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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

T0 review · 2 major / 5 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read A single calibrated formula gives the dark-matter halo mass function from planetary mass to clusters, from z=30 to today, at a few-percent accuracy.

desk verdict Solid, usable empirical HMF that actually covers planetary-to-cluster masses from z=30 to 0, with a clever subsampling fix and public code; the EPS mapping is the softest link but is cross-checked where it matters. read the letter →

arxiv 2607.05505 v1 pith:TL4SXDIT submitted 2026-07-06 astro-ph.CO astro-ph.GA

Haonan Zheng , Sownak Bose , Carlos S. Frenk , Liang Gao , Adrian Jenkins , Shihong Liao , Yizhou Liu , Volker Springel
show 2 more authors
Jie Wang Simon D. M. White
This is my paper
classification astro-ph.COastro-ph.GA
keywords halomassfunctionΛCDMstructureformationN-bodysimulationsextendedPress-Schechterfittingformula
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Dark-matter haloes are the basic building blocks of cosmic structure. Their abundance as a function of mass and time sets the expected number of galaxy clusters, the density of the tiniest mini-haloes that dominate possible annihilation signals, and the supply of gas for the first stars. Earlier analytic formulae were accurate only over limited ranges of mass or redshift. This paper measures the halo mass function across twelve orders of magnitude in mass and from redshift 30 to the present by combining nested void zoom simulations with large-volume boxes, then recovers the cosmic-mean abundance from the biased underdense zooms via a density-dependent transformation. The resulting empirical correction to an existing fitting form reproduces the measurements to roughly 2–3 percent at low redshift and about 7 percent at high redshift, and remains usable for modest changes in cosmological parameters. The practical payoff is a single, publicly coded expression that can be dropped into forecasts for cluster cosmology, high-redshift galaxies, and dark-matter indirect detection.

What carries the argument

The extended Press–Schechter transformation ν=(δ_1−δ_0)/√(σ_1^{2}−σ_0^{2}) together with weighted-median subsampling of higher-density subvolumes inside the underdense VVV zooms; this maps local, environmentally biased counts onto the global f(ν) that the fitting formula then matches.

What would settle it

A new unbiased simulation that resolves the same low-mass, high-redshift regime without voids, or an independent large-volume run at intermediate mass, that yields an f(ν) differing from the calibrated formula by more than the quoted few-percent residuals in the overlapping ν range.

Watch

Extended reading notes

Core claim

An empirical correction to the Reed et al. (2007) formula, calibrated on the combined VVV and large-volume simulations, reproduces the measured halo mass function f(ν) to ∼2–3 percent at z<2 and ∼7 percent at z≳5 across the full range ln ν ∈ [−2,1.8] and 10^{-6}–10^{15.5} M_⊙, and remains accurate for modest variations of cosmological parameters.

Load-bearing premise

That extracting denser subvolumes from the extremely underdense void zooms and converting them with the extended Press–Schechter ν formula truly recovers the cosmic-mean halo abundance, with only limited direct overlap against unbiased large boxes to check.

Editorial extensions

If this is right

  • Cluster-count forecasts for surveys can use one continuous expression from 10^{14} M_⊙ upward without stitching separate high-mass fits.
  • Indirect-detection calculations can integrate the annihilation luminosity down to the planetary-mass free-streaming cut-off with a controlled abundance uncertainty.
  • High-redshift galaxy and reionization models can adopt a consistent low-mass halo supply from z=30 to reionization without switching formulae.
  • The same functional form remains usable, after modest re-calibration of σ(M), for cosmologies whose parameters differ only slightly from the Planck values used here.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the formula is already written in ν-space, any future change in the linear power spectrum (warm dark matter cut-offs, running spectral index) can be absorbed by recomputing σ(M) alone, provided the collapse barrier itself does not change.
  • The residual redshift dependence that required the new Γ(ν,z) term hints that a single universal barrier is incomplete once the effective spectral index approaches −3; a scale-dependent barrier calibrated at high z could eliminate the extra free functions.
  • Public release of the fitting code makes it straightforward to re-fit the same functional form to any new suite of simulations that span a comparable dynamic range, turning the present calibration into a living standard.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper measures the dark-matter halo mass function over an unprecedented range (planetary-mass 10^{-6} M_⊙ to cluster-mass 10^{15.5} M_⊙, z=0–30) by combining the nested VVV void zooms with P-Millennium and a new large-box VVV2.8 run. An EPS-motivated weighted-median subsampling of higher-density subvolumes inside the underdense VVV regions is used to reconstruct the global f(ν). The authors then introduce an empirical, redshift-dependent correction (Eqs. 12–17) to the Reed et al. (2007) formula that reduces residuals to ∼2–3 % at z<2 and ∼7 % at z≳5. Appendices test the conversion between mass definitions and modest changes in cosmological parameters; a public Python implementation is provided.

Significance. A single, publicly coded fitting formula that is accurate from planetary to cluster scales and from z=30 to the present is a practical tool for dark-matter annihilation forecasts, high-z galaxy modelling, and cluster cosmology. The multi-simulation consistency checks (Figs. 1–3, Table 3), the explicit quantification of Poisson and cosmic-variance scatter, the mass-definition conversion test (Appendix A), and the modest cosmology-variation test (Appendix B) give the result a solid empirical foundation within ΛCDM. The public code further raises the utility of the work.

major comments (2)
  1. The free coefficients in Eqs. (12)–(17) (p_this work, c_this work, the amplitude and centre of G_this work, and the polynomials α(z), β(z) that define Γ) are fitted to the same multi-redshift, multi-environment suite that is later used to quote the residuals in Table 3. The comparison is therefore partly a self-consistency check rather than a fully independent validation. A short leave-one-redshift-out or leave-one-level-out exercise (or an explicit statement that the functional form was frozen before the final residual evaluation) would strengthen the claim that the ∼2–7 % accuracy is not an over-fit.
  2. The decisive cross-check of the EPS-motivated subsampling (Eq. 1 + §2.3) is the recovery of the unbiased P-Millennium (and VVV2.8) f(ν) to ≲0.05 dex in the overlapping ln ν interval (Figs. 1–3). That check is reassuring but limited in dynamic range. A quantitative statement of how the residual 0.05 dex scatter, once folded through the weighted-median procedure, propagates into the final ε_fit values in Table 3 would make the error budget fully transparent.
minor comments (5)
  1. Table 3 caption and surrounding text: clarify that ε_simulation is the floor set by inter-run scatter and that ε_fit is obtained by subtracting this floor in quadrature (or whatever procedure is actually used).
  2. Eq. (20): the Gaussian correction terms N(μ,ς²) are written with log M; state the base of the logarithm explicitly for reproducibility.
  3. Figure 4 right panel: the vertical axis label is crowded; a simpler ratio label would improve readability.
  4. Appendix A: the conversion to M_200c relies on the Ludlow et al. (2016) mass–concentration relation with α=0.18; a one-sentence note on the sensitivity of the transformed f(ν) to that choice would be useful.
  5. A few minor typos (e.g., “thesetsofsimulations”, “halomassfunction”) remain in the compiled text and should be cleaned.

Circularity Check

1 steps flagged · score 4.0 of 10

Empirical correction to Reed et al. (2007) is fitted to the VVV+PMILL+VVV2.8 suite and then compared to the same suite; residual accuracy is partly by construction of the parametric fit, though the form is constrained across mass/redshift/environment and checked on external cosmologies.

  1. fitted input called prediction [Section 3.2, Eqs. (12)–(17); Table 3; bottom panels of Figs. 1–3]
    "Motivated by the observed evolution of the deviations, we propose an empirical correction to the Reed et al. (2007) formula: f_this work(ν)=… where p_this work(z)=0.33, c_this work=1.04, G_this work(ν,z)=… and Γ_this work(ν,z)=… We thereby correct the original R07 formula… as corroborated by the average deviation ε_fit at a ∼2−7% level in Table 3, highlighting the great performance of our fit across such a broad range of ln ν ∈ [−2,1.8] and z ∈ [0,30]."

    The free parameters and z-dependent functions that define the correction are chosen expressly to reduce the residuals between the formula and the simulation measurements of f(ν). The subsequent ratios and the tabulated ε_fit therefore quantify how well the chosen parametric form interpolates the same data set used for calibration; the small residuals are statistically forced once the coefficients have been adjusted. (External checks on WMAP cosmologies and the multi-environment consistency partially mitigate, but do not remove, this character.)

full rationale

The paper is transparent that it supplies a calibrated fitting formula rather than a first-principles derivation. The functional skeleton is taken from the external Reed et al. (2007) expression; a small number of free coefficients and z-dependent functions (p, c, G amplitude/width/centre, α(z), β(z)) are then adjusted so that the modified f(ν) matches the measured halo abundances. The quoted 2–7 % deviations (Table 3, bottom panels of Figs. 1–3) are therefore residuals of that fit, not independent predictions. This is the classic “fitted input called prediction” pattern and warrants a moderate score. It is not pure circularity: the same parametric form must simultaneously describe an enormous dynamic range (ln ν ∈ [−2, 1.8], z = 0–30, multiple environments via EPS-motivated subsampling), the overlap with the unbiased P-Millennium box supplies an internal cross-check, and Appendix B tests the identical formula on independent WMAP-1 simulations (Millennium-2, Millennium-XXL). No self-definitional loop, uniqueness theorem, or load-bearing self-citation of an unverified prior result is present. The EPS transformation (Eq. 1) is an assumption whose validity is checked rather than assumed by fiat. Overall circularity is therefore limited to the ordinary residual-of-fit character of an empirical HMF formula.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard ΛCDM N-body methodology plus an empirical recalibration of an existing fitting form. The free parameters are the handful of numerical coefficients introduced in Eqs. (12)–(17); the main domain assumptions are the validity of the EPS ν-mapping for underdense regions and the universality of the corrected f(ν) under modest cosmology changes. No new physical entities are postulated.

free parameters (4)
  • p_this work = 0.33
    Fixed to 0.33 in the power-law prefactor of Eq. (12); chosen to improve the low-mass fit relative to the original Sheth–Tormen value.
  • c_this work = 1.04
    Exponential cut-off coefficient set to 1.04 (versus Reed’s 1.08) to adjust the high-mass end.
  • G_this work amplitude and centre = polynomial/min expressions
    Redshift-dependent Gaussian correction amplitude min(2.70, 2.35−0.36z+0.07z²) and centre min(0.5+0.05z,0.6) in Eq. (13); free functions fitted to high-ν residuals.
  • α(z), β(z) in Γ = 0.252+1.53θ−2.028θ²+1.382θ³ ; 4.10−2.10θ
    Coefficients of the hyperbolic-tangent suppression term (Eqs. 14–17) that control the low-ν reduction; polynomials in θ(z)=exp(−(1+z)/10) fitted to high-z, low-mass residuals.
assumptions (4)
  • domain assumption Extended Press–Schechter mapping ν=(δ₁−δ₀)/√(σ₁²−σ₀²) converts local underdense counts into the global mass function.
    Invoked throughout Section 2.3 and used to justify the entire subsampling procedure; validated only by overlap with P-Millennium.
  • domain assumption Spherical-collapse barrier δ_c=1.68647 and linear growth factor D(z) remain adequate after empirical corrections.
    Standard EPS ingredients retained in Eqs. (1) and (18).
  • domain assumption Friends-of-friends + SUBFIND with M_200m definition yields a mass function that can be mapped to other common definitions via NFW/Einasto + Ludlow concentration.
    Halo catalogue construction (Section 2.1) and Appendix A conversion.
  • domain assumption Planck 2014 cosmological parameters and a BBKS small-scale power spectrum with no free-streaming cut-off for the L7 analysis.
    Simulation setup (Section 2.1).

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Cite this review

Pith. "Pith review of 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." pith.science (2026). https://pith.science/paper/TL4SXDIT

@misc{pith2026260705505,
  author       = {Pith},
  title        = {Pith review of: The dark matter halo mass function in the $\Lambda\mathrmCDM$ cosmology at all times and over all scales -- from planetary to galaxy cluster masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TL4SXDIT}},
  note         = {Machine review of arXiv:2607.05505}
}
abstract

The dark matter halo mass function is one of the most fundamental predictions of structure formation theory and cosmological simulations. We present the full halo mass function in the $\Lambda$ cold dark matter ($\Lambda\mathrm{CDM}$) model, ranging from a planetary mass ($10^{-6}\,\mathrm{M}_\odot$; the thermal cutoff in the initial power spectrum for a fiducial CDM particle mass of $100\,\mathrm{GeV}$) to the mass of a rich galaxy cluster ($10^{15.5}\,\mathrm{M}_\odot$), and from redshift, $z=30$ to the present. To span this very large dynamic range, we combine our earlier Voids-within-Voids-within-Voids (VVV) set of simulations (Wang et al) with large volume, lower resolution cosmological simulations. We develop a subsampling method to extract subvolumes from the original simulations, allowing us to reconstruct the global halo mass function from the biased underdense VVV regions. We show that the results agree reasonably well among the sets of simulations on different scales and environments. We provide a fitting formula for the dark matter halo mass function based on the work of Reed et al. calibrated with our simulations, such that it can be applied at all scales, all environments and all times, with deviations of $\sim2-3\%$ at $z < 2$ and $\sim 7\%$ at higher redshift $z \gtrsim 5$. This formula is also accurate at least for a restricted set of models we tested with modest deviations from $\Lambda\mathrm{CDM}$ in the values of some of the cosmological parameters. A python code is publicly available at https://github.com/haonan-zheng/hmfc.

Figures

Figures reproduced from arXiv: 2607.05505 by the authors.

Figure 1
Figure 1. Halo mass functions at 𝑧 = 0 and 𝑧 ∼ 1. In the top panels, thick solid lines show the halo mass functions with Poisson errors from the original simulation volumes (i.e., VVV2.8, PMILL, L1-L7), while the thin solid lines show those from subvolumes subsampled from L4-L7 with different local densities (see Section 2.3 for details) with the 16th-84th percentiles shown. The dashed lines show the predictions of different … view at source ↗
Figure 2
Figure 2. Same as [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Same as [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Left: The evolution of the halo mass function for different masses (10−6 −1015 M⊙, as represented by each set of curves labelled with the corresponding log(𝑀/M⊙ )) in the whole universe, predicted by different formulae (gray: PS, black: ST, blue: R07, red: this work); …

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Reference graph

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.