Gravitational Lensing By a Massive Object in a Dark Matter Halo. I.
  Critical Curves and Caustics
  We study the gravitational lensing properties of a massive object in a dark
matter halo, concentrating on the critical curves and caustics of the combined
lens. We model the system in the simplest approximation by a point mass
embedded in a spherical Navarro$-$Frenk$-$White density profile. The low number
of parameters of such a model permits a systematic exploration of its parameter
space. We present galleries of critical curves and caustics for different
masses and positions of the point in the halo. We demonstrate the existence of
a critical mass, above which the gravitational influence of the centrally
positioned point is strong enough to eliminate the radial critical curve and
caustic of the halo. In the point-mass parameter space we identify the
boundaries at which critical-curve transitions and corresponding caustic
metamorphoses occur. The number of transitions as a function of position of the
point is surprisingly high, ranging from three for higher masses to as many as
eight for lower masses. On the caustics we identify the occurrence of six
different types of caustic metamorphoses. We illustrate the peculiar properties
of the single radial critical curve and caustic appearing in an additional
unusual non-local metamorphosis for a critical mass positioned at the halo
center. Although we constructed the model primarily to study the lensing
influence of individual galaxies in a galaxy cluster, it can also be used to
study the lensing by dwarf satellite galaxies in the halo of a host galaxy, as
well as (super)massive black holes at a general position in a galactic halo.
