Probing gravity at large scales through CMB lensing

We describe a methodology to probe gravity with the cosmic microwave background (CMB) lensing convergence $\kappa$, specifically by measuring $E_G$, the ratio of the Laplacian of the gravitational scalar potential difference with 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 RSD 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% with galaxies and 8% 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 $<1$% measurement of $E_G$ requires a 10% precision in $\beta$ from Euclid or LSST, currently achievable with a spectroscopic survey but difficult with only a photometric survey.