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On the redshift evolution of the spin parameter in cosmological simulations

This paper uses high-resolution $\Lambda$ CDM N-body simulations to characterize the redshift evolution of dark matter halo spin parameters and provides closed-form fitting functions to replace the common assumption of constant spin in galaxy formation models.

Original authors: Tomas Riera, Alexander Knebe, Chris Power, Robert Adriel Mostoghiu Paun, Adam Ussing

Published 2026-04-27

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CC BY 4.0

Original authors: Tomas Riera, Alexander Knebe, Chris Power, Robert Adriel Mostoghiu Paun, Adam Ussing

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

The Cosmic Spin Cycle: Why Galaxies Aren't Just Spinning in Circles

Imagine you are watching a massive, swirling dance marathon that has been going on for billions of years. The dancers are "dark matter halos"—giant, invisible clouds of matter that act as the gravitational stages upon which galaxies are built.

As these dancers move, they don't just drift; they spin. Some spin wildly like figure skaters, while others move more sluggishly. Scientists have long known that these "dancers" follow a general pattern of spinning, but they haven't quite figured out how the "choreography" changes as the universe gets older.

This paper, written by a team of international researchers, finally provides the "instruction manual" for how that cosmic spin changes over time.

1. The Two Ways to Measure a Spin

In the world of physics, there isn't just one way to say "how fast something is spinning." The researchers looked at two different "rulers":

The Peebles Ruler ( $\lambda$ ): Think of this like measuring a dancer by looking at their total energy. It takes into account how much they are moving and how much gravity is pulling them inward. It’s a deep, complex measurement.
The Bullock Ruler ( $\lambda'$ ): Think of this like measuring a dancer by their speed at the edge of the dance floor. It’s much easier and faster to calculate because you only need to look at how fast they are moving at their outer boundary.

The researchers discovered something fascinating: these two rulers tell different stories.

2. The "Growing Pains" of the Universe

The core discovery of the paper is that the "average spin" of these dark matter clouds isn't a constant number. It changes as the universe ages (what scientists call "redshift evolution").

The Steady Spinner (Peebles): Using the first ruler, the researchers found that as the universe gets older, the spin increases in a very steady, predictable way. It’s like a spinning top that slowly picks up speed as it settles down.
The Rollercoaster (Bullock): Using the second ruler, the story is much more dramatic. The spin increases for a while, but then—around the time the universe was about 7 to 10 billion years old—it actually started to slow down. It’s like a dancer who starts with high energy, hits a peak, and then begins to tire out.

Why the difference? The researchers explain that as the universe ages, these dark matter clouds become more "concentrated" (denser in the middle). Because the Peebles ruler cares about total energy and the Bullock ruler only cares about the edges, they react differently to this "thickening" of the clouds.

3. The "Chaos Factor" (Dispersion)

The researchers also looked at the variety of spins. In a room full of dancers, some are doing pirouettes while others are just swaying.

They found that as the universe gets older, the "dance floor" gets more chaotic. The difference between the fastest spinners and the slowest spinners grows wider. This is likely because, as time goes on, these clouds are constantly crashing into each other and merging, which makes the "dance" much more unpredictable.

4. Why Does This Matter? (The "Recipe" for Galaxies)

You might ask: "Who cares how much invisible dark matter is spinning?"

Well, dark matter is the "scaffolding" of the universe. Inside these spinning clouds, regular matter (the stuff that makes up stars, planets, and you) settles down to form galaxies.

If the dark matter is spinning fast, the galaxy will likely be a wide, beautiful spiral (like our Milky Way). If the spin is low, the galaxy might look very different.

By providing mathematical formulas (the "fitting functions" mentioned in the paper), the researchers have given other scientists a universal recipe. Now, when astronomers try to simulate how the universe grew, they don't have to guess how much spin to include—they can use these formulas to plug in the exact "spin speed" for any moment in cosmic history.

Summary in a Nutshell

The Old Assumption: "All dark matter clouds spin at roughly the same average speed, regardless of age."

The New Reality: "The spin changes! One way of measuring shows a steady increase, while the other shows a peak and a decline. And as the universe ages, the 'spin styles' become much more diverse and chaotic."

Technical Summary: On the redshift evolution of the spin parameter in cosmological simulations

1. Problem Statement

The angular momentum of dark matter halos is a fundamental property that dictates the formation and structural properties (such as size and surface density) of the galaxies they host. While it is well-established that the spin parameter follows a log-normal distribution at any fixed epoch, the quantitative evolution of this distribution across cosmic time—specifically how both the mean ( $\mu$ ) and the dispersion ( $\sigma$ ) change with redshift ( $z$ )—has remained poorly constrained.

Previous studies have faced two primary limitations:

Resolution Issues: Some analyses lacked the mass resolution necessary to establish reliable evolutionary trends, leading to potentially biased results.
Lack of Analytical Tools: Existing studies often failed to provide closed-form, analytical fitting functions, making it difficult for researchers to implement redshift-dependent spin into semi-analytic models (SAMs) or forward-modeling frameworks without running expensive new simulations.

2. Methodology

The authors utilized a suite of four dissipationless $\Lambda$ CDM N-body simulations (the "B-suite") run using the GADGET-4 code.

Simulation Design: To ensure statistical robustness across a wide range of scales, they used varying box sizes ( $B = 20, 40, 80, 160 \, h^{-1}\text{Mpc}$ ) with a constant minimum particle number per halo to maintain uniform resolution.
Halo Identification: Halos were identified using the Amiga Halo Finder (AHF), focusing on the $R_{200}$ definition.
Selection Criteria: To ensure high-quality data, the sample was restricted to "field halos" (excluding substructures) containing at least 500 particles.
Spin Definitions: The study characterizes two distinct dimensionless spin parameters:
1. Peebles ( $\lambda$ ): Based on total energy ( $E$ ), angular momentum ( $J$ ), and mass ( $M$ ).
2. Bullock ( $\lambda'$ ): Based on circular velocity ( $V$ ), angular momentum ( $J$ ), and mass ( $M$ ).
Statistical Analysis: The authors modeled the evolution of the mean ( $\ln \lambda$ ) and standard deviation ( $\sigma$ ) of these parameters from $z=5$ to $z=0$ .

3. Key Results

The study reveals a striking divergence in how the two spin definitions evolve:

Evolution of the Mean ( $\mu$ ):
- Peebles ( $\lambda$ ): Shows a mild, statistically robust linear increase in $\ln \lambda$ as redshift decreases (moving toward the present day).
- Bullock ( $\lambda'$ ): Exhibits non-monotonic behavior. The mean $\ln \lambda'$ increases as redshift decreases until $z \approx 1\text{--}2$ , after which it begins to decline toward $z=0$ .
- Physical Origin: The authors attribute this difference to the fact that $\lambda$ is sensitive to the halo's gravitational potential and concentration ( $c$ ), whereas $\lambda'$ is more dependent on the circular velocity at the halo boundary. As halo concentrations increase over time, $\lambda$ and $\lambda'$ respond differently.
Evolution of the Dispersion ( $\sigma$ ):
- Both $\ln \lambda$ and $\ln \lambda'$ show a weak linear increase in dispersion as redshift decreases. This suggests that as the universe evolves, the complexity of halo assembly (via mergers and accretion) increases the variety of spin values in the population.
- $\lambda'$ consistently shows a higher dispersion and a steeper evolutionary slope than $\lambda$ .
Validation: The authors demonstrated that their findings are robust against changes in mass resolution and halo boundary definitions (e.g., $R_{200}$ vs. $R_{vir}$ ).

4. Key Contributions & Significance

The paper provides three major contributions to the field of computational cosmology:

Closed-Form Fitting Functions: The authors provide analytical expressions (Equations 6, 7, and 8 in the text) that allow modellers to sample physically motivated spin values at any redshift within the range $0 \leq z \leq 5$ . This replaces the common, but potentially inaccurate, assumption of a constant spin distribution.
Resolution-Validated Trends: By performing controlled resolution tests, they reconciled previous discrepancies in the literature (notably with Hetznecker & Burkert 2006), proving that earlier reported inconsistencies were largely artifacts of insufficient mass resolution.
Practical Utility for Galaxy Formation Models: These functions serve as a vital input for Semi-Analytic Models (SAMs) and Semi-Empirical Models. While the study focuses on dark matter, the results provide the necessary "angular momentum budget" required to predict the scale lengths and structural evolution of galactic discs across cosmic time.