2.1. Axially symmetric cold collapse
Caustics are singularities in phase space (e.g. Hogan 1999). For example, it may be possible to observe a radial caustic in the galaxy distribution surrounding clusters as material on the turnaround shell is projected onto the zero velocity surface. Physical singularities in density can occur, for example as shells of material turnaround during the collapse of a dark matter halo. Spherical symmetry gives rise to shells and a singularity at the halo center whereas axial symmetry produces a sequence of caustic rings.
Under certain conditions dynamically significant caustics are a theoretical possibility. Sikivie has calculated the radial positions and structure of caustic rings that occur during cold and axially symmetric collapse. Predictions from this model are often used to calculate rates for direct and indirect detection experiments therefore its assumptions should be critically examined.
Cold initial conditions are essential for the formation of narrow caustics otherwise they will be smeared out over a radial scalelength R200 infall / halo. Here R200 is the virialised extent of the halo and infall is the velocity dispersion of infalling material. In order to achieve a significant density enhancement, the infalling shells must have a velocity dispersion < 10% of the velocity dispersion of the final halo. Unfortunately such a cold collapse is globally unstable to the radial orbit instability (Aguilar and Merritt 1990). This is demonstrated in Figure 1 which shows the gravitational collapse of 107 particles with initial velocity dispersion 10% of the final velocity dispersion. As the central region virialises the bar instability creates a prolate mass distribution with an axial ratio larger than 2:1. This is incompatible with observational constraints on halo shapes.
Figure 1. The collapse of a cold, collisionless sphere. The final structure is a highly flattened prolate system that results from a radial orbit instability.