  Gravitational lensing by the dual cusp catastrophes of the cold dark matter
(CDM) caustic rings at cosmological distances may provide the tantalizing
opportunity to detect CDM indirectly, and discriminate between axions and
weakly interacting massive particles (WIMPs). Caustics are places where the CDM
particles are naturally focussed. Our focus is upon the caustic rings which are
closed tubes whose cross-section is an elliptic umbilic catastrophe with three
dual cusps. A caustic ring has a specific density profile, a specific geometry
and, therefore, precisely calculable gravitational lensing signatures. The
magnification monotonically increases as the line of sight approaches to the
cusps where it diverges in the limit of zero velocity dispersion. In this
limit, we find 37% magnification at a sample point near the outer cusps of the
CDM caustic rings at cosmological distances. In the presence of finite velocity
dispersion, the lower and upper bounds of the effective velocity dispersions of
the axion and WIMP flows in galactic halos may be used to constrain the lensing
effects at the cusps. For a cosmological axion caustic ring, we find that the
magnification may range between 3% and 2800% at the outer cusp, and between 2%
and 46% at the non-planer cusps. For a cosmological WIMP caustic ring, on the
other hand, we constrain the magnification between 3% and 28% at the outer
cusp, and between 2% and 5% at the non-planer cusps. The images of extended
sources may also show distortions that can be unambiguously attributed to
lensing by dark matter caustics. Finally, we derive the Catastrophe Function of
the triaxial caustic rings. We obtain the flow equations as the equilibrium
points of this Catastrophe Function. The analysis of the Stability (Hessian)
Matrix show that the caustic rings are structurally stable.
