  We simulate the growth of isolated dark matter haloes from self-similar and
spherically symmetric initial conditions. Our N-body code integrates the
geodesic deviation equation in order to track the streams and caustics
associated with individual simulation particles. The radial orbit instability
causes our haloes to develop major-to-minor axis ratios approaching 10 to 1 in
their inner regions. They grow similarly in time and have similar density
profiles to the spherical similarity solution, but their detailed structure is
very different. The higher dimensionality of the orbits causes their stream and
caustic densities to drop much more rapidly than in the similarity solution.
This results in a corresponding increase in the number of streams at each
point. At 1% of the turnaround radius (corresponding roughly to the Sun's
position in the Milky Way) we find of order 10^6 streams in our simulations, as
compared to 10^2 in the similarity solution. The number of caustics in the
inner halo increases by a factor of several, because a typical orbit has six
turning points rather than one, but caustic densities drop by a much larger
factor. This reduces the caustic contribution to the annihilation radiation.
For the region between 1% and 50% of the turnaround radius, this is 4% of the
total in our simulated haloes, as compared to 6.5% in the similarity solution.
Caustics contribute much less at smaller radii. These numbers assume a 100 GeV
c^-2 neutralino with present-day velocity dispersion 0.03 cm s^-1, but reducing
the dispersion by ten orders of magnitude only doubles the caustic luminosity.
We conclude that caustics will be unobservable in the inner parts of haloes.
Only the outermost caustic might potentially be detectable.
