeq 9$–$16$, focusing on the impact of spatial fluctuations in the HI spin temperature. This higher-order statistic avoids line-of-sight cancellations in the kSZ signal, while squaring the 21-cm field enables non-vanishing correlations with pairs of equal and opposite high-$k_{\parallel }$ modes accessible outside the foreground wedge. We forecast the cumulative signal-to-noise ratio ($\mathrm{S}\mathrm{/}\mathrm{N}$) for several combinations of 21-cm and CMB experiments, including realistic instrumental noise. For a fiducial heating history and an optimistic foreground-avoidance scenario ($k_{\parallel }\le 0.1h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$), we find $\mathrm{S}\mathrm{/}\mathrm{N}\simeq 5.89$ for SKA-Low AA${}^{\ast }\times$CMB-HD and $\simeq 3.62$ for HERA$\times$CMB-HD. If the foreground wedge extends to the horizon, these decrease to $2.94$ and $2.85$, respectively. Our results demonstrate that ${\mathrm{k}\mathrm{S}\mathrm{Z}}^2\times 21{\mathrm{c}\mathrm{m}}^2$ is sensitive to the thermal evolution of the IGM as well as its ionization structure, providing complementary constraints on the high-redshift thermal and reionization histories.