Probing Gravity at Large Scales with kSZ-Reconstructed Velocities and CMB Lensing

We present a new method for measuring the $E_G$ statistic that combines two CMB secondaries -- the kinematic Sunyaev-Zeldovich (kSZ) effect and CMB lensing -- for the first time to probe gravity on linear scales. The $E_G$ statistic is a discriminating tool for modified gravity theories, which leave imprints in lensing observables and peculiar velocities. Existing $E_G$ measurements rely on redshift space distortions (RSD) to infer the velocity field. Here, we employ kSZ velocity-reconstruction instead of RSD, a complementary technique that constrains the largest-scale modes better than the galaxy survey it uses. We construct a novel $\widehat{V}_G$ estimator that involves a ratio between cross-correlations of a galaxy sample with a CMB convergence map and that with a 3D kSZ-reconstructed velocity field. We forecast for current and upcoming CMB maps from the Atacama Cosmology Telescope (ACT) and the Simons Observatory (SO), respectively, in combination with three spectroscopic galaxy samples from the Dark Energy Spectroscopic Instrument (DESI). We find cumulative detection significances in the range $S/N \sim 20-55$, which can robustly test the scale-independent $E_G$ prediction under general relativity (GR) at different effective redshifts of the galaxy samples ($z\approx 0.73, 1.33, 1.84$). In particular, the SO$\times$DESI LRG measurement would be able to distinguish between GR and certain modified gravity models, including Hu-Sawicki $f(R)$ and Chameleon theories, with high confidence. The proposed $\widehat{V}_G$ estimator opens up a new avenue for stress-testing gravity and the $\Lambda$CDM+GR model at the largest observable scales.