Detectability of the $\tau_{es}$-21cm cross-correlation: a tomographic probe of patchy reionization

The cross-correlation between fluctuations in the electron scattering optical depth $\tau_{\rm es}$ as probed by future Cosmic Microwave Background (CMB) experiments, and fluctuations in the 21cm differential brightness temperature $\Delta T_{\rm 21cm}$ as probed by ground-based radio interferometers, will trace the reionization history of the Universe. In particular, the $\tau_{\rm es}-$21cm cross-correlation should yield a determination of the characteristic bubble size distribution and ionization fraction as a function of redshift. When assuming that the cross-correlation signal is limited by instrumental noise rather than by foregrounds, we estimate its potential detectability by upcoming experiments. Specifically, the combination of HERA and Simons Observatory, CMB-S4 and PICO should yield a signal-to-noise ratio around 3 - 6, while and the exploitation of the SKA should increase it to 10-20. Finally, we have discussed how such levels of detectability can be affected when (simply modeled) 21cm foregrounds are present. For the most promising PICO$\times$SKA configuration, an efficiency of foreground removal to a level of $7\times 10^{-4}$ is needed to achieve a $5\sigma$ detection of the cross-correlation signal; in addition, safe avoidance of foreground contamination in the line-of-sight Fourier modes above $0.03 \,h\,\rm Mpc^{-1}$ would guarantee a detection significance around $3\sigma$.