General relativity predicts that the rotational momentum flux of matter twists the spacetime via a vector gravitomagnetic (frame-dragging) field, which so far remains undetected in cosmology. This vector field induces an additional gravitational lensing effect; at the same time, the momentum field sources the kinetic Sunyaev-Zeldovich (kSZ) effect. The common origin of these two effects allows us to probe the gravitomagnetic signal via their cross-correlations. In this paper, we explore the possibility of detecting the gravitomagnetic field in $Lambda$CDM by cross-correlating the weak lensing convergence field with the CMB temperature map, which is imprinted with the kSZ signal. This approach allows us to extract the gravitomagnetic effect because the cross correlation between the standard Newtonian contribution to the weak-lensing convergence field, $kappa_Phi$, and the kSZ effect is expected to vanish. We study the cross correlations with a suite of large-volume Newtonian $N$-body simulations and a small-volume, high-resolution, general-relativistic $N$-body simulation counterpart. We show that insufficient simulation resolution can introduce significant spurious correlations between $kappa_Phi$ and kSZ. Based on the high-resolution simulation, we find that the cumulative signal-to-noise ratio (SNR) of the kSZ-gravitomagnetic convergence field can reach almost 15 (30) at $ellsimeq5000$ ($10^4$) if only cosmic variance is considered. We make forecast for next-generation lensing surveys such as textsc{euclid} and textsc{lsst}, and CMB experiments such as Simons Observatory and textsc{cmb}-textsc{s4}, and find that the cumulative SNR can exceed 5 (9) at $ellsimeq5000$ ($10^4$), indicating that the cosmological gravitomagnetic effect can be detected, provided that several foreground contaminations can be reliably removed.
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