Spherical symmetry breaking in cold gravitational collapse of isolated systems

We study, using $N$-body simulation, the shape evolution in gravitational collapse of cold uniform spherical system. The central interest is on how the deviation from spherical symmetry depends on particle number $N$. By revisit of the spherical collapse model, we hypothesize that the departure from spherical symmetry is regulated by the finite-$N$ density fluctuation. Following this assumption, the estimate of the flattening of relaxed structures is derived to be $N^{-1/3}$. In numerical part, we find that the virialized states can be characterized by the core-halo structures and the flattenings of the cores fit reasonably well with the prediction. Moreover the results from large $N$ systems suggest the divergence of relaxation time to the final shapes with $N$.We also find that the intrinsic shapes of the cores are considerably diverse as they vary from nearly spherical, prolate, oblate or completely triaxial in each realization. When $N$ increases, this variation is suppressed as the final shapes do not differ much from initial symmetry. In addition, we observe the stable rotation of the virialized states. Further investigation reveals that the origin of this rotation is related in some way to the initial density fluctuation.