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A redshift-independent theoretical halo mass function validated with Uchuu simulations

This paper presents a new redshift-independent theoretical halo mass function based on a generalized Press & Schechter model with triaxial collapse (GPS+), which achieves high accuracy across a wide range of masses and redshifts (0 < z < 20) when validated against the Uchuu N-body simulations, significantly outperforming the Sheth-Tormen model at high redshifts.

Original authors: Elena Fernández-García, Juan E. Betancort-Rijo, Francisco Prada, Tomoaki Ishiyama, Anatoly Klypin, José Ruedas

Published 2026-02-25

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CC BY 4.0

Original authors: Elena Fernández-García, Juan E. Betancort-Rijo, Francisco Prada, Tomoaki Ishiyama, Anatoly Klypin, José Ruedas

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, cosmic construction site. For billions of years, invisible "dark matter" has been the scaffolding, clumping together to form massive structures called halos. These halos are the invisible skeletons that hold galaxies and galaxy clusters together.

Astronomers have long tried to write a "rulebook" to predict how many of these halos exist at any given time, how big they are, and how that changes as the universe ages. This rulebook is called the Halo Mass Function (HMF).

Until now, the rulebooks scientists used were like old, tattered maps. They worked great for the "neighborhoods" (nearby galaxies) and the "cities" (galaxy clusters) we can see easily, but they fell apart when trying to map the tiny "shacks" (dwarf galaxies) or the massive "megacities" in the distant, early universe.

This paper introduces a brand new, high-definition map called GPS+. Here is the story of how they built it and why it's a game-changer.

1. The Problem: The Old Maps Were Broken

The old rulebook (based on a theory from the 1970s called Press-Schechter) was like a recipe that assumed every cake rises exactly the same way. But in reality, some cakes rise fast, some slow, and some collapse.

The Flaw: The old models tried to guess the number of halos based on how much the universe had expanded (redshift). They assumed the rules changed as the universe got older.
The Result: When scientists looked at the very early universe (high redshift), the old maps were wildly wrong. They predicted too many giant halos and missed the small ones entirely. It was like trying to predict the weather in 1920 using a forecast from 2020; the conditions were just too different.

2. The Solution: The "Uchuu" Super-Computer

To fix the map, you need a better survey. The authors used a massive digital simulation called Uchuu (which means "Universe" in Japanese).

The Analogy: Imagine trying to count every grain of sand on a beach. You can't do it by looking at one bucket. You need to simulate the entire ocean.
The Scale: The Uchuu simulation is like a digital universe so huge and detailed that it contains 300 different versions of the same universe running simultaneously. This allowed the scientists to count the halos with incredible precision, from the tiniest clumps to the most massive structures, across 20 billion years of cosmic history.

3. The New Rulebook: GPS+

Using this massive data, the team created GPS+ (Generalized Press-Schechter with triaxial collapse).

The Big Insight: They realized the old rulebook was overcomplicating things. They found that the number of halos doesn't actually care about time (redshift) directly. Instead, it only cares about how "bumpy" the universe is at that specific moment.
The Analogy: Think of the universe as a pot of boiling water.
- Old View: "The number of bubbles depends on how long the water has been boiling."
- GPS+ View: "The number of bubbles depends only on how much energy is in the water right now."
- Because the "bumpiness" of the universe changes smoothly over time, the GPS+ model can predict the halo count for any time in the universe's history using just one simple formula. It's like having a single key that opens every door in the building, rather than needing a different key for every floor.

4. The Test: How Did It Do?

The scientists tested their new map against the Uchuu simulation data.

The Result: The GPS+ model was incredibly accurate. It matched the simulation data within 10–20% across the entire range of masses and time.
The Comparison: They compared it to the previous best model (Sheth-Tormen).
- At "today" (low redshift): Both models were good.
- In the "distant past" (high redshift): The old model failed spectacularly, missing the mark by 70–80%. It was like predicting a snowstorm in the Sahara. The new GPS+ model stayed accurate, staying within 20%.

5. A Crucial Detail: How We Measure "Size"

The paper also discovered a secret ingredient: How you define the size of a halo matters.

The Issue: Astronomers can define a halo's edge in different ways. One way is to draw a circle where the density is 200 times the average background (like a fixed fence). Another way is to use a "virial" definition, which changes the fence size as the universe expands.
The Discovery: The new GPS+ model works perfectly with the fixed fence (M200m). If you try to use the moving fence (virial mass), the model breaks down, especially for nearby, massive clusters.
Why it matters: This suggests that the way halos form is more stable and universal than we thought. The "moving fence" idea might be a trick of the math, not reality.

The Bottom Line

This paper is a major step forward in understanding how the universe builds itself.

Before: We had different rulebooks for different eras of the universe, and they didn't agree with each other.
Now: We have GPS+, a single, elegant rulebook that works from the dawn of time to today, for everything from tiny dwarf galaxies to the largest galaxy clusters.

It's like upgrading from a patchwork quilt of old maps to a single, seamless, high-definition satellite view of the entire cosmic landscape. This new tool will help astronomers interpret data from future telescopes (like the James Webb Space Telescope) and finally understand the true architecture of our universe.

1. Problem Statement

The Halo Mass Function (HMF), which describes the abundance of dark matter halos as a function of mass and redshift, is a cornerstone of modern cosmology. However, existing theoretical models face significant limitations:

Limited Scope: Current models (e.g., Press–Schechter, Sheth–Tormen) are often calibrated only for specific mass ranges (typically cluster scales) and redshifts ( $z \lesssim 3$ or $z \lesssim 10$ ).
Lack of Universality: No existing model provides a self-consistent description across the full halo mass hierarchy (from dwarf galaxies to massive clusters) and the entire cosmic history ( $0 \le z \le 20$ ).
Redshift Dependence: Many models rely on explicit redshift-dependent fitting parameters, which may not be physically motivated and can lead to large deviations when extrapolated to high redshifts.
Data Gaps: Previous N-body simulations lacked the combination of large cosmological volumes and high mass resolution necessary to measure the HMF with low cosmic variance across such an extended domain.

2. Methodology

A. Simulation Data (Uchuu Suite)

The authors utilized the Uchuu N-body simulation suite, which combines extremely large volumes with high mass resolution.

Simulations Used: A combination of six simulations (Phi-4096, Shin-Uchuu, Uchuu, Mucho-Uchuu-140M, Mucho-Uchuu-1G, Mucho-Uchuu-6G) covering Planck15 and Planck18 cosmologies.
Ensemble Approach: To minimize cosmic variance, the authors used up to 300 independent realizations (specifically for the Mucho-Uchuu-1G and 6G sets).
Domain: The HMF was measured over a mass range of $6.5 \le \log(M_{200m}/[h^{-1}M_\odot]) \le 16$ and redshifts $0 \le z \le 20$ .
Halo Definition: Halos were identified using the RockStar finder. Crucially, the authors adopted the $M_{200m}$ definition (mass within an overdensity of 200 times the mean background density), which is a fixed threshold and thus redshift-independent, as opposed to the evolving virial overdensity ( $M_{vir}$ ).

B. Theoretical Framework: GPS+

The authors propose a new model named GPS+ (Generalized Press–Schechter with triaxial collapse).

Foundation: It builds on the Press–Schechter (PS) formalism but replaces the original center-based collapse criterion with a more physically motivated prescription based on triaxial collapse.
Key Innovation: The model formulates the HMF primarily as a function of the variance of the linear density field, $\sigma(M, z)$ $σ (M, z)$ .
- It includes a weak explicit mass dependence (via functions $b(M)$ and $c(M)$ derived from the power spectrum).
- It has no explicit redshift dependence. Evolution enters solely through $\sigma(M, z)$ .
Mathematical Formulation:
- The mass fraction $F(M)$ is derived using the complementary error function (erfc) involving a modified critical overdensity $\langle \delta_c \rangle$ .
- $\langle \delta_c \rangle$ is a function of mass and $\sigma$ , incorporating triaxial collapse physics via parameters $A$ and $B$ (fitted to simulations) and a theoretical parameter $D$ .
- The model preserves the statistical properties of Gaussian random fields while correcting the normalization and high-mass tail issues inherent in the original PS and Sheth–Tormen (ST) models.

3. Key Contributions

New Theoretical Model (GPS+): A physically motivated HMF model that requires minimal calibration (only two parameters, $A$ and $B$ , fitted to simulations) and naturally reproduces correct normalization and high-mass behavior without empirical fitting functions for redshift.
Validation with Uchuu: The first validation of an HMF model against a dataset spanning $6.5 < \log M < 16$ and $0 < z < 20$ with reduced cosmic variance, enabled by the Uchuu ensemble.
Redshift Independence: Demonstration that a redshift-independent formulation (where evolution is driven only by $\sigma$ ) is superior to models with explicit redshift dependence, provided the correct halo mass definition ( $M_{200m}$ ) is used.
Halo Mass Definition Insight: A critical finding that the choice of halo mass definition is paramount. The model works exceptionally well with $M_{200m}$ but degrades significantly when using the evolving virial mass ( $M_{vir}$ ), suggesting that the virial overdensity $\Delta_{vir}$ is likely nearly constant (or weakly dependent) rather than evolving as strongly as previously assumed (e.g., Bryan & Norman 1998).

4. Results

Accuracy: The GPS+ model matches the Uchuu simulation data to within 10–20% across the entire mass and redshift domain.
Comparison with Sheth–Tormen (ST):
- At low redshift ( $z \lesssim 2$ ), GPS+ performs similarly to the ST model.
- At high redshift ( $z \gtrsim 4$ ), GPS+ significantly outperforms ST.
- At $z \approx 18.37$ , the ST model deviates by 70–80%, whereas GPS+ remains within ~20%.
Mass Definition Impact:
- Using $M_{200m}$ yields a nearly universal HMF that is weakly dependent on redshift.
- Using $M_{vir}$ (with evolving $\Delta_{vir}$ ) causes the model to fail at low redshifts and high masses, with discrepancies reaching ~50%. This supports the hypothesis that the stabilisation process of halos is largely independent of redshift.
Comparison with Dark Emulator: At $z < 1.5$ , GPS+ shows better agreement with simulations than the Dark Emulator, particularly at the low-mass end and high-mass tail.

5. Significance

Cosmological Probes: The GPS+ model provides a robust, first-principles framework for interpreting data from current and future large-scale structure surveys (e.g., DESI, Euclid, LSST, JWST, eROSITA). Its ability to predict HMFs up to $z \sim 20$ is crucial for studying the epoch of reionization and the formation of the first structures.
Theoretical Advancement: By removing explicit redshift dependence and relying on triaxial collapse physics, the model offers a more fundamental understanding of halo formation, suggesting that the "universal" nature of the HMF is best captured when using a fixed overdensity definition ( $M_{200m}$ ).
Resource Availability: The authors have made the GPS+ model code publicly available (via GitHub and Colossus) and provided the combined HMF data from the Uchuu suite, facilitating immediate application by the broader astrophysics community.

In conclusion, this work establishes a new standard for theoretical halo mass functions, validated by the most extensive simulation suite to date, and highlights the critical importance of halo mass definitions in cosmological modeling.