Time delay measurements with Broken Power Law model

TL;DR

This study investigates how assumptions about the lens mass profile affect time-delay cosmography by implementing a Broken Power Law (BPL) profile within Lenstronomy and comparing it to the elliptical power-law (EPL) in both simulated lenses and the real quad WGD 2038--4008. A numerical solver for the Fermat potential validates the BPL implementation against the analytic EPL limit, enabling robust comparisons of time delays under different mass distributions. Applying EPL and BPL to WGD 2038--4008 with imaging, stellar kinematics, and line-of-sight convergence reveals a ~20–25% split in the inferred time-delay distance $D_{\Delta t}$ and corresponding $H_0$ values, highlighting the model-dependence of cosmographic inferences. The results emphasize that inner-mass-profile flexibility and the mass-sheet degeneracy are significant systematics in time-delay cosmography, calling for flexible mass modeling and careful PSF treatment in future hierarchical analyses.

Abstract

One of the key challenges in strong gravitational lensing cosmography is the accurate measurement of time delays between multiple lensed images, which are essential for constraining the Hubble constant ($H_0$). We investigate how lens mass-profile assumptions affect time delays. Specifically, we implement a Broken Power Law (BPL) mass model within the 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 lenses and compare the results with those obtained from the elliptical power-law (EPL) model. We then apply both model families to the quadruply imaged quasar WGD~2038-4008. Both models fit the imaging and kinematic data comparably well, yet the greater radial freedom in the BPL model shifts the inferred time-delay distance -- and thus $H_0$ -- by an amount comparable to the current discrepancy between early- and late-universe $H_0$ estimates. In a flat $Λ$CDM cosmology, the $H_0$ inferred using the BPL lens model is $75^{+23.1}_{-16.3} \ \mathrm{km \ s^{-1} \ Mpc^{-1}},$ while the EPL model gives $H_0 = 61^{+19.2}_{-13.2} \ \mathrm{km \ s^{-1} \ Mpc^{-1}}.$ This difference is largely due to uncertainties in the inner mass profile ($θ<0.2''$), a region where point spread function (PSF) reconstruction is a critical factor -- a finding consistent with results reported in Shajib et al. (2022). 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 rather underscores the importance of flexible mass modeling and PSF modeling.

Figures (20)

• Figure 1: Relative residual between the numerically integrated lens potential and the analytic solution, for an EPL model. We show the relative residual ${\rm Res}(x,y)$ across the image plane for a test lens (Einstein radius $\theta_E = 1\farcs5$, slope $\alpha=1.9$, axis ratio $q=0.8$). The residuals are below $10^{-12}$ everywhere, demonstrating the high accuracy of the integration scheme.
• Figure 2: Validation test: Fitting EPL model to the BPL mock data. Top Left: Simulated data with four quasar images and host galaxy arcs. Top Middle: Best-fitting EPL model reconstruction. Top Right: Normalized residuals. Bottom Left: Reconstructed source in the source plane. Bottom Middle: Convergence map of the best-fit EPL model, showing an elliptical mass distribution. Bottom Right: Magnification map with the four quasar image positions (A–D) labeled. We assume an idealized Gaussian PSF with $\mathrm{FWHM}=0\farcs1$ in both the simulation and the fitting.
• Figure 3: Validation test: Fitting BPL model to the BPL mock data. Panels are analogous to Fig. \ref{['fig:image2']}. The BPL model successfully reproduces the lensed images and arcs, yielding negligible residuals. The input and recovered mass profile parameters agree within uncertainties.
• Figure 4: Color composite of WGD 2038--4008 from HST WFC3 imaging (F160W in red, F814W in green, F475X in blue). The four quasar images A, B, C, D surround the foreground lens galaxy (center). Extended lensed host galaxy features are visible as faint greenish arcs. North is up and east is left; the image cutout is $5"$ on a side.
• Figure 5: Lenstronomy-based lens model and image reconstruction of WGD 2038-4008 using an EPL mass profile. The top row shows, from left to right, the observed RGB composite, the model-predicted RGB composite, the convergence map $(\kappa)$, and the magnification $(\mu)$. Rows 2-4 display, for each HST filter, the observed image, the reconstructed image, the residual, and the reconstructed source plane: F160W (row 2), F814W (row 3), and F475X (row 4). All scale bars correspond to $1^{\prime\prime}$. In the source panels, the star marks the centroid of the quasar host galaxy.