Probing gravity at large scales through CMB lensing

We describe a methodology to probe gravity with the cosmic microwave background (CMB) lensing convergence., specifically by measuring E-G, 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 E-G for general relativity and modified gravity, finding that E-G for f(R) gravity should be scale dependent due to the scale dependence of the growth rate f. Next, we construct an estimator for E-G in terms of the galaxy-CMB lensing and galaxy clustering angular power spectra, along with the redshift-space distortion parameter beta. We also forecast statistical errors for E-G 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 E-G 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 E-G 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 3 sigma (13 sigma). Performing a < 1 per cent measurement of E-G requires a 10 per cent precision in beta from Euclid or Large Synoptic Survey Telescope, currently achievable with a spectroscopic survey but difficult with only a photometric survey.