Abstract: We develop a new theoretical framework for studying the pairwise and cross-pairwise polarised kinetic Sunyaev Zeldovich (pkSZ) effect arising from the transverse peculiar velocity of galaxy clusters. The pkSZ effect is second order in peculiar velocities and has a spectrum that can be decomposed into y-type and blackbody components. We consider pairing of clusters with other clusters as well as cross-pairing of clusters with galaxies from spectroscopic galaxy surveys. We develop and compare estimators of the pairwise pkSZ effect and study the detectability of the pairwise signal with cluster catalogs consisting of a few hundred thousand clusters expected from surveys such as eROSITA and CMB-S4. We find that cross-pairing clusters with galaxies from a large overlapping spectroscopic survey having a few billion galaxies will enable us to detect the pairwise pkSZ effect with CMB-S4. The cross-pairwise pkSZ effect will thus open up a new window into the large-scale structure of the Universe in the coming decades.
