Neutral hydrogen (HI) 21-cm intensity mapping is an effective method to track the distribution of baryonic matter, and extract astrophysical and cosmological information. The 21-cm intensity field has a nonvanishing cross-correlation with the kinetic Sunyaev-Zel’dovich (kSZ) effect that traces the velocity and density perturbations of free electrons. By using the linear perturbation theory, in this paper we calculate analytically, for the first time, the cross-correlation between the squared kSZ field and the projection of the squared HI intensity mapping field with the flat-sky approximation. This statistic remains nonvanishing even after the long-wavelength line-of-sight modes ($k_{\parallel }$) are removed due to foreground contamination. We further forecast for the prospects of detection with the SKA-MID 21-cm intensity mapping experiments (redshifts in range of $0.3<z<1$), and the kSZ maps measured by the Atacama Cosmology Telescope (ACT) and Simons Observatory (SO). The predicted cumulative signal-to-noise ratio is 1.92 for SKA-ACT and 3.99 for SKA-SO. These results show a possible on-the-edge detection on the cross-correlation signal at low redshifts, which in turn could serve as a validation step toward using it for the Epoch of Reionization studies.

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• Cosmic microwave background
• Large scale structure of the Universe
• Radio, microwave, & sub-mm astronomy