for the $\pi^-/\pi^+$ ratio [@Li09; @Feng10], are considered as a point of importance in the present study. The single ratio is just the ratio of neutrons to protons and is represented in the study by $R(N/Z)$, while double ratio is the ratio of the single ratios of any two isotopes of Sn. In order to study the systematics of the isospin effects, the single ratio of the isotope with a greater number of neutrons is always mentioned in the numerator when the double ratio is calculated. Mathematically, the double ratio is represented by $DR(N/Z)$ and is given as $$DR(N/Z)~=~\frac{R(N/Z)^{neutron~rich}}{R(N/Z)^{neutron~weak}} \cdot$$

![\[fig:1\](Color online) Time evolution of the average density for the $^{132}$Sn + $^{132}$Sn reaction with the soft symmetry energy ($\gamma_i = 0.5$) at semicentral geometry. The different lines represent different incident energies ranging from 50 to 600 MeV/nucleon.](figure1.eps){width="90mm"}

To predict the high-density behavior of symmetry energy, the very first point is to understand the time evolution of the average density at different incident energies. With increasing incident energy, the density will be expected to be greater than the normal nuclear matter density in the most compressed region. As we know, the density of the environment surrounding the nucleons of a 