MORPHOLOGY IN COSMOLOGICAL GRAVITATIONAL CLUSTERING AND CATASTROPHE-THEORY

Non-linear growth of cosmological density fluctuations from initial sm all fluctuations with a non-scale-free power spectrum is analyzed in t hree-dimensional systems. It is found that after the first appearance of singularities (caustics) of density the power spectrum of the densi ty fields obeys a power law even on small scales below the cutoff scal e of the initial power spectrum. The value of the power index on these scales is approximately -3, irrespective of the power index n of the initial power spectrum above the cutoff scale of the initial power spe ctrum when the cutoff scale is large. In this case, sheet structures o f the particle distribution (the ''sheet'' shape of caustics of partic le trajectories) clearly appear at random places. It is shown that the power index on small scales below the typical size of the thickness o f sheets is determined only by the type of the singularities (caustics ), irrespective of the detailed initial conditions. The type of the si ngularity is classified in accord with the catastrophe theory. In the above case, the A2 type (sheet structure) of singularities dominates o ther types of singularities after the first appearance of caustics, an d this is the reason the value of the power index is close to -3. This argument can be applied also in one-dimensional and two-dimensional s ystems.