🔭 astrophysics

The dark matter halo mass function in the $\Lambda\mathrm{CDM}$ cosmology at all times and over all scales -- from planetary to galaxy cluster masses

This paper presents a comprehensive fitting formula for the dark matter halo mass function in the $\Lambda$ CDM cosmology, derived from a combination of high-resolution VVV simulations and large-volume cosmological runs, which accurately predicts halo abundances across a vast dynamic range from planetary to galaxy cluster masses and from redshift $z=30$ to the present with deviations of only 2–7%.

Original authors: Haonan Zheng, Sownak Bose, Carlos S. Frenk, Liang Gao, Adrian Jenkins, Shihong Liao, Yizhou Liu, Volker Springel, Jie Wang, Simon D. M. White

Published 2026-07-08

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CC BY 4.0

Original authors: Haonan Zheng, Sownak Bose, Carlos S. Frenk, Liang Gao, Adrian Jenkins, Shihong Liao, Yizhou Liu, Volker Springel, Jie Wang, Simon D. M. White

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 ocean. In this ocean, invisible islands of "dark matter" are constantly forming, growing, and merging. These islands are called dark matter haloes. Some are tiny, like pebbles the size of a planet; others are massive, like entire archipelagos holding thousands of galaxies together.

For decades, scientists have tried to write a single "rulebook" to predict exactly how many of these islands exist at any given time, from the very beginning of the universe (when it was a baby) to today. This rulebook is called the Halo Mass Function.

Here is what this paper does, explained simply:

1. The Problem: The Old Rulebooks Were Flawed

Scientists previously had two main rulebooks:

The "Simple" Rulebook (Press-Schechter): This was like a rough sketch. It worked okay for big islands but was very wrong for the tiny ones and the very old ones. It was too simple to capture the messy reality of how the universe grows.
The "Better" Rulebook (Reed et al. 2007): This was a much more detailed map. It worked great for the islands we see today (like our galaxy and its neighbors). However, when scientists tried to use it to look back in time to the early universe, or to look at the tiniest, planet-sized islands, the map started to get blurry. It predicted too many tiny islands in the early universe, leading to errors of up to 50% or even double the actual number!

2. The Challenge: Seeing the Invisible

To fix this, the authors needed to count these dark matter islands across a massive range:

Size: From the mass of a single planet (tiny) to the mass of a giant galaxy cluster (huge).
Time: From 30 billion years ago (the dawn of time) to right now.

The problem is that you can't run one computer simulation that is detailed enough to see a planet-sized island and big enough to see a galaxy cluster. It's like trying to use a single microscope that can zoom in to see a single cell but also zoom out to see the entire Earth. The computer power required would be impossible.

3. The Solution: The "Subsampling" Trick

The authors used a clever trick called subsampling.

Imagine you have a giant, slightly empty room (a simulation of a "void" in the universe). It's mostly empty space, so there aren't many islands in it.

The Trick: Instead of looking at the whole empty room, they took a magnifying glass and looked at tiny, specific corners of that room.
The Result: Even though the whole room was empty, those tiny corners had a density of islands that looked like the "average" universe. By taking hundreds of these tiny snapshots from different parts of the empty room, they could stitch them together to create a perfect picture of the whole universe.

This allowed them to use their "small room" simulations to accurately count the tiny, planet-sized islands that usually get lost in the noise.

4. The New Rulebook

Using this new method, the authors combined data from several massive computer simulations to create a new, universal rulebook.

What it does: It predicts the number of dark matter islands with incredible accuracy (within 2–3% for the recent universe and about 7% for the very early universe).
The Fix: They took the "Better" rulebook (Reed et al.) and added a few "tweaks" or "correction knobs."
- One knob turns down the number of tiny islands in the early universe (fixing the overestimation).
- Another knob adjusts the count of the giant islands to make sure the edges of the map are smooth.

5. Why It Matters (According to the Paper)

The paper states that having this accurate rulebook is crucial for:

Understanding the First Stars: Knowing how many tiny islands existed helps explain how the first stars and galaxies formed.
Dark Matter Signals: It helps predict signals from dark matter particles (like annihilation or decay) which happen most often in the smallest, densest islands.
Galaxy Clusters: It helps astronomers understand the largest structures in the universe, which are used to measure the universe's expansion.

Summary

Think of this paper as the team that finally finished the ultimate atlas of the universe's invisible islands. They used a clever "zoom-in" trick to count the smallest pebbles and the biggest mountains simultaneously, and they wrote a new guide that tells us exactly how many of these islands exist at any size, at any time, from the dawn of creation to the present day.

Technical Summary: The Dark Matter Halo Mass Function in $\Lambda$ CDM

Problem Statement
The dark matter halo mass function (HMF) is a fundamental statistic linking theoretical dark matter physics to observed galaxy populations. While analytical models like Press-Schechter (PS) and Sheth-Tormen (ST) provide baseline predictions, and empirical fits like Reed et al. (2007, R07) improve accuracy for massive haloes at low redshifts, significant gaps remain. Existing formulae have not been rigorously tested across the full dynamic range of mass (from planetary scales to galaxy clusters) and redshift ( $z=0$ to $z=30$ ), particularly in underdense environments or at high redshifts where deviations from "universal" forms are suspected. Furthermore, the extreme computational cost of resolving planetary-mass haloes ( $10^{-6} M_\odot$ ) alongside cluster-mass haloes ( $10^{15.5} M_\odot$ ) in a single simulation has historically prevented a unified calibration.

Methodology
To address these limitations, the authors combine three distinct $N$ -body simulation datasets within the $\Lambda$ CDM cosmology (Planck 2014 parameters):

VVV (Voids-within-Voids-within-Voids): A nested zoom-in simulation (Wang et al. 2020) targeting underdense regions, resolving haloes down to planetary masses ( $10^{-6} M_\odot$ ) but limited in volume at the high-mass end.
P-Millennium (PMILL): A large-volume, lower-resolution simulation providing statistics for massive haloes.
VVV2.8: A bespoke large-box simulation ( $2.8 \text{ Gpc}^3$ ) to increase sample size at the high-mass end and low redshift.

A critical methodological innovation is the subsampling technique. Since the VVV simulations focus on underdense voids, the global halo abundance is biased low. The authors extract smaller, spherical subvolumes from the high-resolution VVV levels (L1–L7) with specific target overdensities (ranging from cosmic mean to highly underdense). By applying the Extended Press-Schechter (EPS) transformation to map local measurements to the global $f(\nu)$ space, they reconstruct the global HMF from these biased regions. To handle cosmic variance and overlapping subvolumes, a weighted median approach is employed.

Key Contributions and Results

Unified Dynamic Range: The study presents the first HMF covering a mass range from $10^{-6} M_\odot$ (planetary mass, set by the thermal cutoff for 100 GeV WIMPs) to $10^{15.5} M_\odot$ (rich clusters), spanning redshifts from $z=30$ to the present.
Validation of Subsampling: The authors demonstrate that the HMF derived from subsampled underdense regions converges with results from full-volume simulations (like PMILL) in overlapping mass bins, with deviations mostly within $\sim 0.05$ dex. This confirms the validity of using EPS-motivated subsampling to recover global statistics from local voids.
Evaluation of Existing Formulae:
- The Press-Schechter (PS) formula significantly overestimates abundance at low masses and underestimates it at high masses, with deviations exceeding 30% at $z=0$ .
- The Reed et al. (2007, R07) formula performs well at low redshifts but deviates increasingly at high redshifts ( $z \gtrsim 5$ ) and low masses, underestimating the abundance by factors up to $\sim 2$ at $z=30$ .
New Empirical Fitting Formula: The authors propose a modified R07 formula with redshift-dependent corrections:
- A suppression term, $\Gamma(\nu, z)$ , utilizing a hyperbolic tangent function to reduce the predicted abundance of low-mass haloes at high redshifts ( $z \gtrsim 2$ ).
- Adjustments to the Gaussian correction term, $G(\nu, z)$ , and the parameter $c$ to better fit the high-mass end.
- Accuracy: The new formula achieves deviations of $\sim 2-3\%$ for $z < 2$ and $\sim 7\%$ for $z \gtrsim 5$ across the entire mass range.
Cosmological Implications: The authors note that while the formula is calibrated for $\Lambda$ CDM, preliminary tests suggest it remains accurate for models with modest deviations in cosmological parameters. They discuss how non-Gaussianity or warm dark matter cutoffs might be incorporated via existing theoretical extensions (e.g., changing window functions) rather than requiring a fundamental rewrite of the fit.

Significance
The paper claims to provide a "practical and accurate tool" for structure formation studies that bridges the gap between the formation of the first mini-haloes and present-day galaxy clusters. The derived fitting formula is intended for applications requiring high precision across diverse scales and epochs, including:

Forecasting dark matter annihilation or decay signals (dominated by small, high-concentration haloes).
Interpreting constraints from galaxy cluster abundance surveys (e.g., LSST, eROSITA).
Modeling the early UV galaxy luminosity function and reionization timing.
Analyzing strong gravitational lensing and stellar stream gaps to constrain mini-halo demographics.

The authors emphasize that their work calibrates the HMF within the conventional EPS framework, treating the empirical corrections as effective descriptions of non-Markovian random walks and environmental impacts that become significant at extreme scales and redshifts. A Python implementation of the formula is made publicly available.