Gravitational Lensing and Structural Stability of Dark Matter Caustic   Rings
  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. 
V. K. Onemli