Lensing time delays as a substructure constraint: a case study with the
  cluster SDSS~J1004+4112
  Gravitational lensing time delays are well known to depend on cosmological
parameters, but they also depend on the details of the mass distribution of the
lens. It is usual to model the mass distribution and use time-delay
observations to infer cosmological parameters, but it is naturally also
possible to take the cosmological parameters as given and use time delays as
constraints on the mass distribution. This paper develops a method to isolate
what exactly those constraints are, using a principal-components analysis of
ensembles of free-form mass models. We find that time delays provide tighter
constraints on the distribution of matter in the very high dense regions of the
lensing clusters. We apply it to the cluster lens SDSS J1004+4112, whose rich
lensing data includes two time delays. We find, assuming a concordance
cosmology, that the time delays constrain the central region of the cluster to
be rounder and less lopsided than would be allowed by lensed images alone. This
detailed information about the distribution of the matter is very useful for
studying the dense regions of the galaxy clusters which are very difficult to
study with direct measurements. A further time-delay measurement, which is
expected, will make this system even more interesting.
