Strong lensing modeling in galaxy clusters as a promising method to test
  cosmography I. Parametric dark energy models
  In this paper we probe five cosmological models for which the dark energy
equation of state parameter, $w(z)$, is parameterized as a function of redshift
using strong lensing data in the galaxy cluster Abell 1689. We constrain the
parameters of the $w(z)$ functions by reconstructing the lens model under each
one of these cosmologies with strong lensing measurements from two galaxy
clusters: Abell 1689 and a mock cluster, Ares, from the Hubble Frontier Fields
Comparison Challenge, to validate our methodology. To quantify how the
cosmological constraints are biased due to systematic effects in the strong
lensing modeling, we carry out three runs considering the following
uncertainties for the multiple images positions: 0.25", 0.5", and 1.0". With
Ares, we find that larger errors decrease the systematic bias on the estimated
cosmological parameters. With real data, our strong-lensing constraints on
$w(z)$ are consistent those derived from other cosmological probes. We confirm
that strong lensing cosmography with galaxy clusters is a promising method to
constrain $w(z)$ parameterizations. A better understanding of galaxy clusters
and their environment is however needed to improve the SL modeling and hence to
estimate stringent cosmological parameters in alternatives cosmologies.
