Time evolution of elliptic flow and medium density in heavy-ion collisions at intermediate energies

Within the ultrarelativistic quantum molecular dynamics (UrQMD) model, by reversely tracing nucleons emitted within different rapidity (${y}_{z}$) and transverse momentum (${p}_{t}$) windows in the final state, the time evolution of elliptic flow (${v}_{2}$) of these nucleons produced in $\text{Au}+\text{Au}$ collisions with beam energies (${E}_{\text{lab}}$) of 0.4, and 0.8 GeV/nucleon is studied. It is found that at ${E}_{\text{lab}}=0.4$ GeV/nucleon, the rapidity and transverse momentum of nucleons in the final state are correlated with the initial momentum. At ${E}_{\text{lab}}=0.8$ GeV/nucleon, the correlation becomes rather weak. In addition, by studying the medium density and the maximum density defined for each traced nucleon, it is found that the density interval at which the nuclear equation of state is probed by the elliptic flow is about 60--80 % of the maximum average density created in the central region of heavy-ion collision.