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MUSE-DARK-II: 3D morpho-kinematic modelling of lensed galaxies. Tully-Fisher relation of $z \sim 1$ star-forming galaxies

This paper presents and validates the MUSE-DARK-II methodology, which integrates gravitational lensing deflections into 3D forward modelling to study lensed galaxies, revealing that while the stellar Tully-Fisher relation evolves significantly since $z \sim 1$ , the baryonic Tully-Fisher relation remains stable due to the compensating mass contribution of cold gas.

Original authors: Alexandre Jeanneau, Johan Richard, Nicolas F. Bouché, Davor Krajnović, Bianca-Iulia Ciocan, Jonathan Freundlich, Benoît Epinat, Thierry Contini

Published 2026-04-01

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CC BY 4.0

Original authors: Alexandre Jeanneau, Johan Richard, Nicolas F. Bouché, Davor Krajnović, Bianca-Iulia Ciocan, Jonathan Freundlich, Benoît Epinat, Thierry Contini

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

Imagine you are trying to understand how a car engine works by looking at a blurry, distorted photo of a race car speeding past a funhouse mirror. The mirror stretches the car, makes it look bigger in some spots and smaller in others, and blurs the details. If you just look at the photo, you might think the car is huge, slow, or shaped like a banana.

This paper is about building a super-smart "digital decoder" that can look at that distorted photo, understand exactly how the funhouse mirror (a galaxy cluster) warped the image, and reconstruct what the car (a distant galaxy) actually looks like and how fast it's really spinning.

Here is a breakdown of the paper's story, using everyday analogies:

1. The Problem: The Cosmic Funhouse

Astronomers have been studying nearby galaxies for decades. They found a neat rule called the Tully-Fisher Relation. Think of this like a rule for cars: "The bigger the engine (mass), the faster the car goes (rotation speed)." It's a tight, predictable relationship.

But what happens to this rule in the past? When we look at galaxies from 8 billion years ago (around redshift $z \sim 1$ ), they are:

Far away: They look tiny and faint.
Distorted: Many are stretched by "gravitational lensing" (massive galaxy clusters acting as cosmic magnifying glasses).
Messy: They are spinning faster and are more chaotic than today's calm galaxies.

The challenge is that the "magnifying glass" doesn't just make things bigger; it stretches them unevenly. If you try to measure a galaxy's speed without correcting for this stretching, your measurements will be wrong. It's like trying to measure a runner's speed by looking at their shadow on a wavy wall.

2. The Solution: The "GalPaK3D" Decoder

The team developed a new tool, an upgrade to a software called GalPaK3D.

The Old Way: Previous methods tried to "fix" the image first (like un-distorting a photo in Photoshop) and then measure the galaxy. The problem is that this process creates "ghosts" and errors because the pixels get mixed up.
The New Way: This paper's method is like a reverse-engineering simulation. Instead of fixing the photo, the software builds a 3D model of what the galaxy should look like, simulates how the galaxy cluster would distort that model, and then compares the simulation directly to the real, blurry photo.

They tested this with thousands of "fake" galaxies (mock data) and found that their new method is 2 to 4 times more accurate than old methods, even when the magnification isn't huge. It's like having a detective who can solve a crime by simulating the crime scene rather than just looking at the blurry security footage.

3. The Discovery: The "Stellar" vs. "Total" Rule

Once they had their accurate measurements, they checked the Tully-Fisher rule for these ancient galaxies. They looked at two different versions of the rule:

The "Stellar" Rule (sTFR): This only counts the stars.
- Result: They found a significant change. Ancient galaxies with a certain spinning speed had fewer stars than modern galaxies with the same speed.
- The Analogy: Imagine a race car from the 1920s. It has a big engine (gravity) spinning fast, but it's made of lightweight aluminum (fewer stars) compared to a modern car of the same speed which is made of heavy steel.
- Why? The authors suggest this isn't because the physics changed, but because the "ruler" we use to measure the universe changes over time. The density of the universe was different back then, which shifts the numbers slightly.
The "Total" Rule (bTFR): This counts stars PLUS the cold gas (the fuel for making new stars).
- Result: No change at all. The rule for ancient galaxies was exactly the same as for modern ones.
- The Analogy: Going back to the race car. If you weigh the 1920s car including its massive, full gas tank, it weighs exactly the same as the modern car with its smaller gas tank.
- The Takeaway: Ancient galaxies had a lot more "fuel" (cold gas) than modern ones. As the universe aged, galaxies burned through that gas to make stars. The gas turned into stars, keeping the total weight (baryonic mass) constant for a given speed.

4. The Big Picture: A Stable Universe

The most exciting conclusion is that the fundamental laws governing how galaxies spin and weigh themselves haven't really changed in 8 billion years.

The "Stellar" rule looked like it changed, but that was just because the universe's background conditions (like the critical density) shifted.
The "Total" rule stayed perfectly flat. This tells us that galaxies are essentially just converting their gas reserves into stars over time, maintaining a perfect balance between their weight and their spin.

Summary

In simple terms, this paper says: "We built a better microscope to look at distorted, ancient galaxies. We found that while they look different because they have more gas and fewer stars than today, the fundamental relationship between their weight and their spin has remained rock-solid for billions of years."

It's a victory for our understanding of the universe: despite the chaos of the early cosmos, the rules of the game have been consistent all along.

1. Problem Statement

Understanding the evolution of galaxies and their dark matter (DM) haloes requires extending local kinematic studies to earlier cosmic times. However, studying intermediate-redshift ( $0.5 < z < 1.5$ ) star-forming galaxies (SFGs) faces significant observational challenges:

Low Surface Brightness & Resolution: Distant galaxies are intrinsically faint and small, leading to beam smearing (smoothing of velocity gradients) and inflated velocity dispersion measurements.
Observational Biases: Standard kinematic analyses often rely on 2D moment maps or slit spectra, which can introduce systematic errors, particularly in the outer regions of rotation curves (RCs) where DM halo constraints are most critical.
Lensing Complexity: While gravitational lensing by galaxy clusters enhances spatial resolution and sensitivity, it introduces differential magnification and shape distortions. Previous kinematic studies of lensed galaxies often reconstructed the source plane before kinematic modeling or applied simple global corrections, failing to account for the complex, position-dependent nature of lensing deflections. This can lead to biased recovery of intrinsic parameters (mass, size, velocity).

2. Methodology

The authors present MUSE-DARK-II, a study combining the MUSE Lensing Cluster survey with a novel extension of the 3D forward-modelling code GalPaK3D.

A. The GalPaK3D Strong Lensing Extension

The core innovation is the integration of lensing physics directly into the forward-modelling loop, rather than treating lensing as a post-processing step.

Coordinate Transformation: The algorithm generates a 3D coordinate cube in the image frame. It then maps these coordinates to the source plane using the lens equation: $r_{src} = r_{img} - \alpha(r_{img})$ , where $\alpha$ is the deflection angle derived from cluster mass models (Lenstool).
Forward Modelling: Parametric flux and velocity profiles (e.g., exponential discs, arctan rotation curves) are defined in the physical source frame. These are mapped back to the image frame, convolved with the instrumental Point Spread Function (PSF) and Line Spread Function (LSF), and compared directly to the observed MUSE datacubes.
Bayesian Inference: The code uses the MultiNest sampler to explore the posterior distribution of parameters (inclination, position angle, effective radius, asymptotic velocity, velocity dispersion, etc.), naturally correcting for lensing distortions and observational effects like beam smearing.

B. Validation with Mock Datacubes

To validate the method, the authors generated 3,000 synthetic lensed galaxies mimicking the MUSE sample properties.

Comparison: They compared the recovery of parameters using their lensing-aware method against a standard "image-plane fitting" approach (where parameters are fitted in the image frame and simply rescaled by magnification).
Key Finding: The lensing-aware method is significantly more robust. For moderate magnifications ( $\mu < 6$ ), ignoring differential magnification leads to large biases (e.g., +9% bias in effective radius, up to 70° errors in position angle). The new method recovers intrinsic properties with uncertainties comparable to unlensed galaxies at similar effective signal-to-noise ratios ( $S/N_{eff}$ ).

C. Sample Selection

Data: 116 strongly lensed [O II] emitters from the MUSE Atlas of Lensing Clusters (four Frontier Fields clusters: Abell 2744, 370, S1063, MACS0416).
Selection Criteria: Secure redshifts ( $0.5 < z < 1.5$ ), no tidal features, and robust kinematic recovery ( $S/N_{eff} > 10$ , inclination $>30^\circ$ , rotation-dominated $v/\sigma > 1$ ).
Final Sample: 95 rotationally supported galaxies with well-constrained velocities were used for the Tully-Fisher analysis.

3. Key Contributions

Methodological Advancement: The first implementation of a full 3D forward-modelling approach that incorporates cluster lensing deflections directly into the kinematic fitting process for intermediate-redshift galaxies.
Validation: Rigorous demonstration that this method significantly outperforms traditional image-plane fitting, particularly in recovering morphological parameters (size, PA) and velocity dispersion, reducing systematic errors by factors of 2–4.
Tully-Fisher Evolution: A robust measurement of the Stellar-mass (sTFR) and Baryonic-mass (bTFR) Tully-Fisher relations at $z \sim 1$ using a sample with a wide range of magnifications ( $\mu = 1.4 - 12.4$ ) and stellar masses ( $10^{8.1} - 10^{10.3} M_\odot$ ).

4. Key Results

A. Stellar-mass Tully-Fisher Relation (sTFR)

Zero-Point Evolution: The authors find a significant evolution in the sTFR zero-point relative to the local universe ( $z \approx 0$ $z \approx 0$ ).
- $\Delta b_{sTFR} = -0.42 \pm 0.05$ dex in stellar mass.
- This implies that at a fixed rotation velocity, galaxies at $z \sim 1$ have $\sim 2.6\times$ less stellar mass than local galaxies.
Slope: The slope is consistent with local relations ( $a \approx 3.7$ ), suggesting no significant change in the slope evolution.
Interpretation: The observed offset is largely explained by the redshift dependence of halo-defining quantities (critical density $\rho_c$ and overdensity $\Delta_c$ ). The product of the stellar-to-halo mass ratio ( $f_M$ ) and the velocity ratio ( $f_V^{-3}$ ) appears to have evolved only weakly since $z \sim 1$ .

B. Baryonic-mass Tully-Fisher Relation (bTFR)

Zero-Point Evolution: In contrast to the sTFR, the bTFR shows no detectable evolution.
- $\Delta b_{bTFR} = 0.00 \pm 0.06$ dex.
Implication: The evolution seen in the stellar component is fully compensated by the increasing contribution of cold gas mass (atomic and molecular) at higher redshifts. As galaxies evolve, they convert gas into stars, maintaining a constant baryonic mass-velocity relation.
Scatter: The intrinsic scatter of the bTFR is larger ( $\sim 0.16$ dex) than local measurements, likely due to uncertainties in estimating cold gas masses via scaling relations and non-circular motions.

5. Significance

Cosmological Context: The results support the standard $\Lambda$ CDM view where the sTFR evolution is driven primarily by the expansion of the universe (changing critical density) rather than drastic changes in the efficiency of star formation relative to halo mass ( $f_M$ ).
Methodological Benchmark: This work establishes a new standard for analyzing lensed kinematics, proving that 3D forward modelling with lensing corrections is essential for unbiased recovery of galaxy properties, even at moderate magnifications.
Gas Evolution: The lack of bTFR evolution provides strong evidence that the "missing" stellar mass in high-redshift galaxies is accounted for by their larger gas reservoirs, highlighting the critical role of cold gas in galaxy evolution over the last 8 billion years.
Future Outlook: The paper sets the stage for future studies using JWST and ALMA to probe these relations at even higher redshifts ( $z > 2$ ) and with direct HI measurements (via SKA) to reduce uncertainties in the bTFR.

In summary, Jeanneau et al. successfully combine advanced 3D modelling with gravitational lensing to provide a precise, bias-corrected view of galaxy kinematics at $z \sim 1$ , revealing that while stellar mass assembly has evolved significantly, the total baryonic mass budget relative to rotation velocity has remained remarkably stable.