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. |