🔭 astrophysics

Weak lensing higher-order statistics to disentangle modified gravity and massive neutrinos

This study demonstrates that higher-order statistics in weak lensing, derived from N-body simulations of General Relativity and $f(R)$ modified gravity, can successfully break degeneracies to distinguish between these gravitational theories even in the presence of massive neutrinos.

Original authors: Alessandro VadalÃ, Vincenzo Fabrizio Cardone, Simone Vinciguerra, Filippo Bouchè, Marco Baldi, Carlo Giocoli

Published 2026-03-02

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CC BY 4.0

Original authors: Alessandro VadalÃ, Vincenzo Fabrizio Cardone, Simone Vinciguerra, Filippo Bouchè, Marco Baldi, Carlo Giocoli

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 to figure out what the water is made of and how it moves. They have a "Standard Recipe" (called the ΛCDM model) that explains most things very well. This recipe says the universe is made of normal stuff, invisible "dark matter," and a mysterious force pushing things apart called "dark energy."

However, there are some cracks in this recipe. Measurements of how fast the universe is expanding don't quite match up with measurements of how clumpy the universe is. It's like a chef tasting a soup and saying, "This tastes like chicken," while the recipe says, "This is definitely beef."

This paper is about a new way to taste the soup to see if we need to change the recipe entirely.

The Problem: Two Chefs, One Dish

The scientists are investigating two main possibilities for why the soup tastes "off":

Massive Neutrinos: Maybe the "ingredients" (particles called neutrinos) are heavier than we thought. Heavy neutrinos act like anchors, slowing down the formation of clumps in the universe.
Modified Gravity: Maybe the "rules of the kitchen" (gravity) are different. Instead of Einstein's General Relativity, maybe gravity works a bit differently on cosmic scales (like a modified version called f(R) gravity).

Here is the tricky part: Heavy neutrinos and Modified Gravity can cancel each other out.

Imagine Modified Gravity is a strong wind pushing the soup ingredients together to make big clumps.
Imagine Heavy Neutrinos are heavy stones dropping into the soup, stopping those clumps from forming.
If you have just the right amount of wind and just the right amount of stones, the soup looks exactly the same as if you had neither! This is called a "degeneracy." Standard tools can't tell the difference.

The Solution: Looking at the Soup with a Microscope

For a long time, scientists looked at the universe using "second-order statistics." Think of this as looking at the soup from a distance and just counting how many clumps there are on average. It's like looking at a forest and saying, "It's pretty dense."

But this paper argues that we need to look closer. The authors use Higher-Order Statistics (HOS).

The Analogy: If you look at a forest from a plane, you just see a green blob. If you look from a drone, you see the shape of the trees. But if you walk through the forest (HOS), you can feel the texture of the bark, the way branches twist, the holes in the canopy, and the specific patterns of the leaves.
The authors used a supercomputer to simulate the universe under different rules (Standard Gravity vs. Modified Gravity) and with different amounts of heavy neutrinos. They created "convergence maps," which are like heatmaps showing where the invisible dark matter is clumping.

They then applied a "toolkit" of advanced mathematical lenses to these maps:

Peaks: Counting the highest mountains in the landscape.
Betti Numbers & Minkowski Functionals: These are fancy topological tools. Imagine the soup as a sponge. These tools count how many holes are in the sponge, how many separate islands of matter exist, and how connected the whole thing is.
Moments: Measuring how "skewed" or "spiky" the distribution is.

The Experiment: The Great Taste Test

The researchers ran 256 different simulations (like running the same recipe 256 times with slight variations) to see if their new "microscope" could spot the difference between the Standard Gravity recipe and the Modified Gravity recipe, even when heavy neutrinos were trying to hide the difference.

What they found:

The Old Way Failed: If you just look at the average clumpiness (the old method), you often can't tell the difference when heavy neutrinos are present. The "wind" and the "stones" cancel out perfectly.
The New Way Succeeded: When they used the "microscope" (Higher-Order Statistics), they could see the subtle fingerprints left behind. Even when the wind and stones canceled each other out, the shape of the clumps was slightly different.
- Some tools were better at spotting the "wind" (Modified Gravity).
- Some were better at spotting the "stones" (Neutrinos).
- Aperture Mass Peaks (counting specific types of peaks in the map) and Minkowski Functionals (counting holes and shapes) were particularly good detectives.

The Takeaway

The paper concludes that we don't need to choose between "Modified Gravity" and "Massive Neutrinos." Instead, by using these advanced, high-resolution statistical tools, we can untangle them. We can look at the universe's "soup" and say, "Ah, I see the wind and I see the stones, and I know exactly how much of each is there."

Why does this matter?
We are about to launch new, powerful telescopes (like the Euclid mission) that will take pictures of billions of galaxies. This paper is like a training manual for those telescopes. It tells the astronomers: "Don't just count the galaxies. Look at the shapes, the holes, and the peaks. That's where the secrets of the universe are hiding."

In short: We found a way to see the difference between a changed recipe and a changed ingredient, even when they try to look identical, by looking at the food with a much sharper eye.

1. Problem Statement

The standard cosmological model ( $\Lambda$ CDM) faces significant challenges, including the Hubble tension and the $S_8$ tension (clustering amplitude). These discrepancies have spurred interest in Modified Gravity (MG) theories, such as $f(R)$ gravity, which alter the growth of cosmic structures while mimicking the background expansion of General Relativity (GR).

However, a major degeneracy exists between MG effects and the presence of massive neutrinos. Massive neutrinos suppress structure growth via free-streaming, an effect that can counterbalance the enhanced growth caused by MG. Consequently, standard cosmological probes (specifically second-order statistics like the convergence power spectrum) often fail to distinguish between a GR model with massive neutrinos and an MG model with massless (or lighter) neutrinos.

The paper investigates whether Higher-Order Statistics (HOS) applied to Weak Lensing (WL) convergence maps can break this degeneracy and discriminate between GR and MG in the presence of massive neutrinos.

2. Methodology

Data Source: DUSTGRAIN-pathfinder Simulations

The authors utilized the DUSTGRAIN-pathfinder suite of N-body simulations.

Physics: Simulations include both GR and $f(R)$ gravity (Hu-Sawicki model) with varying parameters ( $f_{R0}$ ) and massive neutrino masses ( $M_\nu$ ).
Configurations: 8 distinct cosmological models (combinations of $f_{R0}$ and $M_\nu$ ) plus one reference $\Lambda$ CDM model.
Realizations: 256 light-cone realizations per cosmology, generating square convergence maps ( $5^\circ \times 5^\circ$ ) at source redshifts $z_s \in \{0.5, 1.0, 2.0, 4.0\}$ .
Noise: Realistic shape noise was injected ( $\sigma_\epsilon = 0.26$ , $n_{gal} = 30 \text{ arcmin}^{-2}$ ) to mimic Stage IV surveys (e.g., Euclid).

Statistical Probes (HOS)

The study employed a comprehensive suite of statistics beyond the two-point correlation function (2PCF):

One-Point Statistics:
- 1-Point Probability Distribution Function (1-PDF): The distribution of convergence values.
- Higher-Order Moments (HOM): Skewness ( $\langle \kappa^3 \rangle$ ) and Kurtosis ( $\langle \kappa^4 \rangle$ ).
Peak Statistics:
- Counts of local maxima in the convergence field ( $\kappa$ -peaks).
- Counts of local maxima in the Aperture Mass field ( $M_{ap}$ -peaks), which uses a compensated filter to optimize scale selection.
Topological Statistics:
- Betti Numbers (BNs): Counting topological features (connected components $\beta_0$ and loops $\beta_1$ ) as a function of threshold.
- Minkowski Functionals (MFs): Geometric descriptors (Area $V_0$ , Perimeter $V_1$ , Euler Characteristic $V_2$ ) of excursion sets.

Discrimination Framework

Two distinct methodological approaches were used to quantify the ability of these probes to distinguish GR from MG:

Metrics-Based Approach:
- Calculated distances between the data vectors of GR and MG models.
- Metrics Used: Symmetrized Mean Absolute Percentage Error (SMAPE), Mahalanobis distance ( $d_M$ ), Wasserstein distance ( $d_W$ ), and Hellinger distance ( $d_H$ ).
- Metric: Defined $f_{mod}(O, M)$ as the fraction of configurations where the difference between models exceeds $3\sigma$ .
- Constraint: Required data to pass a Gaussianity test; thus, HOMs were excluded from this specific analysis.
Non-Parametric Hypothesis Testing:
- Compared the full distributions of measurements (histograms of 256 realizations) without assuming Gaussianity.
- Tests Used: Kolmogorov-Smirnov (KS), Cramér-von Mises (CvM), Ansari-Bradley (AB), and Mann-Whitney U (MWU).
- Selection: The CvM test was identified as the most robust, as it captures global distribution differences better than KS (which focuses on single-point deviations) or AB/MWU (which focus on variance/ranks).

3. Key Contributions

Comprehensive HOS Comparison: Unlike previous studies focusing on single probes, this work systematically compares a wide range of HOS (moments, peaks, topology) under identical simulation conditions.
Neutrino Degeneracy Analysis: Explicitly quantifies how massive neutrinos degrade the discriminatory power of MG signatures and identifies which probes remain robust against this cancellation.
Methodological Validation: Demonstrates that non-parametric tests (specifically CvM) are essential for analyzing HOMs, which fail Gaussianity assumptions required by metric-based likelihoods.
Aperture Mass Superiority: Highlights that peak statistics derived from the Aperture Mass map ( $M_{ap}$ ) generally outperform those from raw convergence maps ( $\kappa$ ) due to better noise handling via compensated filters.

4. Key Results

Discriminatory Power of HOS vs. 2PCF

HOS vs. 2PCF: When the 2PCF is measured on unsmoothed maps, it appears superior. However, this is an artifact of accessing small, highly nonlinear scales where MG effects are strongest but baryonic physics is uncertain. When 2PCF is restricted to the same scales as HOS (via smoothing), all HOS probes significantly outperform the 2PCF in distinguishing GR from MG.
Robustness: Even for the weakest MG deviations ( $f_{R6}$ models), at least one HOS probe can detect the deviation, whereas 2PCF often fails.

Impact of Massive Neutrinos

Cancellation Effect: Massive neutrinos can effectively cancel the enhanced clustering of $f(R)$ gravity. For extreme cases (e.g., $f_{R4}$ with $M_\nu = 0.3$ eV), the discriminatory power of most statistics drops to near zero.
Resilient Probes: Despite the cancellation, 1-PDF and Minkowski Functionals (MFs) retain some discriminatory power even in highly compensated scenarios.
Redshift Dependence: The effectiveness of probes varies with source redshift ( $z_s$ $z_{s}$ ).
- At low $z_s$ , 1-PDF and $M_{ap}$ -peaks are often most effective.
- At high $z_s$ , topological statistics (Betti numbers, MFs) and higher-order moments ( $\langle \kappa^4 \rangle$ ) often regain power.

Optimal Probes and Metrics

Best Metric: The Hellinger distance and Mahalanobis distance were found to be the most effective metrics for the Gaussian-based approach.
Best Test: The Cramér-von Mises (CvM) test was the most effective non-parametric method, particularly for detecting global distribution shifts caused by MG.
No Single "Best" Probe: There is no single HOS probe that dominates across all models and redshifts. The optimal choice depends on the specific MG strength, neutrino mass, and redshift.
- Example: For $f_{R5}$ , 1-PDF is best at low $z$ , but MFs become competitive at higher $z$ when neutrinos are present.

5. Significance and Conclusion

This paper establishes that Weak Lensing Higher-Order Statistics are a viable and necessary tool for breaking the degeneracy between Modified Gravity and massive neutrinos, a task impossible for standard second-order statistics alone.

Future Surveys: The results suggest that Stage IV surveys (Euclid, LSST, Roman) will be capable of distinguishing between GR and MG even in the presence of massive neutrinos, provided they utilize a combination of HOS probes.
Strategic Implication: Since no single probe is universally optimal, future analyses should employ a multi-probe approach, combining topological, peak, and moment statistics to maximize sensitivity across different redshifts and cosmological scenarios.
Next Steps: The authors note that while noise is well-modeled, systematic effects (masking, photo-z errors, shear bias) need further investigation to translate these simulation-based results into concrete requirements for real-world data analysis.

In summary, the study confirms that moving beyond the two-point correlation function is not just an incremental improvement but a critical requirement for testing the fundamental nature of gravity in the era of precision cosmology.