Dynamical state of superclusters of galaxies: do superclusters expand or have they started to collapse?

We investigate the dynamical state of superclusters in Lambda cold dark matter ($Lambda$CDM) cosmological models, where the density parameter $Omega_0=0.2-0.4$ and $sigma_8$ (the rms fluctuation on the $8h^{-1}$Mpc scale) is $0.7-0.9$. To study the nonlinear regime, we use N-body simulations. We define superclusters as maxima of the density field smoothed on the scale $R=10h^{-1}$Mpc. Smaller superclusters defined by the density field smoothed on the scale $R=5h^{-1}$Mpc are also investigated. We find the relations between the radially averaged peculiar velocity and the density contrast in the superclusters for different cosmological models. These relations can be used to estimate the dynamical state of a supercluster on the basis of its density contrast. In the simulations studied, all the superclusters defined with the $10h^{-1}$Mpc smoothing are expanding by the present epoch. Only a small fraction of the superclusters defined with $R=5h^{-1}$Mpc has already reached their turnaround radius and these superclusters have started to collapse. In the model with $Omega_0=0.3$ and $sigma_8=0.9$, the number density of objects which have started to collapse is $5 times 10^{-6}h^3$Mpc$^{-3}$. The results for superclusters in the N-body simulations are compared with the spherical collapse model. We find that the radial peculiar velocities in N-body simulations are systematically smaller than those predicted by the spherical collapse model ($sim 25$% for the $R=5h^{-1}$Mpc superclusters).