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In this paper we constrain four alternative models to the late cosmicacceleration in the Universe: Chevallier-Polarski-Linder (CPL), interactingdark energy (IDE), Ricci holographic dark energy (HDE), and modified polytropicCardassian (MPC). Strong lensing (SL) images of background galaxies produced bythe galaxy cluster Abell $1689$ are used to test these models. To perform thisanalysis we modify the LENSTOOL lens modeling code. The value added by thisprobe is compared with other complementary probes: Type Ia supernovae (SNIa),baryon acoustic oscillations (BAO), and cosmic microwave background (CMB). Wefound that the CPL constraints obtained of the SL data are consistent withthose estimated using the other probes. The IDE constraints are consistent withthe complementary bounds only if large errors in the SL measurements areconsidered. The Ricci HDE and MPC constraints are weak but they are similar tothe BAO, SNIa and CMB estimations. We also compute the figure-of-merit as atool to quantify the goodness of fit of the data. Our results suggest that theSL method provides statistically significant constraints on the CPL parametersbut weak for those of the other models. Finally, we show that the use of the SLmeasurements in galaxy clusters is a promising and powerful technique toconstrain cosmological models. The advantage of this method is thatcosmological parameters are estimated by modelling the SL features for eachunderlying cosmology. These estimations could be further improved by SLconstraints coming from other galaxy clusters.

A magnified glance into the dark sector: probing cosmological models with strong lensing in A1689