This paper proposes and forecasts a new way to study the early phases of the Epoch of Reionization. The authors cross-correlate the square of the kinetic Sunyaev–Zel'dovich (kSZ) signal from the cosmic microwave background (CMB) with the square of the 21‑cm emission from neutral hydrogen. Squaring both fields produces a higher‑order statistic that can avoid some of the cancellations and foreground problems that limit simpler cross-correlations. The work focuses on redshifts roughly between 9 and 16, when heating of the intergalactic medium was still patchy.

To make concrete predictions the team created simulated skies with the semi‑numeric code 21cmFASTv4. These simulations include spatial fluctuations in the hydrogen spin temperature — the quantity that controls the strength of the 21‑cm signal — rather than assuming the gas was already very hot everywhere. They also folded in realistic instrument sensitivity for radio interferometers using the 21cmSense tool, including Earth rotation synthesis and baseline coverage, and considered kSZ measurements from future CMB experiments.

Why square the fields? The kSZ effect is a Doppler shift of CMB photons that depends on the line‑of‑sight motion of free electrons. That motion can point toward or away from us, so the kSZ map has positive and negative regions that tend to cancel in a simple cross‑correlation. Squaring the kSZ map removes the sign and preserves the magnitude. Likewise, squaring the 21‑cm map lets the correlation use pairs of modes that would otherwise be lost to the so‑called foreground wedge — a region in spectral Fourier space polluted by spectrally smooth foregrounds. The resulting quantity is effectively a four‑point statistic (a “trispectrum”) that can access high line‑of‑sight wave numbers outside the wedge.

Their forecasts find that detectability depends strongly on how much of Fourier space is usable. For an optimistic foreground‑avoidance choice (keeping modes with k_parallel ≤ 0.1 h Mpc^-1 usable) they forecast a cumulative signal‑to‑noise ratio of about 5.9 for SKA‑Low AA* combined with a future CMB‑HD experiment, and about 3.6 for HERA combined with CMB‑HD. If the 21‑cm foreground wedge is larger and extends out to the horizon, those numbers fall to roughly 2.9 and 2.85, respectively. They also show the signal is sensitive not only to which regions are ionized but also to the thermal, X‑ray heating history of the gas.