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.