Testing General Relativity on cosmological scales at redshift $z \sim 1.5$ with quasar and CMB lensing

We test general relativity (GR) at the effective redshift $\\bar{z} \\sim 1.5$ by estimating the statistic $E_G$, a probe of gravity, on cosmological scales $19 - 190\\,h^{-1}{\\rm Mpc}$. This is the highest-redshift and largest-scale estimation of $E_G$ so far. We use the quasar sample with redshifts $0.8 \u003c z \u003c 2.2$ from Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 (DR16) as the large-scale structure (LSS) tracer, for which the angular power spectrum $C_\\ell^{qq}$ and the redshift-space distortion (RSD) parameter $\\beta$ are estimated. By cross correlating with the $\\textit{Planck}$ 2018 cosmic microwave background (CMB) lensing map, we detect the angular cross-power spectrum $C_\\ell^{\\kappa q}$ signal at $12\\,\\sigma$ significance. Both jackknife resampling and simulations are used to estimate the covariance matrix (CM) of $E_G$ at $5$ bins covering different scales, with the later preferred for its better constraints on the covariances. We find $E_G$ estimates agree with the GR prediction at $1\\,\\sigma$ level over all these scales. With the CM estimated with $300$ simulations, we report a best-fit scale-averaged estimate of $E_G(\\bar{z})=0.30\\pm 0.05$, which is in line with the GR prediction $E_G^{\\rm GR}(\\bar{z})=0.33$ with $\\textit{Planck}$ 2018 CMB+BAO matter density fraction $\\Omega_{\\rm m}=0.31$. The statistical errors of $E_G$ with future LSS surveys at similar redshifts will be reduced by an order of magnitude, which makes it possible to constrain modified gravity models.