Time delay measurements with Broken Power Law model
One challenge in strong gravitational lensing cosmography is the measurement of time delays between multiple lensed images, which are essential for constraining the Hubble constant (\(H_0\)). In this study, we investigate how assumptions about the lens mass profile affect time-delay measurements in lensing systems. Specifically, we implement a Broken Power Law (BPL) mass model within the \textsc{Lenstronomy} framework (Birrer & Amara 2018), which introduces additional flexibility in the radial mass distribution and can phenomenologically capture deviations from a single power-law profile. This model is combined with a numerical approach to compute time delays at the image positions. We validate the BPL implementation using simulated lens systems and compare the results with those from the elliptical power-law (EPL) model. We then apply both model families to the quadruply imaged quasar WGD~2038--4008. Both models provide good fits to the imaging and kinematic data, with a slight preference for the BPL model. When the internal mass-sheet factor is allowed to vary, the inferred Hubble constant in a flat \(Λ\)CDM cosmology with fixed \(Ω_{\rm m}=0.3\) is \(H_0 = 75.3^{+23.1}_{-16.3} \ \mathrm{km \ s^{-1} \ Mpc^{-1}}\) for the BPL model and \(H_0 = 61.1^{+19.2}_{-13.2} \ \mathrm{km \ s^{-1} \ Mpc^{-1}}\) for the EPL model. For comparison, in the diagnostic case with the internal mass-sheet factor fixed to unity under the same setup, we obtain \(H_0 = 74.2^{+20.3}_{-13.8} \ \mathrm{km \ s^{-1} \ Mpc^{-1}}\) for the BPL model and \(H_0 = 66.1^{+18.8}_{-12.8} \ \mathrm{km \ s^{-1} \ Mpc^{-1}}\) for the EPL model. This highlights how time-delay cosmography remains sensitive to assumptions about the lens mass profile. With current precision, this difference does not favor one cosmological scenario over another, but underscores the importance of flexible mass modeling.