⚛️ general relativity

A Universal Bilinear Model for the Dependence of Void Asymmetry Distributions on $\Omega_{m}$ and $\sigma_{8}$

This paper proposes a universal bilinear model based on void asymmetry distributions derived from coherent galaxy rotation, which effectively constrains the matter density parameter ( $\Omega_{m}$ ) and initial density amplitude ( $\sigma_{8}$ ) while breaking their degeneracy independently of the dark energy equation of state ( $w$ ).

Original authors: Geonwoo Kang (Seoul National University), Jounghun Lee (Seoul National University)

Published 2026-07-23

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CC BY 4.0

Original authors: Geonwoo Kang (Seoul National University), Jounghun Lee (Seoul National University)

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 not as a smooth, empty soup, but as a giant, cosmic spiderweb. Astronomers call this the "cosmic web." In this web, galaxies aren't scattered randomly; they clump together in dense knots, stretch out into long, thin filaments, and flatten into vast sheets. But the most interesting parts are the holes in the web: the "voids." These are massive, nearly empty bubbles where very few galaxies live. You might think these empty spaces are boring, but they are actually the universe's most pristine laboratories. Because they are so empty, they haven't been messed up by the messy, violent collisions that happen in crowded galaxy clusters. They hold a secret history of how the universe began.

The big mystery scientists are trying to solve is what the universe is made of and how it's growing. Two of the most important ingredients are the amount of normal matter (called $\Omega_m$ ) and how "clumpy" the universe started out (called $\sigma_8$ ). Think of $\Omega_m$ as the total weight of the cosmic web, and $\sigma_8$ as how much the web was crumpled up in the beginning. The problem is, these two numbers are like a tangled pair of headphones; when you try to measure one, it's hard to tell how much the other is affecting the result. Scientists need a new way to untangle them, and they think the answer might be hiding in the way galaxies spin inside those giant empty bubbles.

The Cosmic Dance of Empty Bubbles

In this paper, astronomers Geonwoo Kang and Jounghun Lee propose a clever new way to weigh the universe's ingredients by watching how galaxies dance inside cosmic voids. They discovered that even though voids are empty, the few galaxies living inside them aren't just sitting still. Instead, they are swirling around in a coordinated, spinning motion, much like a school of fish turning in unison or a group of dancers spinning around a central pole.

The authors realized that if you could see this spin from Earth, you would notice something funny about the galaxies' speeds. Imagine a spinning void. On one side of the spin axis, the galaxies are moving toward us, making them look slightly bluer and faster. On the opposite side, they are moving away, making them look redder and slower. This difference in speed creates a "redshift asymmetry"—a tell-tale sign that the void is rotating.

The challenge is that we can't easily see the 3D spin of these bubbles because we can't measure the exact speed of every galaxy in all directions. So, the authors came up with a trick. They treated the void like a pizza and tried to slice it in every possible direction. They looked for the specific slice (or "bisector") that created the biggest difference in speed between the two halves. That perfect slice points right along the void's spin axis. By measuring this maximum speed difference, they could figure out how fast the void is spinning without needing to know the full 3D motion of every single galaxy.

The Universal Formula

To test this idea, the team didn't just look at one universe; they ran thousands of computer simulations of 33 different types of universes. Some had a standard amount of dark energy, some had a changing amount, and some had different amounts of matter. They filled these simulated universes with billions of dark matter particles and watched them evolve into galaxies and voids.

What they found was surprisingly simple and beautiful. No matter what kind of universe they simulated, the distribution of these "spin speeds" (or redshift asymmetries) followed a specific mathematical pattern called a "generalized Gamma model." It's like finding that every type of snowflake, no matter the temperature or humidity, follows the same basic geometric rules.

More importantly, they discovered that this pattern has two main settings: a "shape" setting and a "scale" setting.

The shape setting was stubborn. It stayed exactly the same, regardless of how much matter ( $\Omega_m$ ) or how clumpy ( $\sigma_8$ ) the universe was. It was also completely unaffected by the type of dark energy ( $w$ ) used in the simulation.
The scale setting, however, was a chameleon. It changed in a perfectly straight line depending on the amount of matter and the clumpiness. If you increased the matter, the scale went up. If you increased the clumpiness, the scale went up.

This led the authors to create a "universal bilinear model." Think of it like a master recipe. The authors found that if you know the amount of matter and the clumpiness of a universe, you can plug those numbers into their simple formula to predict exactly what the void spin distribution should look like. And the best part? This recipe works for all the different types of universes they tested, whether they had standard dark energy or weird, changing dark energy.

Why This Matters

The authors tested their recipe on three new, unseen simulations that they hadn't used to build the formula. When they applied their model, it correctly predicted the values for matter and clumpiness in those new universes, landing right in the "bullseye" of the correct answer. This suggests that if we can measure the redshift asymmetry of real voids in our own universe, we can use this formula to untangle the $\Omega_m$ and $\sigma_8$ mystery.

Unlike other methods that get confused by the type of dark energy, this new "void asymmetry" method seems to ignore dark energy entirely, focusing only on the matter and the initial clumpiness. It's a bit like trying to figure out the weight of a suitcase by listening to how the wheels squeak; the type of floor you're walking on (dark energy) doesn't change the sound, but the weight of the suitcase (matter) does.

The paper concludes that while this method is promising and works perfectly in their simulations, it still needs to be tested on real telescope data. But if it holds up, it could give astronomers a powerful new tool to finally understand the fundamental ingredients of our cosmic web, breaking the long-standing tie between how much matter is in the universe and how it started out.

Technical Summary: A Universal Bilinear Model for the Dependence of Void Asymmetry Distributions on $\Omega_m$ and $\sigma_8$

Problem Statement
The cosmic web consists of knots, filaments, sheets, and voids. While extensive studies have focused on over-dense regions, voids (under-dense regions) are believed to retain pristine imprints of early universe physics and are highly sensitive to large-scale tidal effects. Previous work by Kang & Lee (2025) established that the distribution of void spin magnitudes in 3D real space follows a generalized Gamma distribution, with its parameters depending strongly on the matter density parameter ( $\Omega_m$ ) and the amplitude of initial density fluctuations ( $\sigma_8$ ), while being insensitive to the dark energy equation of state ( $w$ ). However, measuring void spins in 3D real space requires precise 3D peculiar velocities of void galaxies, which are observationally infeasible. The challenge addressed in this paper is to identify an observational proxy for void spins that can be measured in 2D redshift space (using only redshifts) while retaining the same cosmological sensitivity to $\Omega_m$ and $\sigma_8$ and independence from $w$ .

Methodology
The authors utilize data from the AbacusSummit suite of cosmological N-body simulations, covering $\sim 330$ billion dark matter particles across 33 distinct background cosmologies. These cosmologies are divided into three classes: $\Lambda$ CDM, $w$ CDM (constant $w$ ), and $w_0w_a$ CDM (dynamical $w$ ).

Sample Construction: Galaxies are identified as dark matter halos with masses $M \ge 10^{11.5} h^{-1} M_\odot$ at $z=0.1$ , mimicking the flux limits of the Vera C. Rubin Observatory LSST.
Redshift Space Transformation: A mock redshift space galaxy sample is generated using the flat-sky approximation, where the line-of-sight (LOS) position is modified by the peculiar velocity ( $x_{r3} = x_3 + v \cdot \hat{x}_3 / H_0$ ).
Void Identification: The classical Void-Finder algorithm (Hoyle & Vogeley 2002) is applied to the redshift space sample to identify voids. Only voids containing at least 15 member galaxies above the mass cutoff are selected.
Controlled Samples: To eliminate spurious correlations arising from differences in void size distributions across cosmologies, the authors construct $R_v$ -controlled samples. They bin voids by effective radius ( $R_v$ ) and draw equal numbers of voids from each bin for all 33 cosmologies, ensuring identical size distributions.
Asymmetry Measurement: Following the methodology of Wang et al. (2021), the projected spin axis of a void is determined iteratively. A bisector line is rotated through the void center in the plane of the sky, and the redshift asymmetry ( $\Delta z$ ) between the two resulting sectors is calculated. The orientation yielding the maximum asymmetry ( $\Delta z_{\max}$ ) is defined as the projected spin axis.
Statistical Modeling: The probability density distribution of $\Delta z_{\max}$ , denoted $p(\Delta z_{\max})$ , is fitted to a generalized Gamma model. The authors test whether the shape parameter ( $k$ ) and scale parameter ( $\theta$ ) depend on cosmology.
Model Development: A bilinear model is developed to describe the dependence of $\theta$ on $\Omega_m$ and $\sigma_8$ . The universality of this model is tested across the three cosmological classes and against variations in simulation volume, resolution, and phase.

Key Contributions and Results

Generalized Gamma Fit: The void asymmetry distributions in both real and redshift spaces are well approximated by the generalized Gamma model for all 33 cosmologies.
Parameter Sensitivity: The shape parameter ( $k$ ) is found to be almost independent of $\Omega_m$ , $\sigma_8$ , and $w$ . In contrast, the scale parameter ( $\theta$ ) exhibits a strong, nearly linear dependence on both $\Omega_m$ and $\sigma_8$ .
Universality of the Bilinear Model: The authors derive a non-parametric bilinear model for the scale parameter:
$10^5 \theta_{\text{model}}(\Omega_m, \sigma_8) = \eta_1 \Omega_m + \eta_2 \sigma_8 + \eta_3$
The coefficients $\{\eta_i\}$ , determined using $\Lambda$ CDM cosmologies, are found to be universal constants that accurately predict $\theta$ for $w$ CDM and $w_0w_a$ CDM cosmologies as well.
Independence from $w$ : The model's dependence on $\Omega_m$ and $\sigma_8$ is shown to be insensitive to the dark energy equation of state $w$ , distinguishing it from standard Large Scale Structure (LSS) probes that suffer from degeneracies involving $w$ .
Robustness: The bilinear model remains valid across different simulation volumes, particle mass resolutions, and initial phases. Misidentification of void centers (using the center of the largest empty sphere rather than the centroid of galaxies) has a negligible effect on the results.
Real vs. Redshift Space: While the linear dependence holds in real space, the scale parameter $\theta$ is lower in real space than in redshift space. The authors attribute the higher $\theta$ in redshift space to the additional uncertainties introduced by redshift-space distortions.

Significance and Claims
The paper claims that the void asymmetry distribution serves as a robust, independent diagnostic for constraining $\Omega_m$ and $\sigma_8$ . Its primary significance lies in its ability to break the $\Omega_m$ - $\sigma_8$ degeneracy without being affected by the dark energy equation of state ( $w$ ), a limitation common in other LSS probes like cosmic shear or cluster abundance evolution.

The authors emphasize that this method is practically feasible because it relies solely on redshift measurements of void galaxies in the local universe, avoiding the need for difficult-to-measure 3D peculiar velocities. By combining this new diagnostic with standard LSS-based methods (e.g., weak lensing), it is proposed that tighter constraints on cosmological parameters can be achieved. The paper concludes that while the current bilinear model is universal and robust, future work utilizing higher-resolution simulations may be necessary to refine the coefficients for improved efficiency.

Technical Summary: A Universal Bilinear Model for the Dependence of Void Asymmetry Distributions on Ωm\Omega_mΩm​ and σ8\sigma_8σ8​

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Technical Summary: A Universal Bilinear Model for the Dependence of Void Asymmetry Distributions on $\Omega_m$ and $\sigma_8$