NONLINEAR GROWTH OF 2-DIMENSIONAL COSMOLOGICAL DENSITY-FLUCTUATIONS AND TYPE OF SINGULARITY IN ACCORD WITH THE CATASTROPHE-THEORY

Nonlinear growth of cosmological density fluctuations from the initial small fluctuations with a non-scale-free or scale-free power spectrum is analyzed in the two-dimensional system by the N-body simulations. It is found that after the first appearance of singularities of densit y (caustics), the power spectrum of the density fields obeys the power law even on small scales below the cutoff scale of the initial power spectrum. The value of the power index on these scales is almost -2 ir respective of the detailed initial conditions in the case that the ini tial power index n of the power spectrum on large scales is small and/ or the cutoff scale is large. In these cases, the filament structures of the particle distribution (the ''filament'' shape of caustics of pa rticle trajectories) clearly appear everywhere. It is shown that the p ower index on the small scales below the typical size of the thickness of filaments is determined only by the type of the singularity (caust ics) irrespective of the detailed initial conditions. The type of the singularity is classified in accord with the catastrophe theory. In th e above case, singularities of A2 type stably appear isotropically eve rywhere after the first appearance of caustics, and this is the reason why the value of the power index is almost -2.