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Cosmological constraining power of the redshifts, heights, and angular clustering of weak gravitational lensing peaks

This study demonstrates that combining the redshift distribution, height distribution, and angular clustering of weak gravitational lensing peaks within a Bayesian framework yields tighter cosmological constraints than traditional two-point statistics, effectively probing a ten-dimensional cosmological parameter space including dark energy, neutrino mass, and dark matter properties.

Original authors: Jeger C. Broxterman, Matthieu Schaller, Ian G. McCarthy, Willem Elbers, John Helly, Henk Hoekstra, Konrad Kuijken, Jaime Salcido, Joop Schaye, Naomi Schutte, Elena Sellentin

Published 2026-08-03

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CC0 1.0

Original authors: Jeger C. Broxterman, Matthieu Schaller, Ian G. McCarthy, Willem Elbers, John Helly, Henk Hoekstra, Konrad Kuijken, Jaime Salcido, Joop Schaye, Naomi Schutte, Elena Sellentin

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

Imagine the universe as a giant, invisible web made of dark matter, stretching across billions of light-years. We can't see this web directly, but we can see how it bends the light from distant galaxies, much like a funhouse mirror distorts a reflection. This bending is called "weak gravitational lensing." By studying these tiny distortions, astronomers try to figure out the universe's recipe: how much dark matter and dark energy are in the mix, and how the universe is expanding. For a long time, scientists have looked at the average distortion across the sky, like measuring the average temperature of a room. But just as a room might have a hot stove and a cold draft that average out to a "comfortable" temperature, the universe has hidden details that get lost in the average. This paper explores a new way to look at the universe: instead of the average, they look at the "peaks"—the specific, high points in the distortion map where the dark matter is most clumped together. It's like realizing that to understand a mountain range, you shouldn't just measure the average elevation of the whole country, but instead count the tallest peaks, measure their heights, and see how they are clustered together.

The authors of this paper, led by Jeger C. Broxterman, decided to test a new strategy for a future space telescope called Euclid. They wanted to know if looking at these "peaks" in three different ways—how tall they are, where they are located in time (redshift), and how they group together in the sky—could give better answers than the traditional methods. To do this, they didn't just look at the real sky; they built a massive digital universe inside a computer. They created a "hypercube" of 100 different universes, each with slightly different rules for how dark energy works, how heavy neutrinos are, and how fast the universe is expanding. They then simulated what a telescope would see in each of these fake universes and compared the results.

The team found that these three ways of looking at the peaks are like three different detectives solving the same mystery, each good at spotting different clues. The "redshift distribution" (where the peaks are in time) turned out to be the best detective for figuring out how much matter is in the universe and how dark energy is behaving. The "height distribution" (how tall the peaks are) and "angular clustering" (how they group together) were the best at measuring the initial strength of the universe's density fluctuations. When they combined all three detectives, they could even solve for the Hubble parameter (how fast the universe is expanding) and the density of normal matter (baryons), which are usually hard to pin down.

Crucially, the paper shows that this new "peak" method is actually better than the old standard method (measuring the average distortion) for most of these questions. In their simulations, the peak method provided tighter, more precise constraints on the universe's composition. However, the authors are careful to note that this is a forecast based on computer simulations. They found that while the peaks are great at finding the "tallest" structures, they aren't as good at measuring the mass of neutrinos or the specific shape of the early universe's power spectrum as the traditional methods are. But when they combined the new peak method with the old traditional method, the results were even stronger, suggesting that the future of cosmology lies in using every tool in the toolbox at once.

The researchers also played with the "zoom level" of their simulations, changing the smoothing scale (how blurry the image is) to see if looking at the peaks with a different magnification helped. They found that looking at the smallest scales (the sharpest images) usually gave the best results, and combining multiple scales improved the accuracy by about a factor of two. While they didn't include every possible real-world complication like the messy physics of gas and black holes in their simulation, their results suggest that for the upcoming Euclid mission, focusing on these cosmic "peaks" could unlock a deeper understanding of the dark universe than we have today.

Technical Summary: Cosmological constraining power of the redshifts, heights, and angular clustering of weak gravitational lensing peaks

Problem Statement
Current cosmological surveys, including Stage III (KiDS, DES, HSC) and upcoming Stage IV (Euclid, Rubin/LSST, Roman) missions, are reaching a precision where potential deviations from the standard $\Lambda$ CDM model are emerging, particularly regarding the nature of dark energy (DE) and the $S_8$ tension. While cosmic shear surveys traditionally rely on two-point statistics (e.g., shear two-point correlation functions, 2PCFs) which capture Gaussian information, these methods may not fully exploit the non-Gaussian information inherent in the large-scale distribution of matter. Weak gravitational lensing (WL) peaks—local maxima in the convergence field—represent the most massive structures and contain higher-order statistical information. This paper investigates the cosmological constraining power of three specific WL peak statistics: the height distribution, the redshift distribution, and the angular clustering. The study aims to determine how these statistics complement one another and how they compare to standard shear 2PCFs in a Euclid-like survey context, specifically within a high-dimensional cosmological parameter space.

Methodology
The authors employ a Bayesian inference framework mimicking a Euclid analysis, utilizing a new set of dark-matter-only (DMO) simulations.

Simulations: The study uses a "hypercube" of 100 simulations (50 in a 10D space and 50 in a 9D subspace) generated via Latin hypercube sampling. The 10D parameter space varies: matter density ( $\Omega_{m}$ ), baryon fraction ( $f_b$ ), Hubble parameter ( $h$ ), scalar spectral index ( $n_s$ ), primordial power spectrum amplitude ( $\ln(10^{10}A_s)$ ), dark energy equation-of-state parameters ( $w_0, w_a$ ), neutrino mass sum ( $\sum m_\nu$ ), running of the spectral index ( $\alpha_s$ ), and the decay rate of decaying dark matter ( $\Gamma_{dcdm}$ ). The simulations use the swift code in $700^3$ cMpc $^3$ volumes with $1260^3$ dark matter particles.
Signal Extraction: Full-sky WL convergence maps are constructed from spherical matter shells using a virtual observer. The maps include shape noise mimicking Euclid specifications ( $n_{gal} = 30$ arcmin $^{-2}$ , $\sigma_\epsilon = 0.26$ ). Peaks are defined as pixels exceeding their eight nearest neighbors.
Statistics: Three statistics are measured for peaks with Signal-to-Noise Ratio (SNR) > 3:
1. Redshift distribution ($dn/dz$): The primary redshift of the overdensity contributing most to the peak is assigned, distorted by a photometric redshift error model ( $\sigma_z = 0.02(1+z)$ ).
2. Height distribution ( $dn/d\kappa$ ): The distribution of peak convergence values.
3. Angular clustering ( $\omega(\theta)$ ): The two-point correlation function of peak positions.
Inference Pipeline: Gaussian Process (GP) emulators are trained on the simulation outputs to predict statistics across the 10D parameter space. A Bayesian likelihood analysis is performed using a multivariate $t$ -distribution to account for covariance estimation from 80 additional $\Lambda$ CDM simulations. The analysis focuses on the Figure of Merit (FoM) and Figure of Bias (FoB) for parameter constraints.
Smoothing: The fiducial analysis uses a Gaussian smoothing scale of 1 arcmin, with additional tests at 0.5 and 2.0 arcmin to assess scale dependence.

Key Contributions and Results

Complementarity of Statistics: The three WL peak statistics are found to be highly complementary.
- The redshift distribution provides the tightest constraints on the matter density ( $\Omega_m$ ) and the dark energy equation-of-state parameters ( $w_0, w_a$ ). It effectively breaks degeneracies present in other statistics by probing the evolution of structure formation over cosmic time.
- The height distribution and angular clustering are most sensitive to the amplitude of the primordial power spectrum ( $\ln(10^{10}A_s)$ ).
- Combined Analysis: Combining all three statistics allows for the probing of the baryon density ( $\Omega_b$ ) and the Hubble parameter ( $h$ ), which are poorly constrained by individual statistics.
Performance vs. Standard Statistics: When compared to the shear two-point correlation functions ( $\xi_{+/−}$ ), the combined WL peak statistics outperform the 2PCFs for $\Omega_m$ , $\Omega_b$ , $w_0$ , and $w_a$ . The 2PCFs retain superior constraining power for the neutrino mass ( $\sum m_\nu$ ), scalar spectral index ( $n_s$ ), and its running ( $\alpha_s$ ). However, the joint inference (peaks + 2PCFs) yields the tightest overall constraints, demonstrating that peaks provide non-Gaussian information that breaks degeneracies in the 2PCF posterior.
Smoothing Scale Dependence: The study finds that smaller smoothing scales generally yield tighter constraints. Combining multiple smoothing scales (0.5, 1.0, and 2.0 arcmin) improves the FoM by a factor of approximately 2 for most statistics. The redshift distribution shows the most significant improvement when combining scales.
Parameter Constraints: In the combined 10D forecast, the method recovers the fiducial D3A cosmology (based on DES Year 3 results) with high fidelity for well-constrained parameters ( $\Omega_m, \Omega_b, h, w_0, w_a, \ln(10^{10}A_s)$ ), with an average Figure of Bias of $0.59\sigma$ . Parameters such as neutrino mass, $\alpha_s$ , and the DDM decay rate remain weakly constrained or recover the prior, consistent with the limitations of WL peak statistics for these specific parameters.

Significance and Claims
The paper claims that WL peak statistics offer a powerful, complementary avenue for cosmological inference that captures non-Gaussian information missed by traditional two-point statistics. Specifically, the authors demonstrate that:

Redshift distribution is critical: Unlike previous studies focusing primarily on peak heights or clustering, this work highlights that the redshift distribution of WL peaks is the dominant driver for constraining dark energy parameters ( $w_0, w_a$ ) and matter density ( $\Omega_m$ ), outperforming both angular clustering and height distributions in these planes.
Synergy over individual metrics: While individual peak statistics have limitations, their combination significantly enhances constraining power, particularly for breaking degeneracies between $\Omega_m$ and $h$ , and for constraining $\Omega_b$ .
Future Survey Potential: The results suggest that for Stage IV surveys like Euclid, incorporating the redshift distribution and angular clustering of WL peaks alongside standard shear 2PCFs will provide a more robust cosmological inference, potentially resolving current tensions and refining constraints on dark energy evolution.

The authors note that these results are based on an idealized inference (ignoring baryonic physics, intrinsic alignments, and other systematics) and that future work must incorporate these effects to validate the precision of these constraints for real observational data.

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