A revised test of cosmic curvature at high redshifts: the distance sum rule

Ultra-compact structure in radio quasars, with milliarcsecond angular sizes measured by very-long-baseline interferometry (VLBI), provides an important source of angular diameter distances that can be observed up to higher redshifts. In this paper, with the latest catalog of galactic-scale strong gravitational lensing systems and the VLBI observation of milliarcsecond compact structure in intermediate-luminosity quasars, we place constraints on the curvature of the universe through the well-known distance sum rule, without assuming any fiducial cosmological model. Assuming power-law density profiles for the total mass density of lensing galaxies ($\rho\sim r^{-\gamma}$), we find that, although the zero cosmic curvature is still included within $2 \sigma$ confidence level, a closed universe is seemed to be more favored in our analysis. In addition, in the framework of a more general lens model which allows the luminosity density profile $\rho\sim r^{-\alpha}$ to be different from that of the total-mass density profile $\nu\sim r^{-\delta}$, a weaker constraint on the curvature ($\Omega_k<0.197$ at 68\% confidence level)is obtained indicates that a more general lens model does have a significant impact on the measurement of cosmic curvature. Finally, based on the mock samples of strong gravitational lenses and quasars with the current measurement accuracy, we find that with about 16000 strong lensing events (observed by the forthcoming LSST survey) combined with the distance information provided by 500 compact uv-coverage, one can constrain the cosmic curvature with an accuracy of $\Delta \Omega_k\simeq 10^{-3}$, which is comparable to the precision of Planck 2015 results.