Caustics of $1/r^n$ binary gravitational lenses: from galactic haloes to
  exotic matter
  We investigate the caustic topologies for binary gravitational lenses made up
of two objects whose gravitational potential declines as $1/r^n$. With $n<1$
this corresponds to power-law dust distributions like the singular isothermal
sphere. The $n>1$ regime can be obtained with some violations of the energy
conditions, one famous example being the Ellis wormhole. Gravitational lensing
provides a natural arena to distinguish and identify such exotic objects in our
Universe. We find that there are still three topologies for caustics as in the
standard Schwarzschild binary lens, with the main novelty coming from the
secondary caustics of the close topology, which become huge at higher $n$.
After drawing caustics by numerical methods, we derive a large amount of
analytical formulae in all limits that are useful to provide deeper insight in
the mathematics of the problem. Our study is useful to better understand the
phenomenology of galaxy lensing in clusters as well as the distinct signatures
of exotic matter in complex systems.
