Cosmological test using strong gravitational lensing systems

As one of the probes of universe, strong gravitational lensing systems allow us to compare different cosmological models and constrain vital cosmological parameters. This purpose can be reached from the dynamic and geometry properties of strong gravitational lensing systems, for instance, time-delay $\Delta\tau$ of images, the velocity dispersion $\sigma$ of the lensing galaxies and the combination of these two effects, $\Delta\tau/\sigma^2$. In this paper, in order to carry out one-on-one comparisons between $\Lambda$CDM universe and $R_h=ct$ universe, we use a sample containing 36 strong lensing systems with the measurement of velocity dispersion from the SLACS and LSD survey. Concerning the time-delay effect, 12 two-image lensing systems with $\Delta\tau$ are also used. In addition, Monte Carlo (MC) simulations are used to compare the efficiency of the three methods as mentioned above. From simulations, we estimate the number of lenses required to rule out one model at the $99.7\%$ confidence level. Comparing with constraints from $\Delta\tau$ and the velocity dispersion $\sigma$, we find that using $\Delta\tau/\sigma^2$ can improve the discrimination between cosmological models. Despite the independence tests of these methods reveal a correlation between $\Delta\tau/\sigma^2$ and $\sigma$, $\Delta\tau/\sigma^2$ could be considered as an improved method of $\sigma$ if more data samples are available.