Probing Early Phases of the Epoch of Reionization with the $\mathrm{kSZ}^{2}\times21\,\mathrm{cm}^{2}$ Cross-Correlation

Abstract

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 ($\mathrm{kSZ}^{2}\times21\mathrm{cm}^{2}$) over $z\simeq9$--$16$, 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 ($\mathrm{S/N}$) for several combinations of 21-cm and CMB experiments, including realistic instrumental noise. For a fiducial heating history and an optimistic foreground-avoidance scenario ($k_\parallel\leq0.1\,h\,\mathrm{Mpc}^{-1}$), we find $\mathrm{S/N}\simeq5.89$ for SKA-Low AA$^\ast\times$CMB-HD and $\simeq3.62$ 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 $\mathrm{kSZ}^{2}\times21\mathrm{cm}^{2}$ 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.