Probing gravity at large scales through CMB lensing

We describe a methodology to probe gravity with the cosmic microwave background (CMB) lensing convergence κ, specifically by measuring EG, the ratio of the Laplacian of the gravitational scalar potential difference to the velocity divergence. Using CMB lensing instead of galaxy-galaxy lensing avoids intrinsic alignments while also lacking a hard limit on the lens redshift and significant uncertainties in the source plane. We model EG for general relativity and modified gravity, finding that EG for f(R) gravity should be scale dependent due to the scale dependence of the growth rate f. Next, we construct an estimator for EG in terms of the galaxy-CMB lensing and galaxy clustering angular power spectra, along with the redshiftspace distortion parameter β.We also forecast statistical errors for EG from the current Planck CMB lensing map and the spectroscopic galaxy and quasar samples from the Sloan Digital Sky Survey Data Release 11, being 9 per cent with galaxies and 8 per cent when quasars are included. We also find that upcoming spectroscopic and photometric surveys, combined with the final Planck lensing map, can measure precisely the redshift- and scale dependence of EG out to redshifts z = 2 and higher, with photometric surveys having an advantage due to their high number densities. Advanced ACTPol's lensing map will increase the EG sensitivity even further. Finally, we find that Advanced ACTPol cross-correlated with spectroscopic (photometric) surveys can differentiate between general relativity and f(R) gravity at the level of 3σ (13σ). Performing a <1 per cent measurement of EG requires a 10 per cent precision in β from Euclid or Large Synoptic Survey Telescope, currently achievable with a spectroscopic survey but difficult with only a photometric survey.