Constraining the ΛCDM and wCDM Models of the Universe using Gravitational Lensing

We investigate late-time cosmology using a compiled sample of 85 galaxy-scale strong gravitational lenses, exploiting the fact that lensing directly probes geometry through the distance ratio D ls /D s while remaining effectively independent of the Hubble constant H 0 .For each system we infer D ls /D s from the observed Einstein radius and stellar velocity dispersion under a standard, spherically symmetric mass description of the lens galaxy.We then confront these measurements with the predictions of two dark-energy scenarios in a spatially flat universe: CDM (cosmological constant) and wCDM with a constant, yet free, equationof-state parameter w.The theoretical distance ratios are computed from the expansion history E(z) and line-of-sight integrals between the lens and source redshifts.Parameter estimation proceeds via a straightforward 2 minimization built from per-lens uncertainties, yielding (i) profiled one-dimensional likelihood curves for m0 and w, and (ii) joint confidence contours in the ( m0 , w) plane.The resulting constraints demonstrate that strong lensing alone provides precise, H 0 -independent information on the matter density and dark-energy phenomenology.In CDM, the best-fit matter density and its confidence interval are obtained from the profiled 2 curve; in wCDM, the twoparameter contours capture the expected degeneracy between m0 and w while still delivering competitive bounds on each.Overall, our analysis shows that a uniform treatment of galaxy-scale lenses-linking well-measured imaging and stellar kinematics to distance-ratio predictions-offers a clean and efficient route to cosmological constraints.The framework is simple, reproducible, and readily extensible: it can incorporate alternative lens models, larger samples, or be combined with complementary probes (e.g., supernovae, BAO, CMB) to further sharpen constraints and test the robustness of the late-time expansion.