Flux-limited strong gravitational lensing and dark energy

In the standard flat cosmological constant ($\Lambda$) cold dark matter (CDM) cosmology, a model of two populations of lens halos for strong gravitational lensing can reproduce the results of the Jodrell-Bank VLA Astrometric Survey (JVAS) and the Cosmic Lens All-Sky Survey (CLASS) radio survey. In such a model, lensing probabilities are sensitive to three parameters: the concentration parameter $c_1$, the cooling mass scale $M_\mathrm{c}$ and the value of the CDM power spectrum normalization parameter $\sigma_8$. The value ranges of these parameters are constrained by various observations. However, we found that predicted lensing probabilities are also quite sensitive to the flux density (brightness) ratio $q_{\mathrm{r}}$ of the multiple lensing images, which has been, in fact, a very important selection criterion of a sample in any lensing survey experiments. We re-examine the above mentioned model by considering the flux ratio and galactic central Super Massive Black Holes (SMBHs), in flat, low-density cosmological models with different cosmic equations of state $\omega$, and find that the predicted lensing probabilities without considering $q_{\mathrm{r}}$ are over-estimated. A low value of $q_\mathrm{r}$ can be compensated by raising the cooling mass scale $M_\mathrm{c}$ in fitting the predicted lensing probabilities to JVAS/CLASS observations. In order to determine the cosmic equation of state $\omega$, the uncertainty in $M_\mathrm{c}$ must be resolved. The effects of SMBHs cannot be detected by strong gravitational lensing method when $q_{\mathrm{r}}\leq 10$.