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Time delay measurements with Broken Power Law model

This study demonstrates that implementing a flexible Broken Power Law mass model within the Lenstronomy framework for the lensed quasar WGD 2038--4008 yields Hubble constant estimates that differ significantly from those obtained using the standard Elliptical Power Law model, thereby highlighting the critical sensitivity of time-delay cosmography to assumptions about the lens mass profile.

Original authors: Guanhua Rui, Bin Hu, Wei Du

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Guanhua Rui, Bin Hu, Wei Du

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Hubble Tension" Puzzle

Imagine the universe is a giant balloon inflating. Astronomers want to know exactly how fast it's expanding right now. This speed is called the Hubble Constant ( $H_0$ ).

Here's the problem: We have two different ways to measure this speed, and they give us two different answers.

Method A (The Baby Picture): Looking at the oldest light in the universe (the Cosmic Microwave Background) suggests the balloon is expanding at about 67 km/s.
Method B (The Local Map): Measuring nearby exploding stars and galaxies suggests it's expanding faster, at about 73 km/s.

This disagreement is called the "Hubble Tension." It's like two doctors looking at the same patient and giving different diagnoses. One says, "He's fine," and the other says, "He has a fever." Someone has to be wrong, or we are missing something important.

The Tool: Gravitational Lensing as a Cosmic Stopwatch

This paper focuses on a third, independent way to measure the expansion rate: Strong Gravitational Lensing.

Imagine a massive galaxy sitting between us and a distant, flickering quasar (a super-bright black hole). The galaxy's gravity acts like a giant, cosmic funhouse mirror. It bends the light from the quasar, creating multiple images of the same object around the galaxy.

Because the light takes different paths to get to us, the images flicker at slightly different times. If the quasar flares up, we might see Image A brighten today, and Image B brighten three days later.

By measuring this time delay, we can calculate the distance to the galaxy and the quasar. Since we know the distance, we can figure out how fast the universe is expanding. It's like using the echo of a shout to measure the size of a canyon.

The Problem: The "Funhouse Mirror" is Distorted

To get an accurate time measurement, we need to know exactly how the "mirror" (the galaxy) is shaped. We need to know how the mass is distributed inside that galaxy.

The Old Way (EPL Model): For years, astronomers assumed these galaxies were like simple, smooth balls of mass, getting denser toward the center in a predictable way. They used a "Single Power Law" model. Think of this like assuming every orange has a perfectly smooth, uniform rind.
The New Idea (BPL Model): This paper suggests that assumption might be too simple. Real galaxies might have a "core" in the middle—a region where the density is flatter, like an orange that has a soft, mushy center rather than a hard pit. They call this the Broken Power Law (BPL) model. It's like saying, "Wait, maybe the rind is smooth on the outside, but the inside is squishy."

What the Scientists Did

The authors took a specific lens system called WGD 2038–4008 (a galaxy with four images of a quasar around it) and ran it through two different computer simulations:

Simulation A: They assumed the galaxy was a simple, smooth ball (the old way).
Simulation B: They assumed the galaxy had a squishy, flat core in the middle (the new way).

They also added in data about how the stars inside the galaxy are moving (stellar kinematics) to help pin down the mass.

The Results: The Mirror Matters

Here is the punchline: The shape of the "mirror" changed the answer.

Using the Simple Model (Smooth Orange): They calculated the universe is expanding at 61 km/s.
Using the Flexible Model (Squishy Core): They calculated the universe is expanding at 75 km/s.

That is a huge difference! It's the difference between the "Baby Picture" answer and the "Local Map" answer.

The Analogy: The Baking Cake

Imagine you are trying to figure out how much flour is in a cake by looking at how much it rises in the oven.

Scenario 1: You assume the cake is a perfect cylinder. You measure the rise and calculate the flour.
Scenario 2: You realize the cake is actually a cylinder with a hollow center (a donut shape). You measure the rise again.

Even though the cake looks the same from the outside, your calculation of the flour (the Hubble Constant) changes completely because your assumption about the inside was different.

In this paper, the "flour" is the expansion rate of the universe. The "cake shape" is the mass distribution of the lensing galaxy.

Why This Matters

The authors aren't saying, "The squishy core is definitely real!" or "The smooth ball is definitely real!"

Instead, they are saying: "We don't know the exact shape of the galaxy's core yet, and that uncertainty is messing up our measurement of the universe's expansion."

The data we have (the pictures of the lens) isn't sharp enough to tell us if the core is squishy or smooth.
Because we don't know, we have to guess.
Depending on which guess we make, our answer for the Hubble Constant swings wildly between 61 and 75.

The Conclusion

This paper is a warning label for cosmologists. It says that to solve the "Hubble Tension," we can't just get better telescopes; we need better models of how galaxies are built inside.

If we keep assuming galaxies are simple smooth balls when they might actually have complex, squishy cores, we will never get an accurate measurement of the universe's expansion rate. We need to build more flexible models (like the Broken Power Law) that admit we don't know the exact shape of the galaxy's heart, and then see how that uncertainty affects our final answer.

In short: The universe is expanding, but we can't agree on how fast because we aren't sure what the "lenses" we are looking through actually look like on the inside. Until we figure that out, the mystery remains.

1. Problem Statement

The measurement of the Hubble constant ( $H_0$ ) via strong gravitational lensing time-delay cosmography is currently limited by systematic uncertainties, particularly those arising from the Mass-Sheet Degeneracy (MSD) and assumptions regarding the radial mass profile of the lens galaxy.

The Core Issue: Standard lens models often assume a single power-law (SPL) or Elliptical Power-Law (EPL) density profile. However, physical processes like baryonic feedback or Self-Interacting Dark Matter (SIDM) may create "cored" density profiles (flatter inner slopes) rather than the steep cusps predicted by Cold Dark Matter (CDM).
The Consequence: If the true mass profile deviates from the assumed model (e.g., a core exists but is modeled as a single power law), the inferred time delays and the resulting $H_0$ can be biased. Current imaging data alone often cannot distinguish between a cored profile and a single power law, making the choice of model a significant source of systematic error.

2. Methodology

The authors developed a framework to test the impact of flexible mass profiles on time-delay cosmography using the lens system WGD 2038–4008.

A. Implementation of the Broken Power Law (BPL) Model

Model Definition: They implemented a Broken Power Law (BPL) mass profile within the Lenstronomy framework. Unlike the standard EPL, the BPL allows for a transition in the logarithmic slope of the density profile at a break radius ( $r_c$ $r_{c}$ ).
- Inner slope: $\alpha_c$
- Outer slope: $\alpha$
- This allows the model to phenomenologically capture both cored (flat inner) and cuspy (steep inner) structures.
Numerical Solver: Since the Fermat potential for the BPL model has no closed-form analytic solution, the authors developed a numerical solver to compute the deflection potential ( $\psi$ ) by integrating the deflection field over the image plane using a 2D Green function method in elliptical coordinates.
Validation: The numerical solver was validated against the analytic EPL solution, showing residuals $< 10^{-12}$ , confirming machine-precision accuracy.

B. Data and Modeling Setup

Target: The quadruply imaged quasar WGD 2038–4008 ( $z_l=0.228$ , $z_s=0.777$ ).
Data: Multi-band HST imaging (F160W, F814W, F475X) and stellar kinematics (velocity dispersion $\sigma_{los} = 296 \pm 19$ km/s).
Comparative Approach: The system was modeled using two families:
1. EPL: Standard elliptical power-law (single slope).
2. BPL: Broken power-law (variable inner/outer slopes).
Constraints: Both models were constrained by imaging data, stellar kinematics (via the Jeans equation), and line-of-sight (LoS) external convergence ( $\kappa_{ext}$ ) derived from the galaxy environment.
Bayesian Analysis: The authors used Particle Swarm Optimization (PSO) followed by MCMC sampling. They employed a robust Bayesian Information Criterion (BIC) weighting scheme to average over different model configurations (e.g., varying source reconstruction modes) to mitigate numerical noise.

3. Key Contributions

Code Implementation: Successfully integrated the BPL mass profile into Lenstronomy, including a high-precision numerical solver for the Fermat potential, making flexible mass modeling accessible for time-delay cosmography.
Systematic Quantification: Demonstrated that the choice between a single power-law and a broken power-law model induces significant shifts in the inferred time delays and $H_0$ , even when both models fit the imaging data equally well.
Diagnostic Analysis: Provided a detailed breakdown of why the results differ. They showed that while the lens potential ( $\psi$ ) at image positions is similar for both models, the deflection angles differ significantly due to the different inner mass distributions. These deflection angle differences alter the geometric term of the Fermat potential, driving the time-delay discrepancy.

4. Key Results

The study applied both models to WGD 2038–4008 under a flat $\Lambda$ CDM cosmology ( $\Omega_m = 0.3$ ).

Imaging Fit Quality: Both EPL and BPL models reproduced the observed lensed morphology and image configurations with comparable statistical quality (reduced $\chi^2 \approx 0.9$ ). The imaging data alone could not distinguish between the two profiles.
Mass Profile Differences:
- EPL: Inferred a single slope $\gamma \approx 2.30$ .
- BPL: Inferred an outer slope $\alpha \approx 2.80$ and a significantly flatter inner slope $\alpha_c \approx 1.26$ with a break radius $r_c \approx 0.40''$ .
Time Delay Predictions:
- The BPL model predicted systematically larger absolute time delays ( $|\Delta t|$ ) than the EPL model (differences of 9–16% in the image-only case, growing to 20–24% after kinematic/LoS corrections).
- This is because the flatter inner core in the BPL model redistributes mass, altering the deflection angles and the geometric path length component of the Fermat potential.
Hubble Constant ( $H_0$ ) Inference:
- With Internal Mass-Sheet Factor ( $\lambda_{int}$ ) allowed to vary:
  - EPL Model: $H_0 = 61.1^{+19.2}_{-13.2}$ km s $^{-1}$ Mpc $^{-1}$
  - BPL Model: $H_0 = 75.3^{+23.1}_{-16.3}$ km s $^{-1}$ Mpc $^{-1}$
- Diagnostic Case ( $\lambda_{int} = 1$ fixed):
  - EPL Model: $H_0 = 66.1^{+18.8}_{-12.8}$ km s $^{-1}$ Mpc $^{-1}$
  - BPL Model: $H_0 = 74.2^{+20.3}_{-13.8}$ km s $^{-1}$ Mpc $^{-1}$
Mass-Sheet Factor: The inferred internal mass-sheet factor ( $\lambda_{int}$ ) differed between models (EPL $\approx 0.90$ , BPL $\approx 1.03$ ), demonstrating how the assumed mass profile couples with kinematic data to break the degeneracy in a model-dependent way.

5. Significance and Conclusion

Model Dependence: The results highlight that time-delay cosmography is highly sensitive to assumptions about the lens mass profile. The difference in $H_0$ between the EPL and BPL models (approx. 14 km s $^{-1}$ Mpc $^{-1}$ ) is comparable to the current "Hubble Tension" itself.
No Definitive Core Detection: The authors emphasize that the current data (lack of a detected central image) does not definitively prove the existence of a core. The BPL model simply offers more flexibility, allowing the data to explore a wider range of mass distributions consistent with the observations.
Future Implications:
- The shift in $H_0$ is driven by how different mass profiles interact with the mass-sheet degeneracy when constrained by stellar kinematics.
- To achieve precise cosmological constraints, future analyses must explicitly marginalize over mass-profile uncertainties (using flexible models like BPL) rather than relying on rigid single-power-law assumptions.
- This study underscores that for individual lenses, systematic uncertainties from mass modeling remain a dominant error source, necessitating hierarchical analyses of multiple lenses to average out these profile-dependent biases.

In summary, this paper provides a critical technical demonstration that flexible mass modeling is essential for robust time-delay cosmography, as rigid assumptions can introduce systematic biases in $H_0$ that rival the magnitude of the cosmological tension itself.