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Measurement of the Weyl Potential Evolution and $E_G$ Statistic from KiDS-1000, BOSS and 2dFLenS

This paper applies a model-independent approach to KiDS-1000 and BOSS/2dFLenS data to measure the Weyl potential and $E_G$ statistic, finding that while results are generally consistent with $\Lambda$ CDM, a mild $1.52\sigma$ deviation appears in the high-redshift bin when using Planck18 priors, likely driven by the "Lensing is low" problem in the CMASS sample.

Original authors: Xuwei Zhang, Ming Zhang, Yunliang Ren, Xiaofeng Yang

Published 2026-04-28

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CC0 1.0

Original authors: Xuwei Zhang, Ming Zhang, Yunliang Ren, Xiaofeng Yang

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 Cosmic Tug-of-War: A Simple Guide to the "Lensing is Low" Mystery

Imagine you are trying to figure out how much weight a massive, invisible trampoline holds. You can’t see the trampoline itself, but you can see how much it dips when you roll marbles across it. In astronomy, the "trampoline" is the fabric of space-time, the "marbles" are galaxies, and the "dip" is something called gravitational lensing.

This paper is essentially a report from a team of scientists who just finished a massive "weight check" of the universe to see if our current rulebook for physics—called $\Lambda$ CDM (pronounced Lambda-CDM)—is still accurate.

1. The Rulebook: $\Lambda$ CDM

Think of $\Lambda$ CDM as the "Standard Operating Manual" for the universe. It says two main things:

Dark Energy ( $\Lambda$ ): The universe is expanding like a balloon being blown up faster and faster.
Dark Matter (CDM): There is invisible "glue" holding galaxies together.

For decades, this manual has worked perfectly. But lately, astronomers have noticed some "glitches." It’s like following a recipe perfectly, but the cake keeps coming out slightly too flat. This is the $S_8$ tension—a disagreement between how "clumpy" the universe should be (based on the early universe) and how "clumpy" it actually looks (based on recent observations).

2. The Tool: The Weyl Potential (The "Dip" Measurement)

The researchers used a clever, model-independent way to measure the Weyl Potential.

The Analogy: Imagine you are looking at a city through a wavy glass window. By studying how much the streetlights look distorted, you can calculate how thick and warped the glass is. In space, gravity warps light. By measuring this "warp," scientists can tell if gravity is behaving exactly as Einstein predicted (General Relativity) or if there is some "extra" or "missing" force at play.

3. The Discovery: A Mild Hiccup in the High Redshift

The team looked at data from two major sources: KiDS-1000 (a massive map of light distortion) and BOSS (a map of galaxy positions). They split their observations into two time periods:

The "Recent" Universe (Low Redshift): Everything looks normal. The "trampoline" is dipping exactly as expected.
The "Distant" Universe (High Redshift): Here, they found a slight mystery. When they compared this distant data to the "Gold Standard" rules from the early universe (the Planck satellite data), the gravity seemed a bit weaker than it should be.

It’s like checking a map of a highway: the first 10 miles are perfect, but at the 50-mile mark, the road seems to be slightly shallower than the blueprint promised. This deviation was about 1.52 $\sigma$ —in science-speak, this means "it’s a suspicious hint, but not quite a smoking gun."

4. The Culprit: Is it New Physics or Just a Bad Map?

When scientists see a glitch, they ask: "Is the universe broken, or is our equipment just acting up?"

The researchers suggest it might be the latter. They point to a known problem called "Lensing is Low." It turns out that the specific group of distant galaxies they studied (the CMASS sample) might be tricky to measure. It’s like trying to measure the depth of a pool, but the water is a bit murky, making the bottom look shallower than it actually is.

The Bottom Line

The universe isn't necessarily breaking the laws of physics (yet!), but there is a persistent "flatness" in the distant parts of our cosmic map that doesn't quite match the early blueprints.

The takeaway: We don't need to throw away Einstein's rulebook just yet, but we definitely need to go back and double-check our measurements to see if we're missing a piece of the cosmic puzzle.

Technical Summary: Measurement of the Weyl Potential Evolution and EG Statistic from KiDS-1000, BOSS, and 2dFLenS

1. Problem Statement

The $\Lambda$ CDM (Lambda Cold Dark Matter) model, while highly successful, faces significant observational tensions. Most notably, the $S_8$ tension (a discrepancy in the amplitude of matter clustering between early-universe CMB measurements and late-universe weak lensing) and the $H_0$ tension suggest potential physics beyond the standard model. These discrepancies could stem from either unaccounted systematic errors or a breakdown of General Relativity (GR) on cosmological scales.

Testing these paradigms is difficult because:

The theoretical landscape of modified gravity is vast.
Observational degeneracies exist between modified gravity effects and other phenomena (e.g., dark matter fifth forces vs. galaxy bias).
Standard tests often rely on specific model assumptions, which can reintroduce bias.

2. Methodology

The authors employ a model-independent approach to isolate gravitational signatures by focusing on two specific observables:

Weyl Potential Amplitude ( $\hat{J}$ ): Instead of testing specific modified gravity theories, they parameterize the evolution of the Weyl potential ( $\Psi_W = (\Phi + \Psi)/2$ ). They introduce a dimensionless observable $\hat{J}(k, z)$ that captures potential modifications to the Poisson equation and anisotropic stress without requiring a specific Lagrangian.
EG Statistic: This is a projected estimator used to probe gravitational slip (the difference between the two metric potentials). It is defined by the ratio of the lensing signal to the galaxy clustering signal, making it sensitive to the relationship between the growth of structure and the gravitational potential.

Data and Pipeline:

Datasets: The study performs a joint analysis of KiDS-1000 (weak lensing shear catalogs) and BOSS/2dFLenS (spectroscopic galaxy clustering).
Redshift Binning: The data is split into two effective redshift bins: LOWZ ( $z_{eff} \approx 0.38$ ) and CMASS ( $z_{eff} \approx 0.61$ ).
Analysis Framework: They use the KiDS Cosmology Analysis Pipeline (KCAP). They test two configurations:
1. "No Prior": Using only late-time LSS data.
2. "With Prior": Supplementing the analysis with Planck18 CMB priors to anchor the background cosmology.

3. Key Contributions

Model-Independent Testing: By using $\hat{J}$ and $EG$, the authors provide a way to test gravity that does not depend on choosing a specific modified gravity theory (like Horndeski or EFTofDE) a priori.
Breaking Degeneracies: The joint analysis of galaxy-galaxy lensing (GGL) and galaxy clustering (GC) helps decouple the effects of galaxy bias from the actual gravitational signal.
Comparative Analysis: The study provides a direct comparison with previous DES Y3 results, helping to contextualize whether observed tensions are universal or survey-specific.

4. Results

Consistency in "No Prior" Case: When using only late-time data, both the Weyl potential ( $\hat{J}$ ) and the $EG$ statistic are consistent with $\Lambda$ CDM and GR within $1\sigma$ . The data alone do not show evidence for modified gravity.
Tension in "With Prior" Case: When Planck18 priors are applied, a mild tension emerges in the high-redshift (CMASS) bin. Specifically:
- The Weyl potential $\hat{J}$ shows a $1.52\sigma$ deviation (it is lower than predicted by $\Lambda$ CDM).
- The $EG$ statistic also shows a suppression ( $\approx 1.5\sigma$ deviation).
Modified Gravity Preference: Phenomenological models (using $\mu-\Sigma$ parameterization) show a mild preference for a suppressed Weyl potential at high redshift, but the statistical significance is insufficient to claim a breakdown of GR.
Identification of the "Lensing is Low" Problem: The authors conclude that the deviation is primarily driven by the CMASS sample, which aligns with the known "Lensing is low" anomaly (where lensing signals appear weaker than expected at specific redshifts).

5. Significance

This research clarifies that the perceived "breakdown" of gravity in recent studies may not be a fundamental failure of $\Lambda$ CDM, but rather a result of comparing late-time lensing measurements (which favor a lower $S_8$ ) with high-amplitude early-universe CMB predictions.

The study suggests that the observed tension is likely tied to systematics or modeling limitations within specific galaxy samples (like the BOSS CMASS sample) rather than new physics. This highlights the need for more sophisticated non-linear bias modeling and finer tomographic binning in future surveys like Euclid, LSST, and CSST to distinguish between true modified gravity and astrophysical systematics.