The amplitude of mass fluctuations and mass density of the Universe constrained by strong gravitational lensing

We investigate the linear amplitude of mass fluctuations in the universe, ${\sigma }_8$, and the present mass density parameter of the Universe, ${\mathrm{\Omega }}_{\mathrm{m}}$, from the statistical strong gravitational lensing. We use the two populations of lens halos model with fixed cooling mass scale $M_{\mathrm{c}}=3\times {10}^{13}{h}^{-1}{M}_{\text{sun}}$ to match the observed lensing probabilities, and leave ${\sigma }_8$ or ${\mathrm{\Omega }}_{\mathrm{m}}$ as a free parameter to be constrained by data. Another varying parameter is the equation of state of dark energy $\omega$, and its typical values of -1, -2/3, -1/2 and -1/3 are investigated. We find that ${\sigma }_8$ is degenerate with ${\mathrm{\Omega }}_{\mathrm{m}}$ in a way similar to that suggested by present day cluster abundance as well as cosmic shear lensing measurements: $\sigma_8\Omega_$\mathrm{m}$^{0.6}\approx 0.33(Bahcall & Bode\cite{bahcall03a} and references therein). However, both\sigma_8\leq 0.7$and$\Omega_$\mathrm{m}$\leq 0.2$ can be safely ruled out, the best value is when ${\sigma }_8=1.0$, ${\mathrm{\Omega }}_{\mathrm{m}}=0.3$ and $\omega =-1$. This result is different from that obtained by Bahcall & Bode (\cite{bahcall03a}), who gives ${\sigma }_8=0.98\pm 0.1$ and $\Omega_m =0.17\pm 0.05$\mathrm{.}For$\sigma_8=1.0$,highervalueof$\Omega_$\mathrm{m}$=0.35$requires$\omega=-2/3$and$\Omega_$\mathrm{m}$=0.40$requires$\omega=-1/2$.