The moving lens effect: analytical modelling and foreground suppression

The moving lens (ML) effect is a secondary anisotropy of the cosmic microwave background (CMB) generated by the transverse motion of gravitational potentials, providing a direct probe of the large-scale cosmic velocity field. Its detection relies on cross-correlating a CMB map with the transverse galaxy momentum field, constructed from the galaxy overdensity and a velocity field reconstructed from it. Restricting the reconstruction to large-scale modes strongly suppresses contamination from small-scale astrophysical foregrounds while preserving the ML signal. In this work, we develop a theoretical framework for the ML estimator and its foreground contamination. We show that the signal and foregrounds have a distinct dependence on the direction of the long-wavelength mode represented by the reconstructed galaxy velocity. In the squeezed limit enforced by the velocity reconstruction filter, the bispectrum sourcing the foreground correlation becomes independent of this direction, causing its leading contribution to vanish after angular averaging. In turn, the ML signal survives by matching this directional dependence, with its amplitude reduced only by the filtered velocity variance. Using the halo model, we derive the leading corrections beyond the squeezed limit and show that the residual contamination remains parametrically suppressed. Finally, we model the cross-correlation exactly in the curved sky, and show that it is sourced solely by the longitudinal component of the galaxy momentum field, in the form of a spin-1 E-mode, with all other contributions either strongly suppressed on small scales or exactly zero.