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KiDS-Legacy: Constraints on Horndeski gravity from weak lensing combined with galaxy clustering and cosmic microwave background anisotropies

This paper presents the first cosmological analysis of the Horndeski modified gravity framework using the final KiDS-Legacy weak lensing data combined with DESI, eBOSS, and Planck observations, revealing a mild preference for modified gravity over General Relativity at the $1.4\sigma$ level while deriving stable constraints on key parameters and the structure growth parameter $S_8$ .

Original authors: Benjamin Stölzner, Robert Reischke, Matteo Grasso, Matteo Cataneo, Benjamin Joachimi, Arthur Loureiro, Alessio Spurio Mancini, Angus H. Wright, Marika Asgari, Maciej Bilicki, Andrej Dvornik, Christos

Published 2026-03-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Benjamin Stölzner, Robert Reischke, Matteo Grasso, Matteo Cataneo, Benjamin Joachimi, Arthur Loureiro, Alessio Spurio Mancini, Angus H. Wright, Marika Asgari, Maciej Bilicki, Andrej Dvornik, Christos Georgiou, Benjamin Giblin, Catherine Heymans, Hendrik Hildebrandt, Shahab Joudaki, Konrad Kuijken, Shun-Sheng Li, Laila Linke, Constance Mahony, Lauro Moscardini, Lucas Porth, Mario Radovich, Tilman Tröster, Maximilian von Wietersheim-Kramsta, Ziang Yan, Mijin Yoon, Yun-Hao Zhang

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 the universe as a giant, invisible ocean. For decades, scientists have been trying to understand the currents and waves in this ocean using a map called $\Lambda$ CDM (Lambda Cold Dark Matter). This map assumes that gravity works exactly the way Einstein described it (General Relativity) and that the universe is expanding because of a mysterious, constant force called "Dark Energy."

However, just like a map might have small errors when you zoom in on a specific coastline, recent measurements of the universe's "waves" (galaxy clusters and light bending) have shown slight mismatches with the map. This has led scientists to ask: Is our map wrong, or is there a hidden current we haven't accounted for?

This paper, "KiDS-Legacy: Constraints on Horndeski gravity," is a massive detective story where the authors try to see if the rules of gravity need a rewrite.

The Detective Team and the Crime Scene

The team combined data from four major "surveillance cameras" watching the universe:

KiDS-Legacy: A powerful telescope survey that maps how light from distant galaxies bends (weak lensing). Think of this as looking at the distortion of a funhouse mirror to see how heavy the air is.
DESI & eBOSS: Surveys that measure how galaxies are moving away from us (clustering and distortions).
Planck: A satellite that took the "baby picture" of the universe (the Cosmic Microwave Background).

The Suspect: Horndeski Gravity

The main suspect is a theory called Horndeski gravity.

The Analogy: Imagine General Relativity is a standard, rigid trampoline. If you put a bowling ball on it, it curves in a specific, predictable way.
The Twist: Horndeski gravity suggests the trampoline isn't just rubber; it has a hidden, invisible spring or a fluid underneath it (a "scalar field") that changes how the trampoline reacts to weight. This extra ingredient could explain why the universe is expanding faster than expected without needing a constant "Dark Energy" force.

The New Tool: The "Stable" Parameter

In the past, testing these theories was like trying to balance a house of cards in a hurricane. Many versions of Horndeski gravity were mathematically "unstable"—meaning if you tried to calculate them, the numbers would explode into infinity (ghosts and tachyons).

The authors used a new, clever tool (the mochi_class code) that acts like a safety harness. Instead of testing random, chaotic versions of the theory, they built a "stable basis." This ensures they only test versions of gravity that are physically possible and won't break the laws of physics. It's like testing only the cars that have working brakes, rather than testing every random engine design.

The Investigation Results

The team ran their "safety-harnessed" theory against the data from all four telescopes. Here is what they found:

The "Funhouse Mirror" Holds Up: The data from the bending light (weak lensing) was incredibly precise. It acted as a strong filter.
No Smoking Gun: While the Horndeski theory could fit the data slightly better than Einstein's standard gravity, the improvement was tiny.
- The Analogy: Imagine you are trying to guess the weight of a package. Einstein's scale says 10 lbs. The new theory says 10.1 lbs. The scale is so precise that 10.1 lbs is technically "better," but it's not enough to prove the package is actually something else entirely.
The Verdict: The data is 1.4 sigma in favor of the new theory. In the world of physics, you usually need 5 sigma (a 1 in 3.5 million chance of being wrong) to declare a discovery. A 1.4 sigma result is like hearing a faint whisper in a noisy room; it's interesting, but you can't be sure it's a voice or just the wind.

The "S8" Tension

One of the biggest mysteries in cosmology right now is the $S_8$ tension.

The Problem: The "baby picture" (Planck) predicts the universe should be clumpy in a certain way. The "adult picture" (KiDS) sees the universe as slightly less clumpy.
The Result: This paper found that even with the new Horndeski gravity, the universe still looks about as clumpy as the standard model predicts. The new theory didn't solve the mystery; it just confirmed that the standard model is still doing a great job.

The "Dark Energy" Twist

The authors also asked: What if Dark Energy isn't constant, but changes over time?

They let the "Dark Energy" variable wiggle.
Result: The universe still looks mostly like the standard model. However, there is a very slight hint (1.57 sigma) that Dark Energy might be changing. But again, it's not a slam-dunk proof.

The Bottom Line

This paper is a triumph of precision. By using a new, stable way to test gravity and combining data from the best telescopes in the world, the authors have tightened the noose around alternative theories.

The takeaway: Einstein's gravity is still the champion. The universe is behaving very much like the standard map predicts. While there are tiny cracks in the map that might hint at new physics (like a hidden spring in the trampoline), we don't have enough evidence yet to throw out the old map. We need more data, sharper telescopes, and perhaps a bit more luck to see if the "whisper" becomes a shout.

In short: The universe is weird, but it's not that weird. General Relativity is still holding the fort.

1. Problem Statement

The standard $\Lambda$ CDM model, while successful, faces theoretical challenges (e.g., the cosmological constant problem) and observational tensions (e.g., the $H_0$ and $S_8$ tensions). Modified gravity (MG) theories, particularly the Horndeski class (the most general scalar-tensor theory with second-order equations of motion), offer a potential alternative to General Relativity (GR) and dark energy.

However, testing Horndeski gravity is challenging due to:

Stability Issues: Many regions of the Horndeski parameter space suffer from ghost, gradient, or tachyonic instabilities, making Markov Chain Monte Carlo (MCMC) sampling inefficient as most proposed models are unphysical.
Parameter Degeneracies: Traditional parameterizations (using $\alpha$ functions) often suffer from strong degeneracies and numerical instabilities.
Data Consistency: Previous analyses of weak lensing data (e.g., KiDS-1000) showed tensions with Planck CMB data regarding the structure growth parameter $S_8$ . The final KiDS-Legacy dataset resolves the $S_8$ tension with Planck, necessitating a re-evaluation of MG constraints using this consistent dataset.

2. Methodology

The authors perform a joint cosmological analysis combining multiple probes to constrain Horndeski gravity.

Data Sources:

KiDS-Legacy: The final (5th) data release of the Kilo-Degree Survey, providing cosmic shear (weak lensing) measurements.
DESI DR2: Baryon Acoustic Oscillation (BAO) measurements.
eBOSS DR16: Redshift Space Distortion (RSD) measurements.
Planck 2018: CMB temperature and polarization power spectra (TT, TE, EE).

Theoretical Framework:

Effective Field Theory (EFT): The authors employ an EFT approach to describe linear perturbations on top of a background cosmology.
Stable Parameterization (Key Innovation): Instead of the traditional $\alpha_i(t)$ $α_{i} (t)$ functions, the analysis uses the inherently stable basis proposed by Kennedy et al. (2018) and implemented in the mochi_class code. This basis replaces the $\alpha$ $α$ functions with:
1. De-mixed kinetic term ( $D_{kin}$ ): Ensures the absence of ghost instabilities.
2. Effective sound speed squared ( $c_s^2$ ): Ensures the absence of gradient instabilities.
3. Effective Planck mass ( $M_*^2$ ): Parameterized as a deviation from the GR value ( $\Delta M_*^2$ ).
4. Initial braiding condition ( $\alpha_{B,0}$ ): Determined via an ODE to satisfy stability.
- Constraint: The analysis enforces $D_{kin} > 0$ , $M_*^2 > 0$ , and $c_s^2 > 0$ by construction, ensuring all sampled models are physically viable.
Non-linear Modeling: The authors use the HMCode2020 augmented halo model and a Vainshtein screening mechanism to recover GR on small scales (high $k$ ).
Analysis Setup: Two primary stable parameterizations were tested:
1. Varying $\hat{D}_{kin}$ and $\Delta \hat{M}_*^2$ (fixing $c_s^2$ ).
2. Varying $\hat{D}_{kin}$ and $\hat{c}_s^2$ (fixing $\Delta \hat{M}_*^2 = 0$ ).
3. A comparison using the traditional $\hat{\alpha}_B$ and $\hat{\alpha}_M$ parameterization.

3. Key Contributions

First Stable Basis Analysis: This is the first cosmological analysis of Horndeski gravity using an inherently stable parameter basis that automatically excludes unphysical models (ghosts/gradient instabilities) during the sampling process.
KiDS-Legacy + Multi-Probe: Utilizes the final KiDS-Legacy dataset, which is consistent with Planck, allowing for a robust joint analysis without the $S_8$ tension that plagued previous KiDS-Planck comparisons.
Decoupling Background and Perturbations: The study demonstrates that constraints on modified gravity perturbations are robust even when the background cosmology is extended to include dynamical dark energy ( $w_0 w_a$ CDM).
Screening Mechanism: Implementation of a phenomenological Vainshtein screening to ensure GR recovery on small scales, critical for interpreting weak lensing data.

4. Key Results

Constraints on Structure Growth ( $S_8$ ):

The combined analysis yields $S_8 = 0.813^{+0.008}_{-0.011}$ .
This value is consistent with the $\Lambda$ CDM constraint at $0.54\sigma$ , confirming that KiDS-Legacy data is fully compatible with the standard model.
The precision on $S_8$ degrades only slightly (by ~1-2%) when allowing for modified gravity, indicating that cosmic shear remains a powerful probe even in extended models.

Constraints on Horndeski Parameters (Stable Basis):

Planck Mass Deviation: $\Delta \hat{M}_*^2 = 0.32^{+0.07}_{-0.21}$ . This deviates from GR ( $\Delta M_*^2 = 0$ ) at $1.5\sigma$ .
De-mixed Kinetic Term: $\hat{D}_{kin} = 3.74^{+0.69}_{-1.92}$ .
Braiding Term: The derived braiding term at $z=0$ is $\alpha_{B,0} = 1.44^{+0.39}_{-0.76}$ , showing a preference for positive values (deviating from GR) at $1.9\sigma$ .
Effective Newtonian Coupling: The deviation from GR is $\Delta \mu_{\infty, eff} = 0.066 \pm 0.023$ , corresponding to a $2.9\sigma$ significance.

Model Comparison:

While modified gravity models provide a slightly better fit to the data (lower $\chi^2$ ) than $\Lambda$ CDM, the improvement is not statistically significant.
The Bayesian model comparison shows only a weak preference for modified gravity at the $1.4\sigma$ level.
The data remains highly consistent with General Relativity.

Dynamical Dark Energy ( $w_0 w_a$ CDM):

When the background cosmology is allowed to evolve dynamically ( $w_0, w_a$ ), the constraints on the modified gravity parameters remain stable.
There is a mild preference ( $1.57\sigma$ ) for dynamical dark energy over a cosmological constant, but this is driven largely by the lensing amplitude parameter ( $A_L$ ) and not the modified gravity sector itself.

Data Drivers:

Cosmic Shear (KiDS) and CMB (Planck) are the primary drivers of constraints on Horndeski parameters.
BAO and RSD data primarily constrain the background expansion and standard parameters, contributing little to the MG constraints directly, though they help break degeneracies.

5. Significance

This paper represents a major step forward in testing gravity on cosmological scales by:

Solving Numerical Instability: By using the stable basis, the authors avoid the inefficiency of sampling unphysical models, providing a more reliable exploration of the Horndeski parameter space.
Validating $\Lambda$ CDM: Despite the flexibility of the Horndeski framework, the data (KiDS-Legacy + DESI + eBOSS + Planck) strongly favors the standard $\Lambda$ CDM model. The observed deviations from GR (e.g., in $\mu_{\infty}$ ) are not statistically robust enough to claim a detection of new physics.
Future-Proofing: The methodology establishes a robust framework for future Stage-IV surveys (like Euclid and LSST), demonstrating that weak lensing combined with CMB data can tightly constrain modified gravity even when the background cosmology is complex.

In conclusion, the KiDS-Legacy dataset, when combined with other major cosmological probes, finds no compelling evidence for Horndeski modified gravity, reinforcing the validity of General Relativity and the $\Lambda$ CDM model on large scales.